--- a/src/HOL/Analysis/Analysis.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Analysis.thy Sat Nov 30 13:47:33 2019 +0100
@@ -35,16 +35,14 @@
Weierstrass_Theorems
Polytope
Jordan_Curve
- Winding_Numbers
- Riemann_Mapping
Poly_Roots
- Conformal_Mappings
- FPS_Convergence
Generalised_Binomial_Theorem
Gamma_Function
Change_Of_Vars
Multivariate_Analysis
Simplex_Content
+ FPS_Convergence
+ Smooth_Paths
begin
end
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Wed Nov 27 16:54:33 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,7847 +0,0 @@
-section \<open>Complex Path Integrals and Cauchy's Integral Theorem\<close>
-
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2015)\<close>
-
-theory Cauchy_Integral_Theorem
-imports
- Complex_Transcendental
- Henstock_Kurzweil_Integration
- Weierstrass_Theorems
- Retracts
-begin
-
-lemma leibniz_rule_holomorphic:
- fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex"
- assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
- assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
- assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
- assumes "convex U"
- shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
- using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
- by (auto simp: holomorphic_on_def)
-
-lemma Ln_measurable [measurable]: "Ln \<in> measurable borel borel"
-proof -
- have *: "Ln (-of_real x) = of_real (ln x) + \<i> * pi" if "x > 0" for x
- using that by (subst Ln_minus) (auto simp: Ln_of_real)
- have **: "Ln (of_real x) = of_real (ln (-x)) + \<i> * pi" if "x < 0" for x
- using *[of "-x"] that by simp
- have cont: "(\<lambda>x. indicat_real (- \<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel_measurable borel"
- by (intro borel_measurable_continuous_on_indicator continuous_intros) auto
- have "(\<lambda>x. if x \<in> \<real>\<^sub>\<le>\<^sub>0 then ln (-Re x) + \<i> * pi else indicator (-\<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel \<rightarrow>\<^sub>M borel"
- (is "?f \<in> _") by (rule measurable_If_set[OF _ cont]) auto
- hence "(\<lambda>x. if x = 0 then Ln 0 else ?f x) \<in> borel \<rightarrow>\<^sub>M borel" by measurable
- also have "(\<lambda>x. if x = 0 then Ln 0 else ?f x) = Ln"
- by (auto simp: fun_eq_iff ** nonpos_Reals_def)
- finally show ?thesis .
-qed
-
-lemma powr_complex_measurable [measurable]:
- assumes [measurable]: "f \<in> measurable M borel" "g \<in> measurable M borel"
- shows "(\<lambda>x. f x powr g x :: complex) \<in> measurable M borel"
- using assms by (simp add: powr_def)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Homeomorphisms of arc images\<close>
-
-lemma homeomorphism_arc:
- fixes g :: "real \<Rightarrow> 'a::t2_space"
- assumes "arc g"
- obtains h where "homeomorphism {0..1} (path_image g) g h"
-using assms by (force simp: arc_def homeomorphism_compact path_def path_image_def)
-
-lemma homeomorphic_arc_image_interval:
- fixes g :: "real \<Rightarrow> 'a::t2_space" and a::real
- assumes "arc g" "a < b"
- shows "(path_image g) homeomorphic {a..b}"
-proof -
- have "(path_image g) homeomorphic {0..1::real}"
- by (meson assms(1) homeomorphic_def homeomorphic_sym homeomorphism_arc)
- also have "\<dots> homeomorphic {a..b}"
- using assms by (force intro: homeomorphic_closed_intervals_real)
- finally show ?thesis .
-qed
-
-lemma homeomorphic_arc_images:
- fixes g :: "real \<Rightarrow> 'a::t2_space" and h :: "real \<Rightarrow> 'b::t2_space"
- assumes "arc g" "arc h"
- shows "(path_image g) homeomorphic (path_image h)"
-proof -
- have "(path_image g) homeomorphic {0..1::real}"
- by (meson assms homeomorphic_def homeomorphic_sym homeomorphism_arc)
- also have "\<dots> homeomorphic (path_image h)"
- by (meson assms homeomorphic_def homeomorphism_arc)
- finally show ?thesis .
-qed
-
-lemma path_connected_arc_complement:
- fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
- assumes "arc \<gamma>" "2 \<le> DIM('a)"
- shows "path_connected(- path_image \<gamma>)"
-proof -
- have "path_image \<gamma> homeomorphic {0..1::real}"
- by (simp add: assms homeomorphic_arc_image_interval)
- then
- show ?thesis
- apply (rule path_connected_complement_homeomorphic_convex_compact)
- apply (auto simp: assms)
- done
-qed
-
-lemma connected_arc_complement:
- fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
- assumes "arc \<gamma>" "2 \<le> DIM('a)"
- shows "connected(- path_image \<gamma>)"
- by (simp add: assms path_connected_arc_complement path_connected_imp_connected)
-
-lemma inside_arc_empty:
- fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
- assumes "arc \<gamma>"
- shows "inside(path_image \<gamma>) = {}"
-proof (cases "DIM('a) = 1")
- case True
- then show ?thesis
- using assms connected_arc_image connected_convex_1_gen inside_convex by blast
-next
- case False
- show ?thesis
- proof (rule inside_bounded_complement_connected_empty)
- show "connected (- path_image \<gamma>)"
- apply (rule connected_arc_complement [OF assms])
- using False
- by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym)
- show "bounded (path_image \<gamma>)"
- by (simp add: assms bounded_arc_image)
- qed
-qed
-
-lemma inside_simple_curve_imp_closed:
- fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
- shows "\<lbrakk>simple_path \<gamma>; x \<in> inside(path_image \<gamma>)\<rbrakk> \<Longrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
- using arc_simple_path inside_arc_empty by blast
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Piecewise differentiable functions\<close>
-
-definition piecewise_differentiable_on
- (infixr "piecewise'_differentiable'_on" 50)
- where "f piecewise_differentiable_on i \<equiv>
- continuous_on i f \<and>
- (\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))"
-
-lemma piecewise_differentiable_on_imp_continuous_on:
- "f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f"
-by (simp add: piecewise_differentiable_on_def)
-
-lemma piecewise_differentiable_on_subset:
- "f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T"
- using continuous_on_subset
- unfolding piecewise_differentiable_on_def
- apply safe
- apply (blast elim: continuous_on_subset)
- by (meson Diff_iff differentiable_within_subset subsetCE)
-
-lemma differentiable_on_imp_piecewise_differentiable:
- fixes a:: "'a::{linorder_topology,real_normed_vector}"
- shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
- apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
- apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
- done
-
-lemma differentiable_imp_piecewise_differentiable:
- "(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S))
- \<Longrightarrow> f piecewise_differentiable_on S"
-by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
- intro: differentiable_within_subset)
-
-lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S"
- by (simp add: differentiable_imp_piecewise_differentiable)
-
-lemma piecewise_differentiable_compose:
- "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S);
- \<And>x. finite (S \<inter> f-`{x})\<rbrakk>
- \<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S"
- apply (simp add: piecewise_differentiable_on_def, safe)
- apply (blast intro: continuous_on_compose2)
- apply (rename_tac A B)
- apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI)
- apply (blast intro!: differentiable_chain_within)
- done
-
-lemma piecewise_differentiable_affine:
- fixes m::real
- assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)"
- shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S"
-proof (cases "m = 0")
- case True
- then show ?thesis
- unfolding o_def
- by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
-next
- case False
- show ?thesis
- apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
- apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
- done
-qed
-
-lemma piecewise_differentiable_cases:
- fixes c::real
- assumes "f piecewise_differentiable_on {a..c}"
- "g piecewise_differentiable_on {c..b}"
- "a \<le> c" "c \<le> b" "f c = g c"
- shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
-proof -
- obtain S T where st: "finite S" "finite T"
- and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}"
- and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}"
- using assms
- by (auto simp: piecewise_differentiable_on_def)
- have finabc: "finite ({a,b,c} \<union> (S \<union> T))"
- by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI)
- have "continuous_on {a..c} f" "continuous_on {c..b} g"
- using assms piecewise_differentiable_on_def by auto
- then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
- using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
- OF closed_real_atLeastAtMost [of c b],
- of f g "\<lambda>x. x\<le>c"] assms
- by (force simp: ivl_disj_un_two_touch)
- moreover
- { fix x
- assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))"
- have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
- proof (cases x c rule: le_cases)
- case le show ?diff_fg
- proof (rule differentiable_transform_within [where d = "dist x c"])
- have "f differentiable at x"
- using x le fd [of x] at_within_interior [of x "{a..c}"] by simp
- then show "f differentiable at x within {a..b}"
- by (simp add: differentiable_at_withinI)
- qed (use x le st dist_real_def in auto)
- next
- case ge show ?diff_fg
- proof (rule differentiable_transform_within [where d = "dist x c"])
- have "g differentiable at x"
- using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp
- then show "g differentiable at x within {a..b}"
- by (simp add: differentiable_at_withinI)
- qed (use x ge st dist_real_def in auto)
- qed
- }
- then have "\<exists>S. finite S \<and>
- (\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
- by (meson finabc)
- ultimately show ?thesis
- by (simp add: piecewise_differentiable_on_def)
-qed
-
-lemma piecewise_differentiable_neg:
- "f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S"
- by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
-
-lemma piecewise_differentiable_add:
- assumes "f piecewise_differentiable_on i"
- "g piecewise_differentiable_on i"
- shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
-proof -
- obtain S T where st: "finite S" "finite T"
- "\<forall>x\<in>i - S. f differentiable at x within i"
- "\<forall>x\<in>i - T. g differentiable at x within i"
- using assms by (auto simp: piecewise_differentiable_on_def)
- then have "finite (S \<union> T) \<and> (\<forall>x\<in>i - (S \<union> T). (\<lambda>x. f x + g x) differentiable at x within i)"
- by auto
- moreover have "continuous_on i f" "continuous_on i g"
- using assms piecewise_differentiable_on_def by auto
- ultimately show ?thesis
- by (auto simp: piecewise_differentiable_on_def continuous_on_add)
-qed
-
-lemma piecewise_differentiable_diff:
- "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on S\<rbrakk>
- \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S"
- unfolding diff_conv_add_uminus
- by (metis piecewise_differentiable_add piecewise_differentiable_neg)
-
-lemma continuous_on_joinpaths_D1:
- "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
- apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> ((*)(inverse 2))"])
- apply (rule continuous_intros | simp)+
- apply (auto elim!: continuous_on_subset simp: joinpaths_def)
- done
-
-lemma continuous_on_joinpaths_D2:
- "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
- apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> (\<lambda>x. inverse 2*x + 1/2)"])
- apply (rule continuous_intros | simp)+
- apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
- done
-
-lemma piecewise_differentiable_D1:
- assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}"
- shows "g1 piecewise_differentiable_on {0..1}"
-proof -
- obtain S where cont: "continuous_on {0..1} g1" and "finite S"
- and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
- using assms unfolding piecewise_differentiable_on_def
- by (blast dest!: continuous_on_joinpaths_D1)
- show ?thesis
- unfolding piecewise_differentiable_on_def
- proof (intro exI conjI ballI cont)
- show "finite (insert 1 (((*)2) ` S))"
- by (simp add: \<open>finite S\<close>)
- show "g1 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
- proof (rule_tac d="dist (x/2) (1/2)" in differentiable_transform_within)
- have "g1 +++ g2 differentiable at (x / 2) within {0..1/2}"
- by (rule differentiable_subset [OF S [of "x/2"]] | use that in force)+
- then show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x within {0..1}"
- using image_affinity_atLeastAtMost_div [of 2 0 "0::real" 1]
- by (auto intro: differentiable_chain_within)
- qed (use that in \<open>auto simp: joinpaths_def\<close>)
- qed
-qed
-
-lemma piecewise_differentiable_D2:
- assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" and eq: "pathfinish g1 = pathstart g2"
- shows "g2 piecewise_differentiable_on {0..1}"
-proof -
- have [simp]: "g1 1 = g2 0"
- using eq by (simp add: pathfinish_def pathstart_def)
- obtain S where cont: "continuous_on {0..1} g2" and "finite S"
- and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
- using assms unfolding piecewise_differentiable_on_def
- by (blast dest!: continuous_on_joinpaths_D2)
- show ?thesis
- unfolding piecewise_differentiable_on_def
- proof (intro exI conjI ballI cont)
- show "finite (insert 0 ((\<lambda>x. 2*x-1)`S))"
- by (simp add: \<open>finite S\<close>)
- show "g2 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1)`S)" for x
- proof (rule_tac d="dist ((x+1)/2) (1/2)" in differentiable_transform_within)
- have x2: "(x + 1) / 2 \<notin> S"
- using that
- apply (clarsimp simp: image_iff)
- by (metis add.commute add_diff_cancel_left' mult_2 field_sum_of_halves)
- have "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
- by (rule differentiable_chain_within differentiable_subset [OF S [of "(x+1)/2"]] | use x2 that in force)+
- then show "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
- by (auto intro: differentiable_chain_within)
- show "(g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" if "x' \<in> {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x'
- proof -
- have [simp]: "(2*x'+2)/2 = x'+1"
- by (simp add: field_split_simps)
- show ?thesis
- using that by (auto simp: joinpaths_def)
- qed
- qed (use that in \<open>auto simp: joinpaths_def\<close>)
- qed
-qed
-
-
-subsection\<open>The concept of continuously differentiable\<close>
-
-text \<open>
-John Harrison writes as follows:
-
-``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
-continuously differentiable, which ensures that the path integral exists at least for any continuous
-f, since all piecewise continuous functions are integrable. However, our notion of validity is
-weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a
-finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
-the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
-can integrate all derivatives.''
-
-"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
-Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
-
-And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
-difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
-asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
-
-definition\<^marker>\<open>tag important\<close> C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
- (infix "C1'_differentiable'_on" 50)
- where
- "f C1_differentiable_on S \<longleftrightarrow>
- (\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)"
-
-lemma C1_differentiable_on_eq:
- "f C1_differentiable_on S \<longleftrightarrow>
- (\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- unfolding C1_differentiable_on_def
- by (metis (no_types, lifting) continuous_on_eq differentiableI_vector vector_derivative_at)
-next
- assume ?rhs
- then show ?lhs
- using C1_differentiable_on_def vector_derivative_works by fastforce
-qed
-
-lemma C1_differentiable_on_subset:
- "f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S"
- unfolding C1_differentiable_on_def continuous_on_eq_continuous_within
- by (blast intro: continuous_within_subset)
-
-lemma C1_differentiable_compose:
- assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
- shows "(g \<circ> f) C1_differentiable_on S"
-proof -
- have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x"
- by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI)
- moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))"
- proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
- show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))"
- using fg
- apply (clarsimp simp add: C1_differentiable_on_eq)
- apply (rule Limits.continuous_on_scaleR, assumption)
- by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def)
- show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)"
- by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at)
- qed
- ultimately show ?thesis
- by (simp add: C1_differentiable_on_eq)
-qed
-
-lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S"
- by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
-
-lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S"
- by (auto simp: C1_differentiable_on_eq)
-
-lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S"
- by (auto simp: C1_differentiable_on_eq)
-
-lemma C1_differentiable_on_add [simp, derivative_intros]:
- "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S"
- unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
-
-lemma C1_differentiable_on_minus [simp, derivative_intros]:
- "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S"
- unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
-
-lemma C1_differentiable_on_diff [simp, derivative_intros]:
- "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S"
- unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
-
-lemma C1_differentiable_on_mult [simp, derivative_intros]:
- fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
- shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S"
- unfolding C1_differentiable_on_eq
- by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
-
-lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
- "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S"
- unfolding C1_differentiable_on_eq
- by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
-
-
-definition\<^marker>\<open>tag important\<close> piecewise_C1_differentiable_on
- (infixr "piecewise'_C1'_differentiable'_on" 50)
- where "f piecewise_C1_differentiable_on i \<equiv>
- continuous_on i f \<and>
- (\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))"
-
-lemma C1_differentiable_imp_piecewise:
- "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
- by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
-
-lemma piecewise_C1_imp_differentiable:
- "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
- by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
- C1_differentiable_on_def differentiable_def has_vector_derivative_def
- intro: has_derivative_at_withinI)
-
-lemma piecewise_C1_differentiable_compose:
- assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
- shows "(g \<circ> f) piecewise_C1_differentiable_on S"
-proof -
- have "continuous_on S (\<lambda>x. g (f x))"
- by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def)
- moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T"
- proof -
- obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S"
- using fg by (auto simp: piecewise_C1_differentiable_on_def)
- obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S"
- using fg by (auto simp: piecewise_C1_differentiable_on_def)
- show ?thesis
- proof (intro exI conjI)
- show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))"
- using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI)
- show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))"
- apply (rule C1_differentiable_compose)
- apply (blast intro: C1_differentiable_on_subset [OF F])
- apply (blast intro: C1_differentiable_on_subset [OF G])
- by (simp add: C1_differentiable_on_subset G Diff_Int_distrib2 fin)
- qed
- qed
- ultimately show ?thesis
- by (simp add: piecewise_C1_differentiable_on_def)
-qed
-
-lemma piecewise_C1_differentiable_on_subset:
- "f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T"
- by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
-
-lemma C1_differentiable_imp_continuous_on:
- "f C1_differentiable_on S \<Longrightarrow> continuous_on S f"
- unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
- using differentiable_at_withinI differentiable_imp_continuous_within by blast
-
-lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
- unfolding C1_differentiable_on_def
- by auto
-
-lemma piecewise_C1_differentiable_affine:
- fixes m::real
- assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)"
- shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S"
-proof (cases "m = 0")
- case True
- then show ?thesis
- unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def)
-next
- case False
- have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})"
- using False not_finite_existsD by fastforce
- show ?thesis
- apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
- apply (rule * assms derivative_intros | simp add: False vimage_def)+
- done
-qed
-
-lemma piecewise_C1_differentiable_cases:
- fixes c::real
- assumes "f piecewise_C1_differentiable_on {a..c}"
- "g piecewise_C1_differentiable_on {c..b}"
- "a \<le> c" "c \<le> b" "f c = g c"
- shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
-proof -
- obtain S T where st: "f C1_differentiable_on ({a..c} - S)"
- "g C1_differentiable_on ({c..b} - T)"
- "finite S" "finite T"
- using assms
- by (force simp: piecewise_C1_differentiable_on_def)
- then have f_diff: "f differentiable_on {a..<c} - S"
- and g_diff: "g differentiable_on {c<..b} - T"
- by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
- have "continuous_on {a..c} f" "continuous_on {c..b} g"
- using assms piecewise_C1_differentiable_on_def by auto
- then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
- using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
- OF closed_real_atLeastAtMost [of c b],
- of f g "\<lambda>x. x\<le>c"] assms
- by (force simp: ivl_disj_un_two_touch)
- { fix x
- assume x: "x \<in> {a..b} - insert c (S \<union> T)"
- have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
- proof (cases x c rule: le_cases)
- case le show ?diff_fg
- apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
- using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
- next
- case ge show ?diff_fg
- apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
- using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
- qed
- }
- then have "(\<forall>x \<in> {a..b} - insert c (S \<union> T). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
- by auto
- moreover
- { assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))"
- and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))"
- have "open ({a<..<c} - S)" "open ({c<..<b} - T)"
- using st by (simp_all add: open_Diff finite_imp_closed)
- moreover have "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
- proof -
- have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x)) (at x)"
- if "a < x" "x < c" "x \<notin> S" for x
- proof -
- have f: "f differentiable at x"
- by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that)
- show ?thesis
- using that
- apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
- apply (auto simp: dist_norm vector_derivative_works [symmetric] f)
- done
- qed
- then show ?thesis
- by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at)
- qed
- moreover have "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
- proof -
- have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x)) (at x)"
- if "c < x" "x < b" "x \<notin> T" for x
- proof -
- have g: "g differentiable at x"
- by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that)
- show ?thesis
- using that
- apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
- apply (auto simp: dist_norm vector_derivative_works [symmetric] g)
- done
- qed
- then show ?thesis
- by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at)
- qed
- ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T))
- (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
- by (rule continuous_on_subset [OF continuous_on_open_Un], auto)
- } note * = this
- have "continuous_on ({a<..<b} - insert c (S \<union> T)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
- using st
- by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
- ultimately have "\<exists>S. finite S \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - S)"
- apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI)
- using st by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
- with cab show ?thesis
- by (simp add: piecewise_C1_differentiable_on_def)
-qed
-
-lemma piecewise_C1_differentiable_neg:
- "f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S"
- unfolding piecewise_C1_differentiable_on_def
- by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
-
-lemma piecewise_C1_differentiable_add:
- assumes "f piecewise_C1_differentiable_on i"
- "g piecewise_C1_differentiable_on i"
- shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
-proof -
- obtain S t where st: "finite S" "finite t"
- "f C1_differentiable_on (i-S)"
- "g C1_differentiable_on (i-t)"
- using assms by (auto simp: piecewise_C1_differentiable_on_def)
- then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)"
- by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
- moreover have "continuous_on i f" "continuous_on i g"
- using assms piecewise_C1_differentiable_on_def by auto
- ultimately show ?thesis
- by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
-qed
-
-lemma piecewise_C1_differentiable_diff:
- "\<lbrakk>f piecewise_C1_differentiable_on S; g piecewise_C1_differentiable_on S\<rbrakk>
- \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
- unfolding diff_conv_add_uminus
- by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
-
-lemma piecewise_C1_differentiable_D1:
- fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
- assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
- shows "g1 piecewise_C1_differentiable_on {0..1}"
-proof -
- obtain S where "finite S"
- and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
- and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
- using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
- proof (rule differentiable_transform_within)
- show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x"
- using that g12D
- apply (simp only: joinpaths_def)
- by (rule differentiable_chain_at derivative_intros | force)+
- show "\<And>x'. \<lbrakk>dist x' x < dist (x/2) (1/2)\<rbrakk>
- \<Longrightarrow> (g1 +++ g2 \<circ> (*) (inverse 2)) x' = g1 x'"
- using that by (auto simp: dist_real_def joinpaths_def)
- qed (use that in \<open>auto simp: dist_real_def\<close>)
- have [simp]: "vector_derivative (g1 \<circ> (*) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
- if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
- apply (subst vector_derivative_chain_at)
- using that
- apply (rule derivative_eq_intros g1D | simp)+
- done
- have "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
- using co12 by (rule continuous_on_subset) force
- then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 \<circ> (*)2) (at x))"
- proof (rule continuous_on_eq [OF _ vector_derivative_at])
- show "(g1 +++ g2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)"
- if "x \<in> {0..1/2} - insert (1/2) S" for x
- proof (rule has_vector_derivative_transform_within)
- show "(g1 \<circ> (*) 2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)"
- using that
- by (force intro: g1D differentiable_chain_at simp: vector_derivative_works [symmetric])
- show "\<And>x'. \<lbrakk>dist x' x < dist x (1/2)\<rbrakk> \<Longrightarrow> (g1 \<circ> (*) 2) x' = (g1 +++ g2) x'"
- using that by (auto simp: dist_norm joinpaths_def)
- qed (use that in \<open>auto simp: dist_norm\<close>)
- qed
- have "continuous_on ({0..1} - insert 1 ((*) 2 ` S))
- ((\<lambda>x. 1/2 * vector_derivative (g1 \<circ> (*)2) (at x)) \<circ> (*)(1/2))"
- apply (rule continuous_intros)+
- using coDhalf
- apply (simp add: scaleR_conv_of_real image_set_diff image_image)
- done
- then have con_g1: "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) (\<lambda>x. vector_derivative g1 (at x))"
- by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
- have "continuous_on {0..1} g1"
- using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
- with \<open>finite S\<close> show ?thesis
- apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- apply (rule_tac x="insert 1 (((*)2)`S)" in exI)
- apply (simp add: g1D con_g1)
- done
-qed
-
-lemma piecewise_C1_differentiable_D2:
- fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
- assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
- shows "g2 piecewise_C1_differentiable_on {0..1}"
-proof -
- obtain S where "finite S"
- and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
- and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
- using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
- proof (rule differentiable_transform_within)
- show "g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2) differentiable at x"
- using g12D that
- apply (simp only: joinpaths_def)
- apply (drule_tac x= "(x+1) / 2" in bspec, force simp: field_split_simps)
- apply (rule differentiable_chain_at derivative_intros | force)+
- done
- show "\<And>x'. dist x' x < dist ((x + 1) / 2) (1/2) \<Longrightarrow> (g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'"
- using that by (auto simp: dist_real_def joinpaths_def field_simps)
- qed (use that in \<open>auto simp: dist_norm\<close>)
- have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
- if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
- using that by (auto simp: vector_derivative_chain_at field_split_simps g2D)
- have "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
- using co12 by (rule continuous_on_subset) force
- then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x))"
- proof (rule continuous_on_eq [OF _ vector_derivative_at])
- show "(g1 +++ g2 has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
- (at x)"
- if "x \<in> {1 / 2..1} - insert (1 / 2) S" for x
- proof (rule_tac f="g2 \<circ> (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
- show "(g2 \<circ> (\<lambda>x. 2 * x - 1) has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
- (at x)"
- using that by (force intro: g2D differentiable_chain_at simp: vector_derivative_works [symmetric])
- show "\<And>x'. \<lbrakk>dist x' x < dist (3 / 4) ((x + 1) / 2)\<rbrakk> \<Longrightarrow> (g2 \<circ> (\<lambda>x. 2 * x - 1)) x' = (g1 +++ g2) x'"
- using that by (auto simp: dist_norm joinpaths_def add_divide_distrib)
- qed (use that in \<open>auto simp: dist_norm\<close>)
- qed
- have [simp]: "((\<lambda>x. (x+1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S))) = ({1/2..1} - insert (1/2) S)"
- apply (simp add: image_set_diff inj_on_def image_image)
- apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
- done
- have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S))
- ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) \<circ> (\<lambda>x. (x+1)/2))"
- by (rule continuous_intros | simp add: coDhalf)+
- then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)) (\<lambda>x. vector_derivative g2 (at x))"
- by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
- have "continuous_on {0..1} g2"
- using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
- with \<open>finite S\<close> show ?thesis
- apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` S)" in exI)
- apply (simp add: g2D con_g2)
- done
-qed
-
-subsection \<open>Valid paths, and their start and finish\<close>
-
-definition\<^marker>\<open>tag important\<close> valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
- where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
-
-definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
- where "closed_path g \<equiv> g 0 = g 1"
-
-text\<open>In particular, all results for paths apply\<close>
-
-lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
- by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
-
-lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
- by (metis connected_path_image valid_path_imp_path)
-
-lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
- by (metis compact_path_image valid_path_imp_path)
-
-lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
- by (metis bounded_path_image valid_path_imp_path)
-
-lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
- by (metis closed_path_image valid_path_imp_path)
-
-lemma valid_path_compose:
- assumes "valid_path g"
- and der: "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable (at x)"
- and con: "continuous_on (path_image g) (deriv f)"
- shows "valid_path (f \<circ> g)"
-proof -
- obtain S where "finite S" and g_diff: "g C1_differentiable_on {0..1} - S"
- using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
- have "f \<circ> g differentiable at t" when "t\<in>{0..1} - S" for t
- proof (rule differentiable_chain_at)
- show "g differentiable at t" using \<open>valid_path g\<close>
- by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - S\<close> that)
- next
- have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
- then show "f differentiable at (g t)"
- using der[THEN field_differentiable_imp_differentiable] by auto
- qed
- moreover have "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
- proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
- rule continuous_intros)
- show "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative g (at x))"
- using g_diff C1_differentiable_on_eq by auto
- next
- have "continuous_on {0..1} (\<lambda>x. deriv f (g x))"
- using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def]
- \<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def
- by blast
- then show "continuous_on ({0..1} - S) (\<lambda>x. deriv f (g x))"
- using continuous_on_subset by blast
- next
- show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
- when "t \<in> {0..1} - S" for t
- proof (rule vector_derivative_chain_at_general[symmetric])
- show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
- next
- have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
- then show "f field_differentiable at (g t)" using der by auto
- qed
- qed
- ultimately have "f \<circ> g C1_differentiable_on {0..1} - S"
- using C1_differentiable_on_eq by blast
- moreover have "path (f \<circ> g)"
- apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]])
- using der
- by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at)
- ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
- using \<open>finite S\<close> by auto
-qed
-
-lemma valid_path_uminus_comp[simp]:
- fixes g::"real \<Rightarrow> 'a ::real_normed_field"
- shows "valid_path (uminus \<circ> g) \<longleftrightarrow> valid_path g"
-proof
- show "valid_path g \<Longrightarrow> valid_path (uminus \<circ> g)" for g::"real \<Rightarrow> 'a"
- by (auto intro!: valid_path_compose derivative_intros simp add: deriv_linear[of "-1",simplified])
- then show "valid_path g" when "valid_path (uminus \<circ> g)"
- by (metis fun.map_comp group_add_class.minus_comp_minus id_comp that)
-qed
-
-lemma valid_path_offset[simp]:
- shows "valid_path (\<lambda>t. g t - z) \<longleftrightarrow> valid_path g"
-proof
- show *: "valid_path (g::real\<Rightarrow>'a) \<Longrightarrow> valid_path (\<lambda>t. g t - z)" for g z
- unfolding valid_path_def
- by (fastforce intro:derivative_intros C1_differentiable_imp_piecewise piecewise_C1_differentiable_diff)
- show "valid_path (\<lambda>t. g t - z) \<Longrightarrow> valid_path g"
- using *[of "\<lambda>t. g t - z" "-z",simplified] .
-qed
-
-
-subsection\<open>Contour Integrals along a path\<close>
-
-text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
-
-text\<open>piecewise differentiable function on [0,1]\<close>
-
-definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
- (infixr "has'_contour'_integral" 50)
- where "(f has_contour_integral i) g \<equiv>
- ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
- has_integral i) {0..1}"
-
-definition\<^marker>\<open>tag important\<close> contour_integrable_on
- (infixr "contour'_integrable'_on" 50)
- where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
-
-definition\<^marker>\<open>tag important\<close> contour_integral
- where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
-
-lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
- unfolding contour_integrable_on_def contour_integral_def by blast
-
-lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
- apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
- using has_integral_unique by blast
-
-lemma has_contour_integral_eqpath:
- "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
- contour_integral p f = contour_integral \<gamma> f\<rbrakk>
- \<Longrightarrow> (f has_contour_integral y) \<gamma>"
-using contour_integrable_on_def contour_integral_unique by auto
-
-lemma has_contour_integral_integral:
- "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
- by (metis contour_integral_unique contour_integrable_on_def)
-
-lemma has_contour_integral_unique:
- "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
- using has_integral_unique
- by (auto simp: has_contour_integral_def)
-
-lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
- using contour_integrable_on_def by blast
-
-text\<open>Show that we can forget about the localized derivative.\<close>
-
-lemma has_integral_localized_vector_derivative:
- "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
- ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
-proof -
- have *: "{a..b} - {a,b} = interior {a..b}"
- by (simp add: atLeastAtMost_diff_ends)
- show ?thesis
- apply (rule has_integral_spike_eq [of "{a,b}"])
- apply (auto simp: at_within_interior [of _ "{a..b}"])
- done
-qed
-
-lemma integrable_on_localized_vector_derivative:
- "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
- (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
- by (simp add: integrable_on_def has_integral_localized_vector_derivative)
-
-lemma has_contour_integral:
- "(f has_contour_integral i) g \<longleftrightarrow>
- ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
- by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
-
-lemma contour_integrable_on:
- "f contour_integrable_on g \<longleftrightarrow>
- (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
- by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close>
-
-lemma valid_path_imp_reverse:
- assumes "valid_path g"
- shows "valid_path(reversepath g)"
-proof -
- obtain S where "finite S" and S: "g C1_differentiable_on ({0..1} - S)"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- then have "finite ((-) 1 ` S)"
- by auto
- moreover have "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` S))"
- unfolding reversepath_def
- apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
- using S
- by (force simp: finite_vimageI inj_on_def C1_differentiable_on_eq elim!: continuous_on_subset)+
- ultimately show ?thesis using assms
- by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
-qed
-
-lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
- using valid_path_imp_reverse by force
-
-lemma has_contour_integral_reversepath:
- assumes "valid_path g" and f: "(f has_contour_integral i) g"
- shows "(f has_contour_integral (-i)) (reversepath g)"
-proof -
- { fix S x
- assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
- have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
- - vector_derivative g (at (1 - x) within {0..1})"
- proof -
- obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
- using xs
- by (force simp: has_vector_derivative_def C1_differentiable_on_def)
- have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
- by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
- then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
- by (simp add: o_def)
- show ?thesis
- using xs
- by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
- qed
- } note * = this
- obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
- {0..1}"
- using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
- by (simp add: has_integral_neg)
- then show ?thesis
- using S
- apply (clarsimp simp: reversepath_def has_contour_integral_def)
- apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
- apply (auto simp: *)
- done
-qed
-
-lemma contour_integrable_reversepath:
- "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
- using has_contour_integral_reversepath contour_integrable_on_def by blast
-
-lemma contour_integrable_reversepath_eq:
- "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
- using contour_integrable_reversepath valid_path_reversepath by fastforce
-
-lemma contour_integral_reversepath:
- assumes "valid_path g"
- shows "contour_integral (reversepath g) f = - (contour_integral g f)"
-proof (cases "f contour_integrable_on g")
- case True then show ?thesis
- by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
-next
- case False then have "\<not> f contour_integrable_on (reversepath g)"
- by (simp add: assms contour_integrable_reversepath_eq)
- with False show ?thesis by (simp add: not_integrable_contour_integral)
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close>
-
-lemma valid_path_join:
- assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
- shows "valid_path(g1 +++ g2)"
-proof -
- have "g1 1 = g2 0"
- using assms by (auto simp: pathfinish_def pathstart_def)
- moreover have "(g1 \<circ> (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
- apply (rule piecewise_C1_differentiable_compose)
- using assms
- apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
- apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
- done
- moreover have "(g2 \<circ> (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
- apply (rule piecewise_C1_differentiable_compose)
- using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
- by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
- simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
- ultimately show ?thesis
- apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
- apply (rule piecewise_C1_differentiable_cases)
- apply (auto simp: o_def)
- done
-qed
-
-lemma valid_path_join_D1:
- fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
- shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
- unfolding valid_path_def
- by (rule piecewise_C1_differentiable_D1)
-
-lemma valid_path_join_D2:
- fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
- shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
- unfolding valid_path_def
- by (rule piecewise_C1_differentiable_D2)
-
-lemma valid_path_join_eq [simp]:
- fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
- shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
- using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
-
-lemma has_contour_integral_join:
- assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
- "valid_path g1" "valid_path g2"
- shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
-proof -
- obtain s1 s2
- where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
- and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
- using assms
- by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
- and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
- using assms
- by (auto simp: has_contour_integral)
- have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
- and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
- using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
- has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
- by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
- have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
- 2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
- apply (simp_all add: dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
- apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- using s1
- apply (auto simp: algebra_simps vector_derivative_works)
- done
- have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
- 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
- apply (simp_all add: dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
- apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- using s2
- apply (auto simp: algebra_simps vector_derivative_works)
- done
- have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
- apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
- using s1
- apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
- apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
- done
- moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
- apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
- using s2
- apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
- apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
- done
- ultimately
- show ?thesis
- apply (simp add: has_contour_integral)
- apply (rule has_integral_combine [where c = "1/2"], auto)
- done
-qed
-
-lemma contour_integrable_joinI:
- assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
- "valid_path g1" "valid_path g2"
- shows "f contour_integrable_on (g1 +++ g2)"
- using assms
- by (meson has_contour_integral_join contour_integrable_on_def)
-
-lemma contour_integrable_joinD1:
- assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
- shows "f contour_integrable_on g1"
-proof -
- obtain s1
- where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
- using assms
- apply (auto simp: contour_integrable_on)
- apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
- apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
- done
- then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
- by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
- have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
- 2 *\<^sub>R vector_derivative g1 (at z)" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
- apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
- using s1
- apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
- done
- show ?thesis
- using s1
- apply (auto simp: contour_integrable_on)
- apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
- apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
- done
-qed
-
-lemma contour_integrable_joinD2:
- assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
- shows "f contour_integrable_on g2"
-proof -
- obtain s2
- where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
- using assms
- apply (auto simp: contour_integrable_on)
- apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
- apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
- apply (simp add: image_affinity_atLeastAtMost_diff)
- done
- then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
- integrable_on {0..1}"
- by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
- have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
- vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
- 2 *\<^sub>R vector_derivative g2 (at z)" for z
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
- apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
- using s2
- apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
- vector_derivative_works add_divide_distrib)
- done
- show ?thesis
- using s2
- apply (auto simp: contour_integrable_on)
- apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
- apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
- done
-qed
-
-lemma contour_integrable_join [simp]:
- shows
- "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
- \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
-using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
-
-lemma contour_integral_join [simp]:
- shows
- "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
- \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
- by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close>
-
-lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
- by (auto simp: shiftpath_def)
-
-lemma valid_path_shiftpath [intro]:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "valid_path(shiftpath a g)"
- using assms
- apply (auto simp: valid_path_def shiftpath_alt_def)
- apply (rule piecewise_C1_differentiable_cases)
- apply (auto simp: algebra_simps)
- apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
- apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
- apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
- apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
- done
-
-lemma has_contour_integral_shiftpath:
- assumes f: "(f has_contour_integral i) g" "valid_path g"
- and a: "a \<in> {0..1}"
- shows "(f has_contour_integral i) (shiftpath a g)"
-proof -
- obtain s
- where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
- using assms by (auto simp: has_contour_integral)
- then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
- integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
- apply (rule has_integral_unique)
- apply (subst add.commute)
- apply (subst integral_combine)
- using assms * integral_unique by auto
- { fix x
- have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
- vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
- unfolding shiftpath_def
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
- apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
- apply (intro derivative_eq_intros | simp)+
- using g
- apply (drule_tac x="x+a" in bspec)
- using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
- done
- } note vd1 = this
- { fix x
- have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
- vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
- unfolding shiftpath_def
- apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
- apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
- apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
- apply (intro derivative_eq_intros | simp)+
- using g
- apply (drule_tac x="x+a-1" in bspec)
- using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
- done
- } note vd2 = this
- have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
- using * a by (fastforce intro: integrable_subinterval_real)
- have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
- apply (rule integrable_subinterval_real)
- using * a by auto
- have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
- has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}"
- apply (rule has_integral_spike_finite
- [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
- using s apply blast
- using a apply (auto simp: algebra_simps vd1)
- apply (force simp: shiftpath_def add.commute)
- using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
- apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
- done
- moreover
- have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
- has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}"
- apply (rule has_integral_spike_finite
- [where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
- using s apply blast
- using a apply (auto simp: algebra_simps vd2)
- apply (force simp: shiftpath_def add.commute)
- using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
- apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
- apply (simp add: algebra_simps)
- done
- ultimately show ?thesis
- using a
- by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
-qed
-
-lemma has_contour_integral_shiftpath_D:
- assumes "(f has_contour_integral i) (shiftpath a g)"
- "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "(f has_contour_integral i) g"
-proof -
- obtain s
- where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- { fix x
- assume x: "0 < x" "x < 1" "x \<notin> s"
- then have gx: "g differentiable at x"
- using g by auto
- have "vector_derivative g (at x within {0..1}) =
- vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
- apply (rule vector_derivative_at_within_ivl
- [OF has_vector_derivative_transform_within_open
- [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
- using s g assms x
- apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
- at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
- apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
- apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
- done
- } note vd = this
- have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
- using assms by (auto intro!: has_contour_integral_shiftpath)
- show ?thesis
- apply (simp add: has_contour_integral_def)
- apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_contour_integral_def]])
- using s assms vd
- apply (auto simp: Path_Connected.shiftpath_shiftpath)
- done
-qed
-
-lemma has_contour_integral_shiftpath_eq:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
- using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
-
-lemma contour_integrable_on_shiftpath_eq:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
-using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
-
-lemma contour_integral_shiftpath:
- assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
- shows "contour_integral (shiftpath a g) f = contour_integral g f"
- using assms
- by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close>
-
-lemma has_vector_derivative_linepath_within:
- "(linepath a b has_vector_derivative (b - a)) (at x within s)"
-apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
-apply (rule derivative_eq_intros | simp)+
-done
-
-lemma vector_derivative_linepath_within:
- "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
- apply (rule vector_derivative_within_cbox [of 0 "1::real", simplified])
- apply (auto simp: has_vector_derivative_linepath_within)
- done
-
-lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
- by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
-
-lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
- apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
- apply (rule_tac x="{}" in exI)
- apply (simp add: differentiable_on_def differentiable_def)
- using has_vector_derivative_def has_vector_derivative_linepath_within
- apply (fastforce simp add: continuous_on_eq_continuous_within)
- done
-
-lemma has_contour_integral_linepath:
- shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
- ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
- by (simp add: has_contour_integral)
-
-lemma linepath_in_path:
- shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
- by (auto simp: segment linepath_def)
-
-lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
- by (auto simp: segment linepath_def)
-
-lemma linepath_in_convex_hull:
- fixes x::real
- assumes a: "a \<in> convex hull s"
- and b: "b \<in> convex hull s"
- and x: "0\<le>x" "x\<le>1"
- shows "linepath a b x \<in> convex hull s"
- apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
- using x
- apply (auto simp: linepath_image_01 [symmetric])
- done
-
-lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
- by (simp add: linepath_def)
-
-lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
- by (simp add: linepath_def)
-
-lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
- by (simp add: has_contour_integral_linepath)
-
-lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
- using has_contour_integral_unique by blast
-
-lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
- using has_contour_integral_trivial contour_integral_unique by blast
-
-lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
- by (auto simp: linepath_def)
-
-lemma bounded_linear_linepath:
- assumes "bounded_linear f"
- shows "f (linepath a b x) = linepath (f a) (f b) x"
-proof -
- interpret f: bounded_linear f by fact
- show ?thesis by (simp add: linepath_def f.add f.scale)
-qed
-
-lemma bounded_linear_linepath':
- assumes "bounded_linear f"
- shows "f \<circ> linepath a b = linepath (f a) (f b)"
- using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)
-
-lemma cnj_linepath: "cnj (linepath a b x) = linepath (cnj a) (cnj b) x"
- by (simp add: linepath_def)
-
-lemma cnj_linepath': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)"
- by (simp add: linepath_def fun_eq_iff)
-
-subsection\<open>Relation to subpath construction\<close>
-
-lemma valid_path_subpath:
- fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
- assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
- shows "valid_path(subpath u v g)"
-proof (cases "v=u")
- case True
- then show ?thesis
- unfolding valid_path_def subpath_def
- by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
-next
- case False
- have "(g \<circ> (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
- apply (rule piecewise_C1_differentiable_compose)
- apply (simp add: C1_differentiable_imp_piecewise)
- apply (simp add: image_affinity_atLeastAtMost)
- using assms False
- apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
- apply (subst Int_commute)
- apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
- done
- then show ?thesis
- by (auto simp: o_def valid_path_def subpath_def)
-qed
-
-lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
- by (simp add: has_contour_integral subpath_def)
-
-lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
- using has_contour_integral_subpath_refl contour_integrable_on_def by blast
-
-lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
- by (simp add: contour_integral_unique)
-
-lemma has_contour_integral_subpath:
- assumes f: "f contour_integrable_on g" and g: "valid_path g"
- and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "(f has_contour_integral integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
- (subpath u v g)"
-proof (cases "v=u")
- case True
- then show ?thesis
- using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
-next
- case False
- obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
- using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
- have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
- has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
- {0..1}"
- using f uv
- apply (simp add: contour_integrable_on subpath_def has_contour_integral)
- apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
- apply (simp_all add: has_integral_integral)
- apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
- apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
- apply (simp add: divide_simps False)
- done
- { fix x
- have "x \<in> {0..1} \<Longrightarrow>
- x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
- vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
- apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
- apply (intro derivative_eq_intros | simp)+
- apply (cut_tac s [of "(v - u) * x + u"])
- using uv mult_left_le [of x "v-u"]
- apply (auto simp: vector_derivative_works)
- done
- } note vd = this
- show ?thesis
- apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
- using fs assms
- apply (simp add: False subpath_def has_contour_integral)
- apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
- apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
- done
-qed
-
-lemma contour_integrable_subpath:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
- shows "f contour_integrable_on (subpath u v g)"
- apply (cases u v rule: linorder_class.le_cases)
- apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
- apply (subst reversepath_subpath [symmetric])
- apply (rule contour_integrable_reversepath)
- using assms apply (blast intro: valid_path_subpath)
- apply (simp add: contour_integrable_on_def)
- using assms apply (blast intro: has_contour_integral_subpath)
- done
-
-lemma has_integral_contour_integral_subpath:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
- has_integral contour_integral (subpath u v g) f) {u..v}"
- using assms
- apply (auto simp: has_integral_integrable_integral)
- apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
- apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
- done
-
-lemma contour_integral_subcontour_integral:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
- shows "contour_integral (subpath u v g) f =
- integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
- using assms has_contour_integral_subpath contour_integral_unique by blast
-
-lemma contour_integral_subpath_combine_less:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
- "u<v" "v<w"
- shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
- contour_integral (subpath u w g) f"
- using assms apply (auto simp: contour_integral_subcontour_integral)
- apply (rule integral_combine, auto)
- apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
- apply (auto simp: contour_integrable_on)
- done
-
-lemma contour_integral_subpath_combine:
- assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
- shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
- contour_integral (subpath u w g) f"
-proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
- case True
- have *: "subpath v u g = reversepath(subpath u v g) \<and>
- subpath w u g = reversepath(subpath u w g) \<and>
- subpath w v g = reversepath(subpath v w g)"
- by (auto simp: reversepath_subpath)
- have "u < v \<and> v < w \<or>
- u < w \<and> w < v \<or>
- v < u \<and> u < w \<or>
- v < w \<and> w < u \<or>
- w < u \<and> u < v \<or>
- w < v \<and> v < u"
- using True assms by linarith
- with assms show ?thesis
- using contour_integral_subpath_combine_less [of f g u v w]
- contour_integral_subpath_combine_less [of f g u w v]
- contour_integral_subpath_combine_less [of f g v u w]
- contour_integral_subpath_combine_less [of f g v w u]
- contour_integral_subpath_combine_less [of f g w u v]
- contour_integral_subpath_combine_less [of f g w v u]
- apply simp
- apply (elim disjE)
- apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
- valid_path_subpath algebra_simps)
- done
-next
- case False
- then show ?thesis
- apply (auto)
- using assms
- by (metis eq_neg_iff_add_eq_0 contour_integral_reversepath reversepath_subpath valid_path_subpath)
-qed
-
-lemma contour_integral_integral:
- "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
- by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
-
-lemma contour_integral_cong:
- assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x"
- shows "contour_integral g f = contour_integral g' f'"
- unfolding contour_integral_integral using assms
- by (intro integral_cong) (auto simp: path_image_def)
-
-
-text \<open>Contour integral along a segment on the real axis\<close>
-
-lemma has_contour_integral_linepath_Reals_iff:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
- ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
-proof -
- from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
- by (simp_all add: complex_eq_iff)
- from assms have "a \<noteq> b" by auto
- have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow>
- ((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}"
- by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
- (insert assms, simp_all add: field_simps scaleR_conv_of_real)
- also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) =
- (\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))"
- using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
- also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow>
- ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
- by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
- also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
- by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
- finally show ?thesis by simp
-qed
-
-lemma contour_integrable_linepath_Reals_iff:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "(f contour_integrable_on linepath a b) \<longleftrightarrow>
- (\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}"
- using has_contour_integral_linepath_Reals_iff[OF assms, of f]
- by (auto simp: contour_integrable_on_def integrable_on_def)
-
-lemma contour_integral_linepath_Reals_eq:
- fixes a b :: complex and f :: "complex \<Rightarrow> complex"
- assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
- shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
-proof (cases "f contour_integrable_on linepath a b")
- case True
- thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
- using has_contour_integral_integral has_contour_integral_unique by blast
-next
- case False
- thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
- by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-
-
-text\<open>Cauchy's theorem where there's a primitive\<close>
-
-lemma contour_integral_primitive_lemma:
- fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
- assumes "a \<le> b"
- and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
- shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
- has_integral (f(g b) - f(g a))) {a..b}"
-proof -
- obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
- using assms by (auto simp: piecewise_differentiable_on_def)
- have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
- apply (rule continuous_on_compose [OF cg, unfolded o_def])
- using assms
- apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
- done
- { fix x::real
- assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
- then have "g differentiable at x within {a..b}"
- using k by (simp add: differentiable_at_withinI)
- then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
- by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
- then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
- by (simp add: has_vector_derivative_def scaleR_conv_of_real)
- have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
- using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
- then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
- by (simp add: has_field_derivative_def)
- have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
- using diff_chain_within [OF gdiff fdiff]
- by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
- } note * = this
- show ?thesis
- apply (rule fundamental_theorem_of_calculus_interior_strong)
- using k assms cfg *
- apply (auto simp: at_within_Icc_at)
- done
-qed
-
-lemma contour_integral_primitive:
- assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "valid_path g" "path_image g \<subseteq> s"
- shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
- using assms
- apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
- apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
- done
-
-corollary Cauchy_theorem_primitive:
- assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
- and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
- shows "(f' has_contour_integral 0) g"
- using assms
- by (metis diff_self contour_integral_primitive)
-
-text\<open>Existence of path integral for continuous function\<close>
-lemma contour_integrable_continuous_linepath:
- assumes "continuous_on (closed_segment a b) f"
- shows "f contour_integrable_on (linepath a b)"
-proof -
- have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
- apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
- apply (rule continuous_intros | simp add: assms)+
- done
- then show ?thesis
- apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
- apply (rule integrable_continuous [of 0 "1::real", simplified])
- apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
- apply (auto simp: vector_derivative_linepath_within)
- done
-qed
-
-lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
- by (rule has_derivative_imp_has_field_derivative)
- (rule derivative_intros | simp)+
-
-lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
- apply (rule contour_integral_unique)
- using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
- apply (auto simp: field_simps has_field_der_id)
- done
-
-lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
- by (simp add: contour_integrable_continuous_linepath)
-
-lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
- by (simp add: contour_integrable_continuous_linepath)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close>
-
-lemma has_contour_integral_neg:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
- by (simp add: has_integral_neg has_contour_integral_def)
-
-lemma has_contour_integral_add:
- "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
- by (simp add: has_integral_add has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_diff:
- "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
- by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
-
-lemma has_contour_integral_lmul:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
-apply (simp add: has_contour_integral_def)
-apply (drule has_integral_mult_right)
-apply (simp add: algebra_simps)
-done
-
-lemma has_contour_integral_rmul:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
-apply (drule has_contour_integral_lmul)
-apply (simp add: mult.commute)
-done
-
-lemma has_contour_integral_div:
- "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
- by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
-
-lemma has_contour_integral_eq:
- "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
-apply (simp add: path_image_def has_contour_integral_def)
-by (metis (no_types, lifting) image_eqI has_integral_eq)
-
-lemma has_contour_integral_bound_linepath:
- assumes "(f has_contour_integral i) (linepath a b)"
- "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
- shows "norm i \<le> B * norm(b - a)"
-proof -
- { fix x::real
- assume x: "0 \<le> x" "x \<le> 1"
- have "norm (f (linepath a b x)) *
- norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
- by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
- } note * = this
- have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
- apply (rule has_integral_bound
- [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
- using assms * unfolding has_contour_integral_def
- apply (auto simp: norm_mult)
- done
- then show ?thesis
- by (auto simp: content_real)
-qed
-
-(*UNUSED
-lemma has_contour_integral_bound_linepath_strong:
- fixes a :: real and f :: "complex \<Rightarrow> real"
- assumes "(f has_contour_integral i) (linepath a b)"
- "finite k"
- "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
- shows "norm i \<le> B*norm(b - a)"
-*)
-
-lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
- unfolding has_contour_integral_linepath
- by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
-
-lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
- by (simp add: has_contour_integral_def)
-
-lemma has_contour_integral_is_0:
- "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
- by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
-
-lemma has_contour_integral_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
- by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close>
-
-lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
- by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
-
-lemma contour_integral_neg:
- "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
-
-lemma contour_integral_add:
- "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
- contour_integral g f1 + contour_integral g f2"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
-
-lemma contour_integral_diff:
- "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
- contour_integral g f1 - contour_integral g f2"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
-
-lemma contour_integral_lmul:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
-
-lemma contour_integral_rmul:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
-
-lemma contour_integral_div:
- shows "f contour_integrable_on g
- \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
- by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
-
-lemma contour_integral_eq:
- "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
- apply (simp add: contour_integral_def)
- using has_contour_integral_eq
- by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
-
-lemma contour_integral_eq_0:
- "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
- by (simp add: has_contour_integral_is_0 contour_integral_unique)
-
-lemma contour_integral_bound_linepath:
- shows
- "\<lbrakk>f contour_integrable_on (linepath a b);
- 0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
- apply (rule has_contour_integral_bound_linepath [of f])
- apply (auto simp: has_contour_integral_integral)
- done
-
-lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
- by (simp add: contour_integral_unique has_contour_integral_0)
-
-lemma contour_integral_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
- \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
- by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
-
-lemma contour_integrable_eq:
- "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
- unfolding contour_integrable_on_def
- by (metis has_contour_integral_eq)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close>
-
-lemma contour_integrable_neg:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
- using has_contour_integral_neg contour_integrable_on_def by blast
-
-lemma contour_integrable_add:
- "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
- using has_contour_integral_add contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_diff:
- "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
- using has_contour_integral_diff contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_lmul:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
- using has_contour_integral_lmul contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_rmul:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
- using has_contour_integral_rmul contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_div:
- "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
- using has_contour_integral_div contour_integrable_on_def
- by fastforce
-
-lemma contour_integrable_sum:
- "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
- \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
- unfolding contour_integrable_on_def
- by (metis has_contour_integral_sum)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close>
-
-lemma has_contour_integral_reverse_linepath:
- "(f has_contour_integral i) (linepath a b)
- \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
- using has_contour_integral_reversepath valid_path_linepath by fastforce
-
-lemma contour_integral_reverse_linepath:
- "continuous_on (closed_segment a b) f
- \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
-apply (rule contour_integral_unique)
-apply (rule has_contour_integral_reverse_linepath)
-by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
-
-
-(* Splitting a path integral in a flat way.*)
-
-lemma has_contour_integral_split:
- assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "(f has_contour_integral (i + j)) (linepath a b)"
-proof (cases "k = 0 \<or> k = 1")
- case True
- then show ?thesis
- using assms by auto
-next
- case False
- then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
- using assms by auto
- have c': "c = k *\<^sub>R (b - a) + a"
- by (metis diff_add_cancel c)
- have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
- by (simp add: algebra_simps c')
- { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
- have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
- using False apply (simp add: c' algebra_simps)
- apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
- done
- have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
- using k has_integral_affinity01 [OF *, of "inverse k" "0"]
- apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
- apply (auto dest: has_integral_cmul [where c = "inverse k"])
- done
- } note fi = this
- { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
- have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
- using k
- apply (simp add: c' field_simps)
- apply (simp add: scaleR_conv_of_real divide_simps)
- apply (simp add: field_simps)
- done
- have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
- using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
- apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
- apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
- done
- } note fj = this
- show ?thesis
- using f k
- apply (simp add: has_contour_integral_linepath)
- apply (simp add: linepath_def)
- apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
- done
-qed
-
-lemma continuous_on_closed_segment_transform:
- assumes f: "continuous_on (closed_segment a b) f"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "continuous_on (closed_segment a c) f"
-proof -
- have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
- using c by (simp add: algebra_simps)
- have "closed_segment a c \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
- then show "continuous_on (closed_segment a c) f"
- by (rule continuous_on_subset [OF f])
-qed
-
-lemma contour_integral_split:
- assumes f: "continuous_on (closed_segment a b) f"
- and k: "0 \<le> k" "k \<le> 1"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
-proof -
- have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
- using c by (simp add: algebra_simps)
- have "closed_segment a c \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
- moreover have "closed_segment c b \<subseteq> closed_segment a b"
- by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
- ultimately
- have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
- by (auto intro: continuous_on_subset [OF f])
- show ?thesis
- by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
-qed
-
-lemma contour_integral_split_linepath:
- assumes f: "continuous_on (closed_segment a b) f"
- and c: "c \<in> closed_segment a b"
- shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
- using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
-
-text\<open>The special case of midpoints used in the main quadrisection\<close>
-
-lemma has_contour_integral_midpoint:
- assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
- "(f has_contour_integral j) (linepath (midpoint a b) b)"
- shows "(f has_contour_integral (i + j)) (linepath a b)"
- apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
- using assms
- apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
- done
-
-lemma contour_integral_midpoint:
- "continuous_on (closed_segment a b) f
- \<Longrightarrow> contour_integral (linepath a b) f =
- contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
- apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
- apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
- done
-
-
-text\<open>A couple of special case lemmas that are useful below\<close>
-
-lemma triangle_linear_has_chain_integral:
- "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
- apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
- apply (auto intro!: derivative_eq_intros)
- done
-
-lemma has_chain_integral_chain_integral3:
- "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
- \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
- apply (subst contour_integral_unique [symmetric], assumption)
- apply (drule has_contour_integral_integrable)
- apply (simp add: valid_path_join)
- done
-
-lemma has_chain_integral_chain_integral4:
- "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
- \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i"
- apply (subst contour_integral_unique [symmetric], assumption)
- apply (drule has_contour_integral_integrable)
- apply (simp add: valid_path_join)
- done
-
-subsection\<open>Reversing the order in a double path integral\<close>
-
-text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
-
-lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
- by (auto simp: cbox_Pair_eq)
-
-lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
- by (auto simp: cbox_Pair_eq)
-
-proposition contour_integral_swap:
- assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
- and vp: "valid_path g" "valid_path h"
- and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
- and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
- shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
- contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
-proof -
- have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
- using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
- by (rule ext) simp
- have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
- by (rule ext) simp
- have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
- by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
- have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
- by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
- have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
- by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
- then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
- using continuous_on_mult gvcon integrable_continuous_real by blast
- have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
- by auto
- then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
- apply (rule ssubst)
- apply (rule continuous_intros | simp add: gvcon)+
- done
- have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
- by auto
- then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
- apply (rule ssubst)
- apply (rule continuous_intros | simp add: hvcon)+
- done
- have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
- by auto
- then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
- apply (rule ssubst)
- apply (rule gcon hcon continuous_intros | simp)+
- apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
- done
- have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
- integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
- proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
- show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
- unfolding contour_integrable_on
- apply (rule integrable_continuous_real)
- apply (rule continuous_on_mult [OF _ hvcon])
- apply (subst fgh1)
- apply (rule fcon_im1 hcon continuous_intros | simp)+
- done
- qed
- also have "\<dots> = integral {0..1}
- (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
- unfolding contour_integral_integral
- apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
- apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
- unfolding integral_mult_left [symmetric]
- apply (simp only: mult_ac)
- done
- also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
- unfolding contour_integral_integral
- apply (rule integral_cong)
- unfolding integral_mult_left [symmetric]
- apply (simp add: algebra_simps)
- done
- finally show ?thesis
- by (simp add: contour_integral_integral)
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The key quadrisection step\<close>
-
-lemma norm_sum_half:
- assumes "norm(a + b) \<ge> e"
- shows "norm a \<ge> e/2 \<or> norm b \<ge> e/2"
-proof -
- have "e \<le> norm (- a - b)"
- by (simp add: add.commute assms norm_minus_commute)
- thus ?thesis
- using norm_triangle_ineq4 order_trans by fastforce
-qed
-
-lemma norm_sum_lemma:
- assumes "e \<le> norm (a + b + c + d)"
- shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
-proof -
- have "e \<le> norm ((a + b) + (c + d))" using assms
- by (simp add: algebra_simps)
- then show ?thesis
- by (auto dest!: norm_sum_half)
-qed
-
-lemma Cauchy_theorem_quadrisection:
- assumes f: "continuous_on (convex hull {a,b,c}) f"
- and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
- and e: "e * K^2 \<le>
- norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
- shows "\<exists>a' b' c'.
- a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
- dist a' b' \<le> K/2 \<and> dist b' c' \<le> K/2 \<and> dist c' a' \<le> K/2 \<and>
- e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
- (is "\<exists>x y z. ?\<Phi> x y z")
-proof -
- note divide_le_eq_numeral1 [simp del]
- define a' where "a' = midpoint b c"
- define b' where "b' = midpoint c a"
- define c' where "c' = midpoint a b"
- have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
- using f continuous_on_subset segments_subset_convex_hull by metis+
- have fcont': "continuous_on (closed_segment c' b') f"
- "continuous_on (closed_segment a' c') f"
- "continuous_on (closed_segment b' a') f"
- unfolding a'_def b'_def c'_def
- by (rule continuous_on_subset [OF f],
- metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
- let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
- have *: "?pathint a b + ?pathint b c + ?pathint c a =
- (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
- (?pathint a' c' + ?pathint c' b + ?pathint b a') +
- (?pathint a' c + ?pathint c b' + ?pathint b' a') +
- (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
- by (simp add: fcont' contour_integral_reverse_linepath) (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
- have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
- by (metis left_diff_distrib mult.commute norm_mult_numeral1)
- have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
- by (simp add: norm_minus_commute)
- consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
- "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
- "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
- "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
- using assms unfolding * by (blast intro: that dest!: norm_sum_lemma)
- then show ?thesis
- proof cases
- case 1 then have "?\<Phi> a c' b'"
- using assms
- apply (clarsimp simp: c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
- done
- then show ?thesis by blast
- next
- case 2 then have "?\<Phi> a' c' b"
- using assms
- apply (clarsimp simp: a'_def c'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
- done
- then show ?thesis by blast
- next
- case 3 then have "?\<Phi> a' c b'"
- using assms
- apply (clarsimp simp: a'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
- done
- then show ?thesis by blast
- next
- case 4 then have "?\<Phi> a' b' c'"
- using assms
- apply (clarsimp simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
- apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
- done
- then show ?thesis by blast
- qed
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for triangles\<close>
-
-lemma triangle_points_closer:
- fixes a::complex
- shows "\<lbrakk>x \<in> convex hull {a,b,c}; y \<in> convex hull {a,b,c}\<rbrakk>
- \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
- norm(x - y) \<le> norm(b - c) \<or>
- norm(x - y) \<le> norm(c - a)"
- using simplex_extremal_le [of "{a,b,c}"]
- by (auto simp: norm_minus_commute)
-
-lemma holomorphic_point_small_triangle:
- assumes x: "x \<in> S"
- and f: "continuous_on S f"
- and cd: "f field_differentiable (at x within S)"
- and e: "0 < e"
- shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and>
- x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> S
- \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
- contour_integral(linepath c a) f)
- \<le> e*(dist a b + dist b c + dist c a)^2"
- (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
-proof -
- have le_of_3: "\<And>a x y z. \<lbrakk>0 \<le> x*y; 0 \<le> x*z; 0 \<le> y*z; a \<le> (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\<rbrakk>
- \<Longrightarrow> a \<le> e*(x + y + z)^2"
- by (simp add: algebra_simps power2_eq_square)
- have disj_le: "\<lbrakk>x \<le> a \<or> x \<le> b \<or> x \<le> c; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> x \<le> a + b + c"
- for x::real and a b c
- by linarith
- have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
- if "convex hull {a, b, c} \<subseteq> S" for a b c
- using segments_subset_convex_hull that
- by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
- note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
- { fix f' a b c d
- assume d: "0 < d"
- and f': "\<And>y. \<lbrakk>cmod (y - x) \<le> d; y \<in> S\<rbrakk> \<Longrightarrow> cmod (f y - f x - f' * (y - x)) \<le> e * cmod (y - x)"
- and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
- and xc: "x \<in> convex hull {a, b, c}"
- and S: "convex hull {a, b, c} \<subseteq> S"
- have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
- contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
- contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
- contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
- apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF S])
- apply (simp add: field_simps)
- done
- { fix y
- assume yc: "y \<in> convex hull {a,b,c}"
- have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
- proof (rule f')
- show "cmod (y - x) \<le> d"
- by (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
- qed (use S yc in blast)
- also have "\<dots> \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
- by (simp add: yc e xc disj_le [OF triangle_points_closer])
- finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
- } note cm_le = this
- have "?normle a b c"
- unfolding dist_norm pa
- apply (rule le_of_3)
- using f' xc S e
- apply simp_all
- apply (intro norm_triangle_le add_mono path_bound)
- apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
- apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
- done
- } note * = this
- show ?thesis
- using cd e
- apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
- apply (clarify dest!: spec mp)
- using * unfolding dist_norm
- apply blast
- done
-qed
-
-
-text\<open>Hence the most basic theorem for a triangle.\<close>
-
-locale Chain =
- fixes x0 At Follows
- assumes At0: "At x0 0"
- and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
-begin
- primrec f where
- "f 0 = x0"
- | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
-
- lemma At: "At (f n) n"
- proof (induct n)
- case 0 show ?case
- by (simp add: At0)
- next
- case (Suc n) show ?case
- by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
- qed
-
- lemma Follows: "Follows (f(Suc n)) (f n)"
- by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
-
- declare f.simps(2) [simp del]
-end
-
-lemma Chain3:
- assumes At0: "At x0 y0 z0 0"
- and AtSuc: "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z"
- obtains f g h where
- "f 0 = x0" "g 0 = y0" "h 0 = z0"
- "\<And>n. At (f n) (g n) (h n) n"
- "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
-proof -
- interpret three: Chain "(x0,y0,z0)" "\<lambda>(x,y,z). At x y z" "\<lambda>(x',y',z'). \<lambda>(x,y,z). Follows x' y' z' x y z"
- apply unfold_locales
- using At0 AtSuc by auto
- show ?thesis
- apply (rule that [of "\<lambda>n. fst (three.f n)" "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
- using three.At three.Follows
- apply simp_all
- apply (simp_all add: split_beta')
- done
-qed
-
-proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle:
- assumes "f holomorphic_on (convex hull {a,b,c})"
- shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
-proof -
- have contf: "continuous_on (convex hull {a,b,c}) f"
- by (metis assms holomorphic_on_imp_continuous_on)
- let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
- { fix y::complex
- assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
- and ynz: "y \<noteq> 0"
- define K where "K = 1 + max (dist a b) (max (dist b c) (dist c a))"
- define e where "e = norm y / K^2"
- have K1: "K \<ge> 1" by (simp add: K_def max.coboundedI1)
- then have K: "K > 0" by linarith
- have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
- by (simp_all add: K_def)
- have e: "e > 0"
- unfolding e_def using ynz K1 by simp
- define At where "At x y z n \<longleftrightarrow>
- convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
- dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
- norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
- for x y z n
- have At0: "At a b c 0"
- using fy
- by (simp add: At_def e_def has_chain_integral_chain_integral3)
- { fix x y z n
- assume At: "At x y z n"
- then have contf': "continuous_on (convex hull {x,y,z}) f"
- using contf At_def continuous_on_subset by metis
- have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
- using At Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
- apply (simp add: At_def algebra_simps)
- apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
- done
- } note AtSuc = this
- obtain fa fb fc
- where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
- and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
- and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
- "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
- "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
- and no: "\<And>n. norm(?pathint (fa n) (fb n) +
- ?pathint (fb n) (fc n) +
- ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
- and conv_le: "\<And>n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \<subseteq> convex hull {fa n, fb n, fc n}"
- apply (rule Chain3 [of At, OF At0 AtSuc])
- apply (auto simp: At_def)
- done
- obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}"
- proof (rule bounded_closed_nest)
- show "\<And>n. closed (convex hull {fa n, fb n, fc n})"
- by (simp add: compact_imp_closed finite_imp_compact_convex_hull)
- show "\<And>m n. m \<le> n \<Longrightarrow> convex hull {fa n, fb n, fc n} \<subseteq> convex hull {fa m, fb m, fc m}"
- by (erule transitive_stepwise_le) (auto simp: conv_le)
- qed (fastforce intro: finite_imp_bounded_convex_hull)+
- then have xin: "x \<in> convex hull {a,b,c}"
- using assms f0 by blast
- then have fx: "f field_differentiable at x within (convex hull {a,b,c})"
- using assms holomorphic_on_def by blast
- { fix k n
- assume k: "0 < k"
- and le:
- "\<And>x' y' z'.
- \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
- x \<in> convex hull {x',y',z'};
- convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
- \<Longrightarrow>
- cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
- \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
- and Kk: "K / k < 2 ^ n"
- have "K / 2 ^ n < k" using Kk k
- by (auto simp: field_simps)
- then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
- using dist [of n] k
- by linarith+
- have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
- \<le> (3 * K / 2 ^ n)\<^sup>2"
- using dist [of n] e K
- by (simp add: abs_le_square_iff [symmetric])
- have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
- by linarith
- have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2"
- using ynz dle e mult_le_cancel_left_pos by blast
- also have "\<dots> <
- cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
- using no [of n] e K
- apply (simp add: e_def field_simps)
- apply (simp only: zero_less_norm_iff [symmetric])
- done
- finally have False
- using le [OF DD x cosb] by auto
- } then
- have ?thesis
- using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
- apply clarsimp
- apply (rule_tac y1="K/k" in exE [OF real_arch_pow[of 2]], force+)
- done
- }
- moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
- by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1)
- segments_subset_convex_hull(3) segments_subset_convex_hull(5))
- ultimately show ?thesis
- using has_contour_integral_integral by fastforce
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Version needing function holomorphic in interior only\<close>
-
-lemma Cauchy_theorem_flat_lemma:
- assumes f: "continuous_on (convex hull {a,b,c}) f"
- and c: "c - a = k *\<^sub>R (b - a)"
- and k: "0 \<le> k"
- shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof -
- have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
- using f continuous_on_subset segments_subset_convex_hull by metis+
- show ?thesis
- proof (cases "k \<le> 1")
- case True show ?thesis
- by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc)
- next
- case False then show ?thesis
- using fabc c
- apply (subst contour_integral_split [of a c f "1/k" b, symmetric])
- apply (metis closed_segment_commute fabc(3))
- apply (auto simp: k contour_integral_reverse_linepath)
- done
- qed
-qed
-
-lemma Cauchy_theorem_flat:
- assumes f: "continuous_on (convex hull {a,b,c}) f"
- and c: "c - a = k *\<^sub>R (b - a)"
- shows "contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof (cases "0 \<le> k")
- case True with assms show ?thesis
- by (blast intro: Cauchy_theorem_flat_lemma)
-next
- case False
- have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
- using f continuous_on_subset segments_subset_convex_hull by metis+
- moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f +
- contour_integral (linepath c b) f = 0"
- apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
- using False c
- apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
- done
- ultimately show ?thesis
- apply (auto simp: contour_integral_reverse_linepath)
- using add_eq_0_iff by force
-qed
-
-lemma Cauchy_theorem_triangle_interior:
- assumes contf: "continuous_on (convex hull {a,b,c}) f"
- and holf: "f holomorphic_on interior (convex hull {a,b,c})"
- shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
-proof -
- have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
- using contf continuous_on_subset segments_subset_convex_hull by metis+
- have "bounded (f ` (convex hull {a,b,c}))"
- by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
- then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
- by (auto simp: dest!: bounded_pos [THEN iffD1])
- have "bounded (convex hull {a,b,c})"
- by (simp add: bounded_convex_hull)
- then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
- using bounded_pos_less by blast
- then have diff_2C: "norm(x - y) \<le> 2*C"
- if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
- proof -
- have "cmod x \<le> C"
- using x by (meson Cno not_le not_less_iff_gr_or_eq)
- hence "cmod (x - y) \<le> C + C"
- using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
- thus "cmod (x - y) \<le> 2 * C"
- by (metis mult_2)
- qed
- have contf': "continuous_on (convex hull {b,a,c}) f"
- using contf by (simp add: insert_commute)
- { fix y::complex
- assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
- and ynz: "y \<noteq> 0"
- have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y"
- by (rule has_chain_integral_chain_integral3 [OF fy])
- have ?thesis
- proof (cases "c=a \<or> a=b \<or> b=c")
- case True then show ?thesis
- using Cauchy_theorem_flat [OF contf, of 0]
- using has_chain_integral_chain_integral3 [OF fy] ynz
- by (force simp: fabc contour_integral_reverse_linepath)
- next
- case False
- then have car3: "card {a, b, c} = Suc (DIM(complex))"
- by auto
- { assume "interior(convex hull {a,b,c}) = {}"
- then have "collinear{a,b,c}"
- using interior_convex_hull_eq_empty [OF car3]
- by (simp add: collinear_3_eq_affine_dependent)
- with False obtain d where "c \<noteq> a" "a \<noteq> b" "b \<noteq> c" "c - b = d *\<^sub>R (a - b)"
- by (auto simp: collinear_3 collinear_lemma)
- then have "False"
- using False Cauchy_theorem_flat [OF contf'] pi_eq_y ynz
- by (simp add: fabc add_eq_0_iff contour_integral_reverse_linepath)
- }
- then obtain d where d: "d \<in> interior (convex hull {a, b, c})"
- by blast
- { fix d1
- assume d1_pos: "0 < d1"
- and d1: "\<And>x x'. \<lbrakk>x\<in>convex hull {a, b, c}; x'\<in>convex hull {a, b, c}; cmod (x' - x) < d1\<rbrakk>
- \<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)"
- define e where "e = min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))"
- define shrink where "shrink x = x - e *\<^sub>R (x - d)" for x
- let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
- have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B"
- using d1_pos \<open>C>0\<close> \<open>B>0\<close> ynz by (simp_all add: e_def)
- then have eCB: "24 * e * C * B \<le> cmod y"
- using \<open>C>0\<close> \<open>B>0\<close> by (simp add: field_simps)
- have e_le_d1: "e * (4 * C) \<le> d1"
- using e \<open>C>0\<close> by (simp add: field_simps)
- have "shrink a \<in> interior(convex hull {a,b,c})"
- "shrink b \<in> interior(convex hull {a,b,c})"
- "shrink c \<in> interior(convex hull {a,b,c})"
- using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
- then have fhp0: "(f has_contour_integral 0)
- (linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))"
- by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal)
- then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0"
- by (simp add: has_chain_integral_chain_integral3)
- have fpi_abc: "f contour_integrable_on linepath (shrink a) (shrink b)"
- "f contour_integrable_on linepath (shrink b) (shrink c)"
- "f contour_integrable_on linepath (shrink c) (shrink a)"
- using fhp0 by (auto simp: valid_path_join dest: has_contour_integral_integrable)
- have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)"
- using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric])
- have sh_eq: "\<And>a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)"
- by (simp add: algebra_simps)
- have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
- using False \<open>C>0\<close> diff_2C [of b a] ynz
- by (auto simp: field_split_simps hull_inc)
- have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u
- apply (cases "x=0", simp add: \<open>0<C\<close>)
- using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
- { fix u v
- assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v"
- and fpi_uv: "f contour_integrable_on linepath (shrink u) (shrink v)"
- have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})"
- "shrink v \<in> interior(convex hull {a,b,c})"
- using d e uv
- by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
- have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B"
- using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD])
- have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y"
- apply (rule order_trans [OF _ eCB])
- using e \<open>B>0\<close> diff_2C [of u v] uv
- by (auto simp: field_simps)
- { fix x::real assume x: "0\<le>x" "x\<le>1"
- have cmod_less_4C: "cmod ((1 - x) *\<^sub>R u - (1 - x) *\<^sub>R d) + cmod (x *\<^sub>R v - x *\<^sub>R d) < (C+C) + (C+C)"
- apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0])
- using uv x d interior_subset
- apply (auto simp: hull_inc intro!: less_C)
- done
- have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))"
- by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real)
- have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1"
- apply (simp only: ll norm_mult scaleR_diff_right)
- using \<open>e>0\<close> cmod_less_4C apply (force intro: norm_triangle_lt less_le_trans [OF _ e_le_d1])
- done
- have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)"
- using x uv shr_uv cmod_less_dt
- by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
- also have "\<dots> \<le> cmod y / cmod (v - u) / 12"
- using False uv \<open>C>0\<close> diff_2C [of v u] ynz
- by (auto simp: field_split_simps hull_inc)
- finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
- by simp
- then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y"
- using uv False by (auto simp: field_simps)
- have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
- cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
- \<le> B * (cmod y / 24 / C / B * 2 * C) + 2 * C * (cmod y / 24 / C)"
- apply (rule add_mono [OF mult_mono])
- using By_uv e \<open>0 < B\<close> \<open>0 < C\<close> x apply (simp_all add: cmod_fuv cmod_shr cmod_12_le)
- apply (simp add: field_simps)
- done
- also have "\<dots> \<le> cmod y / 6"
- by simp
- finally have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
- cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
- \<le> cmod y / 6" .
- } note cmod_diff_le = this
- have f_uv: "continuous_on (closed_segment u v) f"
- by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull])
- have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)"
- by (simp add: algebra_simps)
- have "norm (?pathint (shrink u) (shrink v) - ?pathint u v)
- \<le> (B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)) * content (cbox 0 (1::real))"
- apply (rule has_integral_bound
- [of _ "\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)"
- _ 0 1])
- using ynz \<open>0 < B\<close> \<open>0 < C\<close>
- apply (simp_all del: le_divide_eq_numeral1)
- apply (simp add: has_integral_diff has_contour_integral_linepath [symmetric] has_contour_integral_integral
- fpi_uv f_uv contour_integrable_continuous_linepath)
- apply (auto simp: ** norm_triangle_le norm_mult cmod_diff_le simp del: le_divide_eq_numeral1)
- done
- also have "\<dots> \<le> norm y / 6"
- by simp
- finally have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6" .
- } note * = this
- have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6"
- using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
- moreover
- have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6"
- using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
- moreover
- have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6"
- using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
- ultimately
- have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) +
- (?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a))
- \<le> norm y / 6 + norm y / 6 + norm y / 6"
- by (metis norm_triangle_le add_mono)
- also have "\<dots> = norm y / 2"
- by simp
- finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) -
- (?pathint a b + ?pathint b c + ?pathint c a))
- \<le> norm y / 2"
- by (simp add: algebra_simps)
- then
- have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2"
- by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff)
- then have "False"
- using pi_eq_y ynz by auto
- }
- moreover have "uniformly_continuous_on (convex hull {a,b,c}) f"
- by (simp add: contf compact_convex_hull compact_uniformly_continuous)
- ultimately have "False"
- unfolding uniformly_continuous_on_def
- by (force simp: ynz \<open>0 < C\<close> dist_norm)
- then show ?thesis ..
- qed
- }
- moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
- using fabc contour_integrable_continuous_linepath by auto
- ultimately show ?thesis
- using has_contour_integral_integral by fastforce
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Version allowing finite number of exceptional points\<close>
-
-proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle_cofinite:
- assumes "continuous_on (convex hull {a,b,c}) f"
- and "finite S"
- and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - S \<Longrightarrow> f field_differentiable (at x))"
- shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
-using assms
-proof (induction "card S" arbitrary: a b c S rule: less_induct)
- case (less S a b c)
- show ?case
- proof (cases "S={}")
- case True with less show ?thesis
- by (fastforce simp: holomorphic_on_def field_differentiable_at_within Cauchy_theorem_triangle_interior)
- next
- case False
- then obtain d S' where d: "S = insert d S'" "d \<notin> S'"
- by (meson Set.set_insert all_not_in_conv)
- then show ?thesis
- proof (cases "d \<in> convex hull {a,b,c}")
- case False
- show "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
- proof (rule less.hyps)
- show "\<And>x. x \<in> interior (convex hull {a, b, c}) - S' \<Longrightarrow> f field_differentiable at x"
- using False d interior_subset by (auto intro!: less.prems)
- qed (use d less.prems in auto)
- next
- case True
- have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}"
- by (meson True hull_subset insert_subset convex_hull_subset)
- have abd: "(f has_contour_integral 0) (linepath a b +++ linepath b d +++ linepath d a)"
- proof (rule less.hyps)
- show "\<And>x. x \<in> interior (convex hull {a, b, d}) - S' \<Longrightarrow> f field_differentiable at x"
- using d not_in_interior_convex_hull_3
- by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
- qed (use d continuous_on_subset [OF _ *] less.prems in auto)
- have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}"
- by (meson True hull_subset insert_subset convex_hull_subset)
- have bcd: "(f has_contour_integral 0) (linepath b c +++ linepath c d +++ linepath d b)"
- proof (rule less.hyps)
- show "\<And>x. x \<in> interior (convex hull {b, c, d}) - S' \<Longrightarrow> f field_differentiable at x"
- using d not_in_interior_convex_hull_3
- by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
- qed (use d continuous_on_subset [OF _ *] less.prems in auto)
- have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}"
- by (meson True hull_subset insert_subset convex_hull_subset)
- have cad: "(f has_contour_integral 0) (linepath c a +++ linepath a d +++ linepath d c)"
- proof (rule less.hyps)
- show "\<And>x. x \<in> interior (convex hull {c, a, d}) - S' \<Longrightarrow> f field_differentiable at x"
- using d not_in_interior_convex_hull_3
- by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
- qed (use d continuous_on_subset [OF _ *] less.prems in auto)
- have "f contour_integrable_on linepath a b"
- using less.prems abd contour_integrable_joinD1 contour_integrable_on_def by blast
- moreover have "f contour_integrable_on linepath b c"
- using less.prems bcd contour_integrable_joinD1 contour_integrable_on_def by blast
- moreover have "f contour_integrable_on linepath c a"
- using less.prems cad contour_integrable_joinD1 contour_integrable_on_def by blast
- ultimately have fpi: "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
- by auto
- { fix y::complex
- assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
- and ynz: "y \<noteq> 0"
- have cont_ad: "continuous_on (closed_segment a d) f"
- by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3))
- have cont_bd: "continuous_on (closed_segment b d) f"
- by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1))
- have cont_cd: "continuous_on (closed_segment c d) f"
- by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2))
- have "contour_integral (linepath a b) f = - (contour_integral (linepath b d) f + (contour_integral (linepath d a) f))"
- "contour_integral (linepath b c) f = - (contour_integral (linepath c d) f + (contour_integral (linepath d b) f))"
- "contour_integral (linepath c a) f = - (contour_integral (linepath a d) f + contour_integral (linepath d c) f)"
- using has_chain_integral_chain_integral3 [OF abd]
- has_chain_integral_chain_integral3 [OF bcd]
- has_chain_integral_chain_integral3 [OF cad]
- by (simp_all add: algebra_simps add_eq_0_iff)
- then have ?thesis
- using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 contour_integral_reverse_linepath by fastforce
- }
- then show ?thesis
- using fpi contour_integrable_on_def by blast
- qed
- qed
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for an open starlike set\<close>
-
-lemma starlike_convex_subset:
- assumes S: "a \<in> S" "closed_segment b c \<subseteq> S" and subs: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
- shows "convex hull {a,b,c} \<subseteq> S"
- using S
- apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull)
- apply (meson subs convexD convex_closed_segment ends_in_segment(1) ends_in_segment(2) subsetCE)
- done
-
-lemma triangle_contour_integrals_starlike_primitive:
- assumes contf: "continuous_on S f"
- and S: "a \<in> S" "open S"
- and x: "x \<in> S"
- and subs: "\<And>y. y \<in> S \<Longrightarrow> closed_segment a y \<subseteq> S"
- and zer: "\<And>b c. closed_segment b c \<subseteq> S
- \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x)"
-proof -
- let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
- { fix e y
- assume e: "0 < e" and bxe: "ball x e \<subseteq> S" and close: "cmod (y - x) < e"
- have y: "y \<in> S"
- using bxe close by (force simp: dist_norm norm_minus_commute)
- have cont_ayf: "continuous_on (closed_segment a y) f"
- using contf continuous_on_subset subs y by blast
- have xys: "closed_segment x y \<subseteq> S"
- apply (rule order_trans [OF _ bxe])
- using close
- by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound)
- have "?pathint a y - ?pathint a x = ?pathint x y"
- using zer [OF xys] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
- } note [simp] = this
- { fix e::real
- assume e: "0 < e"
- have cont_atx: "continuous (at x) f"
- using x S contf continuous_on_eq_continuous_at by blast
- then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2"
- unfolding continuous_at Lim_at dist_norm using e
- by (drule_tac x="e/2" in spec) force
- obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> S" using \<open>open S\<close> x
- by (auto simp: open_contains_ball)
- have dpos: "min d1 d2 > 0" using d1 d2 by simp
- { fix y
- assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2"
- have y: "y \<in> S"
- using d2 close by (force simp: dist_norm norm_minus_commute)
- have "closed_segment x y \<subseteq> S"
- using close d2 by (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1))
- then have fxy: "f contour_integrable_on linepath x y"
- by (metis contour_integrable_continuous_linepath continuous_on_subset [OF contf])
- then obtain i where i: "(f has_contour_integral i) (linepath x y)"
- by (auto simp: contour_integrable_on_def)
- then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
- by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
- then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
- proof (rule has_contour_integral_bound_linepath)
- show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2"
- by (meson close d1_less le_less_trans less_imp_le min.strict_boundedE segment_bound1)
- qed (use e in simp)
- also have "\<dots> < e * cmod (y - x)"
- by (simp add: e yx)
- finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
- using i yx by (simp add: contour_integral_unique divide_less_eq)
- }
- then have "\<exists>d>0. \<forall>y. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
- using dpos by blast
- }
- then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) \<midarrow>x\<rightarrow> 0"
- by (simp add: Lim_at dist_norm inverse_eq_divide)
- show ?thesis
- apply (simp add: has_field_derivative_def has_derivative_at2 bounded_linear_mult_right)
- apply (rule Lim_transform [OF * tendsto_eventually])
- using \<open>open S\<close> x apply (force simp: dist_norm open_contains_ball inverse_eq_divide [symmetric] eventually_at)
- done
-qed
-
-(** Existence of a primitive.*)
-lemma holomorphic_starlike_primitive:
- fixes f :: "complex \<Rightarrow> complex"
- assumes contf: "continuous_on S f"
- and S: "starlike S" and os: "open S"
- and k: "finite k"
- and fcd: "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x"
- shows "\<exists>g. \<forall>x \<in> S. (g has_field_derivative f x) (at x)"
-proof -
- obtain a where a: "a\<in>S" and a_cs: "\<And>x. x\<in>S \<Longrightarrow> closed_segment a x \<subseteq> S"
- using S by (auto simp: starlike_def)
- { fix x b c
- assume "x \<in> S" "closed_segment b c \<subseteq> S"
- then have abcs: "convex hull {a, b, c} \<subseteq> S"
- by (simp add: a a_cs starlike_convex_subset)
- then have "continuous_on (convex hull {a, b, c}) f"
- by (simp add: continuous_on_subset [OF contf])
- then have "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
- using abcs interior_subset by (force intro: fcd Cauchy_theorem_triangle_cofinite [OF _ k])
- } note 0 = this
- show ?thesis
- apply (intro exI ballI)
- apply (rule triangle_contour_integrals_starlike_primitive [OF contf a os], assumption)
- apply (metis a_cs)
- apply (metis has_chain_integral_chain_integral3 0)
- done
-qed
-
-lemma Cauchy_theorem_starlike:
- "\<lbrakk>open S; starlike S; finite k; continuous_on S f;
- \<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x;
- valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
- \<Longrightarrow> (f has_contour_integral 0) g"
- by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open)
-
-lemma Cauchy_theorem_starlike_simple:
- "\<lbrakk>open S; starlike S; f holomorphic_on S; valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
- \<Longrightarrow> (f has_contour_integral 0) g"
-apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI])
-apply (simp_all add: holomorphic_on_imp_continuous_on)
-apply (metis at_within_open holomorphic_on_def)
-done
-
-subsection\<open>Cauchy's theorem for a convex set\<close>
-
-text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close>
-
-lemma triangle_contour_integrals_convex_primitive:
- assumes contf: "continuous_on S f"
- and S: "a \<in> S" "convex S"
- and x: "x \<in> S"
- and zer: "\<And>b c. \<lbrakk>b \<in> S; c \<in> S\<rbrakk>
- \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x within S)"
-proof -
- let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
- { fix y
- assume y: "y \<in> S"
- have cont_ayf: "continuous_on (closed_segment a y) f"
- using S y by (meson contf continuous_on_subset convex_contains_segment)
- have xys: "closed_segment x y \<subseteq> S" (*?*)
- using convex_contains_segment S x y by auto
- have "?pathint a y - ?pathint a x = ?pathint x y"
- using zer [OF x y] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
- } note [simp] = this
- { fix e::real
- assume e: "0 < e"
- have cont_atx: "continuous (at x within S) f"
- using x S contf by (simp add: continuous_on_eq_continuous_within)
- then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. \<lbrakk>y \<in> S; cmod (y - x) < d1\<rbrakk> \<Longrightarrow> cmod (f y - f x) < e/2"
- unfolding continuous_within Lim_within dist_norm using e
- by (drule_tac x="e/2" in spec) force
- { fix y
- assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> S"
- have fxy: "f contour_integrable_on linepath x y"
- using convex_contains_segment S x y
- by (blast intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
- then obtain i where i: "(f has_contour_integral i) (linepath x y)"
- by (auto simp: contour_integrable_on_def)
- then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
- by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
- then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
- proof (rule has_contour_integral_bound_linepath)
- show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2"
- by (meson assms(3) close convex_contains_segment d1_less le_less_trans less_imp_le segment_bound1 subset_iff x y)
- qed (use e in simp)
- also have "\<dots> < e * cmod (y - x)"
- by (simp add: e yx)
- finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
- using i yx by (simp add: contour_integral_unique divide_less_eq)
- }
- then have "\<exists>d>0. \<forall>y\<in>S. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
- using d1 by blast
- }
- then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) \<longlongrightarrow> 0) (at x within S)"
- by (simp add: Lim_within dist_norm inverse_eq_divide)
- show ?thesis
- apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
- apply (rule Lim_transform [OF * tendsto_eventually])
- using linordered_field_no_ub
- apply (force simp: inverse_eq_divide [symmetric] eventually_at)
- done
-qed
-
-lemma contour_integral_convex_primitive:
- assumes "convex S" "continuous_on S f"
- "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
- obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
-proof (cases "S={}")
- case False
- with assms that show ?thesis
- by (blast intro: triangle_contour_integrals_convex_primitive has_chain_integral_chain_integral3)
-qed auto
-
-lemma holomorphic_convex_primitive:
- fixes f :: "complex \<Rightarrow> complex"
- assumes "convex S" "finite K" and contf: "continuous_on S f"
- and fd: "\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x"
- obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
-proof (rule contour_integral_convex_primitive [OF \<open>convex S\<close> contf Cauchy_theorem_triangle_cofinite])
- have *: "convex hull {a, b, c} \<subseteq> S" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c
- by (simp add: \<open>convex S\<close> hull_minimal that)
- show "continuous_on (convex hull {a, b, c}) f" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c
- by (meson "*" contf continuous_on_subset that)
- show "f field_differentiable at x" if "a \<in> S" "b \<in> S" "c \<in> S" "x \<in> interior (convex hull {a, b, c}) - K" for a b c x
- by (metis "*" DiffD1 DiffD2 DiffI fd interior_mono subsetCE that)
-qed (use assms in \<open>force+\<close>)
-
-lemma holomorphic_convex_primitive':
- fixes f :: "complex \<Rightarrow> complex"
- assumes "convex S" and "open S" and "f holomorphic_on S"
- obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
-proof (rule holomorphic_convex_primitive)
- fix x assume "x \<in> interior S - {}"
- with assms show "f field_differentiable at x"
- by (auto intro!: holomorphic_on_imp_differentiable_at simp: interior_open)
-qed (use assms in \<open>auto intro: holomorphic_on_imp_continuous_on\<close>)
-
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_convex:
- "\<lbrakk>continuous_on S f; convex S; finite K;
- \<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x;
- valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
- \<Longrightarrow> (f has_contour_integral 0) g"
- by (metis holomorphic_convex_primitive Cauchy_theorem_primitive)
-
-corollary Cauchy_theorem_convex_simple:
- "\<lbrakk>f holomorphic_on S; convex S;
- valid_path g; path_image g \<subseteq> S;
- pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
- apply (rule Cauchy_theorem_convex [where K = "{}"])
- apply (simp_all add: holomorphic_on_imp_continuous_on)
- using at_within_interior holomorphic_on_def interior_subset by fastforce
-
-text\<open>In particular for a disc\<close>
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc:
- "\<lbrakk>finite K; continuous_on (cball a e) f;
- \<And>x. x \<in> ball a e - K \<Longrightarrow> f field_differentiable at x;
- valid_path g; path_image g \<subseteq> cball a e;
- pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
- by (auto intro: Cauchy_theorem_convex)
-
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc_simple:
- "\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e;
- pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
-by (simp add: Cauchy_theorem_convex_simple)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Generalize integrability to local primitives\<close>
-
-lemma contour_integral_local_primitive_lemma:
- fixes f :: "complex\<Rightarrow>complex"
- shows
- "\<lbrakk>g piecewise_differentiable_on {a..b};
- \<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s);
- \<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk>
- \<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b}))
- integrable_on {a..b}"
- apply (cases "cbox a b = {}", force)
- apply (simp add: integrable_on_def)
- apply (rule exI)
- apply (rule contour_integral_primitive_lemma, assumption+)
- using atLeastAtMost_iff by blast
-
-lemma contour_integral_local_primitive_any:
- fixes f :: "complex \<Rightarrow> complex"
- assumes gpd: "g piecewise_differentiable_on {a..b}"
- and dh: "\<And>x. x \<in> s
- \<Longrightarrow> \<exists>d h. 0 < d \<and>
- (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
- and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
- shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}"
-proof -
- { fix x
- assume x: "a \<le> x" "x \<le> b"
- obtain d h where d: "0 < d"
- and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))"
- using x gs dh by (metis atLeastAtMost_iff)
- have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast
- then obtain e where e: "e>0" and lessd: "\<And>x'. x' \<in> {a..b} \<Longrightarrow> \<bar>x' - x\<bar> < e \<Longrightarrow> cmod (g x' - g x) < d"
- using x d
- apply (auto simp: dist_norm continuous_on_iff)
- apply (drule_tac x=x in bspec)
- using x apply simp
- apply (drule_tac x=d in spec, auto)
- done
- have "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> {u..v} \<subseteq> ball x d \<and> (u \<le> v \<longrightarrow> a \<le> u \<and> v \<le> b) \<longrightarrow>
- (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}"
- apply (rule_tac x=e in exI)
- using e
- apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify)
- apply (rule_tac f = h and s = "g ` {u..v}" in contour_integral_local_primitive_lemma)
- apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset)
- apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force)
- done
- } then
- show ?thesis
- by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified])
-qed
-
-lemma contour_integral_local_primitive:
- fixes f :: "complex \<Rightarrow> complex"
- assumes g: "valid_path g" "path_image g \<subseteq> s"
- and dh: "\<And>x. x \<in> s
- \<Longrightarrow> \<exists>d h. 0 < d \<and>
- (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
- shows "f contour_integrable_on g"
- using g
- apply (simp add: valid_path_def path_image_def contour_integrable_on_def has_contour_integral_def
- has_integral_localized_vector_derivative integrable_on_def [symmetric])
- using contour_integral_local_primitive_any [OF _ dh]
- by (meson image_subset_iff piecewise_C1_imp_differentiable)
-
-
-text\<open>In particular if a function is holomorphic\<close>
-
-lemma contour_integrable_holomorphic:
- assumes contf: "continuous_on s f"
- and os: "open s"
- and k: "finite k"
- and g: "valid_path g" "path_image g \<subseteq> s"
- and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x"
- shows "f contour_integrable_on g"
-proof -
- { fix z
- assume z: "z \<in> s"
- obtain d where "d>0" and d: "ball z d \<subseteq> s" using \<open>open s\<close> z
- by (auto simp: open_contains_ball)
- then have contfb: "continuous_on (ball z d) f"
- using contf continuous_on_subset by blast
- obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)"
- by (metis holomorphic_convex_primitive [OF convex_ball k contfb fcd] d interior_subset Diff_iff subsetD)
- then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)"
- by (metis open_ball at_within_open d os subsetCE)
- then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
- by (force simp: dist_norm norm_minus_commute)
- then have "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
- using \<open>0 < d\<close> by blast
- }
- then show ?thesis
- by (rule contour_integral_local_primitive [OF g])
-qed
-
-lemma contour_integrable_holomorphic_simple:
- assumes fh: "f holomorphic_on S"
- and os: "open S"
- and g: "valid_path g" "path_image g \<subseteq> S"
- shows "f contour_integrable_on g"
- apply (rule contour_integrable_holomorphic [OF _ os Finite_Set.finite.emptyI g])
- apply (simp add: fh holomorphic_on_imp_continuous_on)
- using fh by (simp add: field_differentiable_def holomorphic_on_open os)
-
-lemma continuous_on_inversediff:
- fixes z:: "'a::real_normed_field" shows "z \<notin> S \<Longrightarrow> continuous_on S (\<lambda>w. 1 / (w - z))"
- by (rule continuous_intros | force)+
-
-lemma contour_integrable_inversediff:
- "\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) contour_integrable_on g"
-apply (rule contour_integrable_holomorphic_simple [of _ "UNIV-{z}"])
-apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros)
-done
-
-text\<open>Key fact that path integral is the same for a "nearby" path. This is the
- main lemma for the homotopy form of Cauchy's theorem and is also useful
- if we want "without loss of generality" to assume some nice properties of a
- path (e.g. smoothness). It can also be used to define the integrals of
- analytic functions over arbitrary continuous paths. This is just done for
- winding numbers now.
-\<close>
-
-text\<open>A technical definition to avoid duplication of similar proofs,
- for paths joined at the ends versus looping paths\<close>
-definition linked_paths :: "bool \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
- where "linked_paths atends g h ==
- (if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
- else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)"
-
-text\<open>This formulation covers two cases: \<^term>\<open>g\<close> and \<^term>\<open>h\<close> share their
- start and end points; \<^term>\<open>g\<close> and \<^term>\<open>h\<close> both loop upon themselves.\<close>
-lemma contour_integral_nearby:
- assumes os: "open S" and p: "path p" "path_image p \<subseteq> S"
- shows "\<exists>d. 0 < d \<and>
- (\<forall>g h. valid_path g \<and> valid_path h \<and>
- (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
- linked_paths atends g h
- \<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and>
- (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f))"
-proof -
- have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> S"
- using open_contains_ball os p(2) by blast
- then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> S"
- by metis
- define cover where "cover = (\<lambda>z. ball z (ee z/3)) ` (path_image p)"
- have "compact (path_image p)"
- by (metis p(1) compact_path_image)
- moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))"
- using ee by auto
- ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D"
- by (simp add: compact_eq_Heine_Borel cover_def)
- then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D"
- by blast
- then obtain k where k: "k \<subseteq> {0..1}" "finite k" and D_eq: "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k"
- apply (simp add: cover_def path_image_def image_comp)
- apply (blast dest!: finite_subset_image [OF \<open>finite D\<close>])
- done
- then have kne: "k \<noteq> {}"
- using D by auto
- have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p"
- using k by (auto simp: path_image_def)
- then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))"
- by (metis ee)
- define e where "e = Min((ee \<circ> p) ` k)"
- have fin_eep: "finite ((ee \<circ> p) ` k)"
- using k by blast
- have "0 < e"
- using ee k by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi)
- have "uniformly_continuous_on {0..1} p"
- using p by (simp add: path_def compact_uniformly_continuous)
- then obtain d::real where d: "d>0"
- and de: "\<And>x x'. \<bar>x' - x\<bar> < d \<Longrightarrow> x\<in>{0..1} \<Longrightarrow> x'\<in>{0..1} \<Longrightarrow> cmod (p x' - p x) < e/3"
- unfolding uniformly_continuous_on_def dist_norm real_norm_def
- by (metis divide_pos_pos \<open>0 < e\<close> zero_less_numeral)
- then obtain N::nat where N: "N>0" "inverse N < d"
- using real_arch_inverse [of d] by auto
- show ?thesis
- proof (intro exI conjI allI; clarify?)
- show "e/3 > 0"
- using \<open>0 < e\<close> by simp
- fix g h
- assume g: "valid_path g" and ghp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3 \<and> cmod (h t - p t) < e / 3"
- and h: "valid_path h"
- and joins: "linked_paths atends g h"
- { fix t::real
- assume t: "0 \<le> t" "t \<le> 1"
- then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)"
- using \<open>path_image p \<subseteq> \<Union>D\<close> D_eq by (force simp: path_image_def)
- then have ele: "e \<le> ee (p u)" using fin_eep
- by (simp add: e_def)
- have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3"
- using ghp t by auto
- with ele have "cmod (g t - p t) < ee (p u) / 3"
- "cmod (h t - p t) < ee (p u) / 3"
- by linarith+
- then have "g t \<in> ball(p u) (ee(p u))" "h t \<in> ball(p u) (ee(p u))"
- using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"]
- norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u
- by (force simp: dist_norm ball_def norm_minus_commute)+
- then have "g t \<in> S" "h t \<in> S" using ee u k
- by (auto simp: path_image_def ball_def)
- }
- then have ghs: "path_image g \<subseteq> S" "path_image h \<subseteq> S"
- by (auto simp: path_image_def)
- moreover
- { fix f
- assume fhols: "f holomorphic_on S"
- then have fpa: "f contour_integrable_on g" "f contour_integrable_on h"
- using g ghs h holomorphic_on_imp_continuous_on os contour_integrable_holomorphic_simple
- by blast+
- have contf: "continuous_on S f"
- by (simp add: fhols holomorphic_on_imp_continuous_on)
- { fix z
- assume z: "z \<in> path_image p"
- have "f holomorphic_on ball z (ee z)"
- using fhols ee z holomorphic_on_subset by blast
- then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))"
- using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified]
- by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball)
- }
- then obtain ff where ff:
- "\<And>z w. \<lbrakk>z \<in> path_image p; w \<in> ball z (ee z)\<rbrakk> \<Longrightarrow> (ff z has_field_derivative f w) (at w)"
- by metis
- { fix n
- assume n: "n \<le> N"
- then have "contour_integral(subpath 0 (n/N) h) f - contour_integral(subpath 0 (n/N) g) f =
- contour_integral(linepath (g(n/N)) (h(n/N))) f - contour_integral(linepath (g 0) (h 0)) f"
- proof (induct n)
- case 0 show ?case by simp
- next
- case (Suc n)
- obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)"
- using \<open>path_image p \<subseteq> \<Union>D\<close> [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems
- by (force simp: path_image_def)
- then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3"
- by (simp add: dist_norm)
- have e3le: "e/3 \<le> ee (p t) / 3" using fin_eep t
- by (simp add: e_def)
- { fix x
- assume x: "n/N \<le> x" "x \<le> (1 + n)/N"
- then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1"
- using Suc.prems by auto
- then have x01: "0 \<le> x" "x \<le> 1"
- using x by linarith+
- have "cmod (p t - p x) < ee (p t) / 3 + e/3"
- proof (rule norm_diff_triangle_less [OF ptu de])
- show "\<bar>real n / real N - x\<bar> < d"
- using x N by (auto simp: field_simps)
- qed (use x01 Suc.prems in auto)
- then have ptx: "cmod (p t - p x) < 2*ee (p t)/3"
- using e3le eepi [OF t] by simp
- have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 "
- apply (rule norm_diff_triangle_less [OF ptx])
- using ghp x01 by (simp add: norm_minus_commute)
- also have "\<dots> \<le> ee (p t)"
- using e3le eepi [OF t] by simp
- finally have gg: "cmod (p t - g x) < ee (p t)" .
- have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 "
- apply (rule norm_diff_triangle_less [OF ptx])
- using ghp x01 by (simp add: norm_minus_commute)
- also have "\<dots> \<le> ee (p t)"
- using e3le eepi [OF t] by simp
- finally have "cmod (p t - g x) < ee (p t)"
- "cmod (p t - h x) < ee (p t)"
- using gg by auto
- } note ptgh_ee = this
- have "closed_segment (g (real n / real N)) (h (real n / real N)) = path_image (linepath (h (n/N)) (g (n/N)))"
- by (simp add: closed_segment_commute)
- also have pi_hgn: "\<dots> \<subseteq> ball (p t) (ee (p t))"
- using ptgh_ee [of "n/N"] Suc.prems
- by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
- finally have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> S"
- using ee pi t by blast
- have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N))) \<subseteq> ball (p t) (ee (p t))"
- using ptgh_ee [of "(1+n)/N"] Suc.prems
- by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
- then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> S"
- using \<open>N>0\<close> Suc.prems ee pi t
- by (auto simp: Path_Connected.path_image_join field_simps)
- have pi_subset_ball:
- "path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++
- subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N)))
- \<subseteq> ball (p t) (ee (p t))"
- apply (intro subset_path_image_join pi_hgn pi_ghn')
- using \<open>N>0\<close> Suc.prems
- apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
- done
- have pi0: "(f has_contour_integral 0)
- (subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
- subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))"
- apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"])
- apply (metis ff open_ball at_within_open pi t)
- using Suc.prems pi_subset_ball apply (simp_all add: valid_path_join valid_path_subpath g h)
- done
- have fpa1: "f contour_integrable_on subpath (real n / real N) (real (Suc n) / real N) g"
- using Suc.prems by (simp add: contour_integrable_subpath g fpa)
- have fpa2: "f contour_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))"
- using gh_n's
- by (auto intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
- have fpa3: "f contour_integrable_on linepath (h (real n / real N)) (g (real n / real N))"
- using gh_ns
- by (auto simp: closed_segment_commute intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
- have eq0: "contour_integral (subpath (n/N) ((Suc n) / real N) g) f +
- contour_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f +
- contour_integral (subpath ((Suc n) / N) (n/N) h) f +
- contour_integral (linepath (h (n/N)) (g (n/N))) f = 0"
- using contour_integral_unique [OF pi0] Suc.prems
- by (simp add: g h fpa valid_path_subpath contour_integrable_subpath
- fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc)
- have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
- \<lbrakk>hn - gn = ghn - gh0;
- gd + ghn' + he + hgn = (0::complex);
- hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0"
- by (auto simp: algebra_simps)
- have "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
- contour_integral (subpath 0 (n/N) h) f + contour_integral (subpath (n/N) ((Suc n) / N) h) f"
- unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"]
- using Suc.prems by (simp add: h fpa contour_integral_reversepath valid_path_subpath contour_integrable_subpath)
- also have "\<dots> = contour_integral (subpath 0 ((Suc n) / N) h) f"
- using Suc.prems by (simp add: contour_integral_subpath_combine h fpa)
- finally have pi0_eq:
- "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
- contour_integral (subpath 0 ((Suc n) / N) h) f" .
- show ?case
- apply (rule * [OF Suc.hyps eq0 pi0_eq])
- using Suc.prems
- apply (simp_all add: g h fpa contour_integral_subpath_combine
- contour_integral_reversepath [symmetric] contour_integrable_continuous_linepath
- continuous_on_subset [OF contf gh_ns])
- done
- qed
- } note ind = this
- have "contour_integral h f = contour_integral g f"
- using ind [OF order_refl] N joins
- by (simp add: linked_paths_def pathstart_def pathfinish_def split: if_split_asm)
- }
- ultimately
- show "path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)"
- by metis
- qed
-qed
-
-
-lemma
- assumes "open S" "path p" "path_image p \<subseteq> S"
- shows contour_integral_nearby_ends:
- "\<exists>d. 0 < d \<and>
- (\<forall>g h. valid_path g \<and> valid_path h \<and>
- (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
- pathstart h = pathstart g \<and> pathfinish h = pathfinish g
- \<longrightarrow> path_image g \<subseteq> S \<and>
- path_image h \<subseteq> S \<and>
- (\<forall>f. f holomorphic_on S
- \<longrightarrow> contour_integral h f = contour_integral g f))"
- and contour_integral_nearby_loops:
- "\<exists>d. 0 < d \<and>
- (\<forall>g h. valid_path g \<and> valid_path h \<and>
- (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
- pathfinish g = pathstart g \<and> pathfinish h = pathstart h
- \<longrightarrow> path_image g \<subseteq> S \<and>
- path_image h \<subseteq> S \<and>
- (\<forall>f. f holomorphic_on S
- \<longrightarrow> contour_integral h f = contour_integral g f))"
- using contour_integral_nearby [OF assms, where atends=True]
- using contour_integral_nearby [OF assms, where atends=False]
- unfolding linked_paths_def by simp_all
-
-lemma C1_differentiable_polynomial_function:
- fixes p :: "real \<Rightarrow> 'a::euclidean_space"
- shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S"
- by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function)
-
-lemma valid_path_polynomial_function:
- fixes p :: "real \<Rightarrow> 'a::euclidean_space"
- shows "polynomial_function p \<Longrightarrow> valid_path p"
-by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
-
-lemma valid_path_subpath_trivial [simp]:
- fixes g :: "real \<Rightarrow> 'a::euclidean_space"
- shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
- by (simp add: subpath_def valid_path_polynomial_function)
-
-lemma contour_integral_bound_exists:
-assumes S: "open S"
- and g: "valid_path g"
- and pag: "path_image g \<subseteq> S"
- shows "\<exists>L. 0 < L \<and>
- (\<forall>f B. f holomorphic_on S \<and> (\<forall>z \<in> S. norm(f z) \<le> B)
- \<longrightarrow> norm(contour_integral g f) \<le> L*B)"
-proof -
- have "path g" using g
- by (simp add: valid_path_imp_path)
- then obtain d::real and p
- where d: "0 < d"
- and p: "polynomial_function p" "path_image p \<subseteq> S"
- and pi: "\<And>f. f holomorphic_on S \<Longrightarrow> contour_integral g f = contour_integral p f"
- using contour_integral_nearby_ends [OF S \<open>path g\<close> pag]
- apply clarify
- apply (drule_tac x=g in spec)
- apply (simp only: assms)
- apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function)
- done
- then obtain p' where p': "polynomial_function p'"
- "\<And>x. (p has_vector_derivative (p' x)) (at x)"
- by (blast intro: has_vector_derivative_polynomial_function that)
- then have "bounded(p' ` {0..1})"
- using continuous_on_polymonial_function
- by (force simp: intro!: compact_imp_bounded compact_continuous_image)
- then obtain L where L: "L>0" and nop': "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> norm (p' x) \<le> L"
- by (force simp: bounded_pos)
- { fix f B
- assume f: "f holomorphic_on S" and B: "\<And>z. z\<in>S \<Longrightarrow> cmod (f z) \<le> B"
- then have "f contour_integrable_on p \<and> valid_path p"
- using p S
- by (blast intro: valid_path_polynomial_function contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on)
- moreover have "cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B" if "0 \<le> x" "x \<le> 1" for x
- proof (rule mult_mono)
- show "cmod (vector_derivative p (at x)) \<le> L"
- by (metis nop' p'(2) that vector_derivative_at)
- show "cmod (f (p x)) \<le> B"
- by (metis B atLeastAtMost_iff imageI p(2) path_defs(4) subset_eq that)
- qed (use \<open>L>0\<close> in auto)
- ultimately have "cmod (contour_integral g f) \<le> L * B"
- apply (simp only: pi [OF f])
- apply (simp only: contour_integral_integral)
- apply (rule order_trans [OF integral_norm_bound_integral])
- apply (auto simp: mult.commute integral_norm_bound_integral contour_integrable_on [symmetric] norm_mult)
- done
- } then
- show ?thesis
- by (force simp: L contour_integral_integral)
-qed
-
-text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
-
-subsection \<open>Winding Numbers\<close>
-
-definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "winding_number_prop \<gamma> z e p n \<equiv>
- valid_path p \<and> z \<notin> path_image p \<and>
- pathstart p = pathstart \<gamma> \<and>
- pathfinish p = pathfinish \<gamma> \<and>
- (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
-
-definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
- "winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
-
-
-lemma winding_number:
- assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
- shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain d
- where d: "d>0"
- and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
- pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
- path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
- (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
- using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
- (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
- define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
- have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
- proof (rule_tac x=nn in exI, clarify)
- fix e::real
- assume e: "e>0"
- obtain p where p: "polynomial_function p \<and>
- pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
- have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto simp: intro!: holomorphic_intros)
- then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
- apply (rule_tac x=p in exI)
- using pi_eq [of h p] h p d
- apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
- done
- qed
- then show ?thesis
- unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
-qed
-
-lemma winding_number_unique:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
- shows "winding_number \<gamma> z = n"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain e
- where e: "e>0"
- and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
- pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
- contour_integral h2 f = contour_integral h1 f"
- using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- obtain p where p: "winding_number_prop \<gamma> z e p n"
- using pi [OF e] by blast
- obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
- using winding_number [OF \<gamma> e] by blast
- have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
- using p by (auto simp: winding_number_prop_def)
- also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
- proof (rule pi_eq)
- show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto intro!: holomorphic_intros)
- qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
- also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
- using q by (auto simp: winding_number_prop_def)
- finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
- then show ?thesis
- by simp
-qed
-
-(*NB not winding_number_prop here due to the loop in p*)
-lemma winding_number_unique_loop:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and pi:
- "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
- pathfinish p = pathstart p \<and>
- (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
- shows "winding_number \<gamma> z = n"
-proof -
- have "path_image \<gamma> \<subseteq> UNIV - {z}"
- using assms by blast
- then obtain e
- where e: "e>0"
- and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
- pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
- contour_integral h2 f = contour_integral h1 f"
- using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
- obtain p where p:
- "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
- (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
- using pi [OF e] by blast
- obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
- using winding_number [OF \<gamma> e] by blast
- have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
- using p by auto
- also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
- proof (rule pi_eq)
- show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
- by (auto intro!: holomorphic_intros)
- qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
- also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
- using q by (auto simp: winding_number_prop_def)
- finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
- then show ?thesis
- by simp
-qed
-
-proposition winding_number_valid_path:
- assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
- by (rule winding_number_unique)
- (use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
-
-proposition has_contour_integral_winding_number:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
-by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
-
-lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
- by (simp add: winding_number_valid_path)
-
-lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
- by (simp add: path_image_subpath winding_number_valid_path)
-
-lemma winding_number_join:
- assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
- and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
- and "pathfinish \<gamma>1 = pathstart \<gamma>2"
- shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
-proof (rule winding_number_unique)
- show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
- (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
- proof -
- obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
- using \<open>0 < e\<close> \<gamma>1 winding_number by blast
- moreover
- obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
- using \<open>0 < e\<close> \<gamma>2 winding_number by blast
- ultimately
- have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
- using assms
- apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
- apply (auto simp: joinpaths_def)
- done
- then show ?thesis
- by blast
- qed
-qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
-
-lemma winding_number_reversepath:
- assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
- shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
-proof (rule winding_number_unique)
- show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
- proof -
- obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
- using \<open>0 < e\<close> assms winding_number by blast
- then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
- using assms
- apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
- apply (auto simp: reversepath_def)
- done
- then show ?thesis
- by blast
- qed
-qed (use assms in auto)
-
-lemma winding_number_shiftpath:
- assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
- and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
- shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
-proof (rule winding_number_unique_loop)
- show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
- (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
- contour_integral p (\<lambda>w. 1 / (w - z)) =
- complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- if "e > 0" for e
- proof -
- obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
- using \<open>0 < e\<close> assms winding_number by blast
- then show ?thesis
- apply (rule_tac x="shiftpath a p" in exI)
- using assms that
- apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
- apply (simp add: shiftpath_def)
- done
- qed
-qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
-
-lemma winding_number_split_linepath:
- assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
- shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
-proof -
- have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
- using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
- then show ?thesis
- using assms
- by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
-qed
-
-lemma winding_number_cong:
- "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
- by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
-
-lemma winding_number_constI:
- assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
- shows "winding_number g z = 0"
-proof -
- have "winding_number g z = winding_number (linepath c c) z"
- apply (rule winding_number_cong)
- using assms unfolding linepath_def by auto
- moreover have "winding_number (linepath c c) z =0"
- apply (rule winding_number_trivial)
- using assms by auto
- ultimately show ?thesis by auto
-qed
-
-lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
- unfolding winding_number_def
-proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
- fix n e g
- assume "0 < e" and g: "winding_number_prop p z e g n"
- then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
- by (rule_tac x="\<lambda>t. g t - z" in exI)
- (force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
- vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
-next
- fix n e g
- assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
- then show "\<exists>r. winding_number_prop p z e r n"
- apply (rule_tac x="\<lambda>t. g t + z" in exI)
- apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
- piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
- apply (force simp: algebra_simps)
- done
-qed
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close>
-
-lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
- unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
-
-lemma has_contour_integral_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
- shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
-proof -
- obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
- using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
- have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
- using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
- then
- have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
- proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
- show "negligible S"
- by (simp add: \<open>finite S\<close> negligible_finite)
- show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
- - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
- if "x \<in> {0..1} - S" for x
- proof -
- have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
- proof (rule vector_derivative_within_cbox)
- show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
- using that unfolding o_def
- by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
- qed (use that in auto)
- then show ?thesis
- by simp
- qed
- qed
- then show ?thesis by (simp add: has_contour_integral_def)
-qed
-
-lemma winding_number_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
- shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
-proof -
- have "(/) 1 contour_integrable_on \<gamma>"
- using "0" \<gamma> contour_integrable_inversediff by fastforce
- then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
- by (rule has_contour_integral_integral)
- then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
- using has_contour_integral_neg by auto
- then show ?thesis
- using assms
- apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
- apply (simp add: contour_integral_unique has_contour_integral_negatepath)
- done
-qed
-
-lemma contour_integrable_negatepath:
- assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
- shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
- by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
-
-(* A combined theorem deducing several things piecewise.*)
-lemma winding_number_join_pos_combined:
- "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
- valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
- \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
- by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
-
-
-subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
-
-lemma Re_winding_number:
- "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
- \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
-by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
-
-lemma winding_number_pos_le:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
- shows "0 \<le> Re(winding_number \<gamma> z)"
-proof -
- have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
- using ge by (simp add: Complex.Im_divide algebra_simps x)
- let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
- let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
- have hi: "(?vd has_integral ?int z) (cbox 0 1)"
- unfolding box_real
- apply (subst has_contour_integral [symmetric])
- using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
- have "0 \<le> Im (?int z)"
- proof (rule has_integral_component_nonneg [of \<i>, simplified])
- show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
- by (force simp: ge0)
- show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
- by (rule has_integral_spike_interior [OF hi]) simp
- qed
- then show ?thesis
- by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt_lemma:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and e: "0 < e"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- shows "0 < Re(winding_number \<gamma> z)"
-proof -
- let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
- let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
- have hi: "(?vd has_integral ?int z) (cbox 0 1)"
- unfolding box_real
- apply (subst has_contour_integral [symmetric])
- using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
- have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
- proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
- show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}"
- by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
- show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow>
- e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)"
- by (simp add: ge)
- qed (use has_integral_const_real [of _ 0 1] in auto)
- with e show ?thesis
- by (simp add: Re_winding_number [OF \<gamma>] field_simps)
-qed
-
-lemma winding_number_pos_lt:
- assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
- and e: "0 < e"
- and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
- shows "0 < Re (winding_number \<gamma> z)"
-proof -
- have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
- using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
- then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
- using bounded_pos [THEN iffD1, OF bm] by blast
- { fix x::real assume x: "0 < x" "x < 1"
- then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
- by (simp add: path_image_def power2_eq_square mult_mono')
- with x have "\<gamma> x \<noteq> z" using \<gamma>
- using path_image_def by fastforce
- then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
- using B ge [OF x] B2 e
- apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
- apply (auto simp: divide_left_mono divide_right_mono)
- done
- then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
- } note * = this
- show ?thesis
- using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
-qed
-
-subsection\<open>The winding number is an integer\<close>
-
-text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
- Also on page 134 of Serge Lang's book with the name title, etc.\<close>
-
-lemma exp_fg:
- fixes z::complex
- assumes g: "(g has_vector_derivative g') (at x within s)"
- and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
- and z: "g x \<noteq> z"
- shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
-proof -
- have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
- using assms unfolding has_vector_derivative_def scaleR_conv_of_real
- by (auto intro!: derivative_eq_intros)
- show ?thesis
- apply (rule has_vector_derivative_eq_rhs)
- using z
- apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
- done
-qed
-
-lemma winding_number_exp_integral:
- fixes z::complex
- assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
- and ab: "a \<le> b"
- and z: "z \<notin> \<gamma> ` {a..b}"
- shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
- (is "?thesis1")
- "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
- (is "?thesis2")
-proof -
- let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
- have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
- using z by force
- have cong: "continuous_on {a..b} \<gamma>"
- using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
- obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
- using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
- have \<circ>: "open ({a<..<b} - k)"
- using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
- moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
- by force
- ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
- by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
- { fix w
- assume "w \<noteq> z"
- have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
- by (auto simp: dist_norm intro!: continuous_intros)
- moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
- by (auto simp: intro!: derivative_eq_intros)
- ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
- using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
- by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
- }
- then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
- by meson
- have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
- unfolding integrable_on_def [symmetric]
- proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
- show "\<exists>d h. 0 < d \<and>
- (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
- if "w \<in> - {z}" for w
- apply (rule_tac x="norm(w - z)" in exI)
- using that inverse_eq_divide has_field_derivative_at_within h
- by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
- qed simp
- have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
- unfolding box_real [symmetric] divide_inverse_commute
- by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
- with ab show ?thesis1
- by (simp add: divide_inverse_commute integral_def integrable_on_def)
- { fix t
- assume t: "t \<in> {a..b}"
- have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
- using z by (auto intro!: continuous_intros simp: dist_norm)
- have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
- unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
- obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
- (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
- using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
- by simp (auto simp: ball_def dist_norm that)
- { fix x D
- assume x: "x \<notin> k" "a < x" "x < b"
- then have "x \<in> interior ({a..b} - k)"
- using open_subset_interior [OF \<circ>] by fastforce
- then have con: "isCont ?D\<gamma> x"
- using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
- then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
- by (rule continuous_at_imp_continuous_within)
- have gdx: "\<gamma> differentiable at x"
- using x by (simp add: g_diff_at)
- have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
- (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. integral {a..x}
- (\<lambda>x. ?D\<gamma> x /
- (\<gamma> x - z))) has_vector_derivative
- d / (\<gamma> x - z))
- (at x within {a..b})"
- apply (rule has_vector_derivative_eq_rhs)
- apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
- apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
- done
- then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
- (at x within {a..b})"
- using x gdx t
- apply (clarsimp simp add: differentiable_iff_scaleR)
- apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
- apply (simp_all add: has_vector_derivative_def [symmetric])
- done
- } note * = this
- have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
- apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
- using t
- apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+
- done
- }
- with ab show ?thesis2
- by (simp add: divide_inverse_commute integral_def)
-qed
-
-lemma winding_number_exp_2pi:
- "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
- \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
-using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
- by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
-
-lemma integer_winding_number_eq:
- assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
-proof -
- obtain p where p: "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
- and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
- then have wneq: "winding_number \<gamma> z = winding_number p z"
- using eq winding_number_valid_path by force
- have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
- using eq by (simp add: exp_eq_1 complex_is_Int_iff)
- have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
- using p winding_number_exp_integral(2) [of p 0 1 z]
- apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
- by (metis path_image_def pathstart_def pathstart_in_path_image)
- then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
- using p wneq iff by (auto simp: path_defs)
- then show ?thesis using p eq
- by (auto simp: winding_number_valid_path)
-qed
-
-theorem integer_winding_number:
- "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
-by (metis integer_winding_number_eq)
-
-
-text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
- We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
-
-lemma winding_number_pos_meets:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
- and w: "w \<noteq> z"
- shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
- have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
- using z by (auto simp: path_image_def)
- have [simp]: "z \<notin> \<gamma> ` {0..1}"
- using path_image_def z by auto
- have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
- using \<gamma> valid_path_def by blast
- define r where "r = (w - z) / (\<gamma> 0 - z)"
- have [simp]: "r \<noteq> 0"
- using w z by (auto simp: r_def)
- have cont: "continuous_on {0..1}
- (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
- by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
- have "Arg2pi r \<le> 2*pi"
- by (simp add: Arg2pi less_eq_real_def)
- also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
- using 1
- apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
- apply (simp add: Complex.Re_divide field_simps power2_eq_square)
- done
- finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
- then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
- by (simp add: Arg2pi_ge_0 cont IVT')
- then obtain t where t: "t \<in> {0..1}"
- and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
- by blast
- define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
- have iArg: "Arg2pi r = Im i"
- using eqArg by (simp add: i_def)
- have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
- by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
- have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
- unfolding i_def
- apply (rule winding_number_exp_integral [OF gpdt])
- using t z unfolding path_image_def by force+
- then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
- by (simp add: exp_minus field_simps)
- then have "(w - z) = r * (\<gamma> 0 - z)"
- by (simp add: r_def)
- then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
- apply simp
- apply (subst Complex_Transcendental.Arg2pi_eq [of r])
- apply (simp add: iArg)
- using * apply (simp add: exp_eq_polar field_simps)
- done
- with t show ?thesis
- by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
-qed
-
-lemma winding_number_big_meets:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
- and w: "w \<noteq> z"
- shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
-proof -
- { assume "Re (winding_number \<gamma> z) \<le> - 1"
- then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
- by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
- moreover have "valid_path (reversepath \<gamma>)"
- using \<gamma> valid_path_imp_reverse by auto
- moreover have "z \<notin> path_image (reversepath \<gamma>)"
- by (simp add: z)
- ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
- using winding_number_pos_meets w by blast
- then have ?thesis
- by simp
- }
- then show ?thesis
- using assms
- by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
-qed
-
-lemma winding_number_less_1:
- fixes z::complex
- shows
- "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
- \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
- \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
- by (auto simp: not_less dest: winding_number_big_meets)
-
-text\<open>One way of proving that WN=1 for a loop.\<close>
-lemma winding_number_eq_1:
- fixes z::complex
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
- shows "winding_number \<gamma> z = 1"
-proof -
- have "winding_number \<gamma> z \<in> Ints"
- by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
- then show ?thesis
- using 0 2 by (auto simp: Ints_def)
-qed
-
-subsection\<open>Continuity of winding number and invariance on connected sets\<close>
-
-lemma continuous_at_winding_number:
- fixes z::complex
- assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "continuous (at z) (winding_number \<gamma>)"
-proof -
- obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
- using open_contains_cball [of "- path_image \<gamma>"] z
- by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
- then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
- by (force simp: cball_def dist_norm)
- have oc: "open (- cball z (e / 2))"
- by (simp add: closed_def [symmetric])
- obtain d where "d>0" and pi_eq:
- "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
- (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
- pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
- \<Longrightarrow>
- path_image h1 \<subseteq> - cball z (e / 2) \<and>
- path_image h2 \<subseteq> - cball z (e / 2) \<and>
- (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
- using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
- obtain p where p: "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
- and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
- and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
- { fix w
- assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
- then have wnotp: "w \<notin> path_image p"
- using cbg \<open>d>0\<close> \<open>e>0\<close>
- apply (simp add: path_image_def cball_def dist_norm, clarify)
- apply (frule pg)
- apply (drule_tac c="\<gamma> x" in subsetD)
- apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
- done
- have wnotg: "w \<notin> path_image \<gamma>"
- using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
- { fix k::real
- assume k: "k>0"
- then obtain q where q: "valid_path q" "w \<notin> path_image q"
- "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
- and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
- and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
- using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
- by (force simp: min_divide_distrib_right winding_number_prop_def)
- have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
- apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
- apply (frule pg)
- apply (frule qg)
- using p q \<open>d>0\<close> e2
- apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
- done
- then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
- by (simp add: pi qi)
- } note pip = this
- have "path p"
- using p by (simp add: valid_path_imp_path)
- then have "winding_number p w = winding_number \<gamma> w"
- apply (rule winding_number_unique [OF _ wnotp])
- apply (rule_tac x=p in exI)
- apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
- done
- } note wnwn = this
- obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
- using p open_contains_cball [of "- path_image p"]
- by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
- obtain L
- where "L>0"
- and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
- \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
- cmod (contour_integral p f) \<le> L * B"
- using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by force
- { fix e::real and w::complex
- assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
- then have [simp]: "w \<notin> path_image p"
- using cbp p(2) \<open>0 < pe\<close>
- by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
- have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
- contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
- by (simp add: p contour_integrable_inversediff contour_integral_diff)
- { fix x
- assume pe: "3/4 * pe < cmod (z - x)"
- have "cmod (w - x) < pe/4 + cmod (z - x)"
- by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
- then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
- have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
- using norm_diff_triangle_le by blast
- also have "\<dots> < pe/4 + cmod (w - x)"
- using w by (simp add: norm_minus_commute)
- finally have "pe/2 < cmod (w - x)"
- using pe by auto
- then have "(pe/2)^2 < cmod (w - x) ^ 2"
- apply (rule power_strict_mono)
- using \<open>pe>0\<close> by auto
- then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
- by (simp add: power_divide)
- have "8 * L * cmod (w - z) < e * pe\<^sup>2"
- using w \<open>L>0\<close> by (simp add: field_simps)
- also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
- using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
- also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
- using wx
- apply (rule mult_strict_left_mono)
- using pe2 e not_less_iff_gr_or_eq by fastforce
- finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
- by simp
- also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
- using e by simp
- finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
- have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
- apply (cases "x=z \<or> x=w")
- using pe \<open>pe>0\<close> w \<open>L>0\<close>
- apply (force simp: norm_minus_commute)
- using wx w(2) \<open>L>0\<close> pe pe2 Lwz
- apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
- done
- } note L_cmod_le = this
- have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
- apply (rule L)
- using \<open>pe>0\<close> w
- apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
- using \<open>pe>0\<close> w \<open>L>0\<close>
- apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
- done
- have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
- apply simp
- apply (rule le_less_trans [OF *])
- using \<open>L>0\<close> e
- apply (force simp: field_simps)
- done
- then have "cmod (winding_number p w - winding_number p z) < e"
- using pi_ge_two e
- by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
- } note cmod_wn_diff = this
- then have "isCont (winding_number p) z"
- apply (simp add: continuous_at_eps_delta, clarify)
- apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
- using \<open>pe>0\<close> \<open>L>0\<close>
- apply (simp add: dist_norm cmod_wn_diff)
- done
- then show ?thesis
- apply (rule continuous_transform_within [where d = "min d e / 2"])
- apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
- done
-qed
-
-corollary continuous_on_winding_number:
- "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
- by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
-
-lemma winding_number_constant:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
- shows "winding_number \<gamma> constant_on S"
-proof -
- have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
- if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
- proof -
- have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
- using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
- with ne show ?thesis
- by (auto simp: Ints_def simp flip: of_int_diff)
- qed
- have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
- using continuous_on_winding_number [OF \<gamma>] sg
- by (meson continuous_on_subset disjoint_eq_subset_Compl)
- show ?thesis
- using "*" zero_less_one
- by (blast intro: continuous_discrete_range_constant [OF cs cont])
-qed
-
-lemma winding_number_eq:
- "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
- \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
- using winding_number_constant by (metis constant_on_def)
-
-lemma open_winding_number_levelsets:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
-proof -
- have opn: "open (- path_image \<gamma>)"
- by (simp add: closed_path_image \<gamma> open_Compl)
- { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
- obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
- using open_contains_ball [of "- path_image \<gamma>"] opn z
- by blast
- have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
- apply (rule_tac x=e in exI)
- using e apply (simp add: dist_norm ball_def norm_minus_commute)
- apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
- done
- } then
- show ?thesis
- by (auto simp: open_dist)
-qed
-
-subsection\<open>Winding number is zero "outside" a curve\<close>
-
-proposition winding_number_zero_in_outside:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
- shows "winding_number \<gamma> z = 0"
-proof -
- obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
- using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
- obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
- by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
- have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
- apply (rule outside_subset_convex)
- using B subset_ball by auto
- then have wout: "w \<in> outside (path_image \<gamma>)"
- using w by blast
- moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
- using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
- by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
- ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
- by (metis (no_types, hide_lams) constant_on_def z)
- also have "\<dots> = 0"
- proof -
- have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def)
- { fix e::real assume "0<e"
- obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
- and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
- and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
- using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
- have pip: "path_image p \<subseteq> ball 0 (B + 1)"
- using B
- apply (clarsimp simp add: path_image_def dist_norm ball_def)
- apply (frule (1) pg1)
- apply (fastforce dest: norm_add_less)
- done
- then have "w \<notin> path_image p" using w by blast
- then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
- pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
- (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
- apply (rule_tac x=p in exI)
- apply (simp add: p valid_path_polynomial_function)
- apply (intro conjI)
- using pge apply (simp add: norm_minus_commute)
- apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
- apply (rule holomorphic_intros | simp add: dist_norm)+
- using mem_ball_0 w apply blast
- using p apply (simp_all add: valid_path_polynomial_function loop pip)
- done
- }
- then show ?thesis
- by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
- qed
- finally show ?thesis .
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
- by (rule winding_number_zero_in_outside)
- (auto simp: pathfinish_def pathstart_def path_polynomial_function)
-
-corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
- "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
- by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
-
-lemma winding_number_zero_at_infinity:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
-proof -
- obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
- using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
- then show ?thesis
- apply (rule_tac x="B+1" in exI, clarify)
- apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
- apply (meson less_add_one mem_cball_0 not_le order_trans)
- using ball_subset_cball by blast
-qed
-
-lemma winding_number_zero_point:
- "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
- \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
- using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
- by (fastforce simp add: compact_path_image)
-
-
-text\<open>If a path winds round a set, it winds rounds its inside.\<close>
-lemma winding_number_around_inside:
- assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
- and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
- shows "winding_number \<gamma> w = winding_number \<gamma> z"
-proof -
- have ssb: "s \<subseteq> inside(path_image \<gamma>)"
- proof
- fix x :: complex
- assume "x \<in> s"
- hence "x \<notin> path_image \<gamma>"
- by (meson disjoint_iff_not_equal s_disj)
- thus "x \<in> inside (path_image \<gamma>)"
- using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
-qed
- show ?thesis
- apply (rule winding_number_eq [OF \<gamma> loop w])
- using z apply blast
- apply (simp add: cls connected_with_inside cos)
- apply (simp add: Int_Un_distrib2 s_disj, safe)
- by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
- qed
-
-
-text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
-lemma winding_number_subpath_continuous:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
-proof -
- have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
- winding_number (subpath 0 x \<gamma>) z"
- if x: "0 \<le> x" "x \<le> 1" for x
- proof -
- have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
- 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
- using assms x
- apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
- done
- also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
- apply (subst winding_number_valid_path)
- using assms x
- apply (simp_all add: path_image_subpath valid_path_subpath)
- by (force simp: path_image_def)
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule continuous_on_eq
- [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
- integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
- apply (rule continuous_intros)+
- apply (rule indefinite_integral_continuous_1)
- apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
- using assms
- apply (simp add: *)
- done
-qed
-
-lemma winding_number_ivt_pos:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
- shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
- apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
- apply (rule winding_number_subpath_continuous [OF \<gamma> z])
- using assms
- apply (auto simp: path_image_def image_def)
- done
-
-lemma winding_number_ivt_neg:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
- shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
- apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
- apply (rule winding_number_subpath_continuous [OF \<gamma> z])
- using assms
- apply (auto simp: path_image_def image_def)
- done
-
-lemma winding_number_ivt_abs:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
- shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
- using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
- by force
-
-lemma winding_number_lt_half_lemma:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
- shows "Re(winding_number \<gamma> z) < 1/2"
-proof -
- { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
- then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
- using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
- have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
- using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
- apply (simp add: t \<gamma> valid_path_imp_path)
- using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
- have "b < a \<bullet> \<gamma> 0"
- proof -
- have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
- by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
- thus ?thesis
- by blast
- qed
- moreover have "b < a \<bullet> \<gamma> t"
- proof -
- have "\<gamma> t \<in> {c. b < a \<bullet> c}"
- by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
- thus ?thesis
- by blast
- qed
- ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
- by (simp add: inner_diff_right)+
- then have False
- by (simp add: gt inner_mult_right mult_less_0_iff)
- }
- then show ?thesis by force
-qed
-
-lemma winding_number_lt_half:
- assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
- shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
-proof -
- have "z \<notin> path_image \<gamma>" using assms by auto
- with assms show ?thesis
- apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
- apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
- winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
- done
-qed
-
-lemma winding_number_le_half:
- assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
- and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
- shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
-proof -
- { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
- have "isCont (winding_number \<gamma>) z"
- by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
- then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
- using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
- define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
- have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
- unfolding z'_def inner_mult_right' divide_inverse
- apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
- apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
- done
- have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
- using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
- then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
- by simp
- then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
- using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
- then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
- by linarith
- moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
- apply (rule winding_number_lt_half [OF \<gamma> *])
- using azb \<open>d>0\<close> pag
- apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
- done
- ultimately have False
- by simp
- }
- then show ?thesis by force
-qed
-
-lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
- using separating_hyperplane_closed_point [of "closed_segment a b" z]
- apply auto
- apply (simp add: closed_segment_def)
- apply (drule less_imp_le)
- apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
- apply (auto simp: segment)
- done
-
-
-text\<open> Positivity of WN for a linepath.\<close>
-lemma winding_number_linepath_pos_lt:
- assumes "0 < Im ((b - a) * cnj (b - z))"
- shows "0 < Re(winding_number(linepath a b) z)"
-proof -
- have z: "z \<notin> path_image (linepath a b)"
- using assms
- by (simp add: closed_segment_def) (force simp: algebra_simps)
- show ?thesis
- apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
- apply (simp add: linepath_def algebra_simps)
- done
-qed
-
-
-subsection\<open>Cauchy's integral formula, again for a convex enclosing set\<close>
-
-lemma Cauchy_integral_formula_weak:
- assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
- and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
- and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- obtain f' where f': "(f has_field_derivative f') (at z)"
- using fcd [OF z] by (auto simp: field_differentiable_def)
- have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
- have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
- proof (cases "x = z")
- case True then show ?thesis
- apply (simp add: continuous_within)
- apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
- using has_field_derivative_at_within has_field_derivative_iff f'
- apply (fastforce simp add:)+
- done
- next
- case False
- then have dxz: "dist x z > 0" by auto
- have cf: "continuous (at x within s) f"
- using conf continuous_on_eq_continuous_within that by blast
- have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
- by (rule cf continuous_intros | simp add: False)+
- then show ?thesis
- apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
- apply (force simp: dist_commute)
- done
- qed
- have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
- have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
- apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
- using c apply (force simp: continuous_on_eq_continuous_within)
- apply (rename_tac w)
- apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
- apply (simp_all add: dist_pos_lt dist_commute)
- apply (metis less_irrefl)
- apply (rule derivative_intros fcd | simp)+
- done
- show ?thesis
- apply (rule has_contour_integral_eq)
- using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
- apply (auto simp: ac_simps divide_simps)
- done
-qed
-
-theorem Cauchy_integral_formula_convex_simple:
- "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
- pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
- \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
- apply (rule Cauchy_integral_formula_weak [where k = "{}"])
- using holomorphic_on_imp_continuous_on
- by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
-
-subsection\<open>Homotopy forms of Cauchy's theorem\<close>
-
-lemma Cauchy_theorem_homotopic:
- assumes hom: "if atends then homotopic_paths s g h else homotopic_loops s g h"
- and "open s" and f: "f holomorphic_on s"
- and vpg: "valid_path g" and vph: "valid_path h"
- shows "contour_integral g f = contour_integral h f"
-proof -
- have pathsf: "linked_paths atends g h"
- using hom by (auto simp: linked_paths_def homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish homotopic_loops_imp_loop)
- obtain k :: "real \<times> real \<Rightarrow> complex"
- where contk: "continuous_on ({0..1} \<times> {0..1}) k"
- and ks: "k ` ({0..1} \<times> {0..1}) \<subseteq> s"
- and k [simp]: "\<forall>x. k (0, x) = g x" "\<forall>x. k (1, x) = h x"
- and ksf: "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))"
- using hom pathsf by (auto simp: linked_paths_def homotopic_paths_def homotopic_loops_def homotopic_with_def split: if_split_asm)
- have ucontk: "uniformly_continuous_on ({0..1} \<times> {0..1}) k"
- by (blast intro: compact_Times compact_uniformly_continuous [OF contk])
- { fix t::real assume t: "t \<in> {0..1}"
- have pak: "path (k \<circ> (\<lambda>u. (t, u)))"
- unfolding path_def
- apply (rule continuous_intros continuous_on_subset [OF contk])+
- using t by force
- have pik: "path_image (k \<circ> Pair t) \<subseteq> s"
- using ks t by (auto simp: path_image_def)
- obtain e where "e>0" and e:
- "\<And>g h. \<lbrakk>valid_path g; valid_path h;
- \<forall>u\<in>{0..1}. cmod (g u - (k \<circ> Pair t) u) < e \<and> cmod (h u - (k \<circ> Pair t) u) < e;
- linked_paths atends g h\<rbrakk>
- \<Longrightarrow> contour_integral h f = contour_integral g f"
- using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis
- obtain d where "d>0" and d:
- "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> {0..1}; x' \<in> {0..1} \<times> {0..1}; norm (x'-x) < d\<rbrakk> \<Longrightarrow> norm (k x' - k x) < e/4"
- by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>)
- { fix t1 t2
- assume t1: "0 \<le> t1" "t1 \<le> 1" and t2: "0 \<le> t2" "t2 \<le> 1" and ltd: "\<bar>t1 - t\<bar> < d" "\<bar>t2 - t\<bar> < d"
- have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e"
- using \<open>e > 0\<close>
- apply (rule_tac y = k1 in norm_triangle_half_l)
- apply (auto simp: norm_minus_commute intro: order_less_trans)
- done
- have "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
- (\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and>
- linked_paths atends g1 g2 \<longrightarrow>
- contour_integral g2 f = contour_integral g1 f"
- apply (rule_tac x="e/4" in exI)
- using t t1 t2 ltd \<open>e > 0\<close>
- apply (auto intro!: e simp: d no2 simp del: less_divide_eq_numeral1)
- done
- }
- then have "\<exists>e. 0 < e \<and>
- (\<forall>t1 t2. t1 \<in> {0..1} \<and> t2 \<in> {0..1} \<and> \<bar>t1 - t\<bar> < e \<and> \<bar>t2 - t\<bar> < e
- \<longrightarrow> (\<exists>d. 0 < d \<and>
- (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
- (\<forall>u \<in> {0..1}.
- norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
- linked_paths atends g1 g2
- \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
- by (rule_tac x=d in exI) (simp add: \<open>d > 0\<close>)
- }
- then obtain ee where ee:
- "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0 \<and>
- (\<forall>t1 t2. t1 \<in> {0..1} \<longrightarrow> t2 \<in> {0..1} \<longrightarrow> \<bar>t1 - t\<bar> < ee t \<longrightarrow> \<bar>t2 - t\<bar> < ee t
- \<longrightarrow> (\<exists>d. 0 < d \<and>
- (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
- (\<forall>u \<in> {0..1}.
- norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
- linked_paths atends g1 g2
- \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
- by metis
- note ee_rule = ee [THEN conjunct2, rule_format]
- define C where "C = (\<lambda>t. ball t (ee t / 3)) ` {0..1}"
- obtain C' where C': "C' \<subseteq> C" "finite C'" and C'01: "{0..1} \<subseteq> \<Union>C'"
- proof (rule compactE [OF compact_interval])
- show "{0..1} \<subseteq> \<Union>C"
- using ee [THEN conjunct1] by (auto simp: C_def dist_norm)
- qed (use C_def in auto)
- define kk where "kk = {t \<in> {0..1}. ball t (ee t / 3) \<in> C'}"
- have kk01: "kk \<subseteq> {0..1}" by (auto simp: kk_def)
- define e where "e = Min (ee ` kk)"
- have C'_eq: "C' = (\<lambda>t. ball t (ee t / 3)) ` kk"
- using C' by (auto simp: kk_def C_def)
- have ee_pos[simp]: "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0"
- by (simp add: kk_def ee)
- moreover have "finite kk"
- using \<open>finite C'\<close> kk01 by (force simp: C'_eq inj_on_def ball_eq_ball_iff dest: ee_pos finite_imageD)
- moreover have "kk \<noteq> {}" using \<open>{0..1} \<subseteq> \<Union>C'\<close> C'_eq by force
- ultimately have "e > 0"
- using finite_less_Inf_iff [of "ee ` kk" 0] kk01 by (force simp: e_def)
- then obtain N::nat where "N > 0" and N: "1/N < e/3"
- by (meson divide_pos_pos nat_approx_posE zero_less_Suc zero_less_numeral)
- have e_le_ee: "\<And>i. i \<in> kk \<Longrightarrow> e \<le> ee i"
- using \<open>finite kk\<close> by (simp add: e_def Min_le_iff [of "ee ` kk"])
- have plus: "\<exists>t \<in> kk. x \<in> ball t (ee t / 3)" if "x \<in> {0..1}" for x
- using C' subsetD [OF C'01 that] unfolding C'_eq by blast
- have [OF order_refl]:
- "\<exists>d. 0 < d \<and> (\<forall>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (n/N, u)) < d) \<and> linked_paths atends g j
- \<longrightarrow> contour_integral j f = contour_integral g f)"
- if "n \<le> N" for n
- using that
- proof (induct n)
- case 0 show ?case using ee_rule [of 0 0 0]
- apply clarsimp
- apply (rule_tac x=d in exI, safe)
- by (metis diff_self vpg norm_zero)
- next
- case (Suc n)
- then have N01: "n/N \<in> {0..1}" "(Suc n)/N \<in> {0..1}" by auto
- then obtain t where t: "t \<in> kk" "n/N \<in> ball t (ee t / 3)"
- using plus [of "n/N"] by blast
- then have nN_less: "\<bar>n/N - t\<bar> < ee t"
- by (simp add: dist_norm del: less_divide_eq_numeral1)
- have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t"
- using t N \<open>N > 0\<close> e_le_ee [of t]
- by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps)
- have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast
- obtain d1 where "d1 > 0" and d1:
- "\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2;
- \<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1;
- linked_paths atends g1 g2\<rbrakk>
- \<Longrightarrow> contour_integral g2 f = contour_integral g1 f"
- using ee [THEN conjunct2, rule_format, OF t01 N01 nN_less n'N_less] by fastforce
- have "n \<le> N" using Suc.prems by auto
- with Suc.hyps
- obtain d2 where "d2 > 0"
- and d2: "\<And>j. \<lbrakk>valid_path j; \<forall>u\<in>{0..1}. cmod (j u - k (n/N, u)) < d2; linked_paths atends g j\<rbrakk>
- \<Longrightarrow> contour_integral j f = contour_integral g f"
- by auto
- have "continuous_on {0..1} (k \<circ> (\<lambda>u. (n/N, u)))"
- apply (rule continuous_intros continuous_on_subset [OF contk])+
- using N01 by auto
- then have pkn: "path (\<lambda>u. k (n/N, u))"
- by (simp add: path_def)
- have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
- obtain p where "polynomial_function p"
- and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))"
- "pathfinish p = pathfinish (\<lambda>u. k (n/N, u))"
- and pk_le: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (p t - k (n/N, t)) < min d1 d2"
- using path_approx_polynomial_function [OF pkn min12] by blast
- then have vpp: "valid_path p" using valid_path_polynomial_function by blast
- have lpa: "linked_paths atends g p"
- by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf)
- show ?case
- proof (intro exI; safe)
- fix j
- assume "valid_path j" "linked_paths atends g j"
- and "\<forall>u\<in>{0..1}. cmod (j u - k (real (Suc n) / real N, u)) < min d1 d2"
- then have "contour_integral j f = contour_integral p f"
- using pk_le N01(1) ksf by (force intro!: vpp d1 simp add: linked_paths_def psf)
- also have "... = contour_integral g f"
- using pk_le by (force intro!: vpp d2 lpa)
- finally show "contour_integral j f = contour_integral g f" .
- qed (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
- qed
- then obtain d where "0 < d"
- "\<And>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (1,u)) < d) \<and> linked_paths atends g j
- \<Longrightarrow> contour_integral j f = contour_integral g f"
- using \<open>N>0\<close> by auto
- then have "linked_paths atends g h \<Longrightarrow> contour_integral h f = contour_integral g f"
- using \<open>N>0\<close> vph by fastforce
- then show ?thesis
- by (simp add: pathsf)
-qed
-
-proposition Cauchy_theorem_homotopic_paths:
- assumes hom: "homotopic_paths s g h"
- and "open s" and f: "f holomorphic_on s"
- and vpg: "valid_path g" and vph: "valid_path h"
- shows "contour_integral g f = contour_integral h f"
- using Cauchy_theorem_homotopic [of True s g h] assms by simp
-
-proposition Cauchy_theorem_homotopic_loops:
- assumes hom: "homotopic_loops s g h"
- and "open s" and f: "f holomorphic_on s"
- and vpg: "valid_path g" and vph: "valid_path h"
- shows "contour_integral g f = contour_integral h f"
- using Cauchy_theorem_homotopic [of False s g h] assms by simp
-
-lemma has_contour_integral_newpath:
- "\<lbrakk>(f has_contour_integral y) h; f contour_integrable_on g; contour_integral g f = contour_integral h f\<rbrakk>
- \<Longrightarrow> (f has_contour_integral y) g"
- using has_contour_integral_integral contour_integral_unique by auto
-
-lemma Cauchy_theorem_null_homotopic:
- "\<lbrakk>f holomorphic_on s; open s; valid_path g; homotopic_loops s g (linepath a a)\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
- apply (rule has_contour_integral_newpath [where h = "linepath a a"], simp)
- using contour_integrable_holomorphic_simple
- apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
- by (simp add: Cauchy_theorem_homotopic_loops)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
-
-text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
-
-lemma winding_number_homotopic_paths:
- assumes "homotopic_paths (-{z}) g h"
- shows "winding_number g z = winding_number h z"
-proof -
- have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
- moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
- using homotopic_paths_imp_subset [OF assms] by auto
- ultimately obtain d e where "d > 0" "e > 0"
- and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
- \<Longrightarrow> homotopic_paths (-{z}) g p"
- and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
- \<Longrightarrow> homotopic_paths (-{z}) h q"
- using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
- obtain p where p:
- "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart g" "pathfinish p = pathfinish g"
- and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
- and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
- using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
- obtain q where q:
- "valid_path q" "z \<notin> path_image q"
- "pathstart q = pathstart h" "pathfinish q = pathfinish h"
- and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
- and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
- using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
- have "homotopic_paths (- {z}) g p"
- by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
- moreover have "homotopic_paths (- {z}) h q"
- by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
- ultimately have "homotopic_paths (- {z}) p q"
- by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
- then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
- by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
- then show ?thesis
- by (simp add: pap paq)
-qed
-
-lemma winding_number_homotopic_loops:
- assumes "homotopic_loops (-{z}) g h"
- shows "winding_number g z = winding_number h z"
-proof -
- have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
- moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
- using homotopic_loops_imp_subset [OF assms] by auto
- moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
- using homotopic_loops_imp_loop [OF assms] by auto
- ultimately obtain d e where "d > 0" "e > 0"
- and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
- \<Longrightarrow> homotopic_loops (-{z}) g p"
- and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
- \<Longrightarrow> homotopic_loops (-{z}) h q"
- using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
- obtain p where p:
- "valid_path p" "z \<notin> path_image p"
- "pathstart p = pathstart g" "pathfinish p = pathfinish g"
- and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
- and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
- using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
- obtain q where q:
- "valid_path q" "z \<notin> path_image q"
- "pathstart q = pathstart h" "pathfinish q = pathfinish h"
- and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
- and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
- using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
- have gp: "homotopic_loops (- {z}) g p"
- by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
- have hq: "homotopic_loops (- {z}) h q"
- by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
- have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
- proof (rule Cauchy_theorem_homotopic_loops)
- show "homotopic_loops (- {z}) p q"
- by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
- qed (auto intro!: holomorphic_intros simp: p q)
- then show ?thesis
- by (simp add: pap paq)
-qed
-
-lemma winding_number_paths_linear_eq:
- "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
- \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
-
-lemma winding_number_loops_linear_eq:
- "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
- \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
-
-lemma winding_number_nearby_paths_eq:
- "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
- \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
-
-lemma winding_number_nearby_loops_eq:
- "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
- \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
- \<Longrightarrow> winding_number h z = winding_number g z"
- by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
-
-
-lemma winding_number_subpath_combine:
- "\<lbrakk>path g; z \<notin> path_image g;
- u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
- \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
- winding_number (subpath u w g) z"
-apply (rule trans [OF winding_number_join [THEN sym]
- winding_number_homotopic_paths [OF homotopic_join_subpaths]])
- using path_image_subpath_subset by auto
-
-subsection\<open>Partial circle path\<close>
-
-definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
- where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
-
-lemma pathstart_part_circlepath [simp]:
- "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
-by (metis part_circlepath_def pathstart_def pathstart_linepath)
-
-lemma pathfinish_part_circlepath [simp]:
- "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
-by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
-
-lemma reversepath_part_circlepath[simp]:
- "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
- unfolding part_circlepath_def reversepath_def linepath_def
- by (auto simp:algebra_simps)
-
-lemma has_vector_derivative_part_circlepath [derivative_intros]:
- "((part_circlepath z r s t) has_vector_derivative
- (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
- (at x within X)"
- apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
- apply (rule has_vector_derivative_real_field)
- apply (rule derivative_eq_intros | simp)+
- done
-
-lemma differentiable_part_circlepath:
- "part_circlepath c r a b differentiable at x within A"
- using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
-
-lemma vector_derivative_part_circlepath:
- "vector_derivative (part_circlepath z r s t) (at x) =
- \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
- using has_vector_derivative_part_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_part_circlepath01:
- "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
- \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
- \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
- using has_vector_derivative_part_circlepath
- by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
- apply (simp add: valid_path_def)
- apply (rule C1_differentiable_imp_piecewise)
- apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
- intro!: continuous_intros)
- done
-
-lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
- by (simp add: valid_path_imp_path)
-
-proposition path_image_part_circlepath:
- assumes "s \<le> t"
- shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
-proof -
- { fix z::real
- assume "0 \<le> z" "z \<le> 1"
- with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
- apply (rule_tac x="(1 - z) * s + z * t" in exI)
- apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
- apply (rule conjI)
- using mult_right_mono apply blast
- using affine_ineq by (metis "mult.commute")
- }
- moreover
- { fix z
- assume "s \<le> z" "z \<le> t"
- then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
- apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
- apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
- apply (auto simp: field_split_simps)
- done
- }
- ultimately show ?thesis
- by (fastforce simp add: path_image_def part_circlepath_def)
-qed
-
-lemma path_image_part_circlepath':
- "path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t"
-proof -
- have "path_image (part_circlepath z r s t) =
- (\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}"
- by (simp add: image_image path_image_def part_circlepath_def)
- also have "linepath s t ` {0..1} = closed_segment s t"
- by (rule linepath_image_01)
- finally show ?thesis by (simp add: cis_conv_exp)
-qed
-
-lemma path_image_part_circlepath_subset:
- "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
-by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
-
-lemma in_path_image_part_circlepath:
- assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
- shows "norm(w - z) = r"
-proof -
- have "w \<in> {c. dist z c = r}"
- by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
- thus ?thesis
- by (simp add: dist_norm norm_minus_commute)
-qed
-
-lemma path_image_part_circlepath_subset':
- assumes "r \<ge> 0"
- shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
-proof (cases "s \<le> t")
- case True
- thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
-next
- case False
- thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
- by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
-qed
-
-lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
- by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
-
-lemma contour_integral_bound_part_circlepath:
- assumes "f contour_integrable_on part_circlepath c r a b"
- assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B"
- shows "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
-proof -
- let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) *
- exp (\<i> * linepath a b x))"
- have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)"
- proof (rule integral_norm_bound_integral, goal_cases)
- case 1
- with assms(1) show ?case
- by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
- next
- case (3 x)
- with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
- by (intro mult_mono) (auto simp: path_image_def)
- qed auto
- also have "?I = contour_integral (part_circlepath c r a b) f"
- by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
- finally show ?thesis by simp
-qed
-
-lemma has_contour_integral_part_circlepath_iff:
- assumes "a < b"
- shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
- ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}"
-proof -
- have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
- ((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
- (at x within {0..1})) has_integral I) {0..1}"
- unfolding has_contour_integral_def ..
- also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> *
- cis (linepath a b x)) has_integral I) {0..1}"
- by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
- (simp_all add: cis_conv_exp)
- also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) *
- r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) *
- vector_derivative (linepath (of_real a) (of_real b))
- (at x within {0..1})) has_integral I) {0..1}"
- by (intro has_integral_cong, subst vector_derivative_linepath_within)
- (auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
- also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I)
- (linepath (of_real a) (of_real b))"
- by (simp add: has_contour_integral_def)
- also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms
- by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
- finally show ?thesis .
-qed
-
-lemma contour_integrable_part_circlepath_iff:
- assumes "a < b"
- shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow>
- (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (auto simp: contour_integrable_on_def integrable_on_def
- has_contour_integral_part_circlepath_iff)
-
-lemma contour_integral_part_circlepath_eq:
- assumes "a < b"
- shows "contour_integral (part_circlepath c r a b) f =
- integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)"
-proof (cases "f contour_integrable_on part_circlepath c r a b")
- case True
- hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (simp add: contour_integrable_part_circlepath_iff)
- with True show ?thesis
- using has_contour_integral_part_circlepath_iff[OF assms]
- contour_integral_unique has_integral_integrable_integral by blast
-next
- case False
- hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
- using assms by (simp add: contour_integrable_part_circlepath_iff)
- with False show ?thesis
- by (simp add: not_integrable_contour_integral not_integrable_integral)
-qed
-
-lemma contour_integral_part_circlepath_reverse:
- "contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
- by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all
-
-lemma contour_integral_part_circlepath_reverse':
- "b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f =
- -contour_integral (part_circlepath c r b a) f"
- by (rule contour_integral_part_circlepath_reverse)
-
-lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
-proof (cases "w = 0")
- case True then show ?thesis by auto
-next
- case False
- have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
- apply (simp add: norm_mult finite_int_iff_bounded_le)
- apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
- apply (auto simp: field_split_simps le_floor_iff)
- done
- have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
- by blast
- show ?thesis
- apply (subst exp_Ln [OF False, symmetric])
- apply (simp add: exp_eq)
- using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
- done
-qed
-
-lemma finite_bounded_log2:
- fixes a::complex
- assumes "a \<noteq> 0"
- shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
-proof -
- have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
- by (rule finite_imageI [OF finite_bounded_log])
- show ?thesis
- by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
-qed
-
-lemma has_contour_integral_bound_part_circlepath_strong:
- assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
- and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
- and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
- shows "cmod i \<le> B * r * (t - s)"
-proof -
- consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
- then show ?thesis
- proof cases
- case 1 with fi [unfolded has_contour_integral]
- have "i = 0" by (simp add: vector_derivative_part_circlepath)
- with assms show ?thesis by simp
- next
- case 2
- have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
- have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
- by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
- have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
- proof -
- define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
- have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
- apply (rule finite_vimageI [OF finite_bounded_log2])
- using \<open>s < t\<close> apply (auto simp: inj_of_real)
- done
- show ?thesis
- apply (simp add: part_circlepath_def linepath_def vimage_def)
- apply (rule finite_subset [OF _ fin])
- using le
- apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
- done
- qed
- then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
- by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
- have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
- else f(part_circlepath z r s t x) *
- vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}"
- by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto)
- have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
- by (auto intro!: B [unfolded path_image_def image_def, simplified])
- show ?thesis
- apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
- using assms apply force
- apply (simp add: norm_mult vector_derivative_part_circlepath)
- using le * "2" \<open>r > 0\<close> by auto
- qed
-qed
-
-lemma has_contour_integral_bound_part_circlepath:
- "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
- 0 \<le> B; 0 < r; s \<le> t;
- \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*r*(t - s)"
- by (auto intro: has_contour_integral_bound_part_circlepath_strong)
-
-lemma contour_integrable_continuous_part_circlepath:
- "continuous_on (path_image (part_circlepath z r s t)) f
- \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
- apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
- apply (rule integrable_continuous_real)
- apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
- done
-
-proposition winding_number_part_circlepath_pos_less:
- assumes "s < t" and no: "norm(w - z) < r"
- shows "0 < Re (winding_number(part_circlepath z r s t) w)"
-proof -
- have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
- note valid_path_part_circlepath
- moreover have " w \<notin> path_image (part_circlepath z r s t)"
- using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
- moreover have "0 < r * (t - s) * (r - cmod (w - z))"
- using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
- ultimately show ?thesis
- apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
- apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
- apply (rule mult_left_mono)+
- using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
- apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
- using assms \<open>0 < r\<close> by auto
-qed
-
-lemma simple_path_part_circlepath:
- "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
-proof (cases "r = 0 \<or> s = t")
- case True
- then show ?thesis
- unfolding part_circlepath_def simple_path_def
- by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
-next
- case False then have "r \<noteq> 0" "s \<noteq> t" by auto
- have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
- by (simp add: algebra_simps)
- have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
- \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
- by auto
- have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
- (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
- by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
- intro: exI [where x = "-n" for n])
- have 1: "\<bar>s - t\<bar> \<le> 2 * pi"
- if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1"
- proof (rule ccontr)
- assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
- then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
- using False that [of "2*pi / \<bar>t - s\<bar>"]
- by (simp add: abs_minus_commute divide_simps)
- show False
- using * [of 1] * [of "-1"] by auto
- qed
- have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
- proof -
- have "t-s = 2 * (real_of_int n * pi)/x"
- using that by (simp add: field_simps)
- then show ?thesis by (metis abs_minus_commute)
- qed
- have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
- by force
- show ?thesis using False
- apply (simp add: simple_path_def)
- apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff)
- apply (subst abs_away)
- apply (auto simp: 1)
- apply (rule ccontr)
- apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
- done
-qed
-
-lemma arc_part_circlepath:
- assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
- shows "arc (part_circlepath z r s t)"
-proof -
- have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
- and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
- proof (rule ccontr)
- assume "x \<noteq> y"
- have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
- by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
- then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
- by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
- with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))"
- by (force simp: field_simps)
- have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>"
- using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps)
- then show False
- using assms x y st by (auto dest: of_int_lessD)
- qed
- show ?thesis
- using assms
- apply (simp add: arc_def)
- apply (simp add: part_circlepath_def inj_on_def exp_eq)
- apply (blast intro: *)
- done
-qed
-
-subsection\<open>Special case of one complete circle\<close>
-
-definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex"
- where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
-
-lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
- by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
-
-lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
- by (simp add: circlepath_def)
-
-lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
- by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
-
-lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
-proof -
- have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) =
- z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
- by (simp add: divide_simps) (simp add: algebra_simps)
- also have "\<dots> = z - r * exp (2 * pi * \<i> * x)"
- by (simp add: exp_add)
- finally show ?thesis
- by (simp add: circlepath path_image_def sphere_def dist_norm)
-qed
-
-lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
- using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
- by (simp add: add.commute)
-
-lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
- using circlepath_add1 [of z r "x-1/2"]
- by (simp add: add.commute)
-
-lemma path_image_circlepath_minus_subset:
- "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
- apply (simp add: path_image_def image_def circlepath_minus, clarify)
- apply (case_tac "xa \<le> 1/2", force)
- apply (force simp: circlepath_add_half)+
- done
-
-lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
- using path_image_circlepath_minus_subset by fastforce
-
-lemma has_vector_derivative_circlepath [derivative_intros]:
- "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
- (at x within X)"
- apply (simp add: circlepath_def scaleR_conv_of_real)
- apply (rule derivative_eq_intros)
- apply (simp add: algebra_simps)
- done
-
-lemma vector_derivative_circlepath:
- "vector_derivative (circlepath z r) (at x) =
- 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
-using has_vector_derivative_circlepath vector_derivative_at by blast
-
-lemma vector_derivative_circlepath01:
- "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
- \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
- 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
- using has_vector_derivative_circlepath
- by (auto simp: vector_derivative_at_within_ivl)
-
-lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
- by (simp add: circlepath_def)
-
-lemma path_circlepath [simp]: "path (circlepath z r)"
- by (simp add: valid_path_imp_path)
-
-lemma path_image_circlepath_nonneg:
- assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
-proof -
- have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
- proof (cases "x = z")
- case True then show ?thesis by force
- next
- case False
- define w where "w = x - z"
- then have "w \<noteq> 0" by (simp add: False)
- have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
- using cis_conv_exp complex_eq_iff by auto
- show ?thesis
- apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
- apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
- apply (rule_tac x="t / (2*pi)" in image_eqI)
- apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
- using False **
- apply (auto simp: w_def)
- done
- qed
- show ?thesis
- unfolding circlepath path_image_def sphere_def dist_norm
- by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
-qed
-
-lemma path_image_circlepath [simp]:
- "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
- using path_image_circlepath_minus
- by (force simp: path_image_circlepath_nonneg abs_if)
-
-lemma has_contour_integral_bound_circlepath_strong:
- "\<lbrakk>(f has_contour_integral i) (circlepath z r);
- finite k; 0 \<le> B; 0 < r;
- \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*(2*pi*r)"
- unfolding circlepath_def
- by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
-
-lemma has_contour_integral_bound_circlepath:
- "\<lbrakk>(f has_contour_integral i) (circlepath z r);
- 0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
- \<Longrightarrow> norm i \<le> B*(2*pi*r)"
- by (auto intro: has_contour_integral_bound_circlepath_strong)
-
-lemma contour_integrable_continuous_circlepath:
- "continuous_on (path_image (circlepath z r)) f
- \<Longrightarrow> f contour_integrable_on (circlepath z r)"
- by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
-
-lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
- by (simp add: circlepath_def simple_path_part_circlepath)
-
-lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
- by (simp add: sphere_def dist_norm norm_minus_commute)
-
-lemma contour_integral_circlepath:
- assumes "r > 0"
- shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
-proof (rule contour_integral_unique)
- show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)"
- unfolding has_contour_integral_def using assms
- apply (subst has_integral_cong)
- apply (simp add: vector_derivative_circlepath01)
- using has_integral_const_real [of _ 0 1] apply (force simp: circlepath)
- done
-qed
-
-lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
- apply (rule winding_number_unique_loop)
- apply (simp_all add: sphere_def valid_path_imp_path)
- apply (rule_tac x="circlepath z r" in exI)
- apply (simp add: sphere_def contour_integral_circlepath)
- done
-
-proposition winding_number_circlepath:
- assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
-proof (cases "w = z")
- case True then show ?thesis
- using assms winding_number_circlepath_centre by auto
-next
- case False
- have [simp]: "r > 0"
- using assms le_less_trans norm_ge_zero by blast
- define r' where "r' = norm(w - z)"
- have "r' < r"
- by (simp add: assms r'_def)
- have disjo: "cball z r' \<inter> sphere z r = {}"
- using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
- have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
- proof (rule winding_number_around_inside [where s = "cball z r'"])
- show "winding_number (circlepath z r) z \<noteq> 0"
- by (simp add: winding_number_circlepath_centre)
- show "cball z r' \<inter> path_image (circlepath z r) = {}"
- by (simp add: disjo less_eq_real_def)
- qed (auto simp: r'_def dist_norm norm_minus_commute)
- also have "\<dots> = 1"
- by (simp add: winding_number_circlepath_centre)
- finally show ?thesis .
-qed
-
-
-text\<open> Hence the Cauchy formula for points inside a circle.\<close>
-
-theorem Cauchy_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
- shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
- (circlepath z r)"
-proof -
- have "r > 0"
- using assms le_less_trans norm_ge_zero by blast
- have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
- (circlepath z r)"
- proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
- show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
- f field_differentiable at x"
- using holf holomorphic_on_imp_differentiable_at by auto
- have "w \<notin> sphere z r"
- by simp (metis dist_commute dist_norm not_le order_refl wz)
- then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
- using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
- qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
- then show ?thesis
- by (simp add: winding_number_circlepath assms)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
- assumes "f holomorphic_on cball z r" "norm(w - z) < r"
- shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
- (circlepath z r)"
-using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
-
-
-lemma no_bounded_connected_component_imp_winding_number_zero:
- assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
- and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
- shows "winding_number g z = 0"
-apply (rule winding_number_zero_in_outside)
-apply (simp_all add: assms)
-by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
-
-lemma no_bounded_path_component_imp_winding_number_zero:
- assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
- and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
- shows "winding_number g z = 0"
-apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
-by (simp add: bounded_subset nb path_component_subset_connected_component)
-
-
-subsection\<open> Uniform convergence of path integral\<close>
-
-text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
-
-proposition contour_integral_uniform_limit:
- assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
- and ul_f: "uniform_limit (path_image \<gamma>) f l F"
- and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
- and \<gamma>: "valid_path \<gamma>"
- and [simp]: "\<not> trivial_limit F"
- shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
-proof -
- have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
- { fix e::real
- assume "0 < e"
- then have "0 < e / (\<bar>B\<bar> + 1)" by simp
- then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
- using ul_f [unfolded uniform_limit_iff dist_norm] by auto
- with ev_fint
- obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
- and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
- using eventually_happens [OF eventually_conj]
- by (fastforce simp: contour_integrable_on path_image_def)
- have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
- using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
- have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
- proof (intro exI conjI ballI)
- show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
- if "x \<in> {0..1}" for x
- apply (rule order_trans [OF _ Ble])
- using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
- apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
- apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le])
- done
- qed (rule inta)
- }
- then show lintg: "l contour_integrable_on \<gamma>"
- unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
- { fix e::real
- define B' where "B' = B + 1"
- have B': "B' > 0" "B' > B" using \<open>0 \<le> B\<close> by (auto simp: B'_def)
- assume "0 < e"
- then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
- using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
- by (simp add: field_simps)
- have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
- have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
- if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
- proof -
- have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
- using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
- also have "\<dots> < e"
- by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
- finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
- then show ?thesis
- by (simp add: left_diff_distrib [symmetric] norm_mult)
- qed
- have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk>
- \<Longrightarrow> cmod (integral {0..1}
- (\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e"
- apply (rule le_less_trans [OF integral_norm_bound_integral ie])
- apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
- apply (blast intro: *)+
- done
- have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
- apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
- apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
- apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
- done
- }
- then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
- by (rule tendstoI)
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath:
- assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
- and "uniform_limit (sphere z r) f l F"
- and "\<not> trivial_limit F" "0 < r"
- shows "l contour_integrable_on (circlepath z r)"
- "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
- using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
-
-lemma Cauchy_next_derivative:
- assumes "continuous_on (path_image \<gamma>) f'"
- and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
- and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
- and k: "k \<noteq> 0"
- and "open s"
- and \<gamma>: "valid_path \<gamma>"
- and w: "w \<in> s - path_image \<gamma>"
- shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
- and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
- (at w)" (is "?thes2")
-proof -
- have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
- then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
- using open_contains_ball by blast
- have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
- by (metis norm_of_nat of_nat_Suc)
- have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
- \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
- apply (rule contour_integrable_div [OF contour_integrable_diff])
- using int w d
- by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
- have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
- contour_integrable_on \<gamma>"
- unfolding eventually_at
- apply (rule_tac x=d in exI)
- apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
- done
- have bim_g: "bounded (image f' (path_image \<gamma>))"
- by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
- then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
- by (force simp: bounded_pos path_image_def)
- have twom: "\<forall>\<^sub>F n in at w.
- \<forall>x\<in>path_image \<gamma>.
- cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
- if "0 < e" for e
- proof -
- have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e"
- if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
- and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
- for u x
- proof -
- define ff where [abs_def]:
- "ff n w =
- (if n = 0 then inverse(x - w)^k
- else if n = 1 then k / (x - w)^(Suc k)
- else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
- have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
- by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
- have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
- if "z \<in> ball w (d/2)" "i \<le> 1" for i z
- proof -
- have "z \<notin> path_image \<gamma>"
- using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
- then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
- then have neq: "x * x + z * z \<noteq> x * (z * 2)"
- by (blast intro: dest!: sum_sqs_eq)
- with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
- then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
- by (simp add: algebra_simps)
- show ?thesis using \<open>i \<le> 1\<close>
- apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
- apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
- done
- qed
- { fix a::real and b::real assume ab: "a > 0" "b > 0"
- then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
- by (subst mult_le_cancel_left_pos)
- (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
- with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
- by (simp add: field_simps)
- } note canc = this
- have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
- if "v \<in> ball w (d/2)" for v
- proof -
- have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
- by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
- have "d/2 \<le> cmod (x - v)" using d x that
- using lessd d x
- by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
- then have "d \<le> cmod (x - v) * 2"
- by (simp add: field_split_simps)
- then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
- using \<open>0 < d\<close> order_less_imp_le power_mono by blast
- have "x \<noteq> v" using that
- using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
- then show ?thesis
- using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
- using dpow_le apply (simp add: field_split_simps)
- done
- qed
- have ub: "u \<in> ball w (d/2)"
- using uwd by (simp add: dist_commute dist_norm)
- have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
- using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
- by (simp add: ff_def \<open>0 < d\<close>)
- then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
- by (simp add: field_simps)
- then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
- / (cmod (u - w) * real k)
- \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
- using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
- also have "\<dots> < e"
- using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
- finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
- / cmod ((u - w) * real k) < e"
- by (simp add: norm_mult)
- have "x \<noteq> u"
- using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
- show ?thesis
- apply (rule le_less_trans [OF _ e])
- using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
- apply (simp add: field_simps norm_divide [symmetric])
- done
- qed
- show ?thesis
- unfolding eventually_at
- apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
- apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
- done
- qed
- have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix e::real
- assume "0 < e"
- have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
- if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
- and x: "0 \<le> x" "x \<le> 1"
- for u x
- proof (cases "(f' (\<gamma> x)) = 0")
- case True then show ?thesis by (simp add: \<open>0 < e\<close>)
- next
- case False
- have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
- cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
- by (simp add: field_simps)
- also have "\<dots> = cmod (f' (\<gamma> x)) *
- cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
- inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
- by (simp add: norm_mult)
- also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
- using False mult_strict_left_mono [OF ec] by force
- also have "\<dots> \<le> e" using C
- by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
- finally show ?thesis .
- qed
- show "\<forall>\<^sub>F n in at w.
- \<forall>x\<in>path_image \<gamma>.
- cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
- using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def
- by (force intro: * elim: eventually_mono)
- qed
- show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
- by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
- have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
- \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
- by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
- have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
- (f u - f w) / (u - w) / k"
- if "dist u w < d" for u
- proof -
- have u: "u \<in> s - path_image \<gamma>"
- by (metis subsetD d dist_commute mem_ball that)
- show ?thesis
- apply (rule contour_integral_unique)
- apply (simp add: diff_divide_distrib algebra_simps)
- apply (intro has_contour_integral_diff has_contour_integral_div)
- using u w apply (simp_all add: field_simps int)
- done
- qed
- show ?thes2
- apply (simp add: has_field_derivative_iff del: power_Suc)
- apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
- apply (simp add: \<open>k \<noteq> 0\<close> **)
- done
-qed
-
-lemma Cauchy_next_derivative_circlepath:
- assumes contf: "continuous_on (path_image (circlepath z r)) f"
- and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
- and k: "k \<noteq> 0"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
- (is "?thes2")
-proof -
- have "r > 0" using w
- using ball_eq_empty by fastforce
- have wim: "w \<in> ball z r - path_image (circlepath z r)"
- using w by (auto simp: dist_norm)
- show ?thes1 ?thes2
- by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
- auto simp: vector_derivative_circlepath norm_mult)+
-qed
-
-
-text\<open> In particular, the first derivative formula.\<close>
-
-lemma Cauchy_derivative_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
- (is "?thes2")
-proof -
- have [simp]: "r \<ge> 0" using w
- using ball_eq_empty by fastforce
- have f: "continuous_on (path_image (circlepath z r)) f"
- by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
- have int: "\<And>w. dist z w < r \<Longrightarrow>
- ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
- by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
- show ?thes1
- apply (simp add: power2_eq_square)
- apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
- apply (blast intro: int)
- done
- have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
- apply (simp add: power2_eq_square)
- apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
- apply (blast intro: int)
- done
- then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
- by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
- show ?thes2
- by simp (rule fder)
-qed
-
-subsection\<open>Existence of all higher derivatives\<close>
-
-proposition derivative_is_holomorphic:
- assumes "open S"
- and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
- shows "f' holomorphic_on S"
-proof -
- have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
- proof -
- obtain r where "r > 0" and r: "cball z r \<subseteq> S"
- using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
- then have holf_cball: "f holomorphic_on cball z r"
- apply (simp add: holomorphic_on_def)
- using field_differentiable_at_within field_differentiable_def fder by blast
- then have "continuous_on (path_image (circlepath z r)) f"
- using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
- then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
- by (auto intro: continuous_intros)+
- have contf_cball: "continuous_on (cball z r) f" using holf_cball
- by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
- have holf_ball: "f holomorphic_on ball z r" using holf_cball
- using ball_subset_cball holomorphic_on_subset by blast
- { fix w assume w: "w \<in> ball z r"
- have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
- by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
- have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
- (at w)"
- by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
- have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
- using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
- have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
- contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
- (circlepath z r)"
- by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
- then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
- contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
- (circlepath z r)"
- by (simp add: algebra_simps)
- then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
- by (simp add: f'_eq)
- } note * = this
- show ?thesis
- apply (rule exI)
- apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
- apply (simp_all add: \<open>0 < r\<close> * dist_norm)
- done
- qed
- show ?thesis
- by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
-qed
-
-lemma holomorphic_deriv [holomorphic_intros]:
- "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
-by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
-
-lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
- using analytic_on_holomorphic holomorphic_deriv by auto
-
-lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
- by (induction n) (auto simp: holomorphic_deriv)
-
-lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
- unfolding analytic_on_def using holomorphic_higher_deriv by blast
-
-lemma has_field_derivative_higher_deriv:
- "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
- \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
-by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
- funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
-
-lemma valid_path_compose_holomorphic:
- assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
- shows "valid_path (f \<circ> g)"
-proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
- fix x assume "x \<in> path_image g"
- then show "f field_differentiable at x"
- using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
-next
- have "deriv f holomorphic_on S"
- using holomorphic_deriv holo \<open>open S\<close> by auto
- then show "continuous_on (path_image g) (deriv f)"
- using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
-qed
-
-
-subsection\<open>Morera's theorem\<close>
-
-lemma Morera_local_triangle_ball:
- assumes "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
- (\<forall>b c. closed_segment b c \<subseteq> ball a e
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0)"
- shows "f analytic_on S"
-proof -
- { fix z assume "z \<in> S"
- with assms obtain e a where
- "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
- and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
- \<Longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- by fastforce
- have az: "dist a z < e" using mem_ball z by blast
- have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
- by (simp add: dist_commute ball_subset_ball_iff)
- have "\<exists>e>0. f holomorphic_on ball z e"
- proof (intro exI conjI)
- have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
- by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
- show "f holomorphic_on ball z (e - dist a z)"
- apply (rule holomorphic_on_subset [OF _ sb_ball])
- apply (rule derivative_is_holomorphic[OF open_ball])
- apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
- apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
- done
- qed (simp add: az)
- }
- then show ?thesis
- by (simp add: analytic_on_def)
-qed
-
-lemma Morera_local_triangle:
- assumes "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and>
- (\<forall>a b c. convex hull {a,b,c} \<subseteq> t
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0)"
- shows "f analytic_on S"
-proof -
- { fix z assume "z \<in> S"
- with assms obtain t where
- "open t" and z: "z \<in> t" and contf: "continuous_on t f"
- and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t
- \<Longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
- by force
- then obtain e where "e>0" and e: "ball z e \<subseteq> t"
- using open_contains_ball by blast
- have [simp]: "continuous_on (ball z e) f" using contf
- using continuous_on_subset e by blast
- have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
- contour_integral (linepath z b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c z) f = 0"
- by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
- have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
- (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
- contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
- using \<open>e > 0\<close> eq0 by force
- }
- then show ?thesis
- by (simp add: Morera_local_triangle_ball)
-qed
-
-proposition Morera_triangle:
- "\<lbrakk>continuous_on S f; open S;
- \<And>a b c. convex hull {a,b,c} \<subseteq> S
- \<longrightarrow> contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0\<rbrakk>
- \<Longrightarrow> f analytic_on S"
- using Morera_local_triangle by blast
-
-subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
-
-lemma higher_deriv_linear [simp]:
- "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
- by (induction n) auto
-
-lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
- by (induction n) auto
-
-lemma higher_deriv_ident [simp]:
- "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
- apply (induction n, simp)
- apply (metis higher_deriv_linear lambda_one)
- done
-
-lemma higher_deriv_id [simp]:
- "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
- by (simp add: id_def)
-
-lemma has_complex_derivative_funpow_1:
- "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
- apply (induction n, auto)
- apply (simp add: id_def)
- by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
-
-lemma higher_deriv_uminus:
- assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
- apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
- apply (rule derivative_eq_intros | rule * refl assms)+
- apply (auto simp add: Suc)
- done
- then show ?case
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_add:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
- deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
- apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
- apply (rule derivative_eq_intros | rule * refl assms)+
- apply (auto simp add: Suc)
- done
- then show ?case
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_diff:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
- apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
- apply (subst higher_deriv_add)
- using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
- done
-
-lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
- by (cases k) simp_all
-
-lemma higher_deriv_mult:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
- (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
- "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- have sumeq: "(\<Sum>i = 0..n.
- of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
- g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
- apply (simp add: bb algebra_simps sum.distrib)
- apply (subst (4) sum_Suc_reindex)
- apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
- done
- have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
- (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
- (at z)"
- apply (rule has_field_derivative_transform_within_open
- [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
- apply (simp add: algebra_simps)
- apply (rule DERIV_cong [OF DERIV_sum])
- apply (rule DERIV_cmult)
- apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
- done
- then show ?case
- unfolding funpow.simps o_apply
- by (simp add: DERIV_imp_deriv)
-qed
-
-lemma higher_deriv_transform_within_open:
- fixes z::complex
- assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
- and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
- shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
-using z
-by (induction i arbitrary: z)
- (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
-
-lemma higher_deriv_compose_linear:
- fixes z::complex
- assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
- and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
- shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
-using z
-proof (induction n arbitrary: z)
- case 0 then show ?case by simp
-next
- case (Suc n z)
- have holo0: "f holomorphic_on (*) u ` S"
- by (meson fg f holomorphic_on_subset image_subset_iff)
- have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
- by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
- have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
- by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
- have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
- apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
- apply (rule holo0 holomorphic_intros)+
- done
- have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
- apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
- apply (rule holomorphic_higher_deriv [OF holo1 S])
- apply (simp add: Suc.IH)
- done
- also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
- apply (rule deriv_cmult)
- apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
- apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
- apply (simp)
- apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
- apply (blast intro: fg)
- done
- also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
- apply (subst complex_derivative_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
- apply (rule derivative_intros)
- using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
- apply (simp)
- done
- finally show ?case
- by simp
-qed
-
-lemma higher_deriv_add_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_add show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_diff_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_diff show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-lemma higher_deriv_uminus_at:
- "f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
- using higher_deriv_uminus
- by (auto simp: analytic_at)
-
-lemma higher_deriv_mult_at:
- assumes "f analytic_on {z}" "g analytic_on {z}"
- shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
- (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
-proof -
- have "f analytic_on {z} \<and> g analytic_on {z}"
- using assms by blast
- with higher_deriv_mult show ?thesis
- by (auto simp: analytic_at_two)
-qed
-
-
-text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
-
-proposition no_isolated_singularity:
- fixes z::complex
- assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
- shows "f holomorphic_on S"
-proof -
- { fix z
- assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
- have "f field_differentiable at z"
- proof (cases "z \<in> K")
- case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
- next
- case True
- with finite_set_avoid [OF K, of z]
- obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
- by blast
- obtain e where "e>0" and e: "ball z e \<subseteq> S"
- using S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
- have fde: "continuous_on (ball z (min d e)) f"
- by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
- have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
- by (simp add: hull_minimal continuous_on_subset [OF fde])
- have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
- \<Longrightarrow> f field_differentiable at x" for a b c x
- by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
- obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
- apply (rule contour_integral_convex_primitive
- [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
- using cont fd by auto
- then have "f holomorphic_on ball z (min d e)"
- by (metis open_ball at_within_open derivative_is_holomorphic)
- then show ?thesis
- unfolding holomorphic_on_def
- by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
- qed
- }
- with holf S K show ?thesis
- by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
-qed
-
-lemma no_isolated_singularity':
- fixes z::complex
- assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
- and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
- shows "f holomorphic_on S"
-proof (rule no_isolated_singularity[OF _ assms(2-)])
- show "continuous_on S f" unfolding continuous_on_def
- proof
- fix z assume z: "z \<in> S"
- show "(f \<longlongrightarrow> f z) (at z within S)"
- proof (cases "z \<in> K")
- case False
- from holf have "continuous_on (S - K) f"
- by (rule holomorphic_on_imp_continuous_on)
- with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
- by (simp add: continuous_on_def)
- also from z K S False have "at z within (S - K) = at z within S"
- by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
- finally show "(f \<longlongrightarrow> f z) (at z within S)" .
- qed (insert assms z, simp_all)
- qed
-qed
-
-proposition Cauchy_integral_formula_convex:
- assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
- and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
- and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
- unfolding holomorphic_on_open [symmetric] field_differentiable_def
- using no_isolated_singularity [where S = "interior S"]
- by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
- field_differentiable_at_within field_differentiable_def holomorphic_onI
- holomorphic_on_imp_differentiable_at open_interior)
- show ?thesis
- by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
-qed
-
-text\<open> Formula for higher derivatives.\<close>
-
-lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
- (circlepath z r)"
-using w
-proof (induction k arbitrary: w)
- case 0 then show ?case
- using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
-next
- case (Suc k)
- have [simp]: "r > 0" using w
- using ball_eq_empty by fastforce
- have f: "continuous_on (path_image (circlepath z r)) f"
- by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
- obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
- using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
- by (auto simp: contour_integrable_on_def)
- then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
- by (rule contour_integral_unique)
- have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
- using Suc.prems assms has_field_derivative_higher_deriv by auto
- then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
- by (force simp: field_differentiable_def)
- have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
- of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
- by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
- also have "\<dots> = of_nat (Suc k) * X"
- by (simp only: con)
- finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
- then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
- by (metis deriv_cmult dnf_diff)
- then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
- by (simp add: field_simps)
- then show ?case
- using of_nat_eq_0_iff X by fastforce
-qed
-
-lemma Cauchy_higher_derivative_integral_circlepath:
- assumes contf: "continuous_on (cball z r) f"
- and holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
- (is "?thes1")
- and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
- (is "?thes2")
-proof -
- have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
- (circlepath z r)"
- using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
- by simp
- show ?thes1 using *
- using contour_integrable_on_def by blast
- show ?thes2
- unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
-qed
-
-corollary Cauchy_contour_integral_circlepath:
- assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
- shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
-by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
-
-lemma Cauchy_contour_integral_circlepath_2:
- assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
- shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
- using Cauchy_contour_integral_circlepath [OF assms, of 1]
- by (simp add: power2_eq_square)
-
-
-subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
-
-theorem holomorphic_power_series:
- assumes holf: "f holomorphic_on ball z r"
- and w: "w \<in> ball z r"
- shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
-proof -
- \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
- obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
- proof
- have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
- using w by (simp add: dist_commute field_sum_of_halves subset_eq)
- then show "f holomorphic_on cball z ((r + dist w z) / 2)"
- by (rule holomorphic_on_subset [OF holf])
- have "r > 0"
- using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
- then show "0 < (r + dist w z) / 2"
- by simp (use zero_le_dist [of w z] in linarith)
- qed (use w in \<open>auto simp: dist_commute\<close>)
- then have holf: "f holomorphic_on ball z r"
- using ball_subset_cball holomorphic_on_subset by blast
- have contf: "continuous_on (cball z r) f"
- by (simp add: holfc holomorphic_on_imp_continuous_on)
- have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
- by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
- obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
- by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
- obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
- and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
- proof
- show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
- by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
- qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
- have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix e::real
- assume "0 < e"
- have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto
- obtain n where n: "((r - k) / r) ^ n < e / B * k"
- using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
- have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
- if "n \<le> N" and r: "r = dist z u" for N u
- proof -
- have N: "((r - k) / r) ^ N < e / B * k"
- apply (rule le_less_trans [OF power_decreasing n])
- using \<open>n \<le> N\<close> k by auto
- have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
- using \<open>0 < r\<close> r w by auto
- have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
- by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
- have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
- = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
- unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
- also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
- using \<open>0 < B\<close>
- apply (auto simp: geometric_sum [OF wzu_not1])
- apply (simp add: field_simps norm_mult [symmetric])
- done
- also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
- using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
- also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
- by (simp add: algebra_simps)
- also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
- by (simp add: norm_mult norm_power norm_minus_commute)
- also have "\<dots> \<le> (((r - k)/r)^N) * B"
- using \<open>0 < r\<close> w k
- apply (simp add: divide_simps)
- apply (rule mult_mono [OF power_mono])
- apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
- done
- also have "\<dots> < e * k"
- using \<open>0 < B\<close> N by (simp add: divide_simps)
- also have "\<dots> \<le> e * norm (u - w)"
- using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
- finally show ?thesis
- by (simp add: field_split_simps norm_divide del: power_Suc)
- qed
- with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
- norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
- by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
- qed
- have eq: "\<forall>\<^sub>F x in sequentially.
- contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
- (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
- apply (rule eventuallyI)
- apply (subst contour_integral_sum, simp)
- using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
- apply (simp only: contour_integral_lmul cint algebra_simps)
- done
- have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
- apply (intro contour_integrable_sum contour_integrable_lmul, simp)
- using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
- have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
- sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
- unfolding sums_def
- apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
- using \<open>0 < r\<close> apply auto
- done
- then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
- sums (2 * of_real pi * \<i> * f w)"
- using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
- then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
- sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
- by (rule sums_divide)
- then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
- sums f w"
- by (simp add: field_simps)
- then show ?thesis
- by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
-qed
-
-
-subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
-
-text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
-
-lemma Liouville_weak_0:
- assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
- shows "f z = 0"
-proof (rule ccontr)
- assume fz: "f z \<noteq> 0"
- with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
- obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
- by (auto simp: dist_norm)
- define R where "R = 1 + \<bar>B\<bar> + norm z"
- have "R > 0" unfolding R_def
- proof -
- have "0 \<le> cmod z + \<bar>B\<bar>"
- by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
- then show "0 < 1 + \<bar>B\<bar> + cmod z"
- by linarith
- qed
- have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
- apply (rule Cauchy_integral_circlepath)
- using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
- done
- have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
- unfolding R_def
- by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
- with \<open>R > 0\<close> fz show False
- using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
- by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
-qed
-
-proposition Liouville_weak:
- assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
- shows "f z = l"
- using Liouville_weak_0 [of "\<lambda>z. f z - l"]
- by (simp add: assms holomorphic_on_diff LIM_zero)
-
-proposition Liouville_weak_inverse:
- assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
- obtains z where "f z = 0"
-proof -
- { assume f: "\<And>z. f z \<noteq> 0"
- have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
- by (simp add: holomorphic_on_divide assms f)
- have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
- apply (rule tendstoI [OF eventually_mono])
- apply (rule_tac B="2/e" in unbounded)
- apply (simp add: dist_norm norm_divide field_split_simps)
- done
- have False
- using Liouville_weak_0 [OF 1 2] f by simp
- }
- then show ?thesis
- using that by blast
-qed
-
-text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
-
-theorem fundamental_theorem_of_algebra:
- fixes a :: "nat \<Rightarrow> complex"
- assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
- obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
-using assms
-proof (elim disjE bexE)
- assume "a 0 = 0" then show ?thesis
- by (auto simp: that [of 0])
-next
- fix i
- assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
- have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
- by (rule holomorphic_intros)+
- show thesis
- proof (rule Liouville_weak_inverse [OF 1])
- show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
- using i polyfun_extremal nz by force
- qed (use that in auto)
-qed
-
-subsection\<open>Weierstrass convergence theorem\<close>
-
-lemma holomorphic_uniform_limit:
- assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
- and ulim: "uniform_limit (cball z r) f g F"
- and F: "\<not> trivial_limit F"
- obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
-proof (cases r "0::real" rule: linorder_cases)
- case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
-next
- case equal then show ?thesis
- by (force simp: holomorphic_on_def intro: that)
-next
- case greater
- have contg: "continuous_on (cball z r) g"
- using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast
- have "path_image (circlepath z r) \<subseteq> cball z r"
- using \<open>0 < r\<close> by auto
- then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
- by (intro continuous_intros continuous_on_subset [OF contg])
- have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
- if w: "w \<in> ball z r" for w
- proof -
- define d where "d = (r - norm(w - z))"
- have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
- have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
- unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
- have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
- apply (rule eventually_mono [OF cont])
- using w
- apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
- done
- have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
- using greater \<open>0 < d\<close>
- apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
- apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
- apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
- done
- have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
- by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
- have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
- by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
- have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
- proof (rule Lim_transform_eventually)
- show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
- = 2 * of_real pi * \<i> * f x w"
- apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
- using w\<open>0 < d\<close> d_def by auto
- qed (auto simp: cif_tends_cig)
- have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
- by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
- then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
- by (rule tendsto_mult_left [OF tendstoI])
- then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
- using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
- by fastforce
- then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
- using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
- by (force simp: field_simps)
- then show ?thesis
- by (simp add: dist_norm)
- qed
- show ?thesis
- using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
- by (fastforce simp add: holomorphic_on_open contg intro: that)
-qed
-
-
-text\<open> Version showing that the limit is the limit of the derivatives.\<close>
-
-proposition has_complex_derivative_uniform_limit:
- fixes z::complex
- assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
- (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
- and ulim: "uniform_limit (cball z r) f g F"
- and F: "\<not> trivial_limit F" and "0 < r"
- obtains g' where
- "continuous_on (cball z r) g"
- "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
-proof -
- let ?conint = "contour_integral (circlepath z r)"
- have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
- by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
- auto simp: holomorphic_on_open field_differentiable_def)+
- then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
- using DERIV_deriv_iff_has_field_derivative
- by (fastforce simp add: holomorphic_on_open)
- then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
- by (simp add: DERIV_imp_deriv)
- have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
- proof -
- have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
- if cont_fn: "continuous_on (cball z r) (f n)"
- and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
- proof -
- have hol_fn: "f n holomorphic_on ball z r"
- using fnd by (force simp: holomorphic_on_open)
- have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
- by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
- then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
- using DERIV_unique [OF fnd] w by blast
- show ?thesis
- by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
- qed
- define d where "d = (r - norm(w - z))^2"
- have "d > 0"
- using w by (simp add: dist_commute dist_norm d_def)
- have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
- proof -
- have "w \<in> ball z (cmod (z - y))"
- using that w by fastforce
- then have "cmod (w - z) \<le> cmod (z - y)"
- by (simp add: dist_complex_def norm_minus_commute)
- moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
- by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
- ultimately show ?thesis
- using that by (simp add: d_def norm_power power_mono)
- qed
- have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
- by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
- have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
- unfolding uniform_limit_iff
- proof clarify
- fix e::real
- assume "0 < e"
- with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
- apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
- apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
- apply (simp add: \<open>0 < d\<close>)
- apply (force simp: dist_norm dle intro: less_le_trans)
- done
- qed
- have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
- \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
- by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
- then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
- using Lim_null by (force intro!: tendsto_mult_right_zero)
- have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
- apply (rule Lim_transform_eventually [OF tendsto_0])
- apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
- done
- then show ?thesis using Lim_null by blast
- qed
- obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
- by (blast intro: tends_f'n_g' g')
- then show ?thesis using g
- using that by blast
-qed
-
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
-
-lemma holomorphic_uniform_sequence:
- assumes S: "open S"
- and hol_fn: "\<And>n. (f n) holomorphic_on S"
- and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
- shows "g holomorphic_on S"
-proof -
- have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
- proof -
- obtain r where "0 < r" and r: "cball z r \<subseteq> S"
- and ul: "uniform_limit (cball z r) f g sequentially"
- using ulim_g [OF \<open>z \<in> S\<close>] by blast
- have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
- proof (intro eventuallyI conjI)
- show "continuous_on (cball z r) (f x)" for x
- using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
- show "f x holomorphic_on ball z r" for x
- by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
- qed
- show ?thesis
- apply (rule holomorphic_uniform_limit [OF *])
- using \<open>0 < r\<close> centre_in_ball ul
- apply (auto simp: holomorphic_on_open)
- done
- qed
- with S show ?thesis
- by (simp add: holomorphic_on_open)
-qed
-
-lemma has_complex_derivative_uniform_sequence:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
- and ulim_g: "\<And>x. x \<in> S
- \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
- shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
-proof -
- have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
- proof -
- obtain r where "0 < r" and r: "cball z r \<subseteq> S"
- and ul: "uniform_limit (cball z r) f g sequentially"
- using ulim_g [OF \<open>z \<in> S\<close>] by blast
- have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
- (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
- proof (intro eventuallyI conjI ballI)
- show "continuous_on (cball z r) (f x)" for x
- by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
- show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
- using ball_subset_cball hfd r by blast
- qed
- show ?thesis
- by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
- qed
- show ?thesis
- by (rule bchoice) (blast intro: y)
-qed
-
-subsection\<open>On analytic functions defined by a series\<close>
-
-lemma series_and_derivative_comparison:
- fixes S :: "complex set"
- assumes S: "open S"
- and h: "summable h"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
- obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
- obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- using Weierstrass_m_test_ev [OF to_g h] by force
- have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- if "x \<in> S" for x
- proof -
- obtain d where "d>0" and d: "cball x d \<subseteq> S"
- using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
- show ?thesis
- proof (intro conjI exI)
- show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
- using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
- qed (use \<open>d > 0\<close> d in auto)
- qed
- have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
- by (metis tendsto_uniform_limitI [OF g])
- moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
- by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
- ultimately show ?thesis
- by (metis sums_def that)
-qed
-
-text\<open>A version where we only have local uniform/comparative convergence.\<close>
-
-lemma series_and_derivative_comparison_local:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
- shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-proof -
- have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
- if "z \<in> S" for z
- proof -
- obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
- using to_g \<open>z \<in> S\<close> by meson
- then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
- by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
- have 1: "open (ball z d \<inter> S)"
- by (simp add: open_Int S)
- have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- by (auto simp: hfd)
- obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
- ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
- by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
- then have "(\<lambda>n. f' n z) sums g' z"
- by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
- moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
- using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
- by (metis (full_types) Int_iff gg' summable_def that)
- moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
- proof (rule has_field_derivative_transform_within)
- show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
- by (metis subsetD dist_commute gg' mem_ball r sums_unique)
- qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
- ultimately show ?thesis by auto
- qed
- then show ?thesis
- by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
-qed
-
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-
-lemma series_and_derivative_comparison_complex:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
- shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
-apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
-apply (rule ex_forward [OF to_g], assumption)
-apply (erule exE)
-apply (rule_tac x="Re \<circ> h" in exI)
-apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
-done
-
-text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
-lemma series_differentiable_comparison_complex:
- fixes S :: "complex set"
- assumes S: "open S"
- and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
- and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
- obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
-proof -
- have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
- using hfd field_differentiable_derivI by blast
- have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
- by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
- then show ?thesis
- using field_differentiable_def that by blast
-qed
-
-text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
-
-lemma power_series_and_derivative_0:
- fixes a :: "nat \<Rightarrow> complex" and r::real
- assumes "summable (\<lambda>n. a n * r^n)"
- shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
- ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
-proof (cases "0 < r")
- case True
- have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
- by (rule derivative_eq_intros | simp)+
- have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
- using \<open>r > 0\<close>
- apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
- using norm_triangle_ineq2 [of y z]
- apply (simp only: diff_le_eq norm_minus_commute mult_2)
- done
- have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
- using assms \<open>r > 0\<close> by simp
- moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
- using \<open>r > 0\<close>
- by (simp flip: of_real_add)
- ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
- by (rule power_series_conv_imp_absconv_weak)
- have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and>
- (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
- apply (rule series_and_derivative_comparison_complex [OF open_ball der])
- apply (rule_tac x="(r - norm z)/2" in exI)
- apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
- using \<open>r > 0\<close>
- apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
- done
- then show ?thesis
- by (simp add: ball_def)
-next
- case False then show ?thesis
- apply (simp add: not_less)
- using less_le_trans norm_not_less_zero by blast
-qed
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
- fixes a :: "nat \<Rightarrow> complex" and r::real
- assumes "summable (\<lambda>n. a n * r^n)"
- obtains g g' where "\<forall>z \<in> ball w r.
- ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
- (g has_field_derivative g' z) (at z)"
- using power_series_and_derivative_0 [OF assms]
- apply clarify
- apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
- using DERIV_shift [where z="-w"]
- apply (auto simp: norm_minus_commute Ball_def dist_norm)
- done
-
-proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
- assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
- shows "f holomorphic_on ball z r"
-proof -
- have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
- proof -
- have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
- proof -
- have wz: "cmod (w - z) < r" using w
- by (auto simp: field_split_simps dist_norm norm_minus_commute)
- then have "0 \<le> r"
- by (meson less_eq_real_def norm_ge_zero order_trans)
- show ?thesis
- using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
- qed
- have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
- using assms [OF inb] by (force simp: summable_def dist_norm)
- obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
- (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
- (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
- by (rule power_series_and_derivative [OF sum, of z]) fastforce
- have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
- proof -
- have less: "cmod (z - u) * 2 < cmod (z - w) + r"
- using that dist_triangle2 [of z u w]
- by (simp add: dist_norm [symmetric] algebra_simps)
- show ?thesis
- apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
- using gg' [of u] less w
- apply (auto simp: assms dist_norm)
- done
- qed
- have "(f has_field_derivative g' w) (at w)"
- by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
- (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
- then show ?thesis ..
- qed
- then show ?thesis by (simp add: holomorphic_on_open)
-qed
-
-corollary holomorphic_iff_power_series:
- "f holomorphic_on ball z r \<longleftrightarrow>
- (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
- apply (intro iffI ballI holomorphic_power_series, assumption+)
- apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
- done
-
-lemma power_series_analytic:
- "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
- by (force simp: analytic_on_open intro!: power_series_holomorphic)
-
-lemma analytic_iff_power_series:
- "f analytic_on ball z r \<longleftrightarrow>
- (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
- by (simp add: analytic_on_open holomorphic_iff_power_series)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
-
-lemma holomorphic_fun_eq_on_ball:
- "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
- w \<in> ball z r;
- \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
- \<Longrightarrow> f w = g w"
- apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
- apply (auto simp: holomorphic_iff_power_series)
- done
-
-lemma holomorphic_fun_eq_0_on_ball:
- "\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r;
- \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
- \<Longrightarrow> f w = 0"
- apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
- apply (auto simp: holomorphic_iff_power_series)
- done
-
-lemma holomorphic_fun_eq_0_on_connected:
- assumes holf: "f holomorphic_on S" and "open S"
- and cons: "connected S"
- and der: "\<And>n. (deriv ^^ n) f z = 0"
- and "z \<in> S" "w \<in> S"
- shows "f w = 0"
-proof -
- have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
- proof -
- have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
- apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
- apply (rule holomorphic_on_subset [OF holf])
- using that apply simp_all
- by (metis funpow_add o_apply)
- with that show ?thesis by auto
- qed
- have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- apply (rule open_subset, force)
- using \<open>open S\<close>
- apply (simp add: open_contains_ball Ball_def)
- apply (erule all_forward)
- using "*" by auto blast+
- have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
- using assms
- by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
- obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
- then have holfb: "f holomorphic_on ball w e"
- using holf holomorphic_on_subset by blast
- have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
- using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
- show ?thesis
- using cons der \<open>z \<in> S\<close>
- apply (simp add: connected_clopen)
- apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
- apply (auto simp: 1 2 3)
- done
-qed
-
-lemma holomorphic_fun_eq_on_connected:
- assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S"
- and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
- and "z \<in> S" "w \<in> S"
- shows "f w = g w"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
- show "(\<lambda>x. f x - g x) holomorphic_on S"
- by (intro assms holomorphic_intros)
- show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
- using assms higher_deriv_diff by auto
-qed (use assms in auto)
-
-lemma holomorphic_fun_eq_const_on_connected:
- assumes holf: "f holomorphic_on S" and "open S"
- and cons: "connected S"
- and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
- and "z \<in> S" "w \<in> S"
- shows "f w = f z"
-proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
- show "(\<lambda>w. f w - f z) holomorphic_on S"
- by (intro assms holomorphic_intros)
- show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
- by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
-qed (use assms in auto)
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
-
-lemma pole_lemma:
- assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
- shows "(\<lambda>z. if z = a then deriv f a
- else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
-proof -
- have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
- proof -
- have fcd: "f field_differentiable at u within S"
- using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
- have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
- by (rule fcd derivative_intros | simp add: that)+
- have "0 < dist a u" using that dist_nz by blast
- then show ?thesis
- by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
- qed
- have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
- proof -
- have holfb: "f holomorphic_on ball a e"
- by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
- have 2: "?F holomorphic_on ball a e - {a}"
- apply (simp add: holomorphic_on_def flip: field_differentiable_def)
- using mem_ball that
- apply (auto intro: F1 field_differentiable_within_subset)
- done
- have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
- if "dist a x < e" for x
- proof (cases "x=a")
- case True
- then have "f field_differentiable at a"
- using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
- with True show ?thesis
- by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
- elim: rev_iffD1 [OF _ LIM_equal])
- next
- case False with 2 that show ?thesis
- by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
- qed
- then have 1: "continuous_on (ball a e) ?F"
- by (clarsimp simp: continuous_on_eq_continuous_at)
- have "?F holomorphic_on ball a e"
- by (auto intro: no_isolated_singularity [OF 1 2])
- with that show ?thesis
- by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
- field_differentiable_at_within)
- qed
- show ?thesis
- proof
- fix x assume "x \<in> S" show "?F field_differentiable at x within S"
- proof (cases "x=a")
- case True then show ?thesis
- using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
- next
- case False with F1 \<open>x \<in> S\<close>
- show ?thesis by blast
- qed
- qed
-qed
-
-lemma pole_theorem:
- assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) holomorphic_on S"
- using pole_lemma [OF holg a]
- by (rule holomorphic_transform) (simp add: eq field_split_simps)
-
-lemma pole_lemma_open:
- assumes "f holomorphic_on S" "open S"
- shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
-proof (cases "a \<in> S")
- case True with assms interior_eq pole_lemma
- show ?thesis by fastforce
-next
- case False with assms show ?thesis
- apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
- apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
- apply (rule derivative_intros | force)+
- done
-qed
-
-lemma pole_theorem_open:
- assumes holg: "g holomorphic_on S" and S: "open S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) holomorphic_on S"
- using pole_lemma_open [OF holg S]
- by (rule holomorphic_transform) (auto simp: eq divide_simps)
-
-lemma pole_theorem_0:
- assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f holomorphic_on S"
- using pole_theorem [OF holg a eq]
- by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_open_0:
- assumes holg: "g holomorphic_on S" and S: "open S"
- and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f holomorphic_on S"
- using pole_theorem_open [OF holg S eq]
- by (rule holomorphic_transform) (auto simp: eq field_split_simps)
-
-lemma pole_theorem_analytic:
- assumes g: "g analytic_on S"
- and eq: "\<And>z. z \<in> S
- \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
- shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
- unfolding analytic_on_def
-proof
- fix x
- assume "x \<in> S"
- with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
- by (auto simp add: analytic_on_def)
- obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
- using \<open>x \<in> S\<close> eq by blast
- have "?F holomorphic_on ball x (min d e)"
- using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
- then show "\<exists>e>0. ?F holomorphic_on ball x e"
- using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
-qed
-
-lemma pole_theorem_analytic_0:
- assumes g: "g analytic_on S"
- and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f analytic_on S"
-proof -
- have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
- by auto
- show ?thesis
- using pole_theorem_analytic [OF g eq] by simp
-qed
-
-lemma pole_theorem_analytic_open_superset:
- assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
- and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
- shows "(\<lambda>z. if z = a then deriv g a
- else f z - g a/(z - a)) analytic_on S"
-proof (rule pole_theorem_analytic [OF g])
- fix z
- assume "z \<in> S"
- then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
- using assms openE by blast
- then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
- using eq by auto
-qed
-
-lemma pole_theorem_analytic_open_superset_0:
- assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
- and [simp]: "f a = deriv g a" "g a = 0"
- shows "f analytic_on S"
-proof -
- have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
- by auto
- have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
- by (rule pole_theorem_analytic_open_superset [OF g])
- then show ?thesis by simp
-qed
-
-
-subsection\<open>General, homology form of Cauchy's theorem\<close>
-
-text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
-
-lemma contour_integral_continuous_on_linepath_2D:
- assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
- and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
- and abu: "closed_segment a b \<subseteq> U"
- shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
-proof -
- have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
- dist (contour_integral (linepath a b) (F x'))
- (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
- if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
- proof -
- obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
- let ?TZ = "cball w \<delta> \<times> closed_segment a b"
- have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
- proof (rule compact_uniformly_continuous)
- show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
- by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
- show "compact ?TZ"
- by (simp add: compact_Times)
- qed
- then obtain \<eta> where "\<eta>>0"
- and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
- dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
- apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
- using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
- have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b;
- norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
- \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
- for x1 x2 x1' x2'
- using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
- have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
- if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x'
- proof -
- have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
- by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
- then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
- apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
- using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
- done
- also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
- finally show ?thesis .
- qed
- show ?thesis
- apply (rule_tac x="min \<delta> \<eta>" in exI)
- using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
- apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
- done
- qed
- show ?thesis
- proof (cases "a=b")
- case True
- then show ?thesis by simp
- next
- case False
- show ?thesis
- by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
- qed
-qed
-
-text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
-lemma Cauchy_integral_formula_global_weak:
- assumes "open U" and holf: "f holomorphic_on U"
- and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
- using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
- then have "bounded(path_image \<gamma>')"
- by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
- then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
- using bounded_pos by force
- define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
- define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
- have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
- by (auto simp: path_polynomial_function valid_path_polynomial_function)
- then have ov: "open v"
- by (simp add: v_def open_winding_number_levelsets loop)
- have uv_Un: "U \<union> v = UNIV"
- using pasz zero by (auto simp: v_def)
- have conf: "continuous_on U f"
- by (metis holf holomorphic_on_imp_continuous_on)
- have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
- proof -
- have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
- by (simp add: holf pole_lemma_open \<open>open U\<close>)
- then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
- using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
- then have "continuous_on U (d y)"
- apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
- using * holomorphic_on_def
- by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
- moreover have "d y holomorphic_on U - {y}"
- proof -
- have "\<And>w. w \<in> U - {y} \<Longrightarrow>
- (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
- apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
- apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
- using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
- then show ?thesis
- unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
- qed
- ultimately show ?thesis
- by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
- qed
- have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
- proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
- show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
- by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
- show "path_image \<gamma> \<subseteq> U - {y}"
- using pasz that by blast
- qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
- define h where
- "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
- have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
- proof -
- have "d z holomorphic_on U"
- by (simp add: hol_d that)
- with that show ?thesis
- apply (simp add: h_def)
- by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
- qed
- have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
- proof -
- have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
- using v_def z by auto
- then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
- using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
- then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
- using has_contour_integral_lmul by fastforce
- then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
- by (simp add: field_split_simps)
- moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
- using z
- apply (auto simp: v_def)
- apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
- done
- ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
- by (rule has_contour_integral_add)
- have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
- if "z \<in> U"
- using * by (auto simp: divide_simps has_contour_integral_eq)
- moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
- if "z \<notin> U"
- apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
- using U pasz \<open>valid_path \<gamma>\<close> that
- apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
- apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
- done
- ultimately show ?thesis
- using z by (simp add: h_def)
- qed
- have znot: "z \<notin> path_image \<gamma>"
- using pasz by blast
- obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
- using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close>
- by (fastforce simp add: \<open>path \<gamma>\<close> compact_path_image)
- obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
- apply (rule that [of "d0/2"])
- using \<open>0 < d0\<close>
- apply (auto simp: dist_norm dest: d0)
- done
- have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
- apply (rule_tac x=x in exI)
- apply (rule_tac x="x'-x" in exI)
- apply (force simp: dist_norm)
- done
- then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
- apply (clarsimp simp add: mem_interior)
- using \<open>0 < dd\<close>
- apply (rule_tac x="dd/2" in exI, auto)
- done
- obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U"
- apply (rule that [OF _ 1])
- apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
- apply (rule order_trans [OF _ dd])
- using \<open>0 < dd\<close> by fastforce
- obtain L where "L>0"
- and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
- cmod (contour_integral \<gamma> f) \<le> L * B"
- using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
- by blast
- have "bounded(f ` T)"
- by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
- then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D"
- by (auto simp: bounded_pos)
- obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C"
- using \<open>compact T\<close> bounded_pos compact_imp_bounded by force
- have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
- proof -
- have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
- with le have ybig: "norm y > C" by force
- with C have "y \<notin> T" by force
- then have ynot: "y \<notin> path_image \<gamma>"
- using subt interior_subset by blast
- have [simp]: "winding_number \<gamma> y = 0"
- apply (rule winding_number_zero_outside [of _ "cball 0 C"])
- using ybig interior_subset subt
- apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
- done
- have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
- by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
- have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T"
- apply (rule holomorphic_on_divide)
- using holf holomorphic_on_subset interior_subset T apply blast
- apply (rule holomorphic_intros)+
- using \<open>y \<notin> T\<close> interior_subset by auto
- have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z
- proof -
- have "D * L / e + cmod z \<le> cmod y"
- using le C [of z] z using interior_subset by force
- then have DL2: "D * L / e \<le> cmod (z - y)"
- using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
- have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
- by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
- also have "\<dots> \<le> D * (e / L / D)"
- apply (rule mult_mono)
- using that D interior_subset apply blast
- using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
- apply (auto simp: norm_divide field_split_simps)
- done
- finally show ?thesis .
- qed
- have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
- by (simp add: dist_norm)
- also have "\<dots> \<le> L * (D * (e / L / D))"
- by (rule L [OF holint leD])
- also have "\<dots> = e"
- using \<open>L>0\<close> \<open>0 < D\<close> by auto
- finally show ?thesis .
- qed
- then have "(h \<longlongrightarrow> 0) at_infinity"
- by (meson Lim_at_infinityI)
- moreover have "h holomorphic_on UNIV"
- proof -
- have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
- if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z
- using that conf
- apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
- apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
- done
- have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
- by (rule continuous_intros)+
- have open_uu_Id: "open (U \<times> U - Id)"
- apply (rule open_Diff)
- apply (simp add: open_Times \<open>open U\<close>)
- using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
- apply (auto simp: Id_fstsnd_eq algebra_simps)
- done
- have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
- apply (rule continuous_on_interior [of U])
- apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
- by (simp add: interior_open that \<open>open U\<close>)
- have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
- else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
- (at (x, x) within U \<times> U)" if "x \<in> U" for x
- proof (rule Lim_withinI)
- fix e::real assume "0 < e"
- obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
- using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]]
- by (metis UNIV_I dist_norm)
- obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
- by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>)
- have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
- if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
- for x' z'
- proof -
- have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
- apply (drule segment_furthest_le [where y=x])
- by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
- have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
- by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
- have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)"
- by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
- have "closed_segment x' z' \<subseteq> U"
- by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
- then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
- using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp
- then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
- by (rule has_contour_integral_div)
- have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
- apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
- using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
- \<open>e > 0\<close> \<open>z' \<noteq> x'\<close>
- apply (auto simp: norm_divide divide_simps derf_le)
- done
- also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
- finally show ?thesis .
- qed
- show "\<exists>d>0. \<forall>xa\<in>U \<times> U.
- 0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
- dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
- apply (rule_tac x="min k1 k2" in exI)
- using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
- apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
- done
- qed
- have con_pa_f: "continuous_on (path_image \<gamma>) f"
- by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
- have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B"
- apply (rule B)
- using \<gamma>' using path_image_def vector_derivative_at by fastforce
- have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
- by (simp add: V)
- have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)"
- apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
- apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify)
- apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
- using con_ff
- apply (auto simp: continuous_within)
- done
- have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w
- proof -
- have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))"
- by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
- then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
- by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
- have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
- apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
- apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
- done
- show ?thesis
- unfolding d_def
- apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
- apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
- done
- qed
- { fix a b
- assume abu: "closed_segment a b \<subseteq> U"
- then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
- by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
- then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
- apply (auto intro: continuous_on_swap_args cond_uu)
- done
- have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)"
- proof (rule continuous_on_compose)
- show "continuous_on {0..1} \<gamma>"
- using \<open>path \<gamma>\<close> path_def by blast
- show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- using pasz unfolding path_image_def
- by (auto intro!: continuous_on_subset [OF cont_cint_d])
- qed
- have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
- apply (simp add: contour_integrable_on)
- apply (rule integrable_continuous_real)
- apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
- using pf\<gamma>'
- by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
- have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
- using abu by (force simp: h_def intro: contour_integral_eq)
- also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
- apply (rule contour_integral_swap)
- apply (rule continuous_on_subset [OF cond_uu])
- using abu pasz \<open>valid_path \<gamma>\<close>
- apply (auto intro!: continuous_intros)
- by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
- finally have cint_h_eq:
- "contour_integral (linepath a b) h =
- contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
- note cint_cint cint_h_eq
- } note cint_h = this
- have conthu: "continuous_on U h"
- proof (simp add: continuous_on_sequentially, clarify)
- fix a x
- assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x"
- then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
- by (meson U contour_integrable_on_def eventuallyI)
- obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
- have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
- unfolding uniform_limit_iff dist_norm
- proof clarify
- fix ee::real
- assume "0 < ee"
- show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee"
- proof -
- let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
- have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
- apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
- using dd pasz \<open>valid_path \<gamma>\<close>
- apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
- done
- then obtain kk where "kk>0"
- and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
- dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
- by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
- have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
- for w z
- using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
- show ?thesis
- using ax unfolding lim_sequentially eventually_sequentially
- apply (drule_tac x="min dd kk" in spec)
- using \<open>dd > 0\<close> \<open>kk > 0\<close>
- apply (fastforce simp: kk dist_norm)
- done
- qed
- qed
- have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
- by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
- then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
- by (simp add: h_def x)
- then show "(h \<circ> a) \<longlonglongrightarrow> h x"
- by (simp add: h_def x au o_def)
- qed
- show ?thesis
- proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
- fix z0
- consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
- then show "h field_differentiable at z0"
- proof cases
- assume "z0 \<in> v" then show ?thesis
- using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
- by (auto simp: field_differentiable_def v_def)
- next
- assume "z0 \<in> U" then
- obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
- have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
- if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c
- proof -
- have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
- using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close>
- by (auto intro!: contour_integrable_holomorphic_simple)
- have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U"
- using that e segments_subset_convex_hull by fastforce+
- have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
- apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
- apply (rule holomorphic_on_subset [OF hol_dw])
- using e abc_subset by auto
- have "contour_integral \<gamma>
- (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
- (contour_integral (linepath b c) (\<lambda>z. d z x) +
- contour_integral (linepath c a) (\<lambda>z. d z x))) = 0"
- apply (rule contour_integral_eq_0)
- using abc pasz U
- apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
- done
- then show ?thesis
- by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
- qed
- show ?thesis
- using e \<open>e > 0\<close>
- by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
- Morera_triangle continuous_on_subset [OF conthu] *)
- qed
- qed
- qed
- ultimately have [simp]: "h z = 0" for z
- by (meson Liouville_weak)
- have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
- by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
- then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
- by (metis mult.commute has_contour_integral_lmul)
- then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
- by (simp add: field_split_simps)
- moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
- using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
- show ?thesis
- using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib)
-qed
-
-theorem Cauchy_integral_formula_global:
- assumes S: "open S" and holf: "f holomorphic_on S"
- and z: "z \<in> S" and vpg: "valid_path \<gamma>"
- and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
-proof -
- have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
- have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}"
- by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
- then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
- by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
- obtain d where "d>0"
- and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
- pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
- \<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
- using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis
- obtain p where polyp: "polynomial_function p"
- and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
- using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
- then have ploop: "pathfinish p = pathstart p" using loop by auto
- have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast
- have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
- have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
- using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
- have wn_eq: "winding_number p z = winding_number \<gamma> z"
- using vpp paps
- by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
- have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w
- proof -
- have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}"
- using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
- have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
- then show ?thesis
- using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
- qed
- then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0"
- by (simp add: zero)
- show ?thesis
- using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
- by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
-qed
-
-theorem Cauchy_theorem_global:
- assumes S: "open S" and holf: "f holomorphic_on S"
- and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
- and pas: "path_image \<gamma> \<subseteq> S"
- and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
- shows "(f has_contour_integral 0) \<gamma>"
-proof -
- obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>"
- proof -
- have "compact (path_image \<gamma>)"
- using compact_valid_path_image vpg by blast
- then have "path_image \<gamma> \<noteq> S"
- by (metis (no_types) compact_open path_image_nonempty S)
- with pas show ?thesis by (blast intro: that)
- qed
- then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast
- have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S"
- by (rule holomorphic_intros holf)+
- show ?thesis
- using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero]
- by (auto simp: znot elim!: has_contour_integral_eq)
-qed
-
-corollary Cauchy_theorem_global_outside:
- assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S"
- "\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
- shows "(f has_contour_integral 0) \<gamma>"
-by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
-
-lemma simply_connected_imp_winding_number_zero:
- assumes "simply_connected S" "path g"
- "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
- shows "winding_number g z = 0"
-proof -
- have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
- by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
- then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
- by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
- then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
- by (rule winding_number_homotopic_paths)
- also have "\<dots> = 0"
- using assms by (force intro: winding_number_trivial)
- finally show ?thesis .
-qed
-
-lemma Cauchy_theorem_simply_connected:
- assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
- "path_image g \<subseteq> S" "pathfinish g = pathstart g"
- shows "(f has_contour_integral 0) g"
-using assms
-apply (simp add: simply_connected_eq_contractible_path)
-apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"]
- homotopic_paths_imp_homotopic_loops)
-using valid_path_imp_path by blast
-
-proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists:
- assumes A: "convex A" "open A"
- and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
- and z0: "z0 \<in> A"
- obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x"
-proof -
- note f' = holomorphic_derivI [OF f(1) A(2)]
- obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)"
- proof (rule holomorphic_convex_primitive' [OF A])
- show "(\<lambda>x. deriv f x / f x) holomorphic_on A"
- by (intro holomorphic_intros f A)
- qed (auto simp: A at_within_open[of _ A])
- define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)"
- from g and A have g_holo: "g holomorphic_on A"
- by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
- hence h_holo: "h holomorphic_on A"
- by (auto simp: h_def intro!: holomorphic_intros)
- have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c"
- proof (rule has_field_derivative_zero_constant, goal_cases)
- case (2 x)
- note [simp] = at_within_open[OF _ \<open>open A\<close>]
- from 2 and z0 and f show ?case
- by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
- qed fact+
- then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
- by blast
- from c[OF z0] and z0 and f have "c = 0"
- by (simp add: h_def)
- with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp
- from that[OF h_holo this] show ?thesis .
-qed
-
-end
--- a/src/HOL/Analysis/Change_Of_Vars.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Change_Of_Vars.thy Sat Nov 30 13:47:33 2019 +0100
@@ -3388,7 +3388,7 @@
next
case False
then obtain h where h: "\<And>x. x \<in> S \<Longrightarrow> h (g x) = x" "linear h"
- using assms det_nz_iff_inj linear_injective_isomorphism by blast
+ using assms det_nz_iff_inj linear_injective_isomorphism by metis
show ?thesis
proof (rule has_absolute_integral_change_of_variables_invertible)
show "(g has_derivative g) (at x within S)" for x
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy Sat Nov 30 13:47:33 2019 +0100
@@ -114,42 +114,6 @@
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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def 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"
@@ -310,74 +274,11 @@
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)"
- by (simp add: id_def)
-
-lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
- by (metis DERIV_imp_deriv DERIV_const)
-
-lemma deriv_add [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_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"
- by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
- (simp add: field_split_simps power2_eq_square)
-
-lemma deriv_cdivide_right:
- "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
- by (simp add: field_class.field_divide_inverse)
-
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"
@@ -385,19 +286,6 @@
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"
@@ -615,7 +503,7 @@
by (simp add: algebra_simps)
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)
+ by (metis analytic_on_imp_differentiable_at analytic_on_open deriv_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"
--- a/src/HOL/Analysis/Conformal_Mappings.thy Wed Nov 27 16:54:33 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,5085 +0,0 @@
-section \<open>Conformal Mappings and Consequences of Cauchy's Integral Theorem\<close>
-
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\<close>
-
-text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
-
-theory Conformal_Mappings
-imports Cauchy_Integral_Theorem
-
-begin
-
-(* FIXME mv to Cauchy_Integral_Theorem.thy *)
-subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
-
-lemma Cauchy_higher_deriv_bound:
- assumes holf: "f holomorphic_on (ball z r)"
- and contf: "continuous_on (cball z r) f"
- and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
- and "0 < r" and "0 < n"
- shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
-proof -
- have "0 < B0" using \<open>0 < r\<close> fin [of z]
- by (metis ball_eq_empty ex_in_conv fin not_less)
- have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
- apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
- apply (auto simp: \<open>0 < r\<close> dist_norm norm_minus_commute)
- apply (rule continuous_intros contf)+
- using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
- done
- have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
- using \<open>0 < n\<close> by simp
- also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
- by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>)
- finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
- have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
- by (rule contf continuous_intros)+
- have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
- by (simp add: holf holomorphic_on_diff)
- define a where "a = (2 * pi)/(fact n)"
- have "0 < a" by (simp add: a_def)
- have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
- using \<open>0 < r\<close> by (simp add: a_def field_split_simps)
- have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
- using \<open>0 < r\<close> \<open>0 < n\<close>
- by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
- have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
- \<le> (B0/r^(Suc n)) * (2 * pi * r)"
- apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
- using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
- using \<open>0 < B0\<close> \<open>0 < r\<close>
- apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
- done
- then show ?thesis
- using \<open>0 < r\<close>
- by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
-qed
-
-lemma Cauchy_inequality:
- assumes holf: "f holomorphic_on (ball \<xi> r)"
- and contf: "continuous_on (cball \<xi> r) f"
- and "0 < r"
- and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B"
- shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n"
-proof -
- obtain x where "norm (\<xi>-x) = r"
- by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel
- dual_order.strict_implies_order norm_of_real)
- then have "0 \<le> B"
- by (metis nof norm_not_less_zero not_le order_trans)
- have "((\<lambda>u. f u / (u - \<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
- (circlepath \<xi> r)"
- apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
- using \<open>0 < r\<close> by simp
- then have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
- apply (rule has_contour_integral_bound_circlepath)
- using \<open>0 \<le> B\<close> \<open>0 < r\<close>
- apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
- done
- then show ?thesis using \<open>0 < r\<close>
- by (simp add: norm_divide norm_mult field_simps)
-qed
-
-lemma Liouville_polynomial:
- assumes holf: "f holomorphic_on UNIV"
- and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n"
- shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)"
-proof (cases rule: le_less_linear [THEN disjE])
- assume "B \<le> 0"
- then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0"
- by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
- then have f0: "(f \<longlongrightarrow> 0) at_infinity"
- using Lim_at_infinity by force
- then have [simp]: "f = (\<lambda>w. 0)"
- using Liouville_weak [OF holf, of 0]
- by (simp add: eventually_at_infinity f0) meson
- show ?thesis by simp
-next
- assume "0 < B"
- have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)"
- apply (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
- using holf holomorphic_on_subset apply auto
- done
- then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp
- have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k
- proof (cases "(deriv ^^ k) f 0 = 0")
- case True then show ?thesis by simp
- next
- case False
- define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- using \<open>0 < B\<close> by simp
- then have wge1: "1 \<le> norm w"
- by (metis norm_of_real w_def)
- then have "w \<noteq> 0" by auto
- have kB: "0 < fact k * B"
- using \<open>0 < B\<close> by simp
- then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)"
- by simp
- then have wgeA: "A \<le> cmod w"
- by (simp only: w_def norm_of_real)
- have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
- using \<open>0 < B\<close> by simp
- then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
- by (metis norm_of_real w_def)
- then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
- using False by (simp add: field_split_simps mult.commute split: if_split_asm)
- also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
- apply (rule Cauchy_inequality)
- using holf holomorphic_on_subset apply force
- using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
- using \<open>w \<noteq> 0\<close> apply simp
- by (metis nof wgeA dist_0_norm dist_norm)
- also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
- apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
- using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: field_split_simps semiring_normalization_rules)
- done
- also have "... = fact k * B / cmod w ^ (k-n)"
- by simp
- finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
- then have "1 / cmod w < 1 / cmod w ^ (k - n)"
- by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
- then have "cmod w ^ (k - n) < cmod w"
- by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
- with self_le_power [OF wge1] have False
- by (meson diff_is_0_eq not_gr0 not_le that)
- then show ?thesis by blast
- qed
- then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k
- using not_less_eq by blast
- then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0"
- by (rule sums_0)
- with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
- show ?thesis
- using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
-qed
-
-text\<open>Every bounded entire function is a constant function.\<close>
-theorem Liouville_theorem:
- assumes holf: "f holomorphic_on UNIV"
- and bf: "bounded (range f)"
- obtains c where "\<And>z. f z = c"
-proof -
- obtain B where "\<And>z. cmod (f z) \<le> B"
- by (meson bf bounded_pos rangeI)
- then show ?thesis
- using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
-qed
-
-text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close>
-
-lemma powser_0_nonzero:
- fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
- assumes r: "0 < r"
- and sm: "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums (f x)"
- and [simp]: "f \<xi> = 0"
- and m0: "a m \<noteq> 0" and "m>0"
- obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
-proof -
- have "r \<le> conv_radius a"
- using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>])
- obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)"
- apply (rule_tac m = "LEAST n. a n \<noteq> 0" in that)
- using m0
- apply (rule LeastI2)
- apply (fastforce intro: dest!: not_less_Least)+
- done
- define b where "b i = a (i+m) / a m" for i
- define g where "g x = suminf (\<lambda>i. b i * (x - \<xi>) ^ i)" for x
- have [simp]: "b 0 = 1"
- by (simp add: am b_def)
- { fix x::'a
- assume "norm (x - \<xi>) < r"
- then have "(\<lambda>n. (a m * (x - \<xi>)^m) * (b n * (x - \<xi>)^n)) sums (f x)"
- using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x - \<xi>) ^ n)" "f x"]
- by (simp add: b_def monoid_mult_class.power_add algebra_simps)
- then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x - \<xi>)^n) sums (f x / (a m * (x - \<xi>)^m))"
- using am by (simp add: sums_mult_D)
- } note bsums = this
- then have "norm (x - \<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x - \<xi>)^n)" for x
- using sums_summable by (cases "x=\<xi>") auto
- then have "r \<le> conv_radius b"
- by (simp add: le_conv_radius_iff [where \<xi>=\<xi>])
- then have "r/2 < conv_radius b"
- using not_le order_trans r by fastforce
- then have "continuous_on (cball \<xi> (r/2)) g"
- using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def)
- then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2"
- apply (rule continuous_onE [where x=\<xi> and e = "1/2"])
- using r apply (auto simp: norm_minus_commute dist_norm)
- done
- moreover have "g \<xi> = 1"
- by (simp add: g_def)
- ultimately have gnz: "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0"
- by fastforce
- have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x - \<xi>) \<le> s" "norm (x - \<xi>) \<le> r/2" for x
- using bsums [of x] that gnz [of x]
- apply (auto simp: g_def)
- using r sums_iff by fastforce
- then show ?thesis
- apply (rule_tac s="min s (r/2)" in that)
- using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm)
-qed
-
-subsection \<open>Analytic continuation\<close>
-
-proposition isolated_zeros:
- assumes holf: "f holomorphic_on S"
- and "open S" "connected S" "\<xi> \<in> S" "f \<xi> = 0" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
- obtains r where "0 < r" and "ball \<xi> r \<subseteq> S" and
- "\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
-proof -
- obtain r where "0 < r" and r: "ball \<xi> r \<subseteq> S"
- using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_ball_eq by blast
- have powf: "((\<lambda>n. (deriv ^^ n) f \<xi> / (fact n) * (z - \<xi>)^n) sums f z)" if "z \<in> ball \<xi> r" for z
- apply (rule holomorphic_power_series [OF _ that])
- apply (rule holomorphic_on_subset [OF holf r])
- done
- obtain m where m: "(deriv ^^ m) f \<xi> / (fact m) \<noteq> 0"
- using holomorphic_fun_eq_0_on_connected [OF holf \<open>open S\<close> \<open>connected S\<close> _ \<open>\<xi> \<in> S\<close> \<open>\<beta> \<in> S\<close>] \<open>f \<beta> \<noteq> 0\<close>
- by auto
- then have "m \<noteq> 0" using assms(5) funpow_0 by fastforce
- obtain s where "0 < s" and s: "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
- apply (rule powser_0_nonzero [OF \<open>0 < r\<close> powf \<open>f \<xi> = 0\<close> m])
- using \<open>m \<noteq> 0\<close> by (auto simp: dist_commute dist_norm)
- have "0 < min r s" by (simp add: \<open>0 < r\<close> \<open>0 < s\<close>)
- then show ?thesis
- apply (rule that)
- using r s by auto
-qed
-
-proposition analytic_continuation:
- assumes holf: "f holomorphic_on S"
- and "open S" and "connected S"
- and "U \<subseteq> S" and "\<xi> \<in> S"
- and "\<xi> islimpt U"
- and fU0 [simp]: "\<And>z. z \<in> U \<Longrightarrow> f z = 0"
- and "w \<in> S"
- shows "f w = 0"
-proof -
- obtain e where "0 < e" and e: "cball \<xi> e \<subseteq> S"
- using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_cball_eq by blast
- define T where "T = cball \<xi> e \<inter> U"
- have contf: "continuous_on (closure T) f"
- by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
- holomorphic_on_subset inf.cobounded1)
- have fT0 [simp]: "\<And>x. x \<in> T \<Longrightarrow> f x = 0"
- by (simp add: T_def)
- have "\<And>r. \<lbrakk>\<forall>e>0. \<exists>x'\<in>U. x' \<noteq> \<xi> \<and> dist x' \<xi> < e; 0 < r\<rbrakk> \<Longrightarrow> \<exists>x'\<in>cball \<xi> e \<inter> U. x' \<noteq> \<xi> \<and> dist x' \<xi> < r"
- by (metis \<open>0 < e\<close> IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
- then have "\<xi> islimpt T" using \<open>\<xi> islimpt U\<close>
- by (auto simp: T_def islimpt_approachable)
- then have "\<xi> \<in> closure T"
- by (simp add: closure_def)
- then have "f \<xi> = 0"
- by (auto simp: continuous_constant_on_closure [OF contf])
- show ?thesis
- apply (rule ccontr)
- apply (rule isolated_zeros [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>\<xi> \<in> S\<close> \<open>f \<xi> = 0\<close> \<open>w \<in> S\<close>], assumption)
- by (metis open_ball \<open>\<xi> islimpt T\<close> centre_in_ball fT0 insertE insert_Diff islimptE)
-qed
-
-corollary analytic_continuation_open:
- assumes "open s" and "open s'" and "s \<noteq> {}" and "connected s'"
- and "s \<subseteq> s'"
- assumes "f holomorphic_on s'" and "g holomorphic_on s'"
- and "\<And>z. z \<in> s \<Longrightarrow> f z = g z"
- assumes "z \<in> s'"
- shows "f z = g z"
-proof -
- from \<open>s \<noteq> {}\<close> obtain \<xi> where "\<xi> \<in> s" by auto
- with \<open>open s\<close> have \<xi>: "\<xi> islimpt s"
- by (intro interior_limit_point) (auto simp: interior_open)
- have "f z - g z = 0"
- by (rule analytic_continuation[of "\<lambda>z. f z - g z" s' s \<xi>])
- (insert assms \<open>\<xi> \<in> s\<close> \<xi>, auto intro: holomorphic_intros)
- thus ?thesis by simp
-qed
-
-subsection\<open>Open mapping theorem\<close>
-
-lemma holomorphic_contract_to_zero:
- assumes contf: "continuous_on (cball \<xi> r) f"
- and holf: "f holomorphic_on ball \<xi> r"
- and "0 < r"
- and norm_less: "\<And>z. norm(\<xi> - z) = r \<Longrightarrow> norm(f \<xi>) < norm(f z)"
- obtains z where "z \<in> ball \<xi> r" "f z = 0"
-proof -
- { assume fnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w \<noteq> 0"
- then have "0 < norm (f \<xi>)"
- by (simp add: \<open>0 < r\<close>)
- have fnz': "\<And>w. w \<in> cball \<xi> r \<Longrightarrow> f w \<noteq> 0"
- by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero)
- have "frontier(cball \<xi> r) \<noteq> {}"
- using \<open>0 < r\<close> by simp
- define g where [abs_def]: "g z = inverse (f z)" for z
- have contg: "continuous_on (cball \<xi> r) g"
- unfolding g_def using contf continuous_on_inverse fnz' by blast
- have holg: "g holomorphic_on ball \<xi> r"
- unfolding g_def using fnz holf holomorphic_on_inverse by blast
- have "frontier (cball \<xi> r) \<subseteq> cball \<xi> r"
- by (simp add: subset_iff)
- then have contf': "continuous_on (frontier (cball \<xi> r)) f"
- and contg': "continuous_on (frontier (cball \<xi> r)) g"
- by (blast intro: contf contg continuous_on_subset)+
- have froc: "frontier(cball \<xi> r) \<noteq> {}"
- using \<open>0 < r\<close> by simp
- moreover have "continuous_on (frontier (cball \<xi> r)) (norm o f)"
- using contf' continuous_on_compose continuous_on_norm_id by blast
- ultimately obtain w where w: "w \<in> frontier(cball \<xi> r)"
- and now: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (f w) \<le> norm (f x)"
- apply (rule bexE [OF continuous_attains_inf [OF compact_frontier [OF compact_cball]]])
- apply simp
- done
- then have fw: "0 < norm (f w)"
- by (simp add: fnz')
- have "continuous_on (frontier (cball \<xi> r)) (norm o g)"
- using contg' continuous_on_compose continuous_on_norm_id by blast
- then obtain v where v: "v \<in> frontier(cball \<xi> r)"
- and nov: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (g v) \<ge> norm (g x)"
- apply (rule bexE [OF continuous_attains_sup [OF compact_frontier [OF compact_cball] froc]])
- apply simp
- done
- then have fv: "0 < norm (f v)"
- by (simp add: fnz')
- have "norm ((deriv ^^ 0) g \<xi>) \<le> fact 0 * norm (g v) / r ^ 0"
- by (rule Cauchy_inequality [OF holg contg \<open>0 < r\<close>]) (simp add: dist_norm nov)
- then have "cmod (g \<xi>) \<le> norm (g v)"
- by simp
- with w have wr: "norm (\<xi> - w) = r" and nfw: "norm (f w) \<le> norm (f \<xi>)"
- apply (simp_all add: dist_norm)
- by (metis \<open>0 < cmod (f \<xi>)\<close> g_def less_imp_inverse_less norm_inverse not_le now order_trans v)
- with fw have False
- using norm_less by force
- }
- with that show ?thesis by blast
-qed
-
-theorem open_mapping_thm:
- assumes holf: "f holomorphic_on S"
- and S: "open S" and "connected S"
- and "open U" and "U \<subseteq> S"
- and fne: "\<not> f constant_on S"
- shows "open (f ` U)"
-proof -
- have *: "open (f ` U)"
- if "U \<noteq> {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "\<And>x. \<exists>y \<in> U. f y \<noteq> x"
- for U
- proof (clarsimp simp: open_contains_ball)
- fix \<xi> assume \<xi>: "\<xi> \<in> U"
- show "\<exists>e>0. ball (f \<xi>) e \<subseteq> f ` U"
- proof -
- have hol: "(\<lambda>z. f z - f \<xi>) holomorphic_on U"
- by (rule holomorphic_intros that)+
- obtain s where "0 < s" and sbU: "ball \<xi> s \<subseteq> U"
- and sne: "\<And>z. z \<in> ball \<xi> s - {\<xi>} \<Longrightarrow> (\<lambda>z. f z - f \<xi>) z \<noteq> 0"
- using isolated_zeros [OF hol U \<xi>] by (metis fneU right_minus_eq)
- obtain r where "0 < r" and r: "cball \<xi> r \<subseteq> ball \<xi> s"
- apply (rule_tac r="s/2" in that)
- using \<open>0 < s\<close> by auto
- have "cball \<xi> r \<subseteq> U"
- using sbU r by blast
- then have frsbU: "frontier (cball \<xi> r) \<subseteq> U"
- using Diff_subset frontier_def order_trans by fastforce
- then have cof: "compact (frontier(cball \<xi> r))"
- by blast
- have frne: "frontier (cball \<xi> r) \<noteq> {}"
- using \<open>0 < r\<close> by auto
- have contfr: "continuous_on (frontier (cball \<xi> r)) (\<lambda>z. norm (f z - f \<xi>))"
- by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on)
- obtain w where "norm (\<xi> - w) = r"
- and w: "(\<And>z. norm (\<xi> - z) = r \<Longrightarrow> norm (f w - f \<xi>) \<le> norm(f z - f \<xi>))"
- apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]])
- apply (simp add: dist_norm)
- done
- moreover define \<epsilon> where "\<epsilon> \<equiv> norm (f w - f \<xi>) / 3"
- ultimately have "0 < \<epsilon>"
- using \<open>0 < r\<close> dist_complex_def r sne by auto
- have "ball (f \<xi>) \<epsilon> \<subseteq> f ` U"
- proof
- fix \<gamma>
- assume \<gamma>: "\<gamma> \<in> ball (f \<xi>) \<epsilon>"
- have *: "cmod (\<gamma> - f \<xi>) < cmod (\<gamma> - f z)" if "cmod (\<xi> - z) = r" for z
- proof -
- have lt: "cmod (f w - f \<xi>) / 3 < cmod (\<gamma> - f z)"
- using w [OF that] \<gamma>
- using dist_triangle2 [of "f \<xi>" "\<gamma>" "f z"] dist_triangle2 [of "f \<xi>" "f z" \<gamma>]
- by (simp add: \<epsilon>_def dist_norm norm_minus_commute)
- show ?thesis
- by (metis \<epsilon>_def dist_commute dist_norm less_trans lt mem_ball \<gamma>)
- qed
- have "continuous_on (cball \<xi> r) (\<lambda>z. \<gamma> - f z)"
- apply (rule continuous_intros)+
- using \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close>
- apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on)
- done
- moreover have "(\<lambda>z. \<gamma> - f z) holomorphic_on ball \<xi> r"
- apply (rule holomorphic_intros)+
- apply (metis \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close> holomorphic_on_subset interior_cball interior_subset)
- done
- ultimately obtain z where "z \<in> ball \<xi> r" "\<gamma> - f z = 0"
- apply (rule holomorphic_contract_to_zero)
- apply (blast intro!: \<open>0 < r\<close> *)+
- done
- then show "\<gamma> \<in> f ` U"
- using \<open>cball \<xi> r \<subseteq> U\<close> by fastforce
- qed
- then show ?thesis using \<open>0 < \<epsilon>\<close> by blast
- qed
- qed
- have "open (f ` X)" if "X \<in> components U" for X
- proof -
- have holfU: "f holomorphic_on U"
- using \<open>U \<subseteq> S\<close> holf holomorphic_on_subset by blast
- have "X \<noteq> {}"
- using that by (simp add: in_components_nonempty)
- moreover have "open X"
- using that \<open>open U\<close> open_components by auto
- moreover have "connected X"
- using that in_components_maximal by blast
- moreover have "f holomorphic_on X"
- by (meson that holfU holomorphic_on_subset in_components_maximal)
- moreover have "\<exists>y\<in>X. f y \<noteq> x" for x
- proof (rule ccontr)
- assume not: "\<not> (\<exists>y\<in>X. f y \<noteq> x)"
- have "X \<subseteq> S"
- using \<open>U \<subseteq> S\<close> in_components_subset that by blast
- obtain w where w: "w \<in> X" using \<open>X \<noteq> {}\<close> by blast
- have wis: "w islimpt X"
- using w \<open>open X\<close> interior_eq by auto
- have hol: "(\<lambda>z. f z - x) holomorphic_on S"
- by (simp add: holf holomorphic_on_diff)
- with fne [unfolded constant_on_def]
- analytic_continuation[OF hol S \<open>connected S\<close> \<open>X \<subseteq> S\<close> _ wis] not \<open>X \<subseteq> S\<close> w
- show False by auto
- qed
- ultimately show ?thesis
- by (rule *)
- qed
- then have "open (f ` \<Union>(components U))"
- by (metis (no_types, lifting) imageE image_Union open_Union)
- then show ?thesis
- by force
-qed
-
-text\<open>No need for \<^term>\<open>S\<close> to be connected. But the nonconstant condition is stronger.\<close>
-corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm2:
- assumes holf: "f holomorphic_on S"
- and S: "open S"
- and "open U" "U \<subseteq> S"
- and fnc: "\<And>X. \<lbrakk>open X; X \<subseteq> S; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<not> f constant_on X"
- shows "open (f ` U)"
-proof -
- have "S = \<Union>(components S)" by simp
- with \<open>U \<subseteq> S\<close> have "U = (\<Union>C \<in> components S. C \<inter> U)" by auto
- then have "f ` U = (\<Union>C \<in> components S. f ` (C \<inter> U))"
- using image_UN by fastforce
- moreover
- { fix C assume "C \<in> components S"
- with S \<open>C \<in> components S\<close> open_components in_components_connected
- have C: "open C" "connected C" by auto
- have "C \<subseteq> S"
- by (metis \<open>C \<in> components S\<close> in_components_maximal)
- have nf: "\<not> f constant_on C"
- apply (rule fnc)
- using C \<open>C \<subseteq> S\<close> \<open>C \<in> components S\<close> in_components_nonempty by auto
- have "f holomorphic_on C"
- by (metis holf holomorphic_on_subset \<open>C \<subseteq> S\<close>)
- then have "open (f ` (C \<inter> U))"
- apply (rule open_mapping_thm [OF _ C _ _ nf])
- apply (simp add: C \<open>open U\<close> open_Int, blast)
- done
- } ultimately show ?thesis
- by force
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm3:
- assumes holf: "f holomorphic_on S"
- and "open S" and injf: "inj_on f S"
- shows "open (f ` S)"
-apply (rule open_mapping_thm2 [OF holf])
-using assms
-apply (simp_all add:)
-using injective_not_constant subset_inj_on by blast
-
-subsection\<open>Maximum modulus principle\<close>
-
-text\<open>If \<^term>\<open>f\<close> is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
- properly within the domain of \<^term>\<open>f\<close>.\<close>
-
-proposition maximum_modulus_principle:
- assumes holf: "f holomorphic_on S"
- and S: "open S" and "connected S"
- and "open U" and "U \<subseteq> S" and "\<xi> \<in> U"
- and no: "\<And>z. z \<in> U \<Longrightarrow> norm(f z) \<le> norm(f \<xi>)"
- shows "f constant_on S"
-proof (rule ccontr)
- assume "\<not> f constant_on S"
- then have "open (f ` U)"
- using open_mapping_thm assms by blast
- moreover have "\<not> open (f ` U)"
- proof -
- have "\<exists>t. cmod (f \<xi> - t) < e \<and> t \<notin> f ` U" if "0 < e" for e
- apply (rule_tac x="if 0 < Re(f \<xi>) then f \<xi> + (e/2) else f \<xi> - (e/2)" in exI)
- using that
- apply (simp add: dist_norm)
- apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym)
- done
- then show ?thesis
- unfolding open_contains_ball by (metis \<open>\<xi> \<in> U\<close> contra_subsetD dist_norm imageI mem_ball)
- qed
- ultimately show False
- by blast
-qed
-
-proposition maximum_modulus_frontier:
- assumes holf: "f holomorphic_on (interior S)"
- and contf: "continuous_on (closure S) f"
- and bos: "bounded S"
- and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> norm(f z) \<le> B"
- and "\<xi> \<in> S"
- shows "norm(f \<xi>) \<le> B"
-proof -
- have "compact (closure S)" using bos
- by (simp add: bounded_closure compact_eq_bounded_closed)
- moreover have "continuous_on (closure S) (cmod \<circ> f)"
- using contf continuous_on_compose continuous_on_norm_id by blast
- ultimately obtain z where zin: "z \<in> closure S" and z: "\<And>y. y \<in> closure S \<Longrightarrow> (cmod \<circ> f) y \<le> (cmod \<circ> f) z"
- using continuous_attains_sup [of "closure S" "norm o f"] \<open>\<xi> \<in> S\<close> by auto
- then consider "z \<in> frontier S" | "z \<in> interior S" using frontier_def by auto
- then have "norm(f z) \<le> B"
- proof cases
- case 1 then show ?thesis using leB by blast
- next
- case 2
- have zin: "z \<in> connected_component_set (interior S) z"
- by (simp add: 2)
- have "f constant_on (connected_component_set (interior S) z)"
- apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin])
- apply (metis connected_component_subset holf holomorphic_on_subset)
- apply (simp_all add: open_connected_component)
- by (metis closure_subset comp_eq_dest_lhs interior_subset subsetCE z connected_component_in)
- then obtain c where c: "\<And>w. w \<in> connected_component_set (interior S) z \<Longrightarrow> f w = c"
- by (auto simp: constant_on_def)
- have "f ` closure(connected_component_set (interior S) z) \<subseteq> {c}"
- apply (rule image_closure_subset)
- apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
- using c
- apply auto
- done
- then have cc: "\<And>w. w \<in> closure(connected_component_set (interior S) z) \<Longrightarrow> f w = c" by blast
- have "frontier(connected_component_set (interior S) z) \<noteq> {}"
- apply (simp add: frontier_eq_empty)
- by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV)
- then obtain w where w: "w \<in> frontier(connected_component_set (interior S) z)"
- by auto
- then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def)
- also have "... \<le> B"
- apply (rule leB)
- using w
-using frontier_interior_subset frontier_of_connected_component_subset by blast
- finally show ?thesis .
- qed
- then show ?thesis
- using z \<open>\<xi> \<in> S\<close> closure_subset by fastforce
-qed
-
-corollary\<^marker>\<open>tag unimportant\<close> maximum_real_frontier:
- assumes holf: "f holomorphic_on (interior S)"
- and contf: "continuous_on (closure S) f"
- and bos: "bounded S"
- and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> Re(f z) \<le> B"
- and "\<xi> \<in> S"
- shows "Re(f \<xi>) \<le> B"
-using maximum_modulus_frontier [of "exp o f" S "exp B"]
- Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms
-by auto
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Factoring out a zero according to its order\<close>
-
-lemma holomorphic_factor_order_of_zero:
- assumes holf: "f holomorphic_on S"
- and os: "open S"
- and "\<xi> \<in> S" "0 < n"
- and dnz: "(deriv ^^ n) f \<xi> \<noteq> 0"
- and dfz: "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
- obtains g r where "0 < r"
- "g holomorphic_on ball \<xi> r"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
-proof -
- obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
- then have holfb: "f holomorphic_on ball \<xi> r"
- using holf holomorphic_on_subset by blast
- define g where "g w = suminf (\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i)" for w
- have sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
- and feq: "f w - f \<xi> = (w - \<xi>)^n * g w"
- if w: "w \<in> ball \<xi> r" for w
- proof -
- define powf where "powf = (\<lambda>i. (deriv ^^ i) f \<xi>/(fact i) * (w - \<xi>)^i)"
- have sing: "{..<n} - {i. powf i = 0} = (if f \<xi> = 0 then {} else {0})"
- unfolding powf_def using \<open>0 < n\<close> dfz by (auto simp: dfz; metis funpow_0 not_gr0)
- have "powf sums f w"
- unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
- moreover have "(\<Sum>i<n. powf i) = f \<xi>"
- apply (subst Groups_Big.comm_monoid_add_class.sum.setdiff_irrelevant [symmetric])
- apply simp
- apply (simp only: dfz sing)
- apply (simp add: powf_def)
- done
- ultimately have fsums: "(\<lambda>i. powf (i+n)) sums (f w - f \<xi>)"
- using w sums_iff_shift' by metis
- then have *: "summable (\<lambda>i. (w - \<xi>) ^ n * ((deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n)))"
- unfolding powf_def using sums_summable
- by (auto simp: power_add mult_ac)
- have "summable (\<lambda>i. (deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n))"
- proof (cases "w=\<xi>")
- case False then show ?thesis
- using summable_mult [OF *, of "1 / (w - \<xi>) ^ n"] by simp
- next
- case True then show ?thesis
- by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"]
- split: if_split_asm)
- qed
- then show sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
- by (simp add: summable_sums_iff g_def)
- show "f w - f \<xi> = (w - \<xi>)^n * g w"
- apply (rule sums_unique2)
- apply (rule fsums [unfolded powf_def])
- using sums_mult [OF sumsg, of "(w - \<xi>) ^ n"]
- by (auto simp: power_add mult_ac)
- qed
- then have holg: "g holomorphic_on ball \<xi> r"
- by (meson sumsg power_series_holomorphic)
- then have contg: "continuous_on (ball \<xi> r) g"
- by (blast intro: holomorphic_on_imp_continuous_on)
- have "g \<xi> \<noteq> 0"
- using dnz unfolding g_def
- by (subst suminf_finite [of "{0}"]) auto
- obtain d where "0 < d" and d: "\<And>w. w \<in> ball \<xi> d \<Longrightarrow> g w \<noteq> 0"
- apply (rule exE [OF continuous_on_avoid [OF contg _ \<open>g \<xi> \<noteq> 0\<close>]])
- using \<open>0 < r\<close>
- apply force
- by (metis \<open>0 < r\<close> less_trans mem_ball not_less_iff_gr_or_eq)
- show ?thesis
- apply (rule that [where g=g and r ="min r d"])
- using \<open>0 < r\<close> \<open>0 < d\<close> holg
- apply (auto simp: feq holomorphic_on_subset subset_ball d)
- done
-qed
-
-
-lemma holomorphic_factor_order_of_zero_strong:
- assumes holf: "f holomorphic_on S" "open S" "\<xi> \<in> S" "0 < n"
- and "(deriv ^^ n) f \<xi> \<noteq> 0"
- and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
- obtains g r where "0 < r"
- "g holomorphic_on ball \<xi> r"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
-proof -
- obtain g r where "0 < r"
- and holg: "g holomorphic_on ball \<xi> r"
- and feq: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
- and gne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
- by (auto intro: holomorphic_factor_order_of_zero [OF assms])
- have con: "continuous_on (ball \<xi> r) (\<lambda>z. deriv g z / g z)"
- by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
- have cd: "\<And>x. dist \<xi> x < r \<Longrightarrow> (\<lambda>z. deriv g z / g z) field_differentiable at x"
- apply (rule derivative_intros)+
- using holg mem_ball apply (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at)
- apply (metis open_ball at_within_open holg holomorphic_on_def mem_ball)
- using gne mem_ball by blast
- obtain h where h: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> (h has_field_derivative deriv g x / g x) (at x)"
- apply (rule exE [OF holomorphic_convex_primitive [of "ball \<xi> r" "{}" "\<lambda>z. deriv g z / g z"]])
- apply (auto simp: con cd)
- apply (metis open_ball at_within_open mem_ball)
- done
- then have "continuous_on (ball \<xi> r) h"
- by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
- then have con: "continuous_on (ball \<xi> r) (\<lambda>x. exp (h x) / g x)"
- by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
- have 0: "dist \<xi> x < r \<Longrightarrow> ((\<lambda>x. exp (h x) / g x) has_field_derivative 0) (at x)" for x
- apply (rule h derivative_eq_intros | simp)+
- apply (rule DERIV_deriv_iff_field_differentiable [THEN iffD2])
- using holg apply (auto simp: holomorphic_on_imp_differentiable_at gne h)
- done
- obtain c where c: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> exp (h x) / g x = c"
- by (rule DERIV_zero_connected_constant [of "ball \<xi> r" "{}" "\<lambda>x. exp(h x) / g x"]) (auto simp: con 0)
- have hol: "(\<lambda>z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \<xi> r"
- apply (rule holomorphic_on_compose [unfolded o_def, where g = exp])
- apply (rule holomorphic_intros)+
- using h holomorphic_on_open apply blast
- apply (rule holomorphic_intros)+
- using \<open>0 < n\<close> apply simp
- apply (rule holomorphic_intros)+
- done
- show ?thesis
- apply (rule that [where g="\<lambda>z. exp((Ln(inverse c) + h z)/n)" and r =r])
- using \<open>0 < r\<close> \<open>0 < n\<close>
- apply (auto simp: feq power_mult_distrib exp_divide_power_eq c [symmetric])
- apply (rule hol)
- apply (simp add: Transcendental.exp_add gne)
- done
-qed
-
-
-lemma
- fixes k :: "'a::wellorder"
- assumes a_def: "a == LEAST x. P x" and P: "P k"
- shows def_LeastI: "P a" and def_Least_le: "a \<le> k"
-unfolding a_def
-by (rule LeastI Least_le; rule P)+
-
-lemma holomorphic_factor_zero_nonconstant:
- assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
- and "\<xi> \<in> S" "f \<xi> = 0"
- and nonconst: "\<not> f constant_on S"
- obtains g r n
- where "0 < n" "0 < r" "ball \<xi> r \<subseteq> S"
- "g holomorphic_on ball \<xi> r"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>)^n * g w"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
-proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
- case True then show ?thesis
- using holomorphic_fun_eq_const_on_connected [OF holf S _ \<open>\<xi> \<in> S\<close>] nonconst by (simp add: constant_on_def)
-next
- case False
- then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0" by blast
- obtain r0 where "r0 > 0" "ball \<xi> r0 \<subseteq> S" using S openE \<open>\<xi> \<in> S\<close> by auto
- define n where "n \<equiv> LEAST n. (deriv ^^ n) f \<xi> \<noteq> 0"
- have n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
- by (rule def_LeastI [OF n_def]) (rule n0)
- then have "0 < n" using \<open>f \<xi> = 0\<close>
- using funpow_0 by fastforce
- have n_min: "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
- using def_Least_le [OF n_def] not_le by blast
- then obtain g r1
- where "0 < r1" "g holomorphic_on ball \<xi> r1"
- "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> f w = (w - \<xi>) ^ n * g w"
- "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> g w \<noteq> 0"
- by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne] simp: \<open>f \<xi> = 0\<close>)
- then show ?thesis
- apply (rule_tac g=g and r="min r0 r1" and n=n in that)
- using \<open>0 < n\<close> \<open>0 < r0\<close> \<open>0 < r1\<close> \<open>ball \<xi> r0 \<subseteq> S\<close>
- apply (auto simp: subset_ball intro: holomorphic_on_subset)
- done
-qed
-
-
-lemma holomorphic_lower_bound_difference:
- assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
- and "\<xi> \<in> S" and "\<phi> \<in> S"
- and fne: "f \<phi> \<noteq> f \<xi>"
- obtains k n r
- where "0 < k" "0 < r"
- "ball \<xi> r \<subseteq> S"
- "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * norm(w - \<xi>)^n \<le> norm(f w - f \<xi>)"
-proof -
- define n where "n = (LEAST n. 0 < n \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
- obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0"
- using fne holomorphic_fun_eq_const_on_connected [OF holf S] \<open>\<xi> \<in> S\<close> \<open>\<phi> \<in> S\<close> by blast
- then have "0 < n" and n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
- unfolding n_def by (metis (mono_tags, lifting) LeastI)+
- have n_min: "\<And>k. \<lbrakk>0 < k; k < n\<rbrakk> \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
- unfolding n_def by (blast dest: not_less_Least)
- then obtain g r
- where "0 < r" and holg: "g holomorphic_on ball \<xi> r"
- and fne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w"
- and gnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
- by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne])
- obtain e where "e>0" and e: "ball \<xi> e \<subseteq> S" using assms by (blast elim!: openE)
- then have holfb: "f holomorphic_on ball \<xi> e"
- using holf holomorphic_on_subset by blast
- define d where "d = (min e r) / 2"
- have "0 < d" using \<open>0 < r\<close> \<open>0 < e\<close> by (simp add: d_def)
- have "d < r"
- using \<open>0 < r\<close> by (auto simp: d_def)
- then have cbb: "cball \<xi> d \<subseteq> ball \<xi> r"
- by (auto simp: cball_subset_ball_iff)
- then have "g holomorphic_on cball \<xi> d"
- by (rule holomorphic_on_subset [OF holg])
- then have "closed (g ` cball \<xi> d)"
- by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
- moreover have "g ` cball \<xi> d \<noteq> {}"
- using \<open>0 < d\<close> by auto
- ultimately obtain x where x: "x \<in> g ` cball \<xi> d" and "\<And>y. y \<in> g ` cball \<xi> d \<Longrightarrow> dist 0 x \<le> dist 0 y"
- by (rule distance_attains_inf) blast
- then have leg: "\<And>w. w \<in> cball \<xi> d \<Longrightarrow> norm x \<le> norm (g w)"
- by auto
- have "ball \<xi> d \<subseteq> cball \<xi> d" by auto
- also have "... \<subseteq> ball \<xi> e" using \<open>0 < d\<close> d_def by auto
- also have "... \<subseteq> S" by (rule e)
- finally have dS: "ball \<xi> d \<subseteq> S" .
- moreover have "x \<noteq> 0" using gnz x \<open>d < r\<close> by auto
- ultimately show ?thesis
- apply (rule_tac k="norm x" and n=n and r=d in that)
- using \<open>d < r\<close> leg
- apply (auto simp: \<open>0 < d\<close> fne norm_mult norm_power algebra_simps mult_right_mono)
- done
-qed
-
-lemma
- assumes holf: "f holomorphic_on (S - {\<xi>})" and \<xi>: "\<xi> \<in> interior S"
- shows holomorphic_on_extend_lim:
- "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
- ((\<lambda>z. (z - \<xi>) * f z) \<longlongrightarrow> 0) (at \<xi>)"
- (is "?P = ?Q")
- and holomorphic_on_extend_bounded:
- "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
- (\<exists>B. eventually (\<lambda>z. norm(f z) \<le> B) (at \<xi>))"
- (is "?P = ?R")
-proof -
- obtain \<delta> where "0 < \<delta>" and \<delta>: "ball \<xi> \<delta> \<subseteq> S"
- using \<xi> mem_interior by blast
- have "?R" if holg: "g holomorphic_on S" and gf: "\<And>z. z \<in> S - {\<xi>} \<Longrightarrow> g z = f z" for g
- proof -
- have *: "\<forall>\<^sub>F z in at \<xi>. dist (g z) (g \<xi>) < 1 \<longrightarrow> cmod (f z) \<le> cmod (g \<xi>) + 1"
- apply (simp add: eventually_at)
- apply (rule_tac x="\<delta>" in exI)
- using \<delta> \<open>0 < \<delta>\<close>
- apply (clarsimp simp:)
- apply (drule_tac c=x in subsetD)
- apply (simp add: dist_commute)
- by (metis DiffI add.commute diff_le_eq dist_norm gf le_less_trans less_eq_real_def norm_triangle_ineq2 singletonD)
- have "continuous_on (interior S) g"
- by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset)
- then have "\<And>x. x \<in> interior S \<Longrightarrow> (g \<longlongrightarrow> g x) (at x)"
- using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast
- then have "(g \<longlongrightarrow> g \<xi>) (at \<xi>)"
- by (simp add: \<xi>)
- then show ?thesis
- apply (rule_tac x="norm(g \<xi>) + 1" in exI)
- apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
- done
- qed
- moreover have "?Q" if "\<forall>\<^sub>F z in at \<xi>. cmod (f z) \<le> B" for B
- by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
- moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0"
- proof -
- define h where [abs_def]: "h z = (z - \<xi>)^2 * f z" for z
- have h0: "(h has_field_derivative 0) (at \<xi>)"
- apply (simp add: h_def has_field_derivative_iff)
- apply (rule Lim_transform_within [OF that, of 1])
- apply (auto simp: field_split_simps power2_eq_square)
- done
- have holh: "h holomorphic_on S"
- proof (simp add: holomorphic_on_def, clarify)
- fix z assume "z \<in> S"
- show "h field_differentiable at z within S"
- proof (cases "z = \<xi>")
- case True then show ?thesis
- using field_differentiable_at_within field_differentiable_def h0 by blast
- next
- case False
- then have "f field_differentiable at z within S"
- using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close>
- unfolding field_differentiable_def has_field_derivative_iff
- by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at])
- then show ?thesis
- by (simp add: h_def power2_eq_square derivative_intros)
- qed
- qed
- define g where [abs_def]: "g z = (if z = \<xi> then deriv h \<xi> else (h z - h \<xi>) / (z - \<xi>))" for z
- have holg: "g holomorphic_on S"
- unfolding g_def by (rule pole_lemma [OF holh \<xi>])
- show ?thesis
- apply (rule_tac x="\<lambda>z. if z = \<xi> then deriv g \<xi> else (g z - g \<xi>)/(z - \<xi>)" in exI)
- apply (rule conjI)
- apply (rule pole_lemma [OF holg \<xi>])
- apply (auto simp: g_def power2_eq_square divide_simps)
- using h0 apply (simp add: h0 DERIV_imp_deriv h_def power2_eq_square)
- done
- qed
- ultimately show "?P = ?Q" and "?P = ?R"
- by meson+
-qed
-
-lemma pole_at_infinity:
- assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) \<longlongrightarrow> l) at_infinity"
- obtains a n where "\<And>z. f z = (\<Sum>i\<le>n. a i * z^i)"
-proof (cases "l = 0")
- case False
- with tendsto_inverse [OF lim] show ?thesis
- apply (rule_tac a="(\<lambda>n. inverse l)" and n=0 in that)
- apply (simp add: Liouville_weak [OF holf, of "inverse l"])
- done
-next
- case True
- then have [simp]: "l = 0" .
- show ?thesis
- proof (cases "\<exists>r. 0 < r \<and> (\<forall>z \<in> ball 0 r - {0}. f(inverse z) \<noteq> 0)")
- case True
- then obtain r where "0 < r" and r: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> f(inverse z) \<noteq> 0"
- by auto
- have 1: "inverse \<circ> f \<circ> inverse holomorphic_on ball 0 r - {0}"
- by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
- have 2: "0 \<in> interior (ball 0 r)"
- using \<open>0 < r\<close> by simp
- have "\<exists>B. 0<B \<and> eventually (\<lambda>z. cmod ((inverse \<circ> f \<circ> inverse) z) \<le> B) (at 0)"
- apply (rule exI [where x=1])
- apply simp
- using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
- apply (rule eventually_mono)
- apply (simp add: dist_norm)
- done
- with holomorphic_on_extend_bounded [OF 1 2]
- obtain g where holg: "g holomorphic_on ball 0 r"
- and geq: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> g z = (inverse \<circ> f \<circ> inverse) z"
- by meson
- have ifi0: "(inverse \<circ> f \<circ> inverse) \<midarrow>0\<rightarrow> 0"
- using \<open>l = 0\<close> lim lim_at_infinity_0 by blast
- have g2g0: "g \<midarrow>0\<rightarrow> g 0"
- using \<open>0 < r\<close> centre_in_ball continuous_at continuous_on_eq_continuous_at holg
- by (blast intro: holomorphic_on_imp_continuous_on)
- have g2g1: "g \<midarrow>0\<rightarrow> 0"
- apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
- using \<open>0 < r\<close> by (auto simp: geq)
- have [simp]: "g 0 = 0"
- by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
- have "ball 0 r - {0::complex} \<noteq> {}"
- using \<open>0 < r\<close>
- apply (clarsimp simp: ball_def dist_norm)
- apply (drule_tac c="of_real r/2" in subsetD, auto)
- done
- then obtain w::complex where "w \<noteq> 0" and w: "norm w < r" by force
- then have "g w \<noteq> 0" by (simp add: geq r)
- obtain B n e where "0 < B" "0 < e" "e \<le> r"
- and leg: "\<And>w. norm w < e \<Longrightarrow> B * cmod w ^ n \<le> cmod (g w)"
- apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w])
- using \<open>0 < r\<close> w \<open>g w \<noteq> 0\<close> by (auto simp: ball_subset_ball_iff)
- have "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> cmod z" for z
- proof -
- have ize: "inverse z \<in> ball 0 e - {0}" using that \<open>0 < e\<close>
- by (auto simp: norm_divide field_split_simps algebra_simps)
- then have [simp]: "z \<noteq> 0" and izr: "inverse z \<in> ball 0 r - {0}" using \<open>e \<le> r\<close>
- by auto
- then have [simp]: "f z \<noteq> 0"
- using r [of "inverse z"] by simp
- have [simp]: "f z = inverse (g (inverse z))"
- using izr geq [of "inverse z"] by simp
- show ?thesis using ize leg [of "inverse z"] \<open>0 < B\<close> \<open>0 < e\<close>
- by (simp add: field_split_simps norm_divide algebra_simps)
- qed
- then show ?thesis
- apply (rule_tac a = "\<lambda>k. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
- apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp)
- done
- next
- case False
- then have fi0: "\<And>r. r > 0 \<Longrightarrow> \<exists>z\<in>ball 0 r - {0}. f (inverse z) = 0"
- by simp
- have fz0: "f z = 0" if "0 < r" and lt1: "\<And>x. x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> inverse (cmod (f (inverse x))) < 1"
- for z r
- proof -
- have f0: "(f \<longlongrightarrow> 0) at_infinity"
- proof -
- have DIM_complex[intro]: "2 \<le> DIM(complex)" \<comment> \<open>should not be necessary!\<close>
- by simp
- from lt1 have "f (inverse x) \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> 1 < cmod (f (inverse x))" for x
- using one_less_inverse by force
- then have **: "cmod (f (inverse x)) \<le> 1 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> f (inverse x) = 0" for x
- by force
- then have *: "(f \<circ> inverse) ` (ball 0 r - {0}) \<subseteq> {0} \<union> - ball 0 1"
- by force
- have "continuous_on (inverse ` (ball 0 r - {0})) f"
- using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
- then have "connected ((f \<circ> inverse) ` (ball 0 r - {0}))"
- apply (intro connected_continuous_image continuous_intros)
- apply (force intro: connected_punctured_ball)+
- done
- then have "{0} \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {} \<or> - ball 0 1 \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {}"
- by (rule connected_closedD) (use * in auto)
- then have "w \<noteq> 0 \<Longrightarrow> cmod w < r \<Longrightarrow> f (inverse w) = 0" for w
- using fi0 **[of w] \<open>0 < r\<close>
- apply (auto simp add: inf.commute [of "- ball 0 1"] Diff_eq [symmetric] image_subset_iff dest: less_imp_le)
- apply fastforce
- apply (drule bspec [of _ _ w])
- apply (auto dest: less_imp_le)
- done
- then show ?thesis
- apply (simp add: lim_at_infinity_0)
- apply (rule tendsto_eventually)
- apply (simp add: eventually_at)
- apply (rule_tac x=r in exI)
- apply (simp add: \<open>0 < r\<close> dist_norm)
- done
- qed
- obtain w where "w \<in> ball 0 r - {0}" and "f (inverse w) = 0"
- using False \<open>0 < r\<close> by blast
- then show ?thesis
- by (auto simp: f0 Liouville_weak [OF holf, of 0])
- qed
- show ?thesis
- apply (rule that [of "\<lambda>n. 0" 0])
- using lim [unfolded lim_at_infinity_0]
- apply (simp add: Lim_at dist_norm norm_inverse)
- apply (drule_tac x=1 in spec)
- using fz0 apply auto
- done
- qed
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>Entire proper functions are precisely the non-trivial polynomials\<close>
-
-lemma proper_map_polyfun:
- fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
- assumes "closed S" and "compact K" and c: "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
- shows "compact (S \<inter> {z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
-proof -
- obtain B where "B > 0" and B: "\<And>x. x \<in> K \<Longrightarrow> norm x \<le> B"
- by (metis compact_imp_bounded \<open>compact K\<close> bounded_pos)
- have *: "norm x \<le> b"
- if "\<And>x. b \<le> norm x \<Longrightarrow> B + 1 \<le> norm (\<Sum>i\<le>n. c i * x ^ i)"
- "(\<Sum>i\<le>n. c i * x ^ i) \<in> K" for b x
- proof -
- have "norm (\<Sum>i\<le>n. c i * x ^ i) \<le> B"
- using B that by blast
- moreover have "\<not> B + 1 \<le> B"
- by simp
- ultimately show "norm x \<le> b"
- using that by (metis (no_types) less_eq_real_def not_less order_trans)
- qed
- have "bounded {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
- using polyfun_extremal [where c=c and B="B+1", OF c]
- by (auto simp: bounded_pos eventually_at_infinity_pos *)
- moreover have "closed ((\<lambda>z. (\<Sum>i\<le>n. c i * z ^ i)) -` K)"
- apply (intro allI continuous_closed_vimage continuous_intros)
- using \<open>compact K\<close> compact_eq_bounded_closed by blast
- ultimately show ?thesis
- using closed_Int_compact [OF \<open>closed S\<close>] compact_eq_bounded_closed
- by (auto simp add: vimage_def)
-qed
-
-lemma proper_map_polyfun_univ:
- fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
- assumes "compact K" "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
- shows "compact ({z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
- using proper_map_polyfun [of UNIV K c i n] assms by simp
-
-lemma proper_map_polyfun_eq:
- assumes "f holomorphic_on UNIV"
- shows "(\<forall>k. compact k \<longrightarrow> compact {z. f z \<in> k}) \<longleftrightarrow>
- (\<exists>c n. 0 < n \<and> (c n \<noteq> 0) \<and> f = (\<lambda>z. \<Sum>i\<le>n. c i * z^i))"
- (is "?lhs = ?rhs")
-proof
- assume compf [rule_format]: ?lhs
- have 2: "\<exists>k. 0 < k \<and> a k \<noteq> 0 \<and> f = (\<lambda>z. \<Sum>i \<le> k. a i * z ^ i)"
- if "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)" for a n
- proof (cases "\<forall>i\<le>n. 0<i \<longrightarrow> a i = 0")
- case True
- then have [simp]: "\<And>z. f z = a 0"
- by (simp add: that sum.atMost_shift)
- have False using compf [of "{a 0}"] by simp
- then show ?thesis ..
- next
- case False
- then obtain k where k: "0 < k" "k\<le>n" "a k \<noteq> 0" by force
- define m where "m = (GREATEST k. k\<le>n \<and> a k \<noteq> 0)"
- have m: "m\<le>n \<and> a m \<noteq> 0"
- unfolding m_def
- apply (rule GreatestI_nat [where b = n])
- using k apply auto
- done
- have [simp]: "a i = 0" if "m < i" "i \<le> n" for i
- using Greatest_le_nat [where b = "n" and P = "\<lambda>k. k\<le>n \<and> a k \<noteq> 0"]
- using m_def not_le that by auto
- have "k \<le> m"
- unfolding m_def
- apply (rule Greatest_le_nat [where b = "n"])
- using k apply auto
- done
- with k m show ?thesis
- by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
- qed
- have "((inverse \<circ> f) \<longlongrightarrow> 0) at_infinity"
- proof (rule Lim_at_infinityI)
- fix e::real assume "0 < e"
- with compf [of "cball 0 (inverse e)"]
- show "\<exists>B. \<forall>x. B \<le> cmod x \<longrightarrow> dist ((inverse \<circ> f) x) 0 \<le> e"
- apply simp
- apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
- apply (rule_tac x="b+1" in exI)
- apply (metis inverse_inverse_eq less_add_same_cancel2 less_imp_inverse_less add.commute not_le not_less_iff_gr_or_eq order_trans zero_less_one)
- done
- qed
- then show ?rhs
- apply (rule pole_at_infinity [OF assms])
- using 2 apply blast
- done
-next
- assume ?rhs
- then obtain c n where "0 < n" "c n \<noteq> 0" "f = (\<lambda>z. \<Sum>i\<le>n. c i * z ^ i)" by blast
- then have "compact {z. f z \<in> k}" if "compact k" for k
- by (auto intro: proper_map_polyfun_univ [OF that])
- then show ?lhs by blast
-qed
-
-subsection \<open>Relating invertibility and nonvanishing of derivative\<close>
-
-lemma has_complex_derivative_locally_injective:
- assumes holf: "f holomorphic_on S"
- and S: "\<xi> \<in> S" "open S"
- and dnz: "deriv f \<xi> \<noteq> 0"
- obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
-proof -
- have *: "\<exists>d>0. \<forall>x. dist \<xi> x < d \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) < e" if "e > 0" for e
- proof -
- have contdf: "continuous_on S (deriv f)"
- by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open S\<close>)
- obtain \<delta> where "\<delta>>0" and \<delta>: "\<And>x. \<lbrakk>x \<in> S; dist x \<xi> \<le> \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f x - deriv f \<xi>) \<le> e/2"
- using continuous_onE [OF contdf \<open>\<xi> \<in> S\<close>, of "e/2"] \<open>0 < e\<close>
- by (metis dist_complex_def half_gt_zero less_imp_le)
- obtain \<epsilon> where "\<epsilon>>0" "ball \<xi> \<epsilon> \<subseteq> S"
- by (metis openE [OF \<open>open S\<close> \<open>\<xi> \<in> S\<close>])
- with \<open>\<delta>>0\<close> have "\<exists>\<delta>>0. \<forall>x. dist \<xi> x < \<delta> \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) \<le> e/2"
- apply (rule_tac x="min \<delta> \<epsilon>" in exI)
- apply (intro conjI allI impI Operator_Norm.onorm_le)
- apply simp
- apply (simp only: Rings.ring_class.left_diff_distrib [symmetric] norm_mult)
- apply (rule mult_right_mono [OF \<delta>])
- apply (auto simp: dist_commute Rings.ordered_semiring_class.mult_right_mono \<delta>)
- done
- with \<open>e>0\<close> show ?thesis by force
- qed
- have "inj ((*) (deriv f \<xi>))"
- using dnz by simp
- then obtain g' where g': "linear g'" "g' \<circ> (*) (deriv f \<xi>) = id"
- using linear_injective_left_inverse [of "(*) (deriv f \<xi>)"]
- by (auto simp: linear_times)
- show ?thesis
- apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\<lambda>z h. deriv f z * h" and g' = g'])
- using g' *
- apply (simp_all add: linear_conv_bounded_linear that)
- using DERIV_deriv_iff_field_differentiable has_field_derivative_imp_has_derivative holf
- holomorphic_on_imp_differentiable_at \<open>open S\<close> apply blast
- done
-qed
-
-lemma has_complex_derivative_locally_invertible:
- assumes holf: "f holomorphic_on S"
- and S: "\<xi> \<in> S" "open S"
- and dnz: "deriv f \<xi> \<noteq> 0"
- obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "open (f ` (ball \<xi> r))" "inj_on f (ball \<xi> r)"
-proof -
- obtain r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
- by (blast intro: that has_complex_derivative_locally_injective [OF assms])
- then have \<xi>: "\<xi> \<in> ball \<xi> r" by simp
- then have nc: "\<not> f constant_on ball \<xi> r"
- using \<open>inj_on f (ball \<xi> r)\<close> injective_not_constant by fastforce
- have holf': "f holomorphic_on ball \<xi> r"
- using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
- have "open (f ` ball \<xi> r)"
- apply (rule open_mapping_thm [OF holf'])
- using nc apply auto
- done
- then show ?thesis
- using \<open>0 < r\<close> \<open>ball \<xi> r \<subseteq> S\<close> \<open>inj_on f (ball \<xi> r)\<close> that by blast
-qed
-
-lemma holomorphic_injective_imp_regular:
- assumes holf: "f holomorphic_on S"
- and "open S" and injf: "inj_on f S"
- and "\<xi> \<in> S"
- shows "deriv f \<xi> \<noteq> 0"
-proof -
- obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
- have holf': "f holomorphic_on ball \<xi> r"
- using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
- show ?thesis
- proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
- case True
- have fcon: "f w = f \<xi>" if "w \<in> ball \<xi> r" for w
- apply (rule holomorphic_fun_eq_const_on_connected [OF holf'])
- using True \<open>0 < r\<close> that by auto
- have False
- using fcon [of "\<xi> + r/2"] \<open>0 < r\<close> r injf unfolding inj_on_def
- by (metis \<open>\<xi> \<in> S\<close> contra_subsetD dist_commute fcon mem_ball perfect_choose_dist)
- then show ?thesis ..
- next
- case False
- then obtain n0 where n0: "n0 > 0 \<and> (deriv ^^ n0) f \<xi> \<noteq> 0" by blast
- define n where [abs_def]: "n = (LEAST n. n > 0 \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
- have n_ne: "n > 0" "(deriv ^^ n) f \<xi> \<noteq> 0"
- using def_LeastI [OF n_def n0] by auto
- have n_min: "\<And>k. 0 < k \<Longrightarrow> k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
- using def_Least_le [OF n_def] not_le by auto
- obtain g \<delta> where "0 < \<delta>"
- and holg: "g holomorphic_on ball \<xi> \<delta>"
- and fd: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
- and gnz: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> g w \<noteq> 0"
- apply (rule holomorphic_factor_order_of_zero_strong [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> n_ne])
- apply (blast intro: n_min)+
- done
- show ?thesis
- proof (cases "n=1")
- case True
- with n_ne show ?thesis by auto
- next
- case False
- have holgw: "(\<lambda>w. (w - \<xi>) * g w) holomorphic_on ball \<xi> (min r \<delta>)"
- apply (rule holomorphic_intros)+
- using holg by (simp add: holomorphic_on_subset subset_ball)
- have gd: "\<And>w. dist \<xi> w < \<delta> \<Longrightarrow> (g has_field_derivative deriv g w) (at w)"
- using holg
- by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH)
- have *: "\<And>w. w \<in> ball \<xi> (min r \<delta>)
- \<Longrightarrow> ((\<lambda>w. (w - \<xi>) * g w) has_field_derivative ((w - \<xi>) * deriv g w + g w))
- (at w)"
- by (rule gd derivative_eq_intros | simp)+
- have [simp]: "deriv (\<lambda>w. (w - \<xi>) * g w) \<xi> \<noteq> 0"
- using * [of \<xi>] \<open>0 < \<delta>\<close> \<open>0 < r\<close> by (simp add: DERIV_imp_deriv gnz)
- obtain T where "\<xi> \<in> T" "open T" and Tsb: "T \<subseteq> ball \<xi> (min r \<delta>)" and oimT: "open ((\<lambda>w. (w - \<xi>) * g w) ` T)"
- apply (rule has_complex_derivative_locally_invertible [OF holgw, of \<xi>])
- using \<open>0 < r\<close> \<open>0 < \<delta>\<close>
- apply (simp_all add:)
- by (meson open_ball centre_in_ball)
- define U where "U = (\<lambda>w. (w - \<xi>) * g w) ` T"
- have "open U" by (metis oimT U_def)
- have "0 \<in> U"
- apply (auto simp: U_def)
- apply (rule image_eqI [where x = \<xi>])
- apply (auto simp: \<open>\<xi> \<in> T\<close>)
- done
- then obtain \<epsilon> where "\<epsilon>>0" and \<epsilon>: "cball 0 \<epsilon> \<subseteq> U"
- using \<open>open U\<close> open_contains_cball by blast
- then have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> cball 0 \<epsilon>"
- "\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> cball 0 \<epsilon>"
- by (auto simp: norm_mult)
- with \<epsilon> have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> U"
- "\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> U" by blast+
- then obtain y0 y1 where "y0 \<in> T" and y0: "(y0 - \<xi>) * g y0 = \<epsilon> * exp(2 * of_real pi * \<i> * (0/n))"
- and "y1 \<in> T" and y1: "(y1 - \<xi>) * g y1 = \<epsilon> * exp(2 * of_real pi * \<i> * (1/n))"
- by (auto simp: U_def)
- then have "y0 \<in> ball \<xi> \<delta>" "y1 \<in> ball \<xi> \<delta>" using Tsb by auto
- moreover have "y0 \<noteq> y1"
- using y0 y1 \<open>\<epsilon> > 0\<close> complex_root_unity_eq_1 [of n 1] \<open>n > 0\<close> False by auto
- moreover have "T \<subseteq> S"
- by (meson Tsb min.cobounded1 order_trans r subset_ball)
- ultimately have False
- using inj_onD [OF injf, of y0 y1] \<open>y0 \<in> T\<close> \<open>y1 \<in> T\<close>
- using fd [of y0] fd [of y1] complex_root_unity [of n 1] n_ne
- apply (simp add: y0 y1 power_mult_distrib)
- apply (force simp: algebra_simps)
- done
- then show ?thesis ..
- qed
- qed
-qed
-
-text\<open>Hence a nice clean inverse function theorem\<close>
-
-lemma has_field_derivative_inverse_strong:
- fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
- shows "\<lbrakk>DERIV f x :> f'; f' \<noteq> 0; open S; x \<in> S; continuous_on S f;
- \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
- \<Longrightarrow> DERIV g (f x) :> inverse (f')"
- unfolding has_field_derivative_def
- by (rule has_derivative_inverse_strong [of S x f g]) auto
-
-lemma has_field_derivative_inverse_strong_x:
- fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
- shows "\<lbrakk>DERIV f (g y) :> f'; f' \<noteq> 0; open S; continuous_on S f; g y \<in> S; f(g y) = y;
- \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
- \<Longrightarrow> DERIV g y :> inverse (f')"
- unfolding has_field_derivative_def
- by (rule has_derivative_inverse_strong_x [of S g y f]) auto
-
-proposition holomorphic_has_inverse:
- assumes holf: "f holomorphic_on S"
- and "open S" and injf: "inj_on f S"
- obtains g where "g holomorphic_on (f ` S)"
- "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
- "\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
-proof -
- have ofs: "open (f ` S)"
- by (rule open_mapping_thm3 [OF assms])
- have contf: "continuous_on S f"
- by (simp add: holf holomorphic_on_imp_continuous_on)
- have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z \<in> S" for z
- proof -
- have 1: "(f has_field_derivative deriv f z) (at z)"
- using DERIV_deriv_iff_field_differentiable \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_on_imp_differentiable_at
- by blast
- have 2: "deriv f z \<noteq> 0"
- using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
- show ?thesis
- apply (rule has_field_derivative_inverse_strong [OF 1 2 \<open>open S\<close> \<open>z \<in> S\<close>])
- apply (simp add: holf holomorphic_on_imp_continuous_on)
- by (simp add: injf the_inv_into_f_f)
- qed
- show ?thesis
- proof
- show "the_inv_into S f holomorphic_on f ` S"
- by (simp add: holomorphic_on_open ofs) (blast intro: *)
- next
- fix z assume "z \<in> S"
- have "deriv f z \<noteq> 0"
- using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
- then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
- using * [OF \<open>z \<in> S\<close>] by (simp add: DERIV_imp_deriv)
- next
- fix z assume "z \<in> S"
- show "the_inv_into S f (f z) = z"
- by (simp add: \<open>z \<in> S\<close> injf the_inv_into_f_f)
- qed
-qed
-
-subsection\<open>The Schwarz Lemma\<close>
-
-lemma Schwarz1:
- assumes holf: "f holomorphic_on S"
- and contf: "continuous_on (closure S) f"
- and S: "open S" "connected S"
- and boS: "bounded S"
- and "S \<noteq> {}"
- obtains w where "w \<in> frontier S"
- "\<And>z. z \<in> closure S \<Longrightarrow> norm (f z) \<le> norm (f w)"
-proof -
- have connf: "continuous_on (closure S) (norm o f)"
- using contf continuous_on_compose continuous_on_norm_id by blast
- have coc: "compact (closure S)"
- by (simp add: \<open>bounded S\<close> bounded_closure compact_eq_bounded_closed)
- then obtain x where x: "x \<in> closure S" and xmax: "\<And>z. z \<in> closure S \<Longrightarrow> norm(f z) \<le> norm(f x)"
- apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]])
- using \<open>S \<noteq> {}\<close> apply auto
- done
- then show ?thesis
- proof (cases "x \<in> frontier S")
- case True
- then show ?thesis using that xmax by blast
- next
- case False
- then have "x \<in> S"
- using \<open>open S\<close> frontier_def interior_eq x by auto
- then have "f constant_on S"
- apply (rule maximum_modulus_principle [OF holf S \<open>open S\<close> order_refl])
- using closure_subset apply (blast intro: xmax)
- done
- then have "f constant_on (closure S)"
- by (rule constant_on_closureI [OF _ contf])
- then obtain c where c: "\<And>x. x \<in> closure S \<Longrightarrow> f x = c"
- by (meson constant_on_def)
- obtain w where "w \<in> frontier S"
- by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
- then show ?thesis
- by (simp add: c frontier_def that)
- qed
-qed
-
-lemma Schwarz2:
- "\<lbrakk>f holomorphic_on ball 0 r;
- 0 < s; ball w s \<subseteq> ball 0 r;
- \<And>z. norm (w-z) < s \<Longrightarrow> norm(f z) \<le> norm(f w)\<rbrakk>
- \<Longrightarrow> f constant_on ball 0 r"
-by (rule maximum_modulus_principle [where U = "ball w s" and \<xi> = w]) (simp_all add: dist_norm)
-
-lemma Schwarz3:
- assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
- obtains h where "h holomorphic_on (ball 0 r)" and "\<And>z. norm z < r \<Longrightarrow> f z = z * (h z)" and "deriv f 0 = h 0"
-proof -
- define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z
- have d0: "deriv f 0 = h 0"
- by (simp add: h_def)
- moreover have "h holomorphic_on (ball 0 r)"
- by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
- moreover have "norm z < r \<Longrightarrow> f z = z * h z" for z
- by (simp add: h_def)
- ultimately show ?thesis
- using that by blast
-qed
-
-proposition Schwarz_Lemma:
- assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
- and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
- and \<xi>: "norm \<xi> < 1"
- shows "norm (f \<xi>) \<le> norm \<xi>" and "norm(deriv f 0) \<le> 1"
- and "((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
- \<or> norm(deriv f 0) = 1)
- \<Longrightarrow> \<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1"
- (is "?P \<Longrightarrow> ?Q")
-proof -
- obtain h where holh: "h holomorphic_on (ball 0 1)"
- and fz_eq: "\<And>z. norm z < 1 \<Longrightarrow> f z = z * (h z)" and df0: "deriv f 0 = h 0"
- by (rule Schwarz3 [OF holf]) auto
- have noh_le: "norm (h z) \<le> 1" if z: "norm z < 1" for z
- proof -
- have "norm (h z) < a" if a: "1 < a" for a
- proof -
- have "max (inverse a) (norm z) < 1"
- using z a by (simp_all add: inverse_less_1_iff)
- then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
- using Rats_dense_in_real by blast
- then have nzr: "norm z < r" and ira: "inverse r < a"
- using z a less_imp_inverse_less by force+
- then have "0 < r"
- by (meson norm_not_less_zero not_le order.strict_trans2)
- have holh': "h holomorphic_on ball 0 r"
- by (meson holh \<open>r < 1\<close> holomorphic_on_subset less_eq_real_def subset_ball)
- have conth': "continuous_on (cball 0 r) h"
- by (meson \<open>r < 1\<close> dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
- obtain w where w: "norm w = r" and lenw: "\<And>z. norm z < r \<Longrightarrow> norm(h z) \<le> norm(h w)"
- apply (rule Schwarz1 [OF holh']) using conth' \<open>0 < r\<close> by auto
- have "h w = f w / w" using fz_eq \<open>r < 1\<close> nzr w by auto
- then have "cmod (h z) < inverse r"
- by (metis \<open>0 < r\<close> \<open>r < 1\<close> divide_strict_right_mono inverse_eq_divide
- le_less_trans lenw no norm_divide nzr w)
- then show ?thesis using ira by linarith
- qed
- then show "norm (h z) \<le> 1"
- using not_le by blast
- qed
- show "cmod (f \<xi>) \<le> cmod \<xi>"
- proof (cases "\<xi> = 0")
- case True then show ?thesis by auto
- next
- case False
- then show ?thesis
- by (simp add: noh_le fz_eq \<xi> mult_left_le norm_mult)
- qed
- show no_df0: "norm(deriv f 0) \<le> 1"
- by (simp add: \<open>\<And>z. cmod z < 1 \<Longrightarrow> cmod (h z) \<le> 1\<close> df0)
- show "?Q" if "?P"
- using that
- proof
- assume "\<exists>z. cmod z < 1 \<and> z \<noteq> 0 \<and> cmod (f z) = cmod z"
- then obtain \<gamma> where \<gamma>: "cmod \<gamma> < 1" "\<gamma> \<noteq> 0" "cmod (f \<gamma>) = cmod \<gamma>" by blast
- then have [simp]: "norm (h \<gamma>) = 1"
- by (simp add: fz_eq norm_mult)
- have "ball \<gamma> (1 - cmod \<gamma>) \<subseteq> ball 0 1"
- by (simp add: ball_subset_ball_iff)
- moreover have "\<And>z. cmod (\<gamma> - z) < 1 - cmod \<gamma> \<Longrightarrow> cmod (h z) \<le> cmod (h \<gamma>)"
- apply (simp add: algebra_simps)
- by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4)
- ultimately obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
- using Schwarz2 [OF holh, of "1 - norm \<gamma>" \<gamma>, unfolded constant_on_def] \<gamma> by auto
- then have "norm c = 1"
- using \<gamma> by force
- with c show ?thesis
- using fz_eq by auto
- next
- assume [simp]: "cmod (deriv f 0) = 1"
- then obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
- using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
- by auto
- moreover have "norm c = 1" using df0 c by auto
- ultimately show ?thesis
- using fz_eq by auto
- qed
-qed
-
-corollary Schwarz_Lemma':
- assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
- and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
- shows "((\<forall>\<xi>. norm \<xi> < 1 \<longrightarrow> norm (f \<xi>) \<le> norm \<xi>)
- \<and> norm(deriv f 0) \<le> 1)
- \<and> (((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
- \<or> norm(deriv f 0) = 1)
- \<longrightarrow> (\<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1))"
- using Schwarz_Lemma [OF assms]
- by (metis (no_types) norm_eq_zero zero_less_one)
-
-subsection\<open>The Schwarz reflection principle\<close>
-
-lemma hol_pal_lem0:
- assumes "d \<bullet> a \<le> k" "k \<le> d \<bullet> b"
- obtains c where
- "c \<in> closed_segment a b" "d \<bullet> c = k"
- "\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k"
- "\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z"
-proof -
- obtain c where cin: "c \<in> closed_segment a b" and keq: "k = d \<bullet> c"
- using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
- by (auto simp: assms)
- have "closed_segment a c \<subseteq> {z. d \<bullet> z \<le> k}" "closed_segment c b \<subseteq> {z. k \<le> d \<bullet> z}"
- unfolding segment_convex_hull using assms keq
- by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
- then show ?thesis using cin that by fastforce
-qed
-
-lemma hol_pal_lem1:
- assumes "convex S" "open S"
- and abc: "a \<in> S" "b \<in> S" "c \<in> S"
- "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k" "d \<bullet> c \<le> k"
- and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
- and contf: "continuous_on S f"
- shows "contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof -
- have "interior (convex hull {a, b, c}) \<subseteq> interior(S \<inter> {x. d \<bullet> x \<le> k})"
- apply (rule interior_mono)
- apply (rule hull_minimal)
- apply (simp add: abc lek)
- apply (rule convex_Int [OF \<open>convex S\<close> convex_halfspace_le])
- done
- also have "... \<subseteq> {z \<in> S. d \<bullet> z < k}"
- by (force simp: interior_open [OF \<open>open S\<close>] \<open>d \<noteq> 0\<close>)
- finally have *: "interior (convex hull {a, b, c}) \<subseteq> {z \<in> S. d \<bullet> z < k}" .
- have "continuous_on (convex hull {a,b,c}) f"
- using \<open>convex S\<close> contf abc continuous_on_subset subset_hull
- by fastforce
- moreover have "f holomorphic_on interior (convex hull {a,b,c})"
- by (rule holomorphic_on_subset [OF holf1 *])
- ultimately show ?thesis
- using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3
- by blast
-qed
-
-lemma hol_pal_lem2:
- assumes S: "convex S" "open S"
- and abc: "a \<in> S" "b \<in> S" "c \<in> S"
- and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k"
- and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
- and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
- and contf: "continuous_on S f"
- shows "contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof (cases "d \<bullet> c \<le> k")
- case True show ?thesis
- by (rule hol_pal_lem1 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 contf])
-next
- case False
- then have "d \<bullet> c > k" by force
- obtain a' where a': "a' \<in> closed_segment b c" and "d \<bullet> a' = k"
- and ba': "\<And>z. z \<in> closed_segment b a' \<Longrightarrow> d \<bullet> z \<le> k"
- and a'c: "\<And>z. z \<in> closed_segment a' c \<Longrightarrow> k \<le> d \<bullet> z"
- apply (rule hol_pal_lem0 [of d b k c, OF \<open>d \<bullet> b \<le> k\<close>])
- using False by auto
- obtain b' where b': "b' \<in> closed_segment a c" and "d \<bullet> b' = k"
- and ab': "\<And>z. z \<in> closed_segment a b' \<Longrightarrow> d \<bullet> z \<le> k"
- and b'c: "\<And>z. z \<in> closed_segment b' c \<Longrightarrow> k \<le> d \<bullet> z"
- apply (rule hol_pal_lem0 [of d a k c, OF \<open>d \<bullet> a \<le> k\<close>])
- using False by auto
- have a'b': "a' \<in> S \<and> b' \<in> S"
- using a' abc b' convex_contains_segment \<open>convex S\<close> by auto
- have "continuous_on (closed_segment c a) f"
- by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
- then have 1: "contour_integral (linepath c a) f =
- contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
- apply (rule contour_integral_split_linepath)
- using b' by (simp add: closed_segment_commute)
- have "continuous_on (closed_segment b c) f"
- by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
- then have 2: "contour_integral (linepath b c) f =
- contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
- by (rule contour_integral_split_linepath [OF _ a'])
- have 3: "contour_integral (reversepath (linepath b' a')) f =
- - contour_integral (linepath b' a') f"
- by (rule contour_integral_reversepath [OF valid_path_linepath])
- have fcd_le: "f field_differentiable at x"
- if "x \<in> interior S \<and> x \<in> interior {x. d \<bullet> x \<le> k}" for x
- proof -
- have "f holomorphic_on S \<inter> {c. d \<bullet> c < k}"
- by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1)
- then have "\<exists>C D. x \<in> interior C \<inter> interior D \<and> f holomorphic_on interior C \<inter> interior D"
- using that
- by (metis Collect_mem_eq Int_Collect \<open>d \<noteq> 0\<close> interior_halfspace_le interior_open \<open>open S\<close>)
- then show "f field_differentiable at x"
- by (metis at_within_interior holomorphic_on_def interior_Int interior_interior)
- qed
- have ab_le: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> d \<bullet> x \<le> k"
- proof -
- fix x :: complex
- assume "x \<in> closed_segment a b"
- then have "\<And>C. x \<in> C \<or> b \<notin> C \<or> a \<notin> C \<or> \<not> convex C"
- by (meson contra_subsetD convex_contains_segment)
- then show "d \<bullet> x \<le> k"
- by (metis lek convex_halfspace_le mem_Collect_eq)
- qed
- have "continuous_on (S \<inter> {x. d \<bullet> x \<le> k}) f" using contf
- by (simp add: continuous_on_subset)
- then have "(f has_contour_integral 0)
- (linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)"
- apply (rule Cauchy_theorem_convex [where K = "{}"])
- apply (simp_all add: path_image_join convex_Int convex_halfspace_le \<open>convex S\<close> fcd_le ab_le
- closed_segment_subset abc a'b' ba')
- by (metis \<open>d \<bullet> a' = k\<close> \<open>d \<bullet> b' = k\<close> convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl)
- then have 4: "contour_integral (linepath a b) f +
- contour_integral (linepath b a') f +
- contour_integral (linepath a' b') f +
- contour_integral (linepath b' a) f = 0"
- by (rule has_chain_integral_chain_integral4)
- have fcd_ge: "f field_differentiable at x"
- if "x \<in> interior S \<and> x \<in> interior {x. k \<le> d \<bullet> x}" for x
- proof -
- have f2: "f holomorphic_on S \<inter> {c. k < d \<bullet> c}"
- by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2)
- have f3: "interior S = S"
- by (simp add: interior_open \<open>open S\<close>)
- then have "x \<in> S \<inter> interior {c. k \<le> d \<bullet> c}"
- using that by simp
- then show "f field_differentiable at x"
- using f3 f2 unfolding holomorphic_on_def
- by (metis (no_types) \<open>d \<noteq> 0\<close> at_within_interior interior_Int interior_halfspace_ge interior_interior)
- qed
- have "continuous_on (S \<inter> {x. k \<le> d \<bullet> x}) f" using contf
- by (simp add: continuous_on_subset)
- then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')"
- apply (rule Cauchy_theorem_convex [where K = "{}"])
- apply (simp_all add: path_image_join convex_Int convex_halfspace_ge \<open>convex S\<close>
- fcd_ge closed_segment_subset abc a'b' a'c)
- by (metis \<open>d \<bullet> a' = k\<close> b'c closed_segment_commute convex_contains_segment
- convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl)
- then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
- by (rule has_chain_integral_chain_integral3)
- show ?thesis
- using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath)
-qed
-
-lemma hol_pal_lem3:
- assumes S: "convex S" "open S"
- and abc: "a \<in> S" "b \<in> S" "c \<in> S"
- and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k"
- and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
- and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
- and contf: "continuous_on S f"
- shows "contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof (cases "d \<bullet> b \<le> k")
- case True show ?thesis
- by (rule hol_pal_lem2 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 holf2 contf])
-next
- case False
- show ?thesis
- proof (cases "d \<bullet> c \<le> k")
- case True
- have "contour_integral (linepath c a) f +
- contour_integral (linepath a b) f +
- contour_integral (linepath b c) f = 0"
- by (rule hol_pal_lem2 [OF S \<open>c \<in> S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close> \<open>d \<noteq> 0\<close> \<open>d \<bullet> c \<le> k\<close> lek holf1 holf2 contf])
- then show ?thesis
- by (simp add: algebra_simps)
- next
- case False
- have "contour_integral (linepath b c) f +
- contour_integral (linepath c a) f +
- contour_integral (linepath a b) f = 0"
- apply (rule hol_pal_lem2 [OF S \<open>b \<in> S\<close> \<open>c \<in> S\<close> \<open>a \<in> S\<close>, of "-d" "-k"])
- using \<open>d \<noteq> 0\<close> \<open>\<not> d \<bullet> b \<le> k\<close> False by (simp_all add: holf1 holf2 contf)
- then show ?thesis
- by (simp add: algebra_simps)
- qed
-qed
-
-lemma hol_pal_lem4:
- assumes S: "convex S" "open S"
- and abc: "a \<in> S" "b \<in> S" "c \<in> S" and "d \<noteq> 0"
- and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
- and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
- and contf: "continuous_on S f"
- shows "contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0"
-proof (cases "d \<bullet> a \<le> k")
- case True show ?thesis
- by (rule hol_pal_lem3 [OF S abc \<open>d \<noteq> 0\<close> True holf1 holf2 contf])
-next
- case False
- show ?thesis
- apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"])
- using \<open>d \<noteq> 0\<close> False by (simp_all add: holf1 holf2 contf)
-qed
-
-lemma holomorphic_on_paste_across_line:
- assumes S: "open S" and "d \<noteq> 0"
- and holf1: "f holomorphic_on (S \<inter> {z. d \<bullet> z < k})"
- and holf2: "f holomorphic_on (S \<inter> {z. k < d \<bullet> z})"
- and contf: "continuous_on S f"
- shows "f holomorphic_on S"
-proof -
- have *: "\<exists>t. open t \<and> p \<in> t \<and> continuous_on t f \<and>
- (\<forall>a b c. convex hull {a, b, c} \<subseteq> t \<longrightarrow>
- contour_integral (linepath a b) f +
- contour_integral (linepath b c) f +
- contour_integral (linepath c a) f = 0)"
- if "p \<in> S" for p
- proof -
- obtain e where "e>0" and e: "ball p e \<subseteq> S"
- using \<open>p \<in> S\<close> openE S by blast
- then have "continuous_on (ball p e) f"
- using contf continuous_on_subset by blast
- moreover have "f holomorphic_on {z. dist p z < e \<and> d \<bullet> z < k}"
- apply (rule holomorphic_on_subset [OF holf1])
- using e by auto
- moreover have "f holomorphic_on {z. dist p z < e \<and> k < d \<bullet> z}"
- apply (rule holomorphic_on_subset [OF holf2])
- using e by auto
- ultimately show ?thesis
- apply (rule_tac x="ball p e" in exI)
- using \<open>e > 0\<close> e \<open>d \<noteq> 0\<close>
- apply (simp add:, clarify)
- apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k])
- apply (auto simp: subset_hull)
- done
- qed
- show ?thesis
- by (blast intro: * Morera_local_triangle analytic_imp_holomorphic)
-qed
-
-proposition Schwarz_reflection:
- assumes "open S" and cnjs: "cnj ` S \<subseteq> S"
- and holf: "f holomorphic_on (S \<inter> {z. 0 < Im z})"
- and contf: "continuous_on (S \<inter> {z. 0 \<le> Im z}) f"
- and f: "\<And>z. \<lbrakk>z \<in> S; z \<in> \<real>\<rbrakk> \<Longrightarrow> (f z) \<in> \<real>"
- shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj(f(cnj z))) holomorphic_on S"
-proof -
- have 1: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. 0 < Im z})"
- by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf])
- have cont_cfc: "continuous_on (S \<inter> {z. Im z \<le> 0}) (cnj o f o cnj)"
- apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf])
- using cnjs apply auto
- done
- have "cnj \<circ> f \<circ> cnj field_differentiable at x within S \<inter> {z. Im z < 0}"
- if "x \<in> S" "Im x < 0" "f field_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x
- using that
- apply (simp add: field_differentiable_def has_field_derivative_iff Lim_within dist_norm, clarify)
- apply (rule_tac x="cnj f'" in exI)
- apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify)
- apply (drule_tac x="cnj xa" in bspec)
- using cnjs apply force
- apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj)
- done
- then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S \<inter> {z. Im z < 0})"
- using holf cnjs
- by (force simp: holomorphic_on_def)
- have 2: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. Im z < 0})"
- apply (rule iffD1 [OF holomorphic_cong [OF refl]])
- using hol_cfc by auto
- have [simp]: "(S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}) = S"
- by force
- have "continuous_on ((S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}))
- (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
- apply (rule continuous_on_cases_local)
- using cont_cfc contf
- apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
- using f Reals_cnj_iff complex_is_Real_iff apply auto
- done
- then have 3: "continuous_on S (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
- by force
- show ?thesis
- apply (rule holomorphic_on_paste_across_line [OF \<open>open S\<close>, of "- \<i>" _ 0])
- using 1 2 3
- apply auto
- done
-qed
-
-subsection\<open>Bloch's theorem\<close>
-
-lemma Bloch_lemma_0:
- assumes holf: "f holomorphic_on cball 0 r" and "0 < r"
- and [simp]: "f 0 = 0"
- and le: "\<And>z. norm z < r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f 0)"
- shows "ball 0 ((3 - 2 * sqrt 2) * r * norm(deriv f 0)) \<subseteq> f ` ball 0 r"
-proof -
- have "sqrt 2 < 3/2"
- by (rule real_less_lsqrt) (auto simp: power2_eq_square)
- then have sq3: "0 < 3 - 2 * sqrt 2" by simp
- show ?thesis
- proof (cases "deriv f 0 = 0")
- case True then show ?thesis by simp
- next
- case False
- define C where "C = 2 * norm(deriv f 0)"
- have "0 < C" using False by (simp add: C_def)
- have holf': "f holomorphic_on ball 0 r" using holf
- using ball_subset_cball holomorphic_on_subset by blast
- then have holdf': "deriv f holomorphic_on ball 0 r"
- by (rule holomorphic_deriv [OF _ open_ball])
- have "Le1": "norm(deriv f z - deriv f 0) \<le> norm z / (r - norm z) * C"
- if "norm z < r" for z
- proof -
- have T1: "norm(deriv f z - deriv f 0) \<le> norm z / (R - norm z) * C"
- if R: "norm z < R" "R < r" for R
- proof -
- have "0 < R" using R
- by (metis less_trans norm_zero zero_less_norm_iff)
- have df_le: "\<And>x. norm x < r \<Longrightarrow> norm (deriv f x) \<le> C"
- using le by (simp add: C_def)
- have hol_df: "deriv f holomorphic_on cball 0 R"
- apply (rule holomorphic_on_subset) using R holdf' by auto
- have *: "((\<lambda>w. deriv f w / (w - z)) has_contour_integral 2 * pi * \<i> * deriv f z) (circlepath 0 R)"
- if "norm z < R" for z
- using \<open>0 < R\<close> that Cauchy_integral_formula_convex_simple [OF convex_cball hol_df, of _ "circlepath 0 R"]
- by (force simp: winding_number_circlepath)
- have **: "((\<lambda>x. deriv f x / (x - z) - deriv f x / x) has_contour_integral
- of_real (2 * pi) * \<i> * (deriv f z - deriv f 0))
- (circlepath 0 R)"
- using has_contour_integral_diff [OF * [of z] * [of 0]] \<open>0 < R\<close> that
- by (simp add: algebra_simps)
- have [simp]: "\<And>x. norm x = R \<Longrightarrow> x \<noteq> z" using that(1) by blast
- have "norm (deriv f x / (x - z) - deriv f x / x)
- \<le> C * norm z / (R * (R - norm z))"
- if "norm x = R" for x
- proof -
- have [simp]: "norm (deriv f x * x - deriv f x * (x - z)) =
- norm (deriv f x) * norm z"
- by (simp add: norm_mult right_diff_distrib')
- show ?thesis
- using \<open>0 < R\<close> \<open>0 < C\<close> R that
- apply (simp add: norm_mult norm_divide divide_simps)
- using df_le norm_triangle_ineq2 \<open>0 < C\<close> apply (auto intro!: mult_mono)
- done
- qed
- then show ?thesis
- using has_contour_integral_bound_circlepath
- [OF **, of "C * norm z/(R*(R - norm z))"]
- \<open>0 < R\<close> \<open>0 < C\<close> R
- apply (simp add: norm_mult norm_divide)
- apply (simp add: divide_simps mult.commute)
- done
- qed
- obtain r' where r': "norm z < r'" "r' < r"
- using Rats_dense_in_real [of "norm z" r] \<open>norm z < r\<close> by blast
- then have [simp]: "closure {r'<..<r} = {r'..r}" by simp
- show ?thesis
- apply (rule continuous_ge_on_closure
- [where f = "\<lambda>r. norm z / (r - norm z) * C" and s = "{r'<..<r}",
- OF _ _ T1])
- apply (intro continuous_intros)
- using that r'
- apply (auto simp: not_le)
- done
- qed
- have "*": "(norm z - norm z^2/(r - norm z)) * norm(deriv f 0) \<le> norm(f z)"
- if r: "norm z < r" for z
- proof -
- have 1: "\<And>x. x \<in> ball 0 r \<Longrightarrow>
- ((\<lambda>z. f z - deriv f 0 * z) has_field_derivative deriv f x - deriv f 0)
- (at x within ball 0 r)"
- by (rule derivative_eq_intros holomorphic_derivI holf' | simp)+
- have 2: "closed_segment 0 z \<subseteq> ball 0 r"
- by (metis \<open>0 < r\<close> convex_ball convex_contains_segment dist_self mem_ball mem_ball_0 that)
- have 3: "(\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C) integrable_on {0..1}"
- apply (rule integrable_on_cmult_right [where 'b=real, simplified])
- apply (rule integrable_on_cdivide [where 'b=real, simplified])
- apply (rule integrable_on_cmult_left [where 'b=real, simplified])
- apply (rule ident_integrable_on)
- done
- have 4: "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm z * norm z * x * C / (r - norm z)"
- if x: "0 \<le> x" "x \<le> 1" for x
- proof -
- have [simp]: "x * norm z < r"
- using r x by (meson le_less_trans mult_le_cancel_right2 norm_not_less_zero)
- have "norm (deriv f (x *\<^sub>R z) - deriv f 0) \<le> norm (x *\<^sub>R z) / (r - norm (x *\<^sub>R z)) * C"
- apply (rule Le1) using r x \<open>0 < r\<close> by simp
- also have "... \<le> norm (x *\<^sub>R z) / (r - norm z) * C"
- using r x \<open>0 < r\<close>
- apply (simp add: field_split_simps)
- by (simp add: \<open>0 < C\<close> mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono)
- finally have "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm (x *\<^sub>R z) / (r - norm z) * C * norm z"
- by (rule mult_right_mono) simp
- with x show ?thesis by (simp add: algebra_simps)
- qed
- have le_norm: "abc \<le> norm d - e \<Longrightarrow> norm(f - d) \<le> e \<Longrightarrow> abc \<le> norm f" for abc d e and f::complex
- by (metis add_diff_cancel_left' add_diff_eq diff_left_mono norm_diff_ineq order_trans)
- have "norm (integral {0..1} (\<lambda>x. (deriv f (x *\<^sub>R z) - deriv f 0) * z))
- \<le> integral {0..1} (\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C)"
- apply (rule integral_norm_bound_integral)
- using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
- apply (simp add: has_contour_integral_linepath has_integral_integrable_integral)
- apply (rule 3)
- apply (simp add: norm_mult power2_eq_square 4)
- done
- then have int_le: "norm (f z - deriv f 0 * z) \<le> (norm z)\<^sup>2 * norm(deriv f 0) / ((r - norm z))"
- using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
- apply (simp add: has_contour_integral_linepath has_integral_integrable_integral C_def)
- done
- show ?thesis
- apply (rule le_norm [OF _ int_le])
- using \<open>norm z < r\<close>
- apply (simp add: power2_eq_square divide_simps C_def norm_mult)
- proof -
- have "norm z * (norm (deriv f 0) * (r - norm z - norm z)) \<le> norm z * (norm (deriv f 0) * (r - norm z) - norm (deriv f 0) * norm z)"
- by (simp add: algebra_simps)
- then show "(norm z * (r - norm z) - norm z * norm z) * norm (deriv f 0) \<le> norm (deriv f 0) * norm z * (r - norm z) - norm z * norm z * norm (deriv f 0)"
- by (simp add: algebra_simps)
- qed
- qed
- have sq201 [simp]: "0 < (1 - sqrt 2 / 2)" "(1 - sqrt 2 / 2) < 1"
- by (auto simp: sqrt2_less_2)
- have 1: "continuous_on (closure (ball 0 ((1 - sqrt 2 / 2) * r))) f"
- apply (rule continuous_on_subset [OF holomorphic_on_imp_continuous_on [OF holf]])
- apply (subst closure_ball)
- using \<open>0 < r\<close> mult_pos_pos sq201
- apply (auto simp: cball_subset_cball_iff)
- done
- have 2: "open (f ` interior (ball 0 ((1 - sqrt 2 / 2) * r)))"
- apply (rule open_mapping_thm [OF holf' open_ball connected_ball], force)
- using \<open>0 < r\<close> mult_pos_pos sq201 apply (simp add: ball_subset_ball_iff)
- using False \<open>0 < r\<close> centre_in_ball holf' holomorphic_nonconstant by blast
- have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv f 0)) =
- ball (f 0) ((3 - 2 * sqrt 2) * r * norm (deriv f 0))"
- by simp
- also have "... \<subseteq> f ` ball 0 ((1 - sqrt 2 / 2) * r)"
- proof -
- have 3: "(3 - 2 * sqrt 2) * r * norm (deriv f 0) \<le> norm (f z)"
- if "norm z = (1 - sqrt 2 / 2) * r" for z
- apply (rule order_trans [OF _ *])
- using \<open>0 < r\<close>
- apply (simp_all add: field_simps power2_eq_square that)
- apply (simp add: mult.assoc [symmetric])
- done
- show ?thesis
- apply (rule ball_subset_open_map_image [OF 1 2 _ bounded_ball])
- using \<open>0 < r\<close> sq201 3 apply simp_all
- using C_def \<open>0 < C\<close> sq3 apply force
- done
- qed
- also have "... \<subseteq> f ` ball 0 r"
- apply (rule image_subsetI [OF imageI], simp)
- apply (erule less_le_trans)
- using \<open>0 < r\<close> apply (auto simp: field_simps)
- done
- finally show ?thesis .
- qed
-qed
-
-lemma Bloch_lemma:
- assumes holf: "f holomorphic_on cball a r" and "0 < r"
- and le: "\<And>z. z \<in> ball a r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f a)"
- shows "ball (f a) ((3 - 2 * sqrt 2) * r * norm(deriv f a)) \<subseteq> f ` ball a r"
-proof -
- have fz: "(\<lambda>z. f (a + z)) = f o (\<lambda>z. (a + z))"
- by (simp add: o_def)
- have hol0: "(\<lambda>z. f (a + z)) holomorphic_on cball 0 r"
- unfolding fz by (intro holomorphic_intros holf holomorphic_on_compose | simp)+
- then have [simp]: "\<And>x. norm x < r \<Longrightarrow> (\<lambda>z. f (a + z)) field_differentiable at x"
- by (metis open_ball at_within_open ball_subset_cball diff_0 dist_norm holomorphic_on_def holomorphic_on_subset mem_ball norm_minus_cancel)
- have [simp]: "\<And>z. norm z < r \<Longrightarrow> f field_differentiable at (a + z)"
- by (metis holf open_ball add_diff_cancel_left' dist_complex_def holomorphic_on_imp_differentiable_at holomorphic_on_subset interior_cball interior_subset mem_ball norm_minus_commute)
- then have [simp]: "f field_differentiable at a"
- by (metis add.comm_neutral \<open>0 < r\<close> norm_eq_zero)
- have hol1: "(\<lambda>z. f (a + z) - f a) holomorphic_on cball 0 r"
- by (intro holomorphic_intros hol0)
- then have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv (\<lambda>z. f (a + z) - f a) 0))
- \<subseteq> (\<lambda>z. f (a + z) - f a) ` ball 0 r"
- apply (rule Bloch_lemma_0)
- apply (simp_all add: \<open>0 < r\<close>)
- apply (simp add: fz complex_derivative_chain)
- apply (simp add: dist_norm le)
- done
- then show ?thesis
- apply clarify
- apply (drule_tac c="x - f a" in subsetD)
- apply (force simp: fz \<open>0 < r\<close> dist_norm complex_derivative_chain field_differentiable_compose)+
- done
-qed
-
-proposition Bloch_unit:
- assumes holf: "f holomorphic_on ball a 1" and [simp]: "deriv f a = 1"
- obtains b r where "1/12 < r" and "ball b r \<subseteq> f ` (ball a 1)"
-proof -
- define r :: real where "r = 249/256"
- have "0 < r" "r < 1" by (auto simp: r_def)
- define g where "g z = deriv f z * of_real(r - norm(z - a))" for z
- have "deriv f holomorphic_on ball a 1"
- by (rule holomorphic_deriv [OF holf open_ball])
- then have "continuous_on (ball a 1) (deriv f)"
- using holomorphic_on_imp_continuous_on by blast
- then have "continuous_on (cball a r) (deriv f)"
- by (rule continuous_on_subset) (simp add: cball_subset_ball_iff \<open>r < 1\<close>)
- then have "continuous_on (cball a r) g"
- by (simp add: g_def continuous_intros)
- then have 1: "compact (g ` cball a r)"
- by (rule compact_continuous_image [OF _ compact_cball])
- have 2: "g ` cball a r \<noteq> {}"
- using \<open>r > 0\<close> by auto
- obtain p where pr: "p \<in> cball a r"
- and pge: "\<And>y. y \<in> cball a r \<Longrightarrow> norm (g y) \<le> norm (g p)"
- using distance_attains_sup [OF 1 2, of 0] by force
- define t where "t = (r - norm(p - a)) / 2"
- have "norm (p - a) \<noteq> r"
- using pge [of a] \<open>r > 0\<close> by (auto simp: g_def norm_mult)
- then have "norm (p - a) < r" using pr
- by (simp add: norm_minus_commute dist_norm)
- then have "0 < t"
- by (simp add: t_def)
- have cpt: "cball p t \<subseteq> ball a r"
- using \<open>0 < t\<close> by (simp add: cball_subset_ball_iff dist_norm t_def field_simps)
- have gen_le_dfp: "norm (deriv f y) * (r - norm (y - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
- if "y \<in> cball a r" for y
- proof -
- have [simp]: "norm (y - a) \<le> r"
- using that by (simp add: dist_norm norm_minus_commute)
- have "norm (g y) \<le> norm (g p)"
- using pge [OF that] by simp
- then have "norm (deriv f y) * abs (r - norm (y - a)) \<le> norm (deriv f p) * abs (r - norm (p - a))"
- by (simp only: dist_norm g_def norm_mult norm_of_real)
- with that \<open>norm (p - a) < r\<close> show ?thesis
- by (simp add: dist_norm field_split_simps)
- qed
- have le_norm_dfp: "r / (r - norm (p - a)) \<le> norm (deriv f p)"
- using gen_le_dfp [of a] \<open>r > 0\<close> by auto
- have 1: "f holomorphic_on cball p t"
- apply (rule holomorphic_on_subset [OF holf])
- using cpt \<open>r < 1\<close> order_subst1 subset_ball by auto
- have 2: "norm (deriv f z) \<le> 2 * norm (deriv f p)" if "z \<in> ball p t" for z
- proof -
- have z: "z \<in> cball a r"
- by (meson ball_subset_cball subsetD cpt that)
- then have "norm(z - a) < r"
- by (metis ball_subset_cball contra_subsetD cpt dist_norm mem_ball norm_minus_commute that)
- have "norm (deriv f z) * (r - norm (z - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
- using gen_le_dfp [OF z] by simp
- with \<open>norm (z - a) < r\<close> \<open>norm (p - a) < r\<close>
- have "norm (deriv f z) \<le> (r - norm (p - a)) / (r - norm (z - a)) * norm (deriv f p)"
- by (simp add: field_simps)
- also have "... \<le> 2 * norm (deriv f p)"
- apply (rule mult_right_mono)
- using that \<open>norm (p - a) < r\<close> \<open>norm(z - a) < r\<close>
- apply (simp_all add: field_simps t_def dist_norm [symmetric])
- using dist_triangle3 [of z a p] by linarith
- finally show ?thesis .
- qed
- have sqrt2: "sqrt 2 < 2113/1494"
- by (rule real_less_lsqrt) (auto simp: power2_eq_square)
- then have sq3: "0 < 3 - 2 * sqrt 2" by simp
- have "1 / 12 / ((3 - 2 * sqrt 2) / 2) < r"
- using sq3 sqrt2 by (auto simp: field_simps r_def)
- also have "... \<le> cmod (deriv f p) * (r - cmod (p - a))"
- using \<open>norm (p - a) < r\<close> le_norm_dfp by (simp add: pos_divide_le_eq)
- finally have "1 / 12 < cmod (deriv f p) * (r - cmod (p - a)) * ((3 - 2 * sqrt 2) / 2)"
- using pos_divide_less_eq half_gt_zero_iff sq3 by blast
- then have **: "1 / 12 < (3 - 2 * sqrt 2) * t * norm (deriv f p)"
- using sq3 by (simp add: mult.commute t_def)
- have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball p t"
- by (rule Bloch_lemma [OF 1 \<open>0 < t\<close> 2])
- also have "... \<subseteq> f ` ball a 1"
- apply (rule image_mono)
- apply (rule order_trans [OF ball_subset_cball])
- apply (rule order_trans [OF cpt])
- using \<open>0 < t\<close> \<open>r < 1\<close> apply (simp add: ball_subset_ball_iff dist_norm)
- done
- finally have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball a 1" .
- with ** show ?thesis
- by (rule that)
-qed
-
-theorem Bloch:
- assumes holf: "f holomorphic_on ball a r" and "0 < r"
- and r': "r' \<le> r * norm (deriv f a) / 12"
- obtains b where "ball b r' \<subseteq> f ` (ball a r)"
-proof (cases "deriv f a = 0")
- case True with r' show ?thesis
- using ball_eq_empty that by fastforce
-next
- case False
- define C where "C = deriv f a"
- have "0 < norm C" using False by (simp add: C_def)
- have dfa: "f field_differentiable at a"
- apply (rule holomorphic_on_imp_differentiable_at [OF holf])
- using \<open>0 < r\<close> by auto
- have fo: "(\<lambda>z. f (a + of_real r * z)) = f o (\<lambda>z. (a + of_real r * z))"
- by (simp add: o_def)
- have holf': "f holomorphic_on (\<lambda>z. a + complex_of_real r * z) ` ball 0 1"
- apply (rule holomorphic_on_subset [OF holf])
- using \<open>0 < r\<close> apply (force simp: dist_norm norm_mult)
- done
- have 1: "(\<lambda>z. f (a + r * z) / (C * r)) holomorphic_on ball 0 1"
- apply (rule holomorphic_intros holomorphic_on_compose holf' | simp add: fo)+
- using \<open>0 < r\<close> by (simp add: C_def False)
- have "((\<lambda>z. f (a + of_real r * z) / (C * of_real r)) has_field_derivative
- (deriv f (a + of_real r * z) / C)) (at z)"
- if "norm z < 1" for z
- proof -
- have *: "((\<lambda>x. f (a + of_real r * x)) has_field_derivative
- (deriv f (a + of_real r * z) * of_real r)) (at z)"
- apply (simp add: fo)
- apply (rule DERIV_chain [OF field_differentiable_derivI])
- apply (rule holomorphic_on_imp_differentiable_at [OF holf], simp)
- using \<open>0 < r\<close> apply (simp add: dist_norm norm_mult that)
- apply (rule derivative_eq_intros | simp)+
- done
- show ?thesis
- apply (rule derivative_eq_intros * | simp)+
- using \<open>0 < r\<close> by (auto simp: C_def False)
- qed
- have 2: "deriv (\<lambda>z. f (a + of_real r * z) / (C * of_real r)) 0 = 1"
- apply (subst deriv_cdivide_right)
- apply (simp add: field_differentiable_def fo)
- apply (rule exI)
- apply (rule DERIV_chain [OF field_differentiable_derivI])
- apply (simp add: dfa)
- apply (rule derivative_eq_intros | simp add: C_def False fo)+
- using \<open>0 < r\<close>
- apply (simp add: C_def False fo)
- apply (simp add: derivative_intros dfa complex_derivative_chain)
- done
- have sb1: "(*) (C * r) ` (\<lambda>z. f (a + of_real r * z) / (C * r)) ` ball 0 1
- \<subseteq> f ` ball a r"
- using \<open>0 < r\<close> by (auto simp: dist_norm norm_mult C_def False)
- have sb2: "ball (C * r * b) r' \<subseteq> (*) (C * r) ` ball b t"
- if "1 / 12 < t" for b t
- proof -
- have *: "r * cmod (deriv f a) / 12 \<le> r * (t * cmod (deriv f a))"
- using that \<open>0 < r\<close> less_eq_real_def mult.commute mult.right_neutral mult_left_mono norm_ge_zero times_divide_eq_right
- by auto
- show ?thesis
- apply clarify
- apply (rule_tac x="x / (C * r)" in image_eqI)
- using \<open>0 < r\<close>
- apply (simp_all add: dist_norm norm_mult norm_divide C_def False field_simps)
- apply (erule less_le_trans)
- apply (rule order_trans [OF r' *])
- done
- qed
- show ?thesis
- apply (rule Bloch_unit [OF 1 2])
- apply (rename_tac t)
- apply (rule_tac b="(C * of_real r) * b" in that)
- apply (drule image_mono [where f = "\<lambda>z. (C * of_real r) * z"])
- using sb1 sb2
- apply force
- done
-qed
-
-corollary Bloch_general:
- assumes holf: "f holomorphic_on s" and "a \<in> s"
- and tle: "\<And>z. z \<in> frontier s \<Longrightarrow> t \<le> dist a z"
- and rle: "r \<le> t * norm(deriv f a) / 12"
- obtains b where "ball b r \<subseteq> f ` s"
-proof -
- consider "r \<le> 0" | "0 < t * norm(deriv f a) / 12" using rle by force
- then show ?thesis
- proof cases
- case 1 then show ?thesis
- by (simp add: ball_empty that)
- next
- case 2
- show ?thesis
- proof (cases "deriv f a = 0")
- case True then show ?thesis
- using rle by (simp add: ball_empty that)
- next
- case False
- then have "t > 0"
- using 2 by (force simp: zero_less_mult_iff)
- have "\<not> ball a t \<subseteq> s \<Longrightarrow> ball a t \<inter> frontier s \<noteq> {}"
- apply (rule connected_Int_frontier [of "ball a t" s], simp_all)
- using \<open>0 < t\<close> \<open>a \<in> s\<close> centre_in_ball apply blast
- done
- with tle have *: "ball a t \<subseteq> s" by fastforce
- then have 1: "f holomorphic_on ball a t"
- using holf using holomorphic_on_subset by blast
- show ?thesis
- apply (rule Bloch [OF 1 \<open>t > 0\<close> rle])
- apply (rule_tac b=b in that)
- using * apply force
- done
- qed
- qed
-qed
-
-subsection \<open>Cauchy's residue theorem\<close>
-
-text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
- Interactive Theorem Proving\<close>
-
-definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
- "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e
- \<longrightarrow> (f has_contour_integral 2*pi* \<i> *int) (circlepath z \<epsilon>))"
-
-lemma Eps_cong:
- assumes "\<And>x. P x = Q x"
- shows "Eps P = Eps Q"
- using ext[of P Q, OF assms] by simp
-
-lemma residue_cong:
- assumes eq: "eventually (\<lambda>z. f z = g z) (at z)" and "z = z'"
- shows "residue f z = residue g z'"
-proof -
- from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
- by (simp add: eq_commute)
- let ?P = "\<lambda>f c e. (\<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow>
- (f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>))"
- have "residue f z = residue g z" unfolding residue_def
- proof (rule Eps_cong)
- fix c :: complex
- have "\<exists>e>0. ?P g c e"
- if "\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
- proof -
- from that(1) obtain e where e: "e > 0" "?P f c e"
- by blast
- from that(2) obtain e' where e': "e' > 0" "\<And>z'. z' \<noteq> z \<Longrightarrow> dist z' z < e' \<Longrightarrow> f z' = g z'"
- unfolding eventually_at by blast
- have "?P g c (min e e')"
- proof (intro allI exI impI, goal_cases)
- case (1 \<epsilon>)
- hence "(f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>)"
- using e(2) by auto
- thus ?case
- proof (rule has_contour_integral_eq)
- fix z' assume "z' \<in> path_image (circlepath z \<epsilon>)"
- hence "dist z' z < e'" and "z' \<noteq> z"
- using 1 by (auto simp: dist_commute)
- with e'(2)[of z'] show "f z' = g z'" by simp
- qed
- qed
- moreover from e and e' have "min e e' > 0" by auto
- ultimately show ?thesis by blast
- qed
- from this[OF _ eq] and this[OF _ eq']
- show "(\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
- by blast
- qed
- with assms show ?thesis by simp
-qed
-
-lemma contour_integral_circlepath_eq:
- assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1\<le>e2"
- and e2_cball:"cball z e2 \<subseteq> s"
- shows
- "f contour_integrable_on circlepath z e1"
- "f contour_integrable_on circlepath z e2"
- "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
-proof -
- define l where "l \<equiv> linepath (z+e2) (z+e1)"
- have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
- have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
- have zl_img:"z\<notin>path_image l"
- proof
- assume "z \<in> path_image l"
- then have "e2 \<le> cmod (e2 - e1)"
- using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
- by (auto simp add:closed_segment_commute)
- thus False using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- apply (subst (asm) norm_of_real)
- by auto
- qed
- define g where "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
- show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
- proof -
- show "f contour_integrable_on circlepath z e2"
- apply (intro contour_integrable_continuous_circlepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- using \<open>e2>0\<close> e2_cball by auto
- show "f contour_integrable_on (circlepath z e1)"
- apply (intro contour_integrable_continuous_circlepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
- qed
- have [simp]:"f contour_integrable_on l"
- proof -
- have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- by (intro closed_segment_subset,auto simp add:dist_norm)
- hence "closed_segment (z + e2) (z + e1) \<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
- by auto
- then show "f contour_integrable_on l" unfolding l_def
- apply (intro contour_integrable_continuous_linepath[OF
- continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
- by auto
- qed
- let ?ig="\<lambda>g. contour_integral g f"
- have "(f has_contour_integral 0) g"
- proof (rule Cauchy_theorem_global[OF _ f_holo])
- show "open (s - {z})" using \<open>open s\<close> by auto
- show "valid_path g" unfolding g_def l_def by auto
- show "pathfinish g = pathstart g" unfolding g_def l_def by auto
- next
- have path_img:"path_image g \<subseteq> cball z e2"
- proof -
- have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
- by (intro closed_segment_subset,auto simp add:dist_norm)
- moreover have "sphere z \<bar>e1\<bar> \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1\<le>e2\<close> \<open>e1>0\<close> by auto
- ultimately show ?thesis unfolding g_def l_def using \<open>e2>0\<close>
- by (simp add: path_image_join closed_segment_commute)
- qed
- show "path_image g \<subseteq> s - {z}"
- proof -
- have "z\<notin>path_image g" using zl_img
- unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
- moreover note \<open>cball z e2 \<subseteq> s\<close> and path_img
- ultimately show ?thesis by auto
- qed
- show "winding_number g w = 0" when"w \<notin> s - {z}" for w
- proof -
- have "winding_number g w = 0" when "w\<notin>s" using that e2_cball
- apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
- by (auto simp add:g_def l_def)
- moreover have "winding_number g z=0"
- proof -
- let ?Wz="\<lambda>g. winding_number g z"
- have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
- + ?Wz (reversepath l)"
- using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_def
- by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
- also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
- using zl_img
- apply (subst (2) winding_number_reversepath)
- by (auto simp add:l_def closed_segment_commute)
- also have "... = 0"
- proof -
- have "?Wz (circlepath z e2) = 1" using \<open>e2>0\<close>
- by (auto intro: winding_number_circlepath_centre)
- moreover have "?Wz (reversepath (circlepath z e1)) = -1" using \<open>e1>0\<close>
- apply (subst winding_number_reversepath)
- by (auto intro: winding_number_circlepath_centre)
- ultimately show ?thesis by auto
- qed
- finally show ?thesis .
- qed
- ultimately show ?thesis using that by auto
- qed
- qed
- then have "0 = ?ig g" using contour_integral_unique by simp
- also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
- + ?ig (reversepath l)"
- unfolding g_def
- by (auto simp add:contour_integrable_reversepath_eq)
- also have "... = ?ig (circlepath z e2) - ?ig (circlepath z e1)"
- by (auto simp add:contour_integral_reversepath)
- finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
- by simp
-qed
-
-lemma base_residue:
- assumes "open s" "z\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
- and r_cball:"cball z r \<subseteq> s"
- shows "(f has_contour_integral 2 * pi * \<i> * (residue f z)) (circlepath z r)"
-proof -
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s"
- using open_contains_cball[of s] \<open>open s\<close> \<open>z\<in>s\<close> by auto
- define c where "c \<equiv> 2 * pi * \<i>"
- define i where "i \<equiv> contour_integral (circlepath z e) f / c"
- have "(f has_contour_integral c*i) (circlepath z \<epsilon>)" when "\<epsilon>>0" "\<epsilon><e" for \<epsilon>
- proof -
- have "contour_integral (circlepath z e) f = contour_integral (circlepath z \<epsilon>) f"
- "f contour_integrable_on circlepath z \<epsilon>"
- "f contour_integrable_on circlepath z e"
- using \<open>\<epsilon><e\<close>
- by (intro contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> _ e_cball],auto)+
- then show ?thesis unfolding i_def c_def
- by (auto intro:has_contour_integral_integral)
- qed
- then have "\<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- unfolding residue_def c_def
- apply (rule_tac someI[of _ i],intro exI[where x=e])
- by (auto simp add:\<open>e>0\<close> c_def)
- then obtain e' where "e'>0"
- and e'_def:"\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- by auto
- let ?int="\<lambda>e. contour_integral (circlepath z e) f"
- define \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2"
- have "\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
- have "(f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
- using e'_def[rule_format,OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><e'\<close>] .
- then show ?thesis unfolding c_def
- using contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball]
- by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z \<epsilon>" "circlepath z r"])
-qed
-
-lemma residue_holo:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s"
- shows "residue f z = 0"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f has_contour_integral c*residue f z) (circlepath z e)"
- using f_holo
- by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- moreover have "(f has_contour_integral 0) (circlepath z e)"
- using f_holo e_cball \<open>e>0\<close>
- by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
- ultimately have "c*residue f z =0"
- using has_contour_integral_unique by blast
- thus ?thesis unfolding c_def by auto
-qed
-
-lemma residue_const:"residue (\<lambda>_. c) z = 0"
- by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)
-
-lemma residue_add:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- and g_holo:"g holomorphic_on s - {z}"
- shows "residue (\<lambda>z. f z + g z) z= residue f z + residue g z"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- define fg where "fg \<equiv> (\<lambda>z. f z+g z)"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
- unfolding fg_def using f_holo g_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- by (auto intro:holomorphic_intros)
- moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
- unfolding fg_def using f_holo g_holo
- by (auto intro: has_contour_integral_add base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- ultimately have "c*(residue f z + residue g z) = c * residue fg z"
- using has_contour_integral_unique by (auto simp add:distrib_left)
- thus ?thesis unfolding fg_def
- by (auto simp add:c_def)
-qed
-
-lemma residue_lmul:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. c * (f z)) z= c * residue f z"
-proof (cases "c=0")
- case True
- thus ?thesis using residue_const by auto
-next
- case False
- define c' where "c' \<equiv> 2 * pi * \<i>"
- define f' where "f' \<equiv> (\<lambda>z. c * (f z))"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
- unfolding f'_def using f_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
- by (auto intro:holomorphic_intros)
- moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
- unfolding f'_def using f_holo
- by (auto intro: has_contour_integral_lmul
- base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
- ultimately have "c' * residue f' z = c * (c' * residue f z)"
- using has_contour_integral_unique by auto
- thus ?thesis unfolding f'_def c'_def using False
- by (auto simp add:field_simps)
-qed
-
-lemma residue_rmul:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. (f z) * c) z= residue f z * c"
-using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)
-
-lemma residue_div:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. (f z) / c) z= residue f z / c "
-using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)
-
-lemma residue_neg:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- shows "residue (\<lambda>z. - (f z)) z= - residue f z"
-using residue_lmul[OF assms,of "-1"] by auto
-
-lemma residue_diff:
- assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
- and g_holo:"g holomorphic_on s - {z}"
- shows "residue (\<lambda>z. f z - g z) z= residue f z - residue g z"
-using residue_add[OF assms(1,2,3),of "\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
-by (auto intro:holomorphic_intros g_holo)
-
-lemma residue_simple:
- assumes "open s" "z\<in>s" and f_holo:"f holomorphic_on s"
- shows "residue (\<lambda>w. f w / (w - z)) z = f z"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
- obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
- using open_contains_cball_eq by blast
- have "(f' has_contour_integral c * f z) (circlepath z e)"
- unfolding f'_def c_def using \<open>e>0\<close> f_holo e_cball
- by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
- moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
- unfolding f'_def using f_holo
- apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
- by (auto intro!:holomorphic_intros)
- ultimately have "c * f z = c * residue f' z"
- using has_contour_integral_unique by blast
- thus ?thesis unfolding c_def f'_def by auto
-qed
-
-lemma residue_simple':
- assumes s: "open s" "z \<in> s" and holo: "f holomorphic_on (s - {z})"
- and lim: "((\<lambda>w. f w * (w - z)) \<longlongrightarrow> c) (at z)"
- shows "residue f z = c"
-proof -
- define g where "g = (\<lambda>w. if w = z then c else f w * (w - z))"
- from holo have "(\<lambda>w. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
- by (force intro: holomorphic_intros)
- also have "?P \<longleftrightarrow> g holomorphic_on (s - {z})"
- by (intro holomorphic_cong refl) (simp_all add: g_def)
- finally have *: "g holomorphic_on (s - {z})" .
-
- note lim
- also have "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z\<rightarrow> g z"
- by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
- finally have **: "g \<midarrow>z\<rightarrow> g z" .
-
- have g_holo: "g holomorphic_on s"
- by (rule no_isolated_singularity'[where K = "{z}"])
- (insert assms * **, simp_all add: at_within_open_NO_MATCH)
- from s and this have "residue (\<lambda>w. g w / (w - z)) z = g z"
- by (rule residue_simple)
- also have "\<forall>\<^sub>F za in at z. g za / (za - z) = f za"
- unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
- hence "residue (\<lambda>w. g w / (w - z)) z = residue f z"
- by (intro residue_cong refl)
- finally show ?thesis
- by (simp add: g_def)
-qed
-
-lemma residue_holomorphic_over_power:
- assumes "open A" "z0 \<in> A" "f holomorphic_on A"
- shows "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
-proof -
- let ?f = "\<lambda>z. f z / (z - z0) ^ Suc n"
- from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A"
- by (auto simp: open_contains_cball)
- have "(?f has_contour_integral 2 * pi * \<i> * residue ?f z0) (circlepath z0 r)"
- using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
- moreover have "(?f has_contour_integral 2 * pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
- using assms r
- by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
- (auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
- ultimately have "2 * pi * \<i> * residue ?f z0 = 2 * pi * \<i> / fact n * (deriv ^^ n) f z0"
- by (rule has_contour_integral_unique)
- thus ?thesis by (simp add: field_simps)
-qed
-
-lemma residue_holomorphic_over_power':
- assumes "open A" "0 \<in> A" "f holomorphic_on A"
- shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
- using residue_holomorphic_over_power[OF assms] by simp
-
-lemma get_integrable_path:
- assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
- obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
- "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
-proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
- case 1
- obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
- using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
- valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
- moreover have "f contour_integrable_on g"
- using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
- \<open>f holomorphic_on s - {}\<close>
- by auto
- ultimately show ?case using "1"(1)[of g] by auto
-next
- case idt:(2 p pts)
- obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
- using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
- \<open>a \<in> s - insert p pts\<close>
- by auto
- define a' where "a' \<equiv> a+e/2"
- have "a'\<in>s-{p} -pts" using e[rule_format,of "a+e/2"] \<open>e>0\<close>
- by (auto simp add:dist_complex_def a'_def)
- then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
- "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
- using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
- by (metis Diff_insert2 open_delete)
- define g where "g \<equiv> linepath a a' +++ g'"
- have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
- moreover have "pathstart g = a" and "pathfinish g = b" unfolding g_def by auto
- moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
- proof (rule subset_path_image_join)
- have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
- by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
- then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
- by auto
- next
- show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
- qed
- moreover have "f contour_integrable_on g"
- proof -
- have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
- by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
- then have "continuous_on (closed_segment a a') f"
- using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
- apply (elim continuous_on_subset)
- by auto
- then have "f contour_integrable_on linepath a a'"
- using contour_integrable_continuous_linepath by auto
- then show ?thesis unfolding g_def
- apply (rule contour_integrable_joinI)
- by (auto simp add: \<open>e>0\<close>)
- qed
- ultimately show ?case using idt.prems(1)[of g] by auto
-qed
-
-lemma Cauchy_theorem_aux:
- assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
- "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
- "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
- "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- using assms
-proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
- case 1
- then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
-next
- case (2 p pts)
- note fin[simp] = \<open>finite (insert p pts)\<close>
- and connected = \<open>connected (s - insert p pts)\<close>
- and valid[simp] = \<open>valid_path g\<close>
- and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
- and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
- and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
- and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
- and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
- have "h p>0" and "p\<in>s"
- and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
- using h \<open>insert p pts \<subseteq> s\<close> by auto
- obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
- "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
- proof -
- have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
- by (simp add: \<open>p \<in> s\<close> dist_norm)
- then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
- by fastforce
- moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
- ultimately show ?thesis
- using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
- by blast
- qed
- obtain n::int where "n=winding_number g p"
- using integer_winding_number[OF _ g_loop,of p] valid path_img
- by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
- define p_circ where "p_circ \<equiv> circlepath p (h p)"
- define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
- define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
- define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
- have n_circ:"valid_path (n_circ k)"
- "winding_number (n_circ k) p = k"
- "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
- "path_image (n_circ k) = (if k=0 then {p + h p} else sphere p (h p))"
- "p \<notin> path_image (n_circ k)"
- "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
- "f contour_integrable_on (n_circ k)"
- "contour_integral (n_circ k) f = k * contour_integral p_circ f"
- for k
- proof (induct k)
- case 0
- show "valid_path (n_circ 0)"
- and "path_image (n_circ 0) = (if 0=0 then {p + h p} else sphere p (h p))"
- and "winding_number (n_circ 0) p = of_nat 0"
- and "pathstart (n_circ 0) = p + h p"
- and "pathfinish (n_circ 0) = p + h p"
- and "p \<notin> path_image (n_circ 0)"
- unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
- by (auto simp add: dist_norm)
- show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
- unfolding n_circ_def p_circ_pt_def
- apply (auto intro!:winding_number_trivial)
- by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
- show "f contour_integrable_on (n_circ 0)"
- unfolding n_circ_def p_circ_pt_def
- by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
- show "contour_integral (n_circ 0) f = of_nat 0 * contour_integral p_circ f"
- unfolding n_circ_def p_circ_pt_def by auto
- next
- case (Suc k)
- have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
- have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
- using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
- have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
- proof -
- have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
- then show ?thesis using h_p pcirc(1) by auto
- qed
- have pcirc_integrable:"f contour_integrable_on p_circ"
- by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
- contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
- holomorphic_on_subset[OF holo])
- show "valid_path (n_circ (Suc k))"
- using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
- show "path_image (n_circ (Suc k))
- = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
- proof -
- have "path_image p_circ = sphere p (h p)"
- unfolding p_circ_def using \<open>0 < h p\<close> by auto
- then show ?thesis unfolding n_Suc using Suc.hyps(5) \<open>h p>0\<close>
- by (auto simp add: path_image_join[OF pcirc(3)] dist_norm)
- qed
- then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
- show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
- proof -
- have "winding_number p_circ p = 1"
- by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
- moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
- then have "winding_number (p_circ +++ n_circ k) p
- = winding_number p_circ p + winding_number (n_circ k) p"
- using valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
- apply (intro winding_number_join)
- by auto
- ultimately show ?thesis using Suc(2) unfolding n_circ_def
- by auto
- qed
- show "pathstart (n_circ (Suc k)) = p + h p"
- by (simp add: n_circ_def p_circ_def)
- show "pathfinish (n_circ (Suc k)) = p + h p"
- using Suc(4) unfolding n_circ_def by auto
- show "winding_number (n_circ (Suc k)) p'=0 \<and> p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
- proof -
- have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
- moreover have "p' \<notin> path_image (n_circ k)"
- using Suc.hyps(7) that by blast
- moreover have "winding_number p_circ p' = 0"
- proof -
- have "path_image p_circ \<subseteq> cball p (h p)"
- using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
- moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
- ultimately show ?thesis unfolding p_circ_def
- apply (intro winding_number_zero_outside)
- by auto
- qed
- ultimately show ?thesis
- unfolding n_Suc
- apply (subst winding_number_join)
- by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
- qed
- show "f contour_integrable_on (n_circ (Suc k))"
- unfolding n_Suc
- by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
- show "contour_integral (n_circ (Suc k)) f = (Suc k) * contour_integral p_circ f"
- unfolding n_Suc
- by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
- Suc(9) algebra_simps)
- qed
- have cp[simp]:"pathstart cp = p + h p" "pathfinish cp = p + h p"
- "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
- "winding_number cp p = - n"
- "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
- "f contour_integrable_on cp"
- "contour_integral cp f = - n * contour_integral p_circ f"
- proof -
- show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
- using n_circ unfolding cp_def by auto
- next
- have "sphere p (h p) \<subseteq> s - insert p pts"
- using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
- moreover have "p + complex_of_real (h p) \<in> s - insert p pts"
- using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
- ultimately show "path_image cp \<subseteq> s - insert p pts" unfolding cp_def
- using n_circ(5) by auto
- next
- show "winding_number cp p = - n"
- unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
- by (auto simp: valid_path_imp_path)
- next
- show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
- unfolding cp_def
- apply (auto)
- apply (subst winding_number_reversepath)
- by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
- next
- show "f contour_integrable_on cp" unfolding cp_def
- using contour_integrable_reversepath_eq n_circ(1,8) by auto
- next
- show "contour_integral cp f = - n * contour_integral p_circ f"
- unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
- by auto
- qed
- define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
- have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
- proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
- show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
- show "open (s - {p})" using \<open>open s\<close> by auto
- show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close> by blast
- show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
- show "valid_path g'"
- unfolding g'_def cp_def using n_circ valid pg g_loop
- by (auto intro!:valid_path_join )
- show "pathfinish g' = pathstart g'"
- unfolding g'_def cp_def using pg(2) by simp
- show "path_image g' \<subseteq> s - {p} - pts"
- proof -
- define s' where "s' \<equiv> s - {p} - pts"
- have s':"s' = s-insert p pts " unfolding s'_def by auto
- then show ?thesis using path_img pg(4) cp(4)
- unfolding g'_def
- apply (fold s'_def s')
- apply (intro subset_path_image_join)
- by auto
- qed
- note path_join_imp[simp]
- show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
- proof clarify
- fix z assume z:"z\<notin>s - {p}"
- have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
- + winding_number (pg +++ cp +++ (reversepath pg)) z"
- proof (rule winding_number_join)
- show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
- show "z \<notin> path_image g" using z path_img by auto
- show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
- by (simp add: valid_path_imp_path)
- next
- have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
- using pg(4) cp(4) by (auto simp:subset_path_image_join)
- then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
- next
- show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
- qed
- also have "... = winding_number g z + (winding_number pg z
- + winding_number (cp +++ (reversepath pg)) z)"
- proof (subst add_left_cancel,rule winding_number_join)
- show "path pg" and "path (cp +++ reversepath pg)"
- and "pathfinish pg = pathstart (cp +++ reversepath pg)"
- by (auto simp add: valid_path_imp_path)
- show "z \<notin> path_image pg" using pg(4) z by blast
- show "z \<notin> path_image (cp +++ reversepath pg)" using z
- by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
- not_in_path_image_join path_image_reversepath singletonD)
- qed
- also have "... = winding_number g z + (winding_number pg z
- + (winding_number cp z + winding_number (reversepath pg) z))"
- apply (auto intro!:winding_number_join simp: valid_path_imp_path)
- apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
- by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
- also have "... = winding_number g z + winding_number cp z"
- apply (subst winding_number_reversepath)
- apply (auto simp: valid_path_imp_path)
- by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
- finally have "winding_number g' z = winding_number g z + winding_number cp z"
- unfolding g'_def .
- moreover have "winding_number g z + winding_number cp z = 0"
- using winding z \<open>n=winding_number g p\<close> by auto
- ultimately show "winding_number g' z = 0" unfolding g'_def by auto
- qed
- show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
- using h by fastforce
- qed
- moreover have "contour_integral g' f = contour_integral g f
- - winding_number g p * contour_integral p_circ f"
- proof -
- have "contour_integral g' f = contour_integral g f
- + contour_integral (pg +++ cp +++ reversepath pg) f"
- unfolding g'_def
- apply (subst contour_integral_join)
- by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
- intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
- contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral pg f
- + contour_integral (cp +++ reversepath pg) f"
- apply (subst contour_integral_join)
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral pg f
- + contour_integral cp f + contour_integral (reversepath pg) f"
- apply (subst contour_integral_join)
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f + contour_integral cp f"
- using contour_integral_reversepath
- by (auto simp add:contour_integrable_reversepath)
- also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
- using \<open>n=winding_number g p\<close> by auto
- finally show ?thesis .
- qed
- moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
- proof -
- have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
- using "2.prems"(8) that
- apply blast
- apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
- by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
- have "winding_number g' p' = winding_number g p'
- + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
- apply (subst winding_number_join)
- apply (simp_all add: valid_path_imp_path)
- apply (intro not_in_path_image_join)
- by auto
- also have "... = winding_number g p' + winding_number pg p'
- + winding_number (cp +++ reversepath pg) p'"
- apply (subst winding_number_join)
- apply (simp_all add: valid_path_imp_path)
- apply (intro not_in_path_image_join)
- by auto
- also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
- + winding_number (reversepath pg) p'"
- apply (subst winding_number_join)
- by (simp_all add: valid_path_imp_path)
- also have "... = winding_number g p' + winding_number cp p'"
- apply (subst winding_number_reversepath)
- by (simp_all add: valid_path_imp_path)
- also have "... = winding_number g p'" using that by auto
- finally show ?thesis .
- qed
- ultimately show ?case unfolding p_circ_def
- apply (subst (asm) sum.cong[OF refl,
- of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
- by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
-qed
-
-lemma Cauchy_theorem_singularities:
- assumes "open s" "connected s" "finite pts" and
- holo:"f holomorphic_on s-pts" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- "path_image g \<subseteq> s-pts" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
- avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- (is "?L=?R")
-proof -
- define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
- define pts1 where "pts1 \<equiv> pts \<inter> s"
- define pts2 where "pts2 \<equiv> pts - pts1"
- have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
- unfolding pts1_def pts2_def by auto
- have "contour_integral g f = (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
- proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
- have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
- then show "connected (s - pts1)"
- using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
- next
- show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
- show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
- show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
- show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
- by (simp add: avoid pts1_def)
- qed
- moreover have "sum circ pts2=0"
- proof -
- have "winding_number g p=0" when "p\<in>pts2" for p
- using \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
- thus ?thesis unfolding circ_def
- apply (intro sum.neutral)
- by auto
- qed
- moreover have "?R=sum circ pts1 + sum circ pts2"
- unfolding circ_def
- using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
- by blast
- ultimately show ?thesis
- apply (fold circ_def)
- by auto
-qed
-
-theorem Residue_theorem:
- fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
- and g::"real \<Rightarrow> complex"
- assumes "open s" "connected s" "finite pts" and
- holo:"f holomorphic_on s-pts" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- "path_image g \<subseteq> s-pts" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
- shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
-proof -
- define c where "c \<equiv> 2 * pi * \<i>"
- obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
- using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
- have "contour_integral g f
- = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
- using Cauchy_theorem_singularities[OF assms avoid] .
- also have "... = (\<Sum>p\<in>pts. c * winding_number g p * residue f p)"
- proof (intro sum.cong)
- show "pts = pts" by simp
- next
- fix x assume "x \<in> pts"
- show "winding_number g x * contour_integral (circlepath x (h x)) f
- = c * winding_number g x * residue f x"
- proof (cases "x\<in>s")
- case False
- then have "winding_number g x=0" using homo by auto
- thus ?thesis by auto
- next
- case True
- have "contour_integral (circlepath x (h x)) f = c* residue f x"
- using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
- apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
- by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
- then show ?thesis by auto
- qed
- qed
- also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
- by (simp add: sum_distrib_left algebra_simps)
- finally show ?thesis unfolding c_def .
-qed
-
-subsection \<open>Non-essential singular points\<close>
-
-definition\<^marker>\<open>tag important\<close> is_pole ::
- "('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where
- "is_pole f a = (LIM x (at a). f x :> at_infinity)"
-
-lemma is_pole_cong:
- assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
- shows "is_pole f a \<longleftrightarrow> is_pole g b"
- unfolding is_pole_def using assms by (intro filterlim_cong,auto)
-
-lemma is_pole_transform:
- assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
- shows "is_pole g b"
- using is_pole_cong assms by auto
-
-lemma is_pole_tendsto:
- fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)"
- shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)"
-unfolding is_pole_def
-by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
-
-lemma is_pole_inverse_holomorphic:
- assumes "open s"
- and f_holo:"f holomorphic_on (s-{z})"
- and pole:"is_pole f z"
- and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0"
- shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s"
-proof -
- define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
- have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
- apply (subst Lim_cong_at[where b=z and y=0 and g="inverse \<circ> f"])
- by (simp_all add:g_def)
- moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
- hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
- by (auto elim!:continuous_on_inverse simp add:non_z)
- hence "continuous_on (s-{z}) g" unfolding g_def
- apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
- by auto
- ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close>
- by (auto simp add:continuous_on_eq_continuous_at)
- moreover have "(inverse o f) holomorphic_on (s-{z})"
- unfolding comp_def using f_holo
- by (auto elim!:holomorphic_on_inverse simp add:non_z)
- hence "g holomorphic_on (s-{z})"
- apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
- by (auto simp add:g_def)
- ultimately show ?thesis unfolding g_def using \<open>open s\<close>
- by (auto elim!: no_isolated_singularity)
-qed
-
-lemma not_is_pole_holomorphic:
- assumes "open A" "x \<in> A" "f holomorphic_on A"
- shows "\<not>is_pole f x"
-proof -
- have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
- with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
- hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def)
- thus "\<not>is_pole f x" unfolding is_pole_def
- using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
-qed
-
-lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a"
- unfolding is_pole_def inverse_eq_divide [symmetric]
- by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
- (auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
-
-lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a"
- using is_pole_inverse_power[of 1 a] by simp
-
-lemma is_pole_divide:
- fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field"
- assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0"
- shows "is_pole (\<lambda>z. f z / g z) z"
-proof -
- have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)"
- by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
- filterlim_compose[OF filterlim_inverse_at_infinity])+
- (insert assms, auto simp: isCont_def)
- thus ?thesis by (simp add: field_split_simps is_pole_def)
-qed
-
-lemma is_pole_basic:
- assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0"
- shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z"
-proof (rule is_pole_divide)
- have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
- with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
- have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)"
- using assms by (auto intro!: tendsto_eq_intros)
- thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)"
- by (intro filterlim_atI tendsto_eq_intros)
- (insert assms, auto simp: eventually_at_filter)
-qed fact+
-
-lemma is_pole_basic':
- assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0"
- shows "is_pole (\<lambda>w. f w / w ^ n) 0"
- using is_pole_basic[of f A 0] assms by simp
-
-text \<open>The proposition
- \<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close>
-can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close>
-(i.e. the singularity is either removable or a pole).\<close>
-definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)"
-
-definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
- "isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})"
-
-named_theorems singularity_intros "introduction rules for singularities"
-
-lemma holomorphic_factor_unique:
- fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int
- assumes "r>0" "g z\<noteq>0" "h z\<noteq>0"
- and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0"
- and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
- shows "n=m"
-proof -
- have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close>
- by (auto simp add:at_within_ball_bot_iff)
- have False when "n>m"
- proof -
- have "(h \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"])
- have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w"
- using \<open>n>m\<close> asm \<open>r>0\<close>
- apply (auto simp add:field_simps powr_diff)
- by force
- then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto
- next
- define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)"
- have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto
- moreover have "continuous F f'" unfolding f'_def F_def continuous_def
- apply (subst Lim_ident_at)
- using \<open>n>m\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
- ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
- by (simp add: continuous_within)
- moreover have "(g \<longlongrightarrow> g z) F"
- using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] \<open>r>0\<close>
- unfolding F_def by auto
- ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
- qed
- moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)"
- using holomorphic_on_imp_continuous_on[OF h_holo]
- by (auto simp add:continuous_on_def \<open>r>0\<close>)
- ultimately have "h z=0" by (auto intro!: tendsto_unique)
- thus False using \<open>h z\<noteq>0\<close> by auto
- qed
- moreover have False when "m>n"
- proof -
- have "(g \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"])
- have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm
- apply (auto simp add:field_simps powr_diff)
- by force
- then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto
- next
- define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)"
- have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto
- moreover have "continuous F f'" unfolding f'_def F_def continuous_def
- apply (subst Lim_ident_at)
- using \<open>m>n\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
- ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
- by (simp add: continuous_within)
- moreover have "(h \<longlongrightarrow> h z) F"
- using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close>
- unfolding F_def by auto
- ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
- qed
- moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)"
- using holomorphic_on_imp_continuous_on[OF g_holo]
- by (auto simp add:continuous_on_def \<open>r>0\<close>)
- ultimately have "g z=0" by (auto intro!: tendsto_unique)
- thus False using \<open>g z\<noteq>0\<close> by auto
- qed
- ultimately show "n=m" by fastforce
-qed
-
-lemma holomorphic_factor_puncture:
- assumes f_iso:"isolated_singularity_at f z"
- and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close>
- and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close>
- shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)"
-proof -
- define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r"
- proof (rule ex_ex1I[OF that])
- fix n1 n2 :: int
- assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2"
- define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0"
- obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0"
- and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
- obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0"
- and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
- define r where "r \<equiv> min r1 r2"
- have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto
- moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0
- \<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0"
- using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def
- by fastforce
- ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close>
- apply (elim holomorphic_factor_unique)
- by (auto simp add:r_def)
- qed
-
- have P_exist:"\<exists> n g r. P h n g r" when
- "\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
- for h
- proof -
- from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
- unfolding isolated_singularity_at_def by auto
- obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto
- define h' where "h'=(\<lambda>x. if x=z then z' else h x)"
- have "h' holomorphic_on ball z r"
- apply (rule no_isolated_singularity'[of "{z}"])
- subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> empty_iff h'_def insert_iff)
- subgoal using \<open>h analytic_on ball z r - {z}\<close> analytic_imp_holomorphic h'_def holomorphic_transform
- by fastforce
- by auto
- have ?thesis when "z'=0"
- proof -
- have "h' z=0" using that unfolding h'_def by auto
- moreover have "\<not> h' constant_on ball z r"
- using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def
- apply simp
- by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that)
- moreover note \<open>h' holomorphic_on ball z r\<close>
- ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and
- g:"g holomorphic_on ball z r1"
- "\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w"
- "\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0"
- using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
- OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>]
- by (auto simp add:dist_commute)
- define rr where "rr=r1/2"
- have "P h' n g rr"
- unfolding P_def rr_def
- using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat)
- then have "P h n g rr"
- unfolding h'_def P_def by auto
- then show ?thesis unfolding P_def by blast
- qed
- moreover have ?thesis when "z'\<noteq>0"
- proof -
- have "h' z\<noteq>0" using that unfolding h'_def by auto
- obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0"
- proof -
- have "isCont h' z" "h' z\<noteq>0"
- by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def)
- then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0"
- using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
- define r1 where "r1=min r2 r / 2"
- have "0 < r1" "cball z r1 \<subseteq> ball z r"
- using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto
- moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0"
- using r2 unfolding r1_def by simp
- ultimately show ?thesis using that by auto
- qed
- then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto
- then have "P h 0 h' r1" unfolding P_def h'_def by auto
- then show ?thesis unfolding P_def by blast
- qed
- ultimately show ?thesis by auto
- qed
-
- have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)"
- apply (rule_tac imp_unique[unfolded P_def])
- using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
- moreover have ?thesis when "is_pole f z"
- proof (rule imp_unique[unfolded P_def])
- obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0"
- proof -
- have "\<forall>\<^sub>F z in at z. f z \<noteq> 0"
- using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
- by auto
- then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0"
- using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute)
- obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
- define e where "e=min e1 e2"
- show ?thesis
- apply (rule that[of e])
- using e1 e2 unfolding e_def by auto
- qed
-
- define h where "h \<equiv> \<lambda>x. inverse (f x)"
-
- have "\<exists>n g r. P h n g r"
- proof -
- have "h \<midarrow>z\<rightarrow> 0"
- using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
- moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
- using non_zero
- apply (elim frequently_rev_mp)
- unfolding h_def eventually_at by (auto intro:exI[where x=1])
- moreover have "isolated_singularity_at h z"
- unfolding isolated_singularity_at_def h_def
- apply (rule exI[where x=e])
- using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open
- holomorphic_on_inverse open_delete)
- ultimately show ?thesis
- using P_exist[of h] by auto
- qed
- then obtain n g r
- where "0 < r" and
- g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and
- g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- unfolding P_def by auto
- have "P f (-n) (inverse o g) r"
- proof -
- have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w
- using g_fac[rule_format,of w] that unfolding h_def
- apply (auto simp add:powr_minus )
- by (metis inverse_inverse_eq inverse_mult_distrib)
- then show ?thesis
- unfolding P_def comp_def
- using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros)
- qed
- then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
- unfolding P_def by blast
- qed
- ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger
-qed
-
-lemma not_essential_transform:
- assumes "not_essential g z"
- assumes "\<forall>\<^sub>F w in (at z). g w = f w"
- shows "not_essential f z"
- using assms unfolding not_essential_def
- by (simp add: filterlim_cong is_pole_cong)
-
-lemma isolated_singularity_at_transform:
- assumes "isolated_singularity_at g z"
- assumes "\<forall>\<^sub>F w in (at z). g w = f w"
- shows "isolated_singularity_at f z"
-proof -
- obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}"
- using assms(1) unfolding isolated_singularity_at_def by auto
- obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x"
- using assms(2) unfolding eventually_at by auto
- define r3 where "r3=min r1 r2"
- have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto
- moreover have "f analytic_on ball z r3 - {z}"
- proof -
- have "g holomorphic_on ball z r3 - {z}"
- using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto)
- then have "f holomorphic_on ball z r3 - {z}"
- using r2 unfolding r3_def
- by (auto simp add:dist_commute elim!:holomorphic_transform)
- then show ?thesis by (subst analytic_on_open,auto)
- qed
- ultimately show ?thesis unfolding isolated_singularity_at_def by auto
-qed
-
-lemma not_essential_powr[singularity_intros]:
- assumes "LIM w (at z). f w :> (at x)"
- shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z"
-proof -
- define fp where "fp=(\<lambda>w. (f w) powr (of_int n))"
- have ?thesis when "n>0"
- proof -
- have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n"
- using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def
- apply (elim Lim_transform_within[where d=1],simp)
- by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power)
- then show ?thesis unfolding not_essential_def fp_def by auto
- qed
- moreover have ?thesis when "n=0"
- proof -
- have "fp \<midarrow>z\<rightarrow> 1 "
- apply (subst tendsto_cong[where g="\<lambda>_.1"])
- using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto
- then show ?thesis unfolding fp_def not_essential_def by auto
- qed
- moreover have ?thesis when "n<0"
- proof (cases "x=0")
- case True
- have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
- apply (subst filterlim_inverse_at_iff[symmetric],simp)
- apply (rule filterlim_atI)
- subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- subgoal using filterlim_at_within_not_equal[OF assms,of 0]
- by (eventually_elim,insert that,auto)
- done
- then have "LIM w (at z). fp w :> at_infinity"
- proof (elim filterlim_mono_eventually)
- show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x"
- using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
- apply eventually_elim
- using powr_of_int that by auto
- qed auto
- then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
- next
- case False
- let ?xx= "inverse (x ^ (nat (-n)))"
- have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx"
- using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
- then have "fp \<midarrow>z\<rightarrow>?xx"
- apply (elim Lim_transform_within[where d=1],simp)
- unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less
- not_le power_eq_0_iff powr_0 powr_of_int that)
- then show ?thesis unfolding fp_def not_essential_def by auto
- qed
- ultimately show ?thesis by linarith
-qed
-
-lemma isolated_singularity_at_powr[singularity_intros]:
- assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
- shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z"
-proof -
- obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
- using assms(1) unfolding isolated_singularity_at_def by auto
- then have r1:"f holomorphic_on ball z r1 - {z}"
- using analytic_on_open[of "ball z r1-{z}" f] by blast
- obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
- using assms(2) unfolding eventually_at by auto
- define r3 where "r3=min r1 r2"
- have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
- apply (rule holomorphic_on_powr_of_int)
- subgoal unfolding r3_def using r1 by auto
- subgoal unfolding r3_def using r2 by (auto simp add:dist_commute)
- done
- moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith
- ultimately show ?thesis unfolding isolated_singularity_at_def
- apply (subst (asm) analytic_on_open[symmetric])
- by auto
-qed
-
-lemma non_zero_neighbour:
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
-proof -
- obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
- have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0"
- using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
- moreover have "(w - z) powr of_int fn \<noteq>0"
- unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto
- ultimately show ?thesis by auto
- qed
- then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto
-qed
-
-lemma non_zero_neighbour_pole:
- assumes "is_pole f z"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
- using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
- unfolding is_pole_def by auto
-
-lemma non_zero_neighbour_alt:
- assumes holo: "f holomorphic_on S"
- and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
- shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S"
-proof (cases "f z = 0")
- case True
- from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>]
- obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis
- then show ?thesis unfolding eventually_at
- apply (rule_tac x=r in exI)
- by (auto simp add:dist_commute)
-next
- case False
- obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0"
- using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
- holo holomorphic_on_imp_continuous_on by blast
- obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S"
- using assms(2) assms(4) openE by blast
- show ?thesis unfolding eventually_at
- apply (rule_tac x="min r1 r2" in exI)
- using r1 r2 by (auto simp add:dist_commute)
-qed
-
-lemma not_essential_times[singularity_intros]:
- assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- shows "not_essential (\<lambda>w. f w * g w) z"
-proof -
- define fg where "fg = (\<lambda>w. f w * g w)"
- have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))"
- proof -
- have "\<forall>\<^sub>Fw in (at z). fg w=0"
- using that[unfolded frequently_def, simplified] unfolding fg_def
- by (auto elim: eventually_rev_mp)
- from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0"
- proof -
- obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
- obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
- and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
- using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
-
- define r1 where "r1=(min fr gr)"
- have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
- when "w\<in>ball z r1 - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
- using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
- qed
-
- have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
- using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close>
- by (meson open_ball ball_subset_cball centre_in_ball
- continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
- holomorphic_on_subset)+
- have ?thesis when "fn+gn>0"
- proof -
- have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0"
- using that by (auto intro!:tendsto_eq_intros)
- then have "fg \<midarrow>z\<rightarrow> 0"
- apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
- by (metis (no_types, hide_lams) Diff_iff cball_trivial dist_commute dist_self
- eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int
- that)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when "fn+gn=0"
- proof -
- have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z"
- using that by (auto intro!:tendsto_eq_intros)
- then have "fg \<midarrow>z\<rightarrow> fp z*gp z"
- apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
- apply (subst fg_times)
- by (auto simp add:dist_commute that)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- moreover have ?thesis when "fn+gn<0"
- proof -
- have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
- apply (rule filterlim_divide_at_infinity)
- apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI)
- using eventually_at_topological by blast
- then have "is_pole fg z" unfolding is_pole_def
- apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp)
- apply (subst fg_times,simp add:dist_commute)
- apply (subst powr_of_int)
- using that by (auto simp add:field_split_simps)
- then show ?thesis unfolding not_essential_def fg_def by auto
- qed
- ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma not_essential_inverse[singularity_intros]:
- assumes f_ness:"not_essential f z"
- assumes f_iso:"isolated_singularity_at f z"
- shows "not_essential (\<lambda>w. inverse (f w)) z"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
- proof -
- have "\<forall>\<^sub>Fw in (at z). f w=0"
- using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
- then have "\<forall>\<^sub>Fw in (at z). vf w=0"
- unfolding vf_def by auto
- from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "is_pole f z"
- proof -
- have "vf \<midarrow>z\<rightarrow>0"
- using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- proof -
- from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto
- have ?thesis when "fz=0"
- proof -
- have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0"
- using fz that unfolding vf_def by auto
- moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0"
- using non_zero_neighbour[OF f_iso f_ness f_nconst]
- unfolding vf_def by auto
- ultimately have "is_pole vf z"
- using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- moreover have ?thesis when "fz\<noteq>0"
- proof -
- have "vf \<midarrow>z\<rightarrow>inverse fz"
- using fz that unfolding vf_def by (auto intro:tendsto_eq_intros)
- then show ?thesis unfolding not_essential_def vf_def by auto
- qed
- ultimately show ?thesis by auto
- qed
- ultimately show ?thesis using f_ness unfolding not_essential_def by auto
-qed
-
-lemma isolated_singularity_at_inverse[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
- proof -
- have "\<forall>\<^sub>Fw in (at z). f w=0"
- using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
- then have "\<forall>\<^sub>Fw in (at z). vf w=0"
- unfolding vf_def by auto
- then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0"
- unfolding eventually_at by auto
- then have "vf holomorphic_on ball z d1-{z}"
- apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"])
- by (auto simp add:dist_commute)
- then have "vf analytic_on ball z d1 - {z}"
- by (simp add: analytic_on_open open_delete)
- then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto
- qed
- moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- proof -
- have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
- then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0"
- unfolding eventually_at by auto
- obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
- using f_iso unfolding isolated_singularity_at_def by auto
- define d3 where "d3=min d1 d2"
- have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
- moreover have "vf analytic_on ball z d3 - {z}"
- unfolding vf_def
- apply (rule analytic_on_inverse)
- subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto
- subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute)
- done
- ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma not_essential_divide[singularity_intros]:
- assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- shows "not_essential (\<lambda>w. f w / g w) z"
-proof -
- have "not_essential (\<lambda>w. f w * inverse (g w)) z"
- apply (rule not_essential_times[where g="\<lambda>w. inverse (g w)"])
- using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse)
- then show ?thesis by (simp add:field_simps)
-qed
-
-lemma
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- shows isolated_singularity_at_times[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. f w * g w) z" and
- isolated_singularity_at_add[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. f w + g w) z"
-proof -
- obtain d1 d2 where "d1>0" "d2>0"
- and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
- using f_iso g_iso unfolding isolated_singularity_at_def by auto
- define d3 where "d3=min d1 d2"
- have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
-
- have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}"
- apply (rule analytic_on_mult)
- using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
- then show "isolated_singularity_at (\<lambda>w. f w * g w) z"
- using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
- have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}"
- apply (rule analytic_on_add)
- using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
- then show "isolated_singularity_at (\<lambda>w. f w + g w) z"
- using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
-qed
-
-lemma isolated_singularity_at_uminus[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- shows "isolated_singularity_at (\<lambda>w. - f w) z"
- using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
-
-lemma isolated_singularity_at_id[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. w) z"
- unfolding isolated_singularity_at_def by (simp add: gt_ex)
-
-lemma isolated_singularity_at_minus[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- shows "isolated_singularity_at (\<lambda>w. f w - g w) z"
- using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"]
- ,OF g_iso] by simp
-
-lemma isolated_singularity_at_divide[singularity_intros]:
- assumes f_iso:"isolated_singularity_at f z"
- and g_iso:"isolated_singularity_at g z"
- and g_ness:"not_essential g z"
- shows "isolated_singularity_at (\<lambda>w. f w / g w) z"
- using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso,
- of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps)
-
-lemma isolated_singularity_at_const[singularity_intros]:
- "isolated_singularity_at (\<lambda>w. c) z"
- unfolding isolated_singularity_at_def by (simp add: gt_ex)
-
-lemma isolated_singularity_at_holomorphic:
- assumes "f holomorphic_on s-{z}" "open s" "z\<in>s"
- shows "isolated_singularity_at f z"
- using assms unfolding isolated_singularity_at_def
- by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
-
-subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
-
-
-definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
- "zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n)
- \<and> h w \<noteq>0)))"
-
-definition\<^marker>\<open>tag important\<close> zor_poly
- ::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where
- "zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
- \<and> h w \<noteq>0))"
-
-lemma zorder_exist:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))"
-proof -
- define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have "\<exists>!n. \<exists>g r. P n g r"
- using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
- then have "\<exists>g r. P n g r"
- unfolding n_def P_def zorder_def
- by (drule_tac theI',argo)
- then have "\<exists>r. P n g r"
- unfolding P_def zor_poly_def g_def n_def
- by (drule_tac someI_ex,argo)
- then obtain r1 where "P n g r1" by auto
- then show ?thesis unfolding P_def by auto
-qed
-
-lemma
- fixes f::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z"
- and f_ness:"not_essential f z"
- and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
- shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z"
- and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w
- = inverse (zor_poly f z w)"
-proof -
- define vf where "vf = (\<lambda>w. inverse (f w))"
- define fn vfn where
- "fn = zorder f z" and "vfn = zorder vf z"
- define fp vfp where
- "fp = zor_poly f z" and "vfp = zor_poly vf z"
-
- obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
- by auto
- have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
- and fr_nz: "inverse (fp w)\<noteq>0"
- when "w\<in>ball z fr - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that by auto
- then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0"
- unfolding vf_def by (auto simp add:powr_minus)
- qed
- obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
- "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)"
- proof -
- have "isolated_singularity_at vf z"
- using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
- moreover have "not_essential vf z"
- using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
- moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0"
- using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
- ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
- qed
-
-
- define r1 where "r1 = min fr vfr"
- have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp
- show "vfn = - fn"
- apply (rule holomorphic_factor_unique[of r1 vfp z "\<lambda>w. inverse (fp w)" vf])
- subgoal using \<open>r1>0\<close> by simp
- subgoal by simp
- subgoal by simp
- subgoal
- proof (rule ballI)
- fix w assume "w \<in> ball z r1 - {z}"
- then have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" unfolding r1_def by auto
- from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)]
- show "vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0
- \<and> vf w = inverse (fp w) * (w - z) powr of_int (- fn) \<and> inverse (fp w) \<noteq> 0" by auto
- qed
- subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros)
- subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros)
- done
-
- have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w
- proof -
- have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto
- from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close>
- show ?thesis by auto
- qed
- then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)"
- unfolding eventually_at using \<open>r1>0\<close>
- apply (rule_tac x=r1 in exI)
- by (auto simp add:dist_commute)
-qed
-
-lemma
- fixes f g::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- and f_ness:"not_essential f z" and g_ness:"not_essential g z"
- and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
- shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and
- zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w
- = zor_poly f z w *zor_poly g z w"
-proof -
- define fg where "fg = (\<lambda>w. f w * g w)"
- define fn gn fgn where
- "fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
- define fp gp fgp where
- "fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
- have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
- using fg_nconst by (auto elim!:frequently_elim1)
- obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
- and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
- using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
- obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
- and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
- using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
- define r1 where "r1=min fr gr"
- have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
- when "w\<in>ball z r1 - {z}" for w
- proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
- using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
- using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
- qed
-
- obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0"
- and fgr: "fgp holomorphic_on cball z fgr"
- "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0"
- proof -
- have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))"
- apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
- subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
- subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
- subgoal unfolding fg_def using fg_nconst .
- done
- then show ?thesis using that by blast
- qed
- define r2 where "r2 = min fgr r1"
- have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp
- show "fgn = fn + gn "
- apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg])
- subgoal using \<open>r2>0\<close> by simp
- subgoal by simp
- subgoal by simp
- subgoal
- proof (rule ballI)
- fix w assume "w \<in> ball z r2 - {z}"
- then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto
- from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
- show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0
- \<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
- qed
- subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
- subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
- done
-
- have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w
- proof -
- have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto
- from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close>
- show ?thesis by auto
- qed
- then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w"
- using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute)
-qed
-
-lemma
- fixes f g::"complex \<Rightarrow> complex" and z::complex
- assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
- and f_ness:"not_essential f z" and g_ness:"not_essential g z"
- and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
- shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and
- zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w
- = zor_poly f z w / zor_poly g z w"
-proof -
- have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
- using fg_nconst by (auto elim!:frequently_elim1)
- define vg where "vg=(\<lambda>w. inverse (g w))"
- have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z"
- apply (rule zorder_times[OF f_iso _ f_ness,of vg])
- subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
- subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
- subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
- done
- then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
- using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
- by (auto simp add:field_simps)
-
- have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
- apply (rule zor_poly_times[OF f_iso _ f_ness,of vg])
- subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
- subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
- subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
- done
- then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w"
- using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
- apply eventually_elim
- by (auto simp add:field_simps)
-qed
-
-lemma zorder_exist_zero:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes holo: "f holomorphic_on s" and
- "open s" "connected s" "z\<in>s"
- and non_const: "\<exists>w\<in>s. f w \<noteq> 0"
- shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))"
-proof -
- obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- proof -
- have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
- proof (rule zorder_exist[of f z,folded g_def n_def])
- show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using holo assms(4,6)
- by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
- show "not_essential f z" unfolding not_essential_def
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
- by fastforce
- have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s"
- proof -
- obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto
- then show ?thesis
- by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>])
- qed
- then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- apply (elim eventually_frequentlyE)
- by auto
- qed
- then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- by auto
- obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
- using assms(4,6) open_contains_cball_eq by blast
- define r3 where "r3=min r1 r2"
- have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
- moreover have "g holomorphic_on cball z r3"
- using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- using r1(2) unfolding r3_def by auto
- ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
- qed
-
- have if_0:"if f z=0 then n > 0 else n=0"
- proof -
- have "f\<midarrow> z \<rightarrow> f z"
- by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
- then have "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z"
- apply (elim Lim_transform_within_open[where s="ball z r"])
- using r by auto
- moreover have "g \<midarrow>z\<rightarrow>g z"
- by (metis (mono_tags, lifting) open_ball at_within_open_subset
- ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
- ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
- apply (rule_tac tendsto_divide)
- using \<open>g z\<noteq>0\<close> by auto
- then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z"
- apply (elim Lim_transform_within_open[where s="ball z r"])
- using r by auto
-
- have ?thesis when "n\<ge>0" "f z=0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto
- apply (elim Lim_transform_within[where d=r])
- by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
- then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp
- moreover have False when "n=0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1"
- using \<open>n=0\<close> by auto
- then show False using * using LIM_unique zero_neq_one by blast
- qed
- ultimately show ?thesis using that by fastforce
- qed
- moreover have ?thesis when "n\<ge>0" "f z\<noteq>0"
- proof -
- have False when "n>0"
- proof -
- have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto
- apply (elim Lim_transform_within[where d=r])
- by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
- moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0"
- using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros)
- ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast
- qed
- then show ?thesis using that by force
- qed
- moreover have False when "n<0"
- proof -
- have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z"
- "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
- subgoal using powr_tendsto powr_of_int that
- by (elim Lim_transform_within_open[where s=UNIV],auto)
- subgoal using that by (auto intro!:tendsto_eq_intros)
- done
- from tendsto_mult[OF this,simplified]
- have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" .
- then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0"
- by (elim Lim_transform_within_open[where s=UNIV],auto)
- then show False using LIM_const_eq by fastforce
- qed
- ultimately show ?thesis by fastforce
- qed
- moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w
- proof (cases "w=z")
- case True
- then have "f \<midarrow>z\<rightarrow>f w"
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce
- then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w"
- proof (elim Lim_transform_within[OF _ \<open>r>0\<close>])
- fix x assume "0 < dist x z" "dist x z < r"
- then have "x \<in> cball z r - {z}" "x\<noteq>z"
- unfolding cball_def by (auto simp add: dist_commute)
- then have "f x = g x * (x - z) powr of_int n"
- using r(4)[rule_format,of x] by simp
- also have "... = g x * (x - z) ^ nat n"
- apply (subst powr_of_int)
- using if_0 \<open>x\<noteq>z\<close> by (auto split:if_splits)
- finally show "f x = g x * (x - z) ^ nat n" .
- qed
- moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n"
- using True apply (auto intro!:tendsto_eq_intros)
- by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball
- continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that)
- ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
- then show ?thesis using \<open>g z\<noteq>0\<close> True by auto
- next
- case False
- then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
- using r(4) that by auto
- then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
- qed
- ultimately show ?thesis using r by auto
-qed
-
-lemma zorder_exist_pole:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
- assumes holo: "f holomorphic_on s-{z}" and
- "open s" "z\<in>s"
- and "is_pole f z"
- shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))"
-proof -
- obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- proof -
- have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
- proof (rule zorder_exist[of f z,folded g_def n_def])
- show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using holo assms(4,5)
- by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
- show "not_essential f z" unfolding not_essential_def
- using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
- by fastforce
- from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- apply (elim eventually_frequentlyE)
- by auto
- qed
- then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- by auto
- obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
- using assms(4,5) open_contains_cball_eq by metis
- define r3 where "r3=min r1 r2"
- have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
- moreover have "g holomorphic_on cball z r3"
- using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
- using r1(2) unfolding r3_def by auto
- ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
- qed
-
- have "n<0"
- proof (rule ccontr)
- assume " \<not> n < 0"
- define c where "c=(if n=0 then g z else 0)"
- have [simp]:"g \<midarrow>z\<rightarrow> g z"
- by (metis open_ball at_within_open ball_subset_cball centre_in_ball
- continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
- have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
- unfolding eventually_at_topological
- apply (rule_tac exI[where x="ball z r"])
- using r powr_of_int \<open>\<not> n < 0\<close> by auto
- moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow>c"
- proof (cases "n=0")
- case True
- then show ?thesis unfolding c_def by simp
- next
- case False
- then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close>
- by (auto intro!:tendsto_eq_intros)
- from tendsto_mult[OF _ this,of g "g z",simplified]
- show ?thesis unfolding c_def using False by simp
- qed
- ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast
- then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity
- unfolding is_pole_def by blast
- qed
- moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0"
- using r(4) \<open>n<0\<close> powr_of_int
- by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
- ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto
-qed
-
-lemma zorder_eqI:
- assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
- assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n"
- shows "zorder f z = n"
-proof -
- have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact
- moreover have "open (-{0::complex})" by auto
- ultimately have "open ((g -` (-{0})) \<inter> s)"
- unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast
- moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto
- ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))"
- unfolding open_contains_cball by blast
-
- let ?gg= "(\<lambda>w. g w * (w - z) powr n)"
- define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
- have "P n g r"
- unfolding P_def using r assms(3,4,5) by auto
- then have "\<exists>g r. P n g r" by auto
- moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def
- proof (rule holomorphic_factor_puncture)
- have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast
- then have "?gg holomorphic_on ball z r-{z}"
- using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros)
- then have "f holomorphic_on ball z r - {z}"
- apply (elim holomorphic_transform)
- using fg_eq \<open>ball z r-{z} \<subseteq> s\<close> by auto
- then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
- using analytic_on_open open_delete r(1) by blast
- next
- have "not_essential ?gg z"
- proof (intro singularity_intros)
- show "not_essential g z"
- by (meson \<open>continuous_on s g\<close> assms(1) assms(2) continuous_on_eq_continuous_at
- isCont_def not_essential_def)
- show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter)
- then show "LIM w at z. w - z :> at 0"
- unfolding filterlim_at by (auto intro:tendsto_eq_intros)
- show "isolated_singularity_at g z"
- by (meson Diff_subset open_ball analytic_on_holomorphic
- assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
- qed
- then show "not_essential f z"
- apply (elim not_essential_transform)
- unfolding eventually_at using assms(1,2) assms(5)[symmetric]
- by (metis dist_commute mem_ball openE subsetCE)
- show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at
- proof (rule,rule)
- fix d::real assume "0 < d"
- define z' where "z'=z+min d r / 2"
- have "z' \<noteq> z" " dist z' z < d "
- unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close>
- by (auto simp add:dist_norm)
- moreover have "f z' \<noteq> 0"
- proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>])
- have "z' \<in> cball z r" unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm)
- then show " z' \<in> s" using r(2) by blast
- show "g z' * (z' - z) powr of_int n \<noteq> 0"
- using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> calculation(1) by auto
- qed
- ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto
- qed
- qed
- ultimately have "(THE n. \<exists>g r. P n g r) = n"
- by (rule_tac the1_equality)
- then show ?thesis unfolding zorder_def P_def by blast
-qed
-
-lemma residue_pole_order:
- fixes f::"complex \<Rightarrow> complex" and z::complex
- defines "n \<equiv> nat (- zorder f z)" and "h \<equiv> zor_poly f z"
- assumes f_iso:"isolated_singularity_at f z"
- and pole:"is_pole f z"
- shows "residue f z = ((deriv ^^ (n - 1)) h z / fact (n-1))"
-proof -
- define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
- obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
- obtain r where "0 < n" "0 < r" and r_cball:"cball z r \<subseteq> ball z e" and h_holo: "h holomorphic_on cball z r"
- and h_divide:"(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
- proof -
- obtain r where r:"zorder f z < 0" "h z \<noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
- using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>,folded n_def h_def] by auto
- have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
- moreover have "(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
- using \<open>h z\<noteq>0\<close> r(6) by blast
- ultimately show ?thesis using r(3,4,5) that by blast
- qed
- have r_nonzero:"\<And>w. w \<in> ball z r - {z} \<Longrightarrow> f w \<noteq> 0"
- using h_divide by simp
- define c where "c \<equiv> 2 * pi * \<i>"
- define der_f where "der_f \<equiv> ((deriv ^^ (n - 1)) h z / fact (n-1))"
- define h' where "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
- have "(h' has_contour_integral c / fact (n - 1) * (deriv ^^ (n - 1)) h z) (circlepath z r)"
- unfolding h'_def
- proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath[of z r h z "n-1",
- folded c_def Suc_pred'[OF \<open>n>0\<close>]])
- show "continuous_on (cball z r) h" using holomorphic_on_imp_continuous_on h_holo by simp
- show "h holomorphic_on ball z r" using h_holo by auto
- show " z \<in> ball z r" using \<open>r>0\<close> by auto
- qed
- then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def by auto
- then have "(f has_contour_integral c * der_f) (circlepath z r)"
- proof (elim has_contour_integral_eq)
- fix x assume "x \<in> path_image (circlepath z r)"
- hence "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
- then show "h' x = f x" using h_divide unfolding h'_def by auto
- qed
- moreover have "(f has_contour_integral c * residue f z) (circlepath z r)"
- using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
- unfolding c_def by simp
- ultimately have "c * der_f = c * residue f z" using has_contour_integral_unique by blast
- hence "der_f = residue f z" unfolding c_def by auto
- thus ?thesis unfolding der_f_def by auto
-qed
-
-lemma simple_zeroI:
- assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
- assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)"
- shows "zorder f z = 1"
- using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto)
-
-lemma higher_deriv_power:
- shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w =
- pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
-proof (induction j arbitrary: w)
- case 0
- thus ?case by auto
-next
- case (Suc j w)
- have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w"
- by simp
- also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) =
- (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
- using Suc by (intro Suc.IH ext)
- also {
- have "(\<dots> has_field_derivative of_nat (n - j) *
- pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
- using Suc.prems by (auto intro!: derivative_eq_intros)
- also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
- pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
- by (cases "Suc j \<le> n", subst pochhammer_rec)
- (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
- finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
- \<dots> * (w - z) ^ (n - Suc j)"
- by (rule DERIV_imp_deriv)
- }
- finally show ?case .
-qed
-
-lemma zorder_zero_eqI:
- assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s"
- assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0"
- assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0"
- shows "zorder f z = n"
-proof -
- obtain r where [simp]:"r>0" and "ball z r \<subseteq> s"
- using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast
- have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0"
- proof (rule ccontr)
- assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)"
- then have "eventually (\<lambda>u. f u = 0) (nhds z)"
- using \<open>r>0\<close> unfolding eventually_nhds
- apply (rule_tac x="ball z r" in exI)
- by auto
- then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z"
- by (intro higher_deriv_cong_ev) auto
- also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0"
- by (induction n) simp_all
- finally show False using nz by contradiction
- qed
-
- define zn g where "zn = zorder f z" and "g = zor_poly f z"
- obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and
- [simp]:"e>0" and "cball z e \<subseteq> ball z r" and
- g_holo:"g holomorphic_on cball z e" and
- e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)"
- proof -
- have "f holomorphic_on ball z r"
- using f_holo \<open>ball z r \<subseteq> s\<close> by auto
- from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
- show ?thesis by blast
- qed
- from this(1,2,5) have "zn\<ge>0" "g z\<noteq>0"
- subgoal by (auto split:if_splits)
- subgoal using \<open>0 < e\<close> ball_subset_cball centre_in_ball e_fac by blast
- done
-
- define A where "A = (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
- have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i
- proof -
- have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)"
- using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto
- hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
- apply eventually_elim
- by (use e_fac in auto)
- hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z"
- by (intro higher_deriv_cong_ev) auto
- also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) *
- (deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
- using g_holo \<open>e>0\<close>
- by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
- also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then
- of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
- proof (intro sum.cong refl, goal_cases)
- case (1 j)
- have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z =
- pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
- by (subst higher_deriv_power) auto
- also have "\<dots> = (if j = nat zn then fact j else 0)"
- by (auto simp: not_less pochhammer_0_left pochhammer_fact)
- also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z =
- (if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
- * (deriv ^^ (i - nat zn)) g z else 0)"
- by simp
- finally show ?case .
- qed
- also have "\<dots> = (if i \<ge> zn then A i else 0)"
- by (auto simp: A_def)
- finally show "(deriv ^^ i) f z = \<dots>" .
- qed
-
- have False when "n<zn"
- proof -
- have "(deriv ^^ nat n) f z = 0"
- using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto
- with nz show False by auto
- qed
- moreover have "n\<le>zn"
- proof -
- have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp
- then have "(deriv ^^ nat zn) f z \<noteq> 0"
- using deriv_A[of "nat zn"] by(auto simp add:A_def)
- then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith
- moreover have "zn\<ge>0" using e_if by (auto split:if_splits)
- ultimately show ?thesis using nat_le_eq_zle by blast
- qed
- ultimately show ?thesis unfolding zn_def by fastforce
-qed
-
-lemma
- assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'"
- shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
-proof -
- define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))"
- have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h
- proof -
- have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g
- proof -
- from that(1) obtain r1 where r1_P:"P f n h r1" by auto
- from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x"
- unfolding eventually_at_le by auto
- define r where "r=min r1 r2"
- have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto
- moreover have "h holomorphic_on cball z r"
- using r1_P unfolding P_def r_def by auto
- moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
- proof -
- have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0"
- using r1_P that unfolding P_def r_def by auto
- moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
- by (simp add: dist_commute)
- ultimately show ?thesis by simp
- qed
- ultimately show ?thesis unfolding P_def by auto
- qed
- from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
- by (simp add: eq_commute)
- show ?thesis
- by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
- qed
- then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
- using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto
-qed
-
-lemma zorder_nonzero_div_power:
- assumes "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" "n > 0"
- shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
- apply (rule zorder_eqI[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>f holomorphic_on s\<close> \<open>f z\<noteq>0\<close>])
- apply (subst powr_of_int)
- using \<open>n>0\<close> by (auto simp add:field_simps)
-
-lemma zor_poly_eq:
- assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
-proof -
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
- using zorder_exist[OF assms] by blast
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
- by (auto simp: field_simps powr_minus)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_zero_eq:
- assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0"
- shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
-proof -
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
- using zorder_exist_zero[OF assms] by auto
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
- by (auto simp: field_simps powr_minus)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_pole_eq:
- assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
- shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
-proof -
- obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
- using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
- obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
- using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto
- then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
- by (auto simp: field_simps)
- have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
- using r eventually_at_ball'[of r z UNIV] by auto
- thus ?thesis by eventually_elim (insert *, auto)
-qed
-
-lemma zor_poly_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
- assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- from zorder_exist[OF assms(2-4)] obtain r where
- r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
- "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n"
- unfolding n_def by blast
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
- by eventually_elim (insert r, auto simp: field_simps powr_minus)
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma zor_poly_zero_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0"
- assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- from zorder_exist_zero[OF assms(2-6)] obtain r where
- r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r"
- "\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n"
- unfolding n_def by blast
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
- by eventually_elim (insert r, auto simp: field_simps)
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma zor_poly_pole_eqI:
- fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
- defines "n \<equiv> zorder f z0"
- assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
- assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F"
- assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
- shows "zor_poly f z0 z0 = c"
-proof -
- obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
- proof -
- have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
- using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE)
- moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp
- ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
- qed
- from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
- using eventually_at_ball'[of r z0 UNIV] by auto
- have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
- using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def .
- moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
- using r by (intro holomorphic_on_imp_continuous_on) auto
- with r(1,2) have "isCont (zor_poly f z0) z0"
- by (auto simp: continuous_on_eq_continuous_at)
- hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- unfolding isCont_def .
- ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
- by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F"
- by (rule filterlim_compose[OF _ g])
- from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
-qed
-
-lemma residue_simple_pole:
- assumes "isolated_singularity_at f z0"
- assumes "is_pole f z0" "zorder f z0 = - 1"
- shows "residue f z0 = zor_poly f z0 z0"
- using assms by (subst residue_pole_order) simp_all
-
-lemma residue_simple_pole_limit:
- assumes "isolated_singularity_at f z0"
- assumes "is_pole f z0" "zorder f z0 = - 1"
- assumes "((\<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
- assumes "filterlim g (at z0) F" "F \<noteq> bot"
- shows "residue f z0 = c"
-proof -
- have "residue f z0 = zor_poly f z0 z0"
- by (rule residue_simple_pole assms)+
- also have "\<dots> = c"
- apply (rule zor_poly_pole_eqI)
- using assms by auto
- finally show ?thesis .
-qed
-
-lemma lhopital_complex_simple:
- assumes "(f has_field_derivative f') (at z)"
- assumes "(g has_field_derivative g') (at z)"
- assumes "f z = 0" "g z = 0" "g' \<noteq> 0" "f' / g' = c"
- shows "((\<lambda>w. f w / g w) \<longlongrightarrow> c) (at z)"
-proof -
- have "eventually (\<lambda>w. w \<noteq> z) (at z)"
- by (auto simp: eventually_at_filter)
- hence "eventually (\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
- by eventually_elim (simp add: assms field_split_simps)
- moreover have "((\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) \<longlongrightarrow> f' / g') (at z)"
- by (intro tendsto_divide has_field_derivativeD assms)
- ultimately have "((\<lambda>w. f w / g w) \<longlongrightarrow> f' / g') (at z)"
- by (blast intro: Lim_transform_eventually)
- with assms show ?thesis by simp
-qed
-
-lemma
- assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
- and "open s" "connected s" "z \<in> s"
- assumes g_deriv:"(g has_field_derivative g') (at z)"
- assumes "f z \<noteq> 0" "g z = 0" "g' \<noteq> 0"
- shows porder_simple_pole_deriv: "zorder (\<lambda>w. f w / g w) z = - 1"
- and residue_simple_pole_deriv: "residue (\<lambda>w. f w / g w) z = f z / g'"
-proof -
- have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
- using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
- by (meson Diff_subset holomorphic_on_subset)+
- have [simp]:"not_essential f z" "not_essential g z"
- unfolding not_essential_def using f_holo g_holo assms(3,5)
- by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
- have g_nconst:"\<exists>\<^sub>F w in at z. g w \<noteq>0 "
- proof (rule ccontr)
- assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
- then have "\<forall>\<^sub>F w in nhds z. g w = 0"
- unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
- by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
- then have "deriv g z = deriv (\<lambda>_. 0) z"
- by (intro deriv_cong_ev) auto
- then have "deriv g z = 0" by auto
- then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
- then show False using \<open>g'\<noteq>0\<close> by auto
- qed
-
- have "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
- proof -
- have "\<forall>\<^sub>F w in at z. f w \<noteq>0 \<and> w\<in>s"
- apply (rule non_zero_neighbour_alt)
- using assms by auto
- with g_nconst have "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
- by (elim frequently_rev_mp eventually_rev_mp,auto)
- then show ?thesis using zorder_divide[of f z g] by auto
- qed
- moreover have "zorder f z=0"
- apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
- using \<open>f z\<noteq>0\<close> by auto
- moreover have "zorder g z=1"
- apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
- subgoal using assms(8) by auto
- subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
- subgoal by simp
- done
- ultimately show "zorder (\<lambda>w. f w / g w) z = - 1" by auto
-
- show "residue (\<lambda>w. f w / g w) z = f z / g'"
- proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
- show "zorder (\<lambda>w. f w / g w) z = - 1" by fact
- show "isolated_singularity_at (\<lambda>w. f w / g w) z"
- by (auto intro: singularity_intros)
- show "is_pole (\<lambda>w. f w / g w) z"
- proof (rule is_pole_divide)
- have "\<forall>\<^sub>F x in at z. g x \<noteq> 0"
- apply (rule non_zero_neighbour)
- using g_nconst by auto
- moreover have "g \<midarrow>z\<rightarrow> 0"
- using DERIV_isCont assms(8) continuous_at g_deriv by force
- ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
- show "isCont f z"
- using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
- by auto
- show "f z \<noteq> 0" by fact
- qed
- show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
- have "((\<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
- proof (rule lhopital_complex_simple)
- show "((\<lambda>w. f w * (w - z)) has_field_derivative f z) (at z)"
- using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
- show "(g has_field_derivative g') (at z)" by fact
- qed (insert assms, auto)
- then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
- by (simp add: field_split_simps)
- qed
-qed
-
-subsection \<open>The argument principle\<close>
-
-theorem argument_principle:
- fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
- defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
- assumes "open s" and
- "connected s" and
- f_holo:"f holomorphic_on s-poles" and
- h_holo:"h holomorphic_on s" and
- "valid_path g" and
- loop:"pathfinish g = pathstart g" and
- path_img:"path_image g \<subseteq> s - pz" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
- finite:"finite pz" and
- poles:"\<forall>p\<in>poles. is_pole f p"
- shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
- (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
- (is "?L=?R")
-proof -
- define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
- define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
- define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
- define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
-
- have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
- proof -
- obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
- using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
- have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
- proof -
- define po where "po \<equiv> zorder f p"
- define pp where "pp \<equiv> zor_poly f p"
- define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
- define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
- obtain r where "pp p\<noteq>0" "r>0" and
- "r<e1" and
- pp_holo:"pp holomorphic_on cball p r" and
- pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
- proof -
- have "isolated_singularity_at f p"
- proof -
- have "f holomorphic_on ball p e1 - {p}"
- apply (intro holomorphic_on_subset[OF f_holo])
- using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
- then show ?thesis unfolding isolated_singularity_at_def
- using \<open>e1>0\<close> analytic_on_open open_delete by blast
- qed
- moreover have "not_essential f p"
- proof (cases "is_pole f p")
- case True
- then show ?thesis unfolding not_essential_def by auto
- next
- case False
- then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
- moreover have "open (s-poles)"
- using \<open>open s\<close>
- apply (elim open_Diff)
- apply (rule finite_imp_closed)
- using finite unfolding pz_def by simp
- ultimately have "isCont f p"
- using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
- by auto
- then show ?thesis unfolding isCont_def not_essential_def by auto
- qed
- moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
- proof (rule ccontr)
- assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
- then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
- then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
- unfolding eventually_at by (auto simp add:dist_commute)
- then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
- moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
- ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
- then have "infinite pz"
- unfolding pz_def infinite_super by auto
- then show False using \<open>finite pz\<close> by auto
- qed
- ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
- "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
- using zorder_exist[of f p,folded po_def pp_def] by auto
- define r1 where "r1=min r e1 / 2"
- have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
- moreover have "r1>0" "pp holomorphic_on cball p r1"
- "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
- unfolding r1_def using \<open>e1>0\<close> r by auto
- ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
- qed
-
- define e2 where "e2 \<equiv> r/2"
- have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
- define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
- define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
- have "((\<lambda>w. prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
- proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
- have "ball p r \<subseteq> s"
- using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
- then have "cball p e2 \<subseteq> s"
- using \<open>r>0\<close> unfolding e2_def by auto
- then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
- using h_holo by (auto intro!: holomorphic_intros)
- then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
- using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
- unfolding prin_def by (auto simp add: mult.assoc)
- have "anal holomorphic_on ball p r" unfolding anal_def
- using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
- by (auto intro!: holomorphic_intros)
- then show "(anal has_contour_integral 0) (circlepath p e2)"
- using e2_def \<open>r>0\<close>
- by (auto elim!: Cauchy_theorem_disc_simple)
- qed
- then have "cont ff' p e2" unfolding cont_def po_def
- proof (elim has_contour_integral_eq)
- fix w assume "w \<in> path_image (circlepath p e2)"
- then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
- define wp where "wp \<equiv> w-p"
- have "wp\<noteq>0" and "pp w \<noteq>0"
- unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
- moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
- proof (rule DERIV_imp_deriv)
- have "(pp has_field_derivative (deriv pp w)) (at w)"
- using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
- by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
- then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
- + deriv pp w * (w - p) powr of_int po) (at w)"
- unfolding f'_def using \<open>w\<noteq>p\<close>
- by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
- qed
- ultimately show "prin w + anal w = ff' w"
- unfolding ff'_def prin_def anal_def
- apply simp
- apply (unfold f'_def)
- apply (fold wp_def)
- apply (auto simp add:field_simps)
- by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
- qed
- then have "cont ff p e2" unfolding cont_def
- proof (elim has_contour_integral_eq)
- fix w assume "w \<in> path_image (circlepath p e2)"
- then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
- have "deriv f' w = deriv f w"
- proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
- show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
- by (auto intro!: holomorphic_intros)
- next
- have "ball p e1 - {p} \<subseteq> s - poles"
- using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
- by auto
- then have "ball p r - {p} \<subseteq> s - poles"
- apply (elim dual_order.trans)
- using \<open>r<e1\<close> by auto
- then show "f holomorphic_on ball p r - {p}" using f_holo
- by auto
- next
- show "open (ball p r - {p})" by auto
- show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
- next
- fix x assume "x \<in> ball p r - {p}"
- then show "f' x = f x"
- using pp_po unfolding f'_def by auto
- qed
- moreover have " f' w = f w "
- using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
- unfolding f'_def by auto
- ultimately show "ff' w = ff w"
- unfolding ff'_def ff_def by simp
- qed
- moreover have "cball p e2 \<subseteq> ball p e1"
- using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
- ultimately show ?thesis using \<open>e2>0\<close> by auto
- qed
- then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
- by auto
- define e4 where "e4 \<equiv> if p\<in>pz then e2 else e1"
- have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
- moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
- moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
- by (auto simp add: e2 e4_def)
- ultimately show ?thesis by auto
- qed
- then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
- \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
- by metis
- define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
- define w where "w \<equiv> \<lambda>p. winding_number g p"
- have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
- proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
- path_img homo])
- have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
- then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
- by (auto intro!: holomorphic_intros simp add:pz_def)
- next
- show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
- using get_e using avoid_def by blast
- qed
- also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
- proof (rule sum.cong[of pz pz,simplified])
- fix p assume "p \<in> pz"
- show "w p * ci p = c * w p * h p * (zorder f p)"
- proof (cases "p\<in>s")
- assume "p \<in> s"
- have "ci p = c * h p * (zorder f p)" unfolding ci_def
- apply (rule contour_integral_unique)
- using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
- thus ?thesis by auto
- next
- assume "p\<notin>s"
- then have "w p=0" using homo unfolding w_def by auto
- then show ?thesis by auto
- qed
- qed
- also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
- unfolding sum_distrib_left by (simp add:algebra_simps)
- finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
- then show ?thesis unfolding ff_def c_def w_def by simp
-qed
-
-subsection \<open>Rouche's theorem \<close>
-
-theorem Rouche_theorem:
- fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
- defines "fg\<equiv>(\<lambda>p. f p + g p)"
- defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
- assumes
- "open s" and "connected s" and
- "finite zeros_fg" and
- "finite zeros_f" and
- f_holo:"f holomorphic_on s" and
- g_holo:"g holomorphic_on s" and
- "valid_path \<gamma>" and
- loop:"pathfinish \<gamma> = pathstart \<gamma>" and
- path_img:"path_image \<gamma> \<subseteq> s " and
- path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
- homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
- shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
- = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
-proof -
- have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
- proof -
- have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
- proof -
- have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
- moreover have "f z = - g z" using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
- then have "cmod (f z) = cmod (g z)" by auto
- ultimately show False by auto
- qed
- then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
- qed
- have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
- proof -
- have False when "z\<in>path_image \<gamma>" and "f z =0" for z
- proof -
- have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
- then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
- then show False by auto
- qed
- then show ?thesis unfolding zeros_f_def using path_img by auto
- qed
- define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
- define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
- define h where "h \<equiv> \<lambda>p. g p / f p + 1"
- obtain spikes
- where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
- using \<open>valid_path \<gamma>\<close>
- by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
- have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
- proof -
- have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
- proof -
- have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
- proof -
- have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
- apply (cases "cmod (f p) = 0")
- by (auto simp add: norm_divide)
- then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
- qed
- then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
- by (simp add: image_subset_iff path_image_compose)
- moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
- ultimately show "?thesis"
- using convex_in_outside[of "ball 1 1" 0] outside_mono by blast
- qed
- have valid_h:"valid_path (h \<circ> \<gamma>)"
- proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
- show "h holomorphic_on s - zeros_f"
- unfolding h_def using f_holo g_holo
- by (auto intro!: holomorphic_intros simp add:zeros_f_def)
- next
- show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
- by auto
- qed
- have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
- proof -
- have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
- then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
- using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
- unfolding c_def by auto
- moreover have "winding_number (h o \<gamma>) 0 = 0"
- proof -
- have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
- moreover have "path (h o \<gamma>)"
- using valid_h by (simp add: valid_path_imp_path)
- moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
- by (simp add: loop pathfinish_compose pathstart_compose)
- ultimately show ?thesis using winding_number_zero_in_outside by auto
- qed
- ultimately show ?thesis by auto
- qed
- moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
- when "x\<in>{0..1} - spikes" for x
- proof (rule vector_derivative_chain_at_general)
- show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
- next
- define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- define t where "t \<equiv> \<gamma> x"
- have "f t\<noteq>0" unfolding zeros_f_def t_def
- by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
- moreover have "t\<in>s"
- using contra_subsetD path_image_def path_fg t_def that by fastforce
- ultimately have "(h has_field_derivative der t) (at t)"
- unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
- by (auto intro!: holomorphic_derivI derivative_eq_intros)
- then show "h field_differentiable at (\<gamma> x)"
- unfolding t_def field_differentiable_def by blast
- qed
- then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
- = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
- unfolding has_contour_integral
- apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
- by auto
- ultimately show ?thesis by auto
- qed
- then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
- using contour_integral_unique by simp
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
- + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
- proof -
- have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
- proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
- show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
- by auto
- then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
- using f_holo
- by (auto intro!: holomorphic_intros simp add:zeros_f_def)
- qed
- moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
- using h_contour
- by (simp add: has_contour_integral_integrable)
- ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
- contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
- using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
- by auto
- moreover have "deriv fg p / fg p = deriv f p / f p + deriv h p / h p"
- when "p\<in> path_image \<gamma>" for p
- proof -
- have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
- by auto
- have "h p\<noteq>0"
- proof (rule ccontr)
- assume "\<not> h p \<noteq> 0"
- then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
- then have "cmod (g p/f p) = 1" by auto
- moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
- apply (cases "cmod (f p) = 0")
- by (auto simp add: norm_divide)
- ultimately show False by auto
- qed
- have der_fg:"deriv fg p = deriv f p + deriv g p" unfolding fg_def
- using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _ \<open>open s\<close>] path_img that
- by auto
- have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- proof -
- define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
- have "p\<in>s" using path_img that by auto
- then have "(h has_field_derivative der p) (at p)"
- unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
- by (auto intro!: derivative_eq_intros holomorphic_derivI)
- then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
- qed
- show ?thesis
- apply (simp only:der_fg der_h)
- apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
- by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
- qed
- then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
- = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
- by (elim contour_integral_eq)
- ultimately show ?thesis by auto
- qed
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
- unfolding c_def zeros_fg_def w_def
- proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
- , of _ "{}" "\<lambda>_. 1",simplified])
- show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
- show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
- show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
- qed
- moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
- unfolding c_def zeros_f_def w_def
- proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
- , of _ "{}" "\<lambda>_. 1",simplified])
- show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
- show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
- show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
- qed
- ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
- by auto
- then show ?thesis unfolding c_def using w_def by auto
-qed
-
-end
--- a/src/HOL/Analysis/Derivative.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Derivative.thy Sat Nov 30 13:47:33 2019 +0100
@@ -9,6 +9,7 @@
imports
Bounded_Linear_Function
Line_Segment
+ Convex_Euclidean_Space
begin
declare bounded_linear_inner_left [intro]
@@ -2297,6 +2298,86 @@
apply (rule vector_derivative_at [OF field_vector_diff_chain_at])
using assms vector_derivative_works by (auto simp: field_differentiable_derivI)
+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 deriv_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_uminus [simp]: "deriv (\<lambda>w. -w) = (\<lambda>z. -1)"
+ using deriv_linear[of "-1"] by (simp del: deriv_linear)
+
+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)"
+ by (simp add: id_def)
+
+lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
+ by (metis DERIV_imp_deriv DERIV_const)
+
+lemma deriv_add [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_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"
+ by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
+ (simp add: field_split_simps power2_eq_square)
+
+lemma deriv_cdivide_right:
+ "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
+ by (simp add: field_class.field_divide_inverse)
+
+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)
+
+
subsection \<open>Relation between convexity and derivative\<close>
(* TODO: Generalise to real vector spaces? *)
@@ -2959,4 +3040,436 @@
qed auto
qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Piecewise differentiable functions\<close>
+
+definition piecewise_differentiable_on
+ (infixr "piecewise'_differentiable'_on" 50)
+ where "f piecewise_differentiable_on i \<equiv>
+ continuous_on i f \<and>
+ (\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))"
+
+lemma piecewise_differentiable_on_imp_continuous_on:
+ "f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f"
+by (simp add: piecewise_differentiable_on_def)
+
+lemma piecewise_differentiable_on_subset:
+ "f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T"
+ using continuous_on_subset
+ unfolding piecewise_differentiable_on_def
+ apply safe
+ apply (blast elim: continuous_on_subset)
+ by (meson Diff_iff differentiable_within_subset subsetCE)
+
+lemma differentiable_on_imp_piecewise_differentiable:
+ fixes a:: "'a::{linorder_topology,real_normed_vector}"
+ shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
+ apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
+ apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
+ done
+
+lemma differentiable_imp_piecewise_differentiable:
+ "(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S))
+ \<Longrightarrow> f piecewise_differentiable_on S"
+by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
+ intro: differentiable_within_subset)
+
+lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S"
+ by (simp add: differentiable_imp_piecewise_differentiable)
+
+lemma piecewise_differentiable_compose:
+ "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S);
+ \<And>x. finite (S \<inter> f-`{x})\<rbrakk>
+ \<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S"
+ apply (simp add: piecewise_differentiable_on_def, safe)
+ apply (blast intro: continuous_on_compose2)
+ apply (rename_tac A B)
+ apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI)
+ apply (blast intro!: differentiable_chain_within)
+ done
+
+lemma piecewise_differentiable_affine:
+ fixes m::real
+ assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)"
+ shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S"
+proof (cases "m = 0")
+ case True
+ then show ?thesis
+ unfolding o_def
+ by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
+next
+ case False
+ show ?thesis
+ apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
+ apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
+ done
+qed
+
+lemma piecewise_differentiable_cases:
+ fixes c::real
+ assumes "f piecewise_differentiable_on {a..c}"
+ "g piecewise_differentiable_on {c..b}"
+ "a \<le> c" "c \<le> b" "f c = g c"
+ shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
+proof -
+ obtain S T where st: "finite S" "finite T"
+ and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}"
+ and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}"
+ using assms
+ by (auto simp: piecewise_differentiable_on_def)
+ have finabc: "finite ({a,b,c} \<union> (S \<union> T))"
+ by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI)
+ have "continuous_on {a..c} f" "continuous_on {c..b} g"
+ using assms piecewise_differentiable_on_def by auto
+ then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
+ using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
+ OF closed_real_atLeastAtMost [of c b],
+ of f g "\<lambda>x. x\<le>c"] assms
+ by (force simp: ivl_disj_un_two_touch)
+ moreover
+ { fix x
+ assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))"
+ have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
+ proof (cases x c rule: le_cases)
+ case le show ?diff_fg
+ proof (rule differentiable_transform_within [where d = "dist x c"])
+ have "f differentiable at x"
+ using x le fd [of x] at_within_interior [of x "{a..c}"] by simp
+ then show "f differentiable at x within {a..b}"
+ by (simp add: differentiable_at_withinI)
+ qed (use x le st dist_real_def in auto)
+ next
+ case ge show ?diff_fg
+ proof (rule differentiable_transform_within [where d = "dist x c"])
+ have "g differentiable at x"
+ using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp
+ then show "g differentiable at x within {a..b}"
+ by (simp add: differentiable_at_withinI)
+ qed (use x ge st dist_real_def in auto)
+ qed
+ }
+ then have "\<exists>S. finite S \<and>
+ (\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
+ by (meson finabc)
+ ultimately show ?thesis
+ by (simp add: piecewise_differentiable_on_def)
+qed
+
+lemma piecewise_differentiable_neg:
+ "f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S"
+ by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
+
+lemma piecewise_differentiable_add:
+ assumes "f piecewise_differentiable_on i"
+ "g piecewise_differentiable_on i"
+ shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
+proof -
+ obtain S T where st: "finite S" "finite T"
+ "\<forall>x\<in>i - S. f differentiable at x within i"
+ "\<forall>x\<in>i - T. g differentiable at x within i"
+ using assms by (auto simp: piecewise_differentiable_on_def)
+ then have "finite (S \<union> T) \<and> (\<forall>x\<in>i - (S \<union> T). (\<lambda>x. f x + g x) differentiable at x within i)"
+ by auto
+ moreover have "continuous_on i f" "continuous_on i g"
+ using assms piecewise_differentiable_on_def by auto
+ ultimately show ?thesis
+ by (auto simp: piecewise_differentiable_on_def continuous_on_add)
+qed
+
+lemma piecewise_differentiable_diff:
+ "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on S\<rbrakk>
+ \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S"
+ unfolding diff_conv_add_uminus
+ by (metis piecewise_differentiable_add piecewise_differentiable_neg)
+
+
+subsection\<open>The concept of continuously differentiable\<close>
+
+text \<open>
+John Harrison writes as follows:
+
+``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
+continuously differentiable, which ensures that the path integral exists at least for any continuous
+f, since all piecewise continuous functions are integrable. However, our notion of validity is
+weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a
+finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
+the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
+can integrate all derivatives.''
+
+"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
+Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
+
+And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
+difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
+asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
+
+definition\<^marker>\<open>tag important\<close> C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
+ (infix "C1'_differentiable'_on" 50)
+ where
+ "f C1_differentiable_on S \<longleftrightarrow>
+ (\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)"
+
+lemma C1_differentiable_on_eq:
+ "f C1_differentiable_on S \<longleftrightarrow>
+ (\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ unfolding C1_differentiable_on_def
+ by (metis (no_types, lifting) continuous_on_eq differentiableI_vector vector_derivative_at)
+next
+ assume ?rhs
+ then show ?lhs
+ using C1_differentiable_on_def vector_derivative_works by fastforce
+qed
+
+lemma C1_differentiable_on_subset:
+ "f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S"
+ unfolding C1_differentiable_on_def continuous_on_eq_continuous_within
+ by (blast intro: continuous_within_subset)
+
+lemma C1_differentiable_compose:
+ assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
+ shows "(g \<circ> f) C1_differentiable_on S"
+proof -
+ have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x"
+ by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI)
+ moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))"
+ proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
+ show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))"
+ using fg
+ apply (clarsimp simp add: C1_differentiable_on_eq)
+ apply (rule Limits.continuous_on_scaleR, assumption)
+ by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def)
+ show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)"
+ by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at)
+ qed
+ ultimately show ?thesis
+ by (simp add: C1_differentiable_on_eq)
+qed
+
+lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S"
+ by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
+
+lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S"
+ by (auto simp: C1_differentiable_on_eq)
+
+lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S"
+ by (auto simp: C1_differentiable_on_eq)
+
+lemma C1_differentiable_on_add [simp, derivative_intros]:
+ "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S"
+ unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
+
+lemma C1_differentiable_on_minus [simp, derivative_intros]:
+ "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S"
+ unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
+
+lemma C1_differentiable_on_diff [simp, derivative_intros]:
+ "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S"
+ unfolding C1_differentiable_on_eq by (auto intro: continuous_intros)
+
+lemma C1_differentiable_on_mult [simp, derivative_intros]:
+ fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
+ shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S"
+ unfolding C1_differentiable_on_eq
+ by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
+
+lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
+ "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S"
+ unfolding C1_differentiable_on_eq
+ by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
+
+
+definition\<^marker>\<open>tag important\<close> piecewise_C1_differentiable_on
+ (infixr "piecewise'_C1'_differentiable'_on" 50)
+ where "f piecewise_C1_differentiable_on i \<equiv>
+ continuous_on i f \<and>
+ (\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))"
+
+lemma C1_differentiable_imp_piecewise:
+ "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
+ by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
+
+lemma piecewise_C1_imp_differentiable:
+ "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
+ by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
+ C1_differentiable_on_def differentiable_def has_vector_derivative_def
+ intro: has_derivative_at_withinI)
+
+lemma piecewise_C1_differentiable_compose:
+ assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
+ shows "(g \<circ> f) piecewise_C1_differentiable_on S"
+proof -
+ have "continuous_on S (\<lambda>x. g (f x))"
+ by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def)
+ moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T"
+ proof -
+ obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S"
+ using fg by (auto simp: piecewise_C1_differentiable_on_def)
+ obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S"
+ using fg by (auto simp: piecewise_C1_differentiable_on_def)
+ show ?thesis
+ proof (intro exI conjI)
+ show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))"
+ using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI)
+ show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))"
+ apply (rule C1_differentiable_compose)
+ apply (blast intro: C1_differentiable_on_subset [OF F])
+ apply (blast intro: C1_differentiable_on_subset [OF G])
+ by (simp add: C1_differentiable_on_subset G Diff_Int_distrib2 fin)
+ qed
+ qed
+ ultimately show ?thesis
+ by (simp add: piecewise_C1_differentiable_on_def)
+qed
+
+lemma piecewise_C1_differentiable_on_subset:
+ "f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T"
+ by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
+
+lemma C1_differentiable_imp_continuous_on:
+ "f C1_differentiable_on S \<Longrightarrow> continuous_on S f"
+ unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
+ using differentiable_at_withinI differentiable_imp_continuous_within by blast
+
+lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
+ unfolding C1_differentiable_on_def
+ by auto
+
+lemma piecewise_C1_differentiable_affine:
+ fixes m::real
+ assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)"
+ shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S"
+proof (cases "m = 0")
+ case True
+ then show ?thesis
+ unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def)
+next
+ case False
+ have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})"
+ using False not_finite_existsD by fastforce
+ show ?thesis
+ apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
+ apply (rule * assms derivative_intros | simp add: False vimage_def)+
+ done
+qed
+
+lemma piecewise_C1_differentiable_cases:
+ fixes c::real
+ assumes "f piecewise_C1_differentiable_on {a..c}"
+ "g piecewise_C1_differentiable_on {c..b}"
+ "a \<le> c" "c \<le> b" "f c = g c"
+ shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
+proof -
+ obtain S T where st: "f C1_differentiable_on ({a..c} - S)"
+ "g C1_differentiable_on ({c..b} - T)"
+ "finite S" "finite T"
+ using assms
+ by (force simp: piecewise_C1_differentiable_on_def)
+ then have f_diff: "f differentiable_on {a..<c} - S"
+ and g_diff: "g differentiable_on {c<..b} - T"
+ by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
+ have "continuous_on {a..c} f" "continuous_on {c..b} g"
+ using assms piecewise_C1_differentiable_on_def by auto
+ then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
+ using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
+ OF closed_real_atLeastAtMost [of c b],
+ of f g "\<lambda>x. x\<le>c"] assms
+ by (force simp: ivl_disj_un_two_touch)
+ { fix x
+ assume x: "x \<in> {a..b} - insert c (S \<union> T)"
+ have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
+ proof (cases x c rule: le_cases)
+ case le show ?diff_fg
+ apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
+ using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
+ next
+ case ge show ?diff_fg
+ apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
+ using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
+ qed
+ }
+ then have "(\<forall>x \<in> {a..b} - insert c (S \<union> T). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
+ by auto
+ moreover
+ { assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))"
+ and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))"
+ have "open ({a<..<c} - S)" "open ({c<..<b} - T)"
+ using st by (simp_all add: open_Diff finite_imp_closed)
+ moreover have "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
+ proof -
+ have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x)) (at x)"
+ if "a < x" "x < c" "x \<notin> S" for x
+ proof -
+ have f: "f differentiable at x"
+ by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that)
+ show ?thesis
+ using that
+ apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
+ apply (auto simp: dist_norm vector_derivative_works [symmetric] f)
+ done
+ qed
+ then show ?thesis
+ by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at)
+ qed
+ moreover have "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
+ proof -
+ have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x)) (at x)"
+ if "c < x" "x < b" "x \<notin> T" for x
+ proof -
+ have g: "g differentiable at x"
+ by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that)
+ show ?thesis
+ using that
+ apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
+ apply (auto simp: dist_norm vector_derivative_works [symmetric] g)
+ done
+ qed
+ then show ?thesis
+ by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at)
+ qed
+ ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T))
+ (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
+ by (rule continuous_on_subset [OF continuous_on_open_Un], auto)
+ } note * = this
+ have "continuous_on ({a<..<b} - insert c (S \<union> T)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
+ using st
+ by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
+ ultimately have "\<exists>S. finite S \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - S)"
+ apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI)
+ using st by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
+ with cab show ?thesis
+ by (simp add: piecewise_C1_differentiable_on_def)
+qed
+
+lemma piecewise_C1_differentiable_neg:
+ "f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S"
+ unfolding piecewise_C1_differentiable_on_def
+ by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
+
+lemma piecewise_C1_differentiable_add:
+ assumes "f piecewise_C1_differentiable_on i"
+ "g piecewise_C1_differentiable_on i"
+ shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
+proof -
+ obtain S t where st: "finite S" "finite t"
+ "f C1_differentiable_on (i-S)"
+ "g C1_differentiable_on (i-t)"
+ using assms by (auto simp: piecewise_C1_differentiable_on_def)
+ then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)"
+ by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
+ moreover have "continuous_on i f" "continuous_on i g"
+ using assms piecewise_C1_differentiable_on_def by auto
+ ultimately show ?thesis
+ by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
+qed
+
+lemma piecewise_C1_differentiable_diff:
+ "\<lbrakk>f piecewise_C1_differentiable_on S; g piecewise_C1_differentiable_on S\<rbrakk>
+ \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
+ unfolding diff_conv_add_uminus
+ by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
+
end
--- a/src/HOL/Analysis/FPS_Convergence.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/FPS_Convergence.thy Sat Nov 30 13:47:33 2019 +0100
@@ -9,13 +9,20 @@
theory FPS_Convergence
imports
- Conformal_Mappings
Generalised_Binomial_Theorem
"HOL-Computational_Algebra.Formal_Power_Series"
begin
+text \<open>
+ In this theory, we will connect formal power series (which are algebraic objects) with analytic
+ functions. This will become more important in complex analysis, and indeed some of the less
+ trivial results will only be proven there.
+\<close>
+
subsection\<^marker>\<open>tag unimportant\<close> \<open>Balls with extended real radius\<close>
+(* TODO: This should probably go somewhere else *)
+
text \<open>
The following is a variant of \<^const>\<open>ball\<close> that also allows an infinite radius.
\<close>
@@ -61,9 +68,6 @@
definition\<^marker>\<open>tag important\<close> eval_fps :: "'a :: {banach, real_normed_div_algebra} fps \<Rightarrow> 'a \<Rightarrow> 'a" where
"eval_fps f z = (\<Sum>n. fps_nth f n * z ^ n)"
-definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where
- "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
-
lemma norm_summable_fps:
fixes f :: "'a :: {banach, real_normed_div_algebra} fps"
shows "norm z < fps_conv_radius f \<Longrightarrow> summable (\<lambda>n. norm (fps_nth f n * z ^ n))"
@@ -81,38 +85,6 @@
using assms unfolding eval_fps_def fps_conv_radius_def
by (intro summable_sums summable_in_conv_radius) simp_all
-lemma
- fixes r :: ereal
- assumes "f holomorphic_on eball z0 r"
- shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r"
- and eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z"
- and eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)"
-proof -
- have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
- if "z \<in> ball z0 r'" "ereal r' < r" for z r'
- proof -
- from that(2) have "ereal r' \<le> r" by simp
- from assms(1) and this have "f holomorphic_on ball z0 r'"
- by (rule holomorphic_on_subset[OF _ ball_eball_mono])
- from holomorphic_power_series [OF this that(1)]
- show ?thesis by (simp add: fps_expansion_def)
- qed
- hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
- if "z \<in> eball z0 r" for z
- using that by (subst (asm) eball_conv_UNION_balls) blast
- show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def
- proof (rule conv_radius_geI_ex)
- fix r' :: real assume r': "r' > 0" "ereal r' < r"
- thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)"
- using *[of "z0 + of_real r'"]
- by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
- qed
- show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z
- using *[OF that] by (simp add: eval_fps_def sums_iff)
- show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
- using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
-qed
-
lemma continuous_on_eval_fps:
fixes f :: "'a :: {banach, real_normed_div_algebra} fps"
shows "continuous_on (eball 0 (fps_conv_radius f)) (eval_fps f)"
@@ -615,181 +587,14 @@
shows "eval_fps (fps_exp c) z = exp (c * z)" unfolding eval_fps_def exp_def
by (simp add: eval_fps_def exp_def scaleR_conv_of_real field_split_simps power_mult_distrib)
-lemma
- fixes f :: "complex fps" and r :: ereal
- assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"
- shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)"
- and eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow>
- eval_fps (inverse f) z = inverse (eval_fps f z)"
-proof -
- define R where "R = min (fps_conv_radius f) r"
- have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and>
- (\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
- proof (cases "min r (fps_conv_radius f) > 0")
- case True
- define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0"
- have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
- using assms by (intro holomorphic_intros) auto
- from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)"
- unfolding f'_def by (rule conv_radius_fps_expansion)
- have eval_f': "eval_fps f' z = inverse (eval_fps f z)"
- if "norm z < fps_conv_radius f" "norm z < r" for z
- using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
-
- have "f * f' = 1"
- proof (rule eval_fps_eqD)
- from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
- by (auto simp: min_def split: if_splits)
- also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
- finally show "\<dots> > 0" .
- next
- from True have "R > 0" by (auto simp: R_def)
- hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)"
- by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
- thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
- proof eventually_elim
- case (elim z)
- hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
- using radius by (intro eval_fps_mult)
- (auto simp: R_def min_def split: if_splits intro: less_trans)
- also have "eval_fps f' z = inverse (eval_fps f z)"
- using elim by (intro eval_f') (auto simp: R_def)
- also from elim have "eval_fps f z \<noteq> 0"
- by (intro assms) (auto simp: R_def)
- hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z"
- by simp
- finally show "eval_fps (f * f') z = eval_fps 1 z" .
- qed
- qed simp_all
- hence "f' = inverse f"
- by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
- with eval_f' and radius show ?thesis by simp
- next
- case False
- hence *: "eball 0 R = {}"
- by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
- show ?thesis
- proof safe
- from False have "min r (fps_conv_radius f) \<le> 0"
- by (simp add: min_def)
- also have "0 \<le> fps_conv_radius (inverse f)"
- by (simp add: fps_conv_radius_def conv_radius_nonneg)
- finally show "min r (fps_conv_radius f) \<le> \<dots>" .
- qed (unfold * [unfolded R_def], auto)
- qed
-
- from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast
- from * show "eval_fps (inverse f) z = inverse (eval_fps f z)"
- if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
- using that by auto
-qed
-
-lemma
- fixes f g :: "complex fps" and r :: ereal
- defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
- assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
- assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"
- shows fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R"
- and eval_fps_divide':
- "ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
-proof -
- from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0"
- by (auto simp: eval_fps_at_0 zero_ereal_def)
- have "R \<le> min r (fps_conv_radius g)"
- by (auto simp: R_def intro: min.coboundedI2)
- also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)"
- by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
- finally have radius: "fps_conv_radius (inverse g) \<ge> R" .
- have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))"
- by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "\<dots> \<le> fps_conv_radius (f * inverse g)"
- by (rule fps_conv_radius_mult)
- also have "f * inverse g = f / g"
- by (intro fps_divide_unit [symmetric] nz')
- finally show "fps_conv_radius (f / g) \<ge> R" .
-
- assume z: "ereal (norm z) < R"
- have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
- using radius by (intro eval_fps_mult less_le_trans[OF z])
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close>
- by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- also have "f * inverse g = f / g" by fact
- finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
-qed
-
-lemma
- fixes f g :: "complex fps" and r :: ereal
- defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
- assumes "subdegree g \<le> subdegree f"
- assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
- assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0"
- shows fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R"
- and eval_fps_divide:
- "ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow>
- eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
-proof -
- define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
- have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
- unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
- have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
- using assms(2) by (simp_all add: f'_def g'_def)
- have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
- by (simp_all add: f'_def g'_def)
- have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
- "fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
- have g_nz: "g \<noteq> 0"
- proof -
- define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))"
- from \<open>r > 0\<close> have "z \<in> eball 0 r"
- by (cases r) (auto simp: z_def eball_def)
- moreover have "z \<noteq> 0" using \<open>r > 0\<close>
- by (cases r) (auto simp: z_def)
- ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6))
- thus "g \<noteq> 0" by auto
- qed
- have fg: "f / g = f' * inverse g'"
- by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
-
- have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z
- proof (cases "z = 0")
- case False
- with assms and z have "eval_fps g z \<noteq> 0" by auto
- also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
- by (subst g_eq) (auto simp: eval_fps_mult)
- finally show ?thesis by auto
- qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0)
-
- have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')"
- by (auto simp: R_def min.coboundedI1 min.coboundedI2)
- also have "\<dots> \<le> fps_conv_radius (inverse g')"
- using g'_nz by (rule fps_conv_radius_inverse)
- finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" .
- hence "R \<le> fps_conv_radius (f' * inverse g')"
- by (intro order.trans[OF _ fps_conv_radius_mult])
- (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
- thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg)
-
- fix z c :: complex assume z: "ereal (norm z) < R"
- assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
- show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
- proof (cases "z = 0")
- case False
- from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
- by simp
- with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
- unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
- also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
- using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
- also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z"
- using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
- finally show ?thesis using False by simp
- qed (simp_all add: eval_fps_at_0 fg field_simps c)
-qed
+text \<open>
+ The case of division is more complicated and will therefore not be handled here.
+ Handling division becomes much more easy using complex analysis, and we will do so once
+ that is available.
+\<close>
-subsection \<open>Power series expansion of complex functions\<close>
+subsection \<open>Power series expansions of analytic functions\<close>
text \<open>
This predicate contains the notion that the given formal power series converges
@@ -831,25 +636,6 @@
finally show ?thesis .
qed
-lemma has_fps_expansion_fps_expansion [intro]:
- assumes "open A" "0 \<in> A" "f holomorphic_on A"
- shows "f has_fps_expansion fps_expansion f 0"
-proof -
- from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A"
- by (auto simp: open_contains_ball)
- have holo: "f holomorphic_on eball 0 (ereal r)"
- using r(2) and assms(3) by auto
- from r(1) have "0 < ereal r" by simp
- also have "r \<le> fps_conv_radius (fps_expansion f 0)"
- using holo by (intro conv_radius_fps_expansion) auto
- finally have "\<dots> > 0" .
- moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)"
- using r(1) by (intro eventually_nhds_in_open) auto
- hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
- by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
- ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def)
-qed
-
lemma has_fps_expansion_imp_continuous:
fixes F :: "'a::{real_normed_field,banach} fps"
assumes "f has_fps_expansion F"
@@ -1146,35 +932,7 @@
finally show ?thesis by simp
qed
-lemma fps_conv_radius_tan:
- fixes c :: complex
- assumes "c \<noteq> 0"
- shows "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)"
-proof -
- have "fps_conv_radius (fps_tan c) \<ge>
- Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
- unfolding fps_tan_def
- proof (rule fps_conv_radius_divide)
- fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
- with cos_eq_zero_imp_norm_ge[of "c*z"] assms
- show "eval_fps (fps_cos c) z \<noteq> 0" by (auto simp: norm_mult field_simps)
- qed (insert assms, auto)
- thus ?thesis by (simp add: min_def)
-qed
-lemma eval_fps_tan:
- fixes c :: complex
- assumes "norm z < pi / (2 * norm c)"
- shows "eval_fps (fps_tan c) z = tan (c * z)"
-proof (cases "c = 0")
- case False
- show ?thesis unfolding fps_tan_def
- proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
- fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
- with cos_eq_zero_imp_norm_ge[of "c*z"] assms
- show "eval_fps (fps_cos c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps)
- qed (insert False assms, auto simp: field_simps tan_def)
-qed simp_all
lemma eval_fps_binomial:
fixes c :: complex
@@ -1295,17 +1053,4 @@
by (intro that[of ?s']) (auto simp: has_fps_expansion_def zero_ereal_def)
qed
-theorem residue_fps_expansion_over_power_at_0:
- assumes "f has_fps_expansion F"
- shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = fps_nth F n"
-proof -
- from has_fps_expansion_imp_holomorphic[OF assms] guess s . note s = this
- have "residue (\<lambda>z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
- using assms s unfolding has_fps_expansion_def
- by (intro residue_holomorphic_over_power[of s]) (auto simp: zero_ereal_def)
- also from assms have "\<dots> = fps_nth F n"
- by (subst fps_nth_fps_expansion) auto
- finally show ?thesis by simp
-qed
-
-end
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Gamma_Function.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Gamma_Function.thy Sat Nov 30 13:47:33 2019 +0100
@@ -6,7 +6,6 @@
theory Gamma_Function
imports
- Conformal_Mappings
Equivalence_Lebesgue_Henstock_Integration
Summation_Tests
Harmonic_Numbers
@@ -2065,7 +2064,208 @@
by (simp add: field_split_simps Gamma_eq_zero_iff ring_distribs exp_diff exp_of_real ac_simps)
qed
-theorem Gamma_reflection_complex:
+text \<open>
+ The following lemma is somewhat annoying. With a little bit of complex analysis
+ (Cauchy's integral theorem, to be exact), this would be completely trivial. However,
+ we want to avoid depending on the complex analysis session at this point, so we prove it
+ the hard way.
+\<close>
+private lemma Gamma_reflection_aux:
+ defines "h \<equiv> \<lambda>z::complex. if z \<in> \<int> then 0 else
+ (of_real pi * cot (of_real pi*z) + Digamma z - Digamma (1 - z))"
+ defines "a \<equiv> complex_of_real pi"
+ obtains h' where "continuous_on UNIV h'" "\<And>z. (h has_field_derivative (h' z)) (at z)"
+proof -
+ define f where "f n = a * of_real (cos_coeff (n+1) - sin_coeff (n+2))" for n
+ define F where "F z = (if z = 0 then 0 else (cos (a*z) - sin (a*z)/(a*z)) / z)" for z
+ define g where "g n = complex_of_real (sin_coeff (n+1))" for n
+ define G where "G z = (if z = 0 then 1 else sin (a*z)/(a*z))" for z
+ have a_nz: "a \<noteq> 0" unfolding a_def by simp
+
+ have "(\<lambda>n. f n * (a*z)^n) sums (F z) \<and> (\<lambda>n. g n * (a*z)^n) sums (G z)"
+ if "abs (Re z) < 1" for z
+ proof (cases "z = 0"; rule conjI)
+ assume "z \<noteq> 0"
+ note z = this that
+
+ from z have sin_nz: "sin (a*z) \<noteq> 0" unfolding a_def by (auto simp: sin_eq_0)
+ have "(\<lambda>n. of_real (sin_coeff n) * (a*z)^n) sums (sin (a*z))" using sin_converges[of "a*z"]
+ by (simp add: scaleR_conv_of_real)
+ from sums_split_initial_segment[OF this, of 1]
+ have "(\<lambda>n. (a*z) * of_real (sin_coeff (n+1)) * (a*z)^n) sums (sin (a*z))" by (simp add: mult_ac)
+ from sums_mult[OF this, of "inverse (a*z)"] z a_nz
+ have A: "(\<lambda>n. g n * (a*z)^n) sums (sin (a*z)/(a*z))"
+ by (simp add: field_simps g_def)
+ with z show "(\<lambda>n. g n * (a*z)^n) sums (G z)" by (simp add: G_def)
+ from A z a_nz sin_nz have g_nz: "(\<Sum>n. g n * (a*z)^n) \<noteq> 0" by (simp add: sums_iff g_def)
+
+ have [simp]: "sin_coeff (Suc 0) = 1" by (simp add: sin_coeff_def)
+ from sums_split_initial_segment[OF sums_diff[OF cos_converges[of "a*z"] A], of 1]
+ have "(\<lambda>n. z * f n * (a*z)^n) sums (cos (a*z) - sin (a*z) / (a*z))"
+ by (simp add: mult_ac scaleR_conv_of_real ring_distribs f_def g_def)
+ from sums_mult[OF this, of "inverse z"] z assms
+ show "(\<lambda>n. f n * (a*z)^n) sums (F z)" by (simp add: divide_simps mult_ac f_def F_def)
+ next
+ assume z: "z = 0"
+ have "(\<lambda>n. f n * (a * z) ^ n) sums f 0" using powser_sums_zero[of f] z by simp
+ with z show "(\<lambda>n. f n * (a * z) ^ n) sums (F z)"
+ by (simp add: f_def F_def sin_coeff_def cos_coeff_def)
+ have "(\<lambda>n. g n * (a * z) ^ n) sums g 0" using powser_sums_zero[of g] z by simp
+ with z show "(\<lambda>n. g n * (a * z) ^ n) sums (G z)"
+ by (simp add: g_def G_def sin_coeff_def cos_coeff_def)
+ qed
+ note sums = conjunct1[OF this] conjunct2[OF this]
+
+ define h2 where [abs_def]:
+ "h2 z = (\<Sum>n. f n * (a*z)^n) / (\<Sum>n. g n * (a*z)^n) + Digamma (1 + z) - Digamma (1 - z)" for z
+ define POWSER where [abs_def]: "POWSER f z = (\<Sum>n. f n * (z^n :: complex))" for f z
+ define POWSER' where [abs_def]: "POWSER' f z = (\<Sum>n. diffs f n * (z^n))" for f and z :: complex
+ define h2' where [abs_def]:
+ "h2' z = a * (POWSER g (a*z) * POWSER' f (a*z) - POWSER f (a*z) * POWSER' g (a*z)) /
+ (POWSER g (a*z))^2 + Polygamma 1 (1 + z) + Polygamma 1 (1 - z)" for z
+
+ have h_eq: "h t = h2 t" if "abs (Re t) < 1" for t
+ proof -
+ from that have t: "t \<in> \<int> \<longleftrightarrow> t = 0" by (auto elim!: Ints_cases simp: dist_0_norm)
+ hence "h t = a*cot (a*t) - 1/t + Digamma (1 + t) - Digamma (1 - t)"
+ unfolding h_def using Digamma_plus1[of t] by (force simp: field_simps a_def)
+ also have "a*cot (a*t) - 1/t = (F t) / (G t)"
+ using t by (auto simp add: divide_simps sin_eq_0 cot_def a_def F_def G_def)
+ also have "\<dots> = (\<Sum>n. f n * (a*t)^n) / (\<Sum>n. g n * (a*t)^n)"
+ using sums[of t] that by (simp add: sums_iff dist_0_norm)
+ finally show "h t = h2 t" by (simp only: h2_def)
+ qed
+
+ let ?A = "{z. abs (Re z) < 1}"
+ have "open ({z. Re z < 1} \<inter> {z. Re z > -1})"
+ using open_halfspace_Re_gt open_halfspace_Re_lt by auto
+ also have "({z. Re z < 1} \<inter> {z. Re z > -1}) = {z. abs (Re z) < 1}" by auto
+ finally have open_A: "open ?A" .
+ hence [simp]: "interior ?A = ?A" by (simp add: interior_open)
+
+ have summable_f: "summable (\<lambda>n. f n * z^n)" for z
+ by (rule powser_inside, rule sums_summable, rule sums[of "\<i> * of_real (norm z + 1) / a"])
+ (simp_all add: norm_mult a_def del: of_real_add)
+ have summable_g: "summable (\<lambda>n. g n * z^n)" for z
+ by (rule powser_inside, rule sums_summable, rule sums[of "\<i> * of_real (norm z + 1) / a"])
+ (simp_all add: norm_mult a_def del: of_real_add)
+ have summable_fg': "summable (\<lambda>n. diffs f n * z^n)" "summable (\<lambda>n. diffs g n * z^n)" for z
+ by (intro termdiff_converges_all summable_f summable_g)+
+ have "(POWSER f has_field_derivative (POWSER' f z)) (at z)"
+ "(POWSER g has_field_derivative (POWSER' g z)) (at z)" for z
+ unfolding POWSER_def POWSER'_def
+ by (intro termdiffs_strong_converges_everywhere summable_f summable_g)+
+ note derivs = this[THEN DERIV_chain2[OF _ DERIV_cmult[OF DERIV_ident]], unfolded POWSER_def]
+ have "isCont (POWSER f) z" "isCont (POWSER g) z" "isCont (POWSER' f) z" "isCont (POWSER' g) z"
+ for z unfolding POWSER_def POWSER'_def
+ by (intro isCont_powser_converges_everywhere summable_f summable_g summable_fg')+
+ note cont = this[THEN isCont_o2[rotated], unfolded POWSER_def POWSER'_def]
+
+ {
+ fix z :: complex assume z: "abs (Re z) < 1"
+ define d where "d = \<i> * of_real (norm z + 1)"
+ have d: "abs (Re d) < 1" "norm z < norm d" by (simp_all add: d_def norm_mult del: of_real_add)
+ have "eventually (\<lambda>z. h z = h2 z) (nhds z)"
+ using eventually_nhds_in_nhd[of z ?A] using h_eq z
+ by (auto elim!: eventually_mono simp: dist_0_norm)
+
+ moreover from sums(2)[OF z] z have nz: "(\<Sum>n. g n * (a * z) ^ n) \<noteq> 0"
+ unfolding G_def by (auto simp: sums_iff sin_eq_0 a_def)
+ have A: "z \<in> \<int> \<longleftrightarrow> z = 0" using z by (auto elim!: Ints_cases)
+ have no_int: "1 + z \<in> \<int> \<longleftrightarrow> z = 0" using z Ints_diff[of "1+z" 1] A
+ by (auto elim!: nonpos_Ints_cases)
+ have no_int': "1 - z \<in> \<int> \<longleftrightarrow> z = 0" using z Ints_diff[of 1 "1-z"] A
+ by (auto elim!: nonpos_Ints_cases)
+ from no_int no_int' have no_int: "1 - z \<notin> \<int>\<^sub>\<le>\<^sub>0" "1 + z \<notin> \<int>\<^sub>\<le>\<^sub>0" by auto
+ have "(h2 has_field_derivative h2' z) (at z)" unfolding h2_def
+ by (rule DERIV_cong, (rule derivative_intros refl derivs[unfolded POWSER_def] nz no_int)+)
+ (auto simp: h2'_def POWSER_def field_simps power2_eq_square)
+ ultimately have deriv: "(h has_field_derivative h2' z) (at z)"
+ by (subst DERIV_cong_ev[OF refl _ refl])
+
+ from sums(2)[OF z] z have "(\<Sum>n. g n * (a * z) ^ n) \<noteq> 0"
+ unfolding G_def by (auto simp: sums_iff a_def sin_eq_0)
+ hence "isCont h2' z" using no_int unfolding h2'_def[abs_def] POWSER_def POWSER'_def
+ by (intro continuous_intros cont
+ continuous_on_compose2[OF _ continuous_on_Polygamma[of "{z. Re z > 0}"]]) auto
+ note deriv and this
+ } note A = this
+
+ interpret h: periodic_fun_simple' h
+ proof
+ fix z :: complex
+ show "h (z + 1) = h z"
+ proof (cases "z \<in> \<int>")
+ assume z: "z \<notin> \<int>"
+ hence A: "z + 1 \<notin> \<int>" "z \<noteq> 0" using Ints_diff[of "z+1" 1] by auto
+ hence "Digamma (z + 1) - Digamma (-z) = Digamma z - Digamma (-z + 1)"
+ by (subst (1 2) Digamma_plus1) simp_all
+ with A z show "h (z + 1) = h z"
+ by (simp add: h_def sin_plus_pi cos_plus_pi ring_distribs cot_def)
+ qed (simp add: h_def)
+ qed
+
+ have h2'_eq: "h2' (z - 1) = h2' z" if z: "Re z > 0" "Re z < 1" for z
+ proof -
+ have "((\<lambda>z. h (z - 1)) has_field_derivative h2' (z - 1)) (at z)"
+ by (rule DERIV_cong, rule DERIV_chain'[OF _ A(1)])
+ (insert z, auto intro!: derivative_eq_intros)
+ hence "(h has_field_derivative h2' (z - 1)) (at z)" by (subst (asm) h.minus_1)
+ moreover from z have "(h has_field_derivative h2' z) (at z)" by (intro A) simp_all
+ ultimately show "h2' (z - 1) = h2' z" by (rule DERIV_unique)
+ qed
+
+ define h2'' where "h2'' z = h2' (z - of_int \<lfloor>Re z\<rfloor>)" for z
+ have deriv: "(h has_field_derivative h2'' z) (at z)" for z
+ proof -
+ fix z :: complex
+ have B: "\<bar>Re z - real_of_int \<lfloor>Re z\<rfloor>\<bar> < 1" by linarith
+ have "((\<lambda>t. h (t - of_int \<lfloor>Re z\<rfloor>)) has_field_derivative h2'' z) (at z)"
+ unfolding h2''_def by (rule DERIV_cong, rule DERIV_chain'[OF _ A(1)])
+ (insert B, auto intro!: derivative_intros)
+ thus "(h has_field_derivative h2'' z) (at z)" by (simp add: h.minus_of_int)
+ qed
+
+ have cont: "continuous_on UNIV h2''"
+ proof (intro continuous_at_imp_continuous_on ballI)
+ fix z :: complex
+ define r where "r = \<lfloor>Re z\<rfloor>"
+ define A where "A = {t. of_int r - 1 < Re t \<and> Re t < of_int r + 1}"
+ have "continuous_on A (\<lambda>t. h2' (t - of_int r))" unfolding A_def
+ by (intro continuous_at_imp_continuous_on isCont_o2[OF _ A(2)] ballI continuous_intros)
+ (simp_all add: abs_real_def)
+ moreover have "h2'' t = h2' (t - of_int r)" if t: "t \<in> A" for t
+ proof (cases "Re t \<ge> of_int r")
+ case True
+ from t have "of_int r - 1 < Re t" "Re t < of_int r + 1" by (simp_all add: A_def)
+ with True have "\<lfloor>Re t\<rfloor> = \<lfloor>Re z\<rfloor>" unfolding r_def by linarith
+ thus ?thesis by (auto simp: r_def h2''_def)
+ next
+ case False
+ from t have t: "of_int r - 1 < Re t" "Re t < of_int r + 1" by (simp_all add: A_def)
+ with False have t': "\<lfloor>Re t\<rfloor> = \<lfloor>Re z\<rfloor> - 1" unfolding r_def by linarith
+ moreover from t False have "h2' (t - of_int r + 1 - 1) = h2' (t - of_int r + 1)"
+ by (intro h2'_eq) simp_all
+ ultimately show ?thesis by (auto simp: r_def h2''_def algebra_simps t')
+ qed
+ ultimately have "continuous_on A h2''" by (subst continuous_on_cong[OF refl])
+ moreover {
+ have "open ({t. of_int r - 1 < Re t} \<inter> {t. of_int r + 1 > Re t})"
+ by (intro open_Int open_halfspace_Re_gt open_halfspace_Re_lt)
+ also have "{t. of_int r - 1 < Re t} \<inter> {t. of_int r + 1 > Re t} = A"
+ unfolding A_def by blast
+ finally have "open A" .
+ }
+ ultimately have C: "isCont h2'' t" if "t \<in> A" for t using that
+ by (subst (asm) continuous_on_eq_continuous_at) auto
+ have "of_int r - 1 < Re z" "Re z < of_int r + 1" unfolding r_def by linarith+
+ thus "isCont h2'' z" by (intro C) (simp_all add: A_def)
+ qed
+
+ from that[OF cont deriv] show ?thesis .
+qed
+
+lemma Gamma_reflection_complex:
fixes z :: complex
shows "Gamma z * Gamma (1 - z) = of_real pi / sin (of_real pi * z)"
proof -
@@ -2074,7 +2274,7 @@
let ?h = "\<lambda>z::complex. (of_real pi * cot (of_real pi*z) + Digamma z - Digamma (1 - z))"
define h where [abs_def]: "h z = (if z \<in> \<int> then 0 else ?h z)" for z :: complex
- \<comment> \<open>\<^term>\<open>g\<close> is periodic with period 1.\<close>
+ \<comment> \<open>@{term g} is periodic with period 1.\<close>
interpret g: periodic_fun_simple' g
proof
fix z :: complex
@@ -2094,8 +2294,8 @@
qed (simp add: g_def)
qed
- \<comment> \<open>\<^term>\<open>g\<close> is entire.\<close>
- have g_g' [derivative_intros]: "(g has_field_derivative (h z * g z)) (at z)" for z :: complex
+ \<comment> \<open>@{term g} is entire.\<close>
+ have g_g': "(g has_field_derivative (h z * g z)) (at z)" for z :: complex
proof (cases "z \<in> \<int>")
let ?h' = "\<lambda>z. Beta z (1 - z) * ((Digamma z - Digamma (1 - z)) * sin (z * of_real pi) +
of_real pi * cos (z * of_real pi))"
@@ -2144,10 +2344,6 @@
finally show "(g has_field_derivative (h z * g z)) (at z)" by (simp add: z h_def)
qed
- have g_holo [holomorphic_intros]: "g holomorphic_on A" for A
- by (rule holomorphic_on_subset[of _ UNIV])
- (force simp: holomorphic_on_open intro!: derivative_intros)+
-
have g_eq: "g (z/2) * g ((z+1)/2) = Gamma (1/2)^2 * g z" if "Re z > -1" "Re z < 2" for z
proof (cases "z \<in> \<int>")
case True
@@ -2208,9 +2404,6 @@
unfolding g_def using Ints_diff[of 1 "1 - z"]
by (auto simp: Gamma_eq_zero_iff sin_eq_0 dest!: nonpos_Ints_Int)
- have h_altdef: "h z = deriv g z / g z" for z :: complex
- using DERIV_imp_deriv[OF g_g', of z] by simp
-
have h_eq: "h z = (h (z/2) + h ((z+1)/2)) / 2" for z
proof -
have "((\<lambda>t. g (t/2) * g ((t+1)/2)) has_field_derivative
@@ -2230,16 +2423,9 @@
thus "h z = (h (z/2) + h ((z+1)/2)) / 2" by simp
qed
- have h_holo [holomorphic_intros]: "h holomorphic_on A" for A
- unfolding h_altdef [abs_def]
- by (rule holomorphic_on_subset[of _ UNIV]) (auto intro!: holomorphic_intros)
- define h' where "h' = deriv h"
- have h_h': "(h has_field_derivative h' z) (at z)" for z unfolding h'_def
- by (auto intro!: holomorphic_derivI[of _ UNIV] holomorphic_intros)
- have h'_holo [holomorphic_intros]: "h' holomorphic_on A" for A unfolding h'_def
- by (rule holomorphic_on_subset[of _ UNIV]) (auto intro!: holomorphic_intros)
- have h'_cont: "continuous_on UNIV h'"
- by (intro holomorphic_on_imp_continuous_on holomorphic_intros)
+ obtain h' where h'_cont: "continuous_on UNIV h'" and
+ h_h': "\<And>z. (h has_field_derivative h' z) (at z)"
+ unfolding h_def by (erule Gamma_reflection_aux)
have h'_eq: "h' z = (h' (z/2) + h' ((z+1)/2)) / 4" for z
proof -
@@ -2307,13 +2493,16 @@
unfolding g_def using that by (auto intro!: Reals_mult Gamma_complex_real)
have h_real: "h z \<in> \<real>" if "z \<in> \<real>" for z
unfolding h_def using that by (auto intro!: Reals_mult Reals_add Reals_diff Polygamma_Real)
+ have g_nz: "g z \<noteq> 0" for z unfolding g_def using Ints_diff[of 1 "1-z"]
+ by (auto simp: Gamma_eq_zero_iff sin_eq_0)
from h'_zero h_h'_2 have "\<exists>c. \<forall>z\<in>UNIV. h z = c"
- by (intro has_field_derivative_zero_constant) simp_all
+ by (intro has_field_derivative_zero_constant) (simp_all add: dist_0_norm)
then obtain c where c: "\<And>z. h z = c" by auto
have "\<exists>u. u \<in> closed_segment 0 1 \<and> Re (g 1) - Re (g 0) = Re (h u * g u * (1 - 0))"
by (intro complex_mvt_line g_g')
- then guess u by (elim exE conjE) note u = this
+ then obtain u where u: "u \<in> closed_segment 0 1" "Re (g 1) - Re (g 0) = Re (h u * g u)"
+ by auto
from u(1) have u': "u \<in> \<real>" unfolding closed_segment_def
by (auto simp: scaleR_conv_of_real)
from u' g_real[of u] g_nz[of u] have "Re (g u) \<noteq> 0" by (auto elim!: Reals_cases)
@@ -2330,7 +2519,7 @@
show ?thesis
proof (cases "z \<in> \<int>")
case False
- with \<open>g z = pi\<close> show ?thesis by (auto simp: g_def field_split_simps)
+ with \<open>g z = pi\<close> show ?thesis by (auto simp: g_def divide_simps)
next
case True
then obtain n where n: "z = of_int n" by (elim Ints_cases)
@@ -2446,20 +2635,6 @@
finally show "(\<lambda>z. inverse (rGamma z / (z + of_nat n))) \<midarrow> (- of_nat n) \<rightarrow> ?c" .
qed
-lemma is_pole_Gamma: "is_pole Gamma (- of_nat n)"
- unfolding is_pole_def using Gamma_poles .
-
-lemma Gamme_residue:
- "residue Gamma (- of_nat n) = (-1) ^ n / fact n"
-proof (rule residue_simple')
- show "open (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
- by (intro open_Compl closed_subset_Ints) auto
- show "Gamma holomorphic_on (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) - {- of_nat n})"
- by (rule holomorphic_Gamma) auto
- show "(\<lambda>w. Gamma w * (w - - of_nat n)) \<midarrow>- of_nat n \<rightarrow> (- 1) ^ n / fact n"
- using Gamma_residues[of n] by simp
-qed auto
-
subsection \<open>Alternative definitions\<close>
--- a/src/HOL/Analysis/Great_Picard.thy Wed Nov 27 16:54:33 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1848 +0,0 @@
-section \<open>The Great Picard Theorem and its Applications\<close>
-
-text\<open>Ported from HOL Light (cauchy.ml) by L C Paulson, 2017\<close>
-
-theory Great_Picard
- imports Conformal_Mappings Further_Topology
-
-begin
-
-subsection\<open>Schottky's theorem\<close>
-
-lemma Schottky_lemma0:
- assumes holf: "f holomorphic_on S" and cons: "contractible S" and "a \<in> S"
- and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> -1"
- obtains g where "g holomorphic_on S"
- "norm(g a) \<le> 1 + norm(f a) / 3"
- "\<And>z. z \<in> S \<Longrightarrow> f z = cos(of_real pi * g z)"
-proof -
- obtain g where holg: "g holomorphic_on S" and g: "norm(g a) \<le> pi + norm(f a)"
- and f_eq_cos: "\<And>z. z \<in> S \<Longrightarrow> f z = cos(g z)"
- using contractible_imp_holomorphic_arccos_bounded [OF assms]
- by blast
- show ?thesis
- proof
- show "(\<lambda>z. g z / pi) holomorphic_on S"
- by (auto intro: holomorphic_intros holg)
- have "3 \<le> pi"
- using pi_approx by force
- have "3 * norm(g a) \<le> 3 * (pi + norm(f a))"
- using g by auto
- also have "... \<le> pi * 3 + pi * cmod (f a)"
- using \<open>3 \<le> pi\<close> by (simp add: mult_right_mono algebra_simps)
- finally show "cmod (g a / complex_of_real pi) \<le> 1 + cmod (f a) / 3"
- by (simp add: field_simps norm_divide)
- show "\<And>z. z \<in> S \<Longrightarrow> f z = cos (complex_of_real pi * (g z / complex_of_real pi))"
- by (simp add: f_eq_cos)
- qed
-qed
-
-
-lemma Schottky_lemma1:
- fixes n::nat
- assumes "0 < n"
- shows "0 < n + sqrt(real n ^ 2 - 1)"
-proof -
- have "(n-1)^2 \<le> n^2 - 1"
- using assms by (simp add: algebra_simps power2_eq_square)
- then have "real (n - 1) \<le> sqrt (real (n\<^sup>2 - 1))"
- by (metis of_nat_le_iff of_nat_power real_le_rsqrt)
- then have "n-1 \<le> sqrt(real n ^ 2 - 1)"
- by (simp add: Suc_leI assms of_nat_diff)
- then show ?thesis
- using assms by linarith
-qed
-
-
-lemma Schottky_lemma2:
- fixes x::real
- assumes "0 \<le> x"
- obtains n where "0 < n" "\<bar>x - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
-proof -
- obtain n0::nat where "0 < n0" "ln(n0 + sqrt(real n0 ^ 2 - 1)) / pi \<le> x"
- proof
- show "ln(real 1 + sqrt(real 1 ^ 2 - 1))/pi \<le> x"
- by (auto simp: assms)
- qed auto
- moreover
- obtain M::nat where "\<And>n. \<lbrakk>0 < n; ln(n + sqrt(real n ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> n \<le> M"
- proof
- fix n::nat
- assume "0 < n" "ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"
- then have "ln (n + sqrt ((real n)\<^sup>2 - 1)) \<le> x * pi"
- by (simp add: field_split_simps)
- then have *: "exp (ln (n + sqrt ((real n)\<^sup>2 - 1))) \<le> exp (x * pi)"
- by blast
- have 0: "0 \<le> sqrt ((real n)\<^sup>2 - 1)"
- using \<open>0 < n\<close> by auto
- have "n + sqrt ((real n)\<^sup>2 - 1) = exp (ln (n + sqrt ((real n)\<^sup>2 - 1)))"
- by (simp add: Suc_leI \<open>0 < n\<close> add_pos_nonneg real_of_nat_ge_one_iff)
- also have "... \<le> exp (x * pi)"
- using "*" by blast
- finally have "real n \<le> exp (x * pi)"
- using 0 by linarith
- then show "n \<le> nat (ceiling (exp(x * pi)))"
- by linarith
- qed
- ultimately obtain n where
- "0 < n" and le_x: "ln(n + sqrt(real n ^ 2 - 1)) / pi \<le> x"
- and le_n: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> n"
- using bounded_Max_nat [of "\<lambda>n. 0<n \<and> ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"] by metis
- define a where "a \<equiv> ln(n + sqrt(real n ^ 2 - 1)) / pi"
- define b where "b \<equiv> ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / pi"
- have le_xa: "a \<le> x"
- and le_na: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> n"
- using le_x le_n by (auto simp: a_def)
- moreover have "x < b"
- using le_n [of "Suc n"] by (force simp: b_def)
- moreover have "b - a < 1"
- proof -
- have "ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) - ln (real n + sqrt ((real n)\<^sup>2 - 1)) =
- ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1)))"
- by (simp add: \<open>0 < n\<close> Schottky_lemma1 add_pos_nonneg ln_div [symmetric])
- also have "... \<le> 3"
- proof (cases "n = 1")
- case True
- have "sqrt 3 \<le> 2"
- by (simp add: real_le_lsqrt)
- then have "(2 + sqrt 3) \<le> 4"
- by simp
- also have "... \<le> exp 3"
- using exp_ge_add_one_self [of "3::real"] by simp
- finally have "ln (2 + sqrt 3) \<le> 3"
- by (metis add_nonneg_nonneg add_pos_nonneg dbl_def dbl_inc_def dbl_inc_simps(3)
- dbl_simps(3) exp_gt_zero ln_exp ln_le_cancel_iff real_sqrt_ge_0_iff zero_le_one zero_less_one)
- then show ?thesis
- by (simp add: True)
- next
- case False with \<open>0 < n\<close> have "1 < n" "2 \<le> n"
- by linarith+
- then have 1: "1 \<le> real n * real n"
- by (metis less_imp_le_nat one_le_power power2_eq_square real_of_nat_ge_one_iff)
- have *: "4 + (m+2) * 2 \<le> (m+2) * ((m+2) * 3)" for m::nat
- by simp
- have "4 + n * 2 \<le> n * (n * 3)"
- using * [of "n-2"] \<open>2 \<le> n\<close>
- by (metis le_add_diff_inverse2)
- then have **: "4 + real n * 2 \<le> real n * (real n * 3)"
- by (metis (mono_tags, hide_lams) of_nat_le_iff of_nat_add of_nat_mult of_nat_numeral)
- have "sqrt ((1 + real n)\<^sup>2 - 1) \<le> 2 * sqrt ((real n)\<^sup>2 - 1)"
- by (auto simp: real_le_lsqrt power2_eq_square algebra_simps 1 **)
- then
- have "((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> 2"
- using Schottky_lemma1 \<open>0 < n\<close> by (simp add: field_split_simps)
- then have "ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> ln 2"
- apply (subst ln_le_cancel_iff)
- using Schottky_lemma1 \<open>0 < n\<close> by auto (force simp: field_split_simps)
- also have "... \<le> 3"
- using ln_add_one_self_le_self [of 1] by auto
- finally show ?thesis .
- qed
- also have "... < pi"
- using pi_approx by simp
- finally show ?thesis
- by (simp add: a_def b_def field_split_simps)
- qed
- ultimately have "\<bar>x - a\<bar> < 1/2 \<or> \<bar>x - b\<bar> < 1/2"
- by (auto simp: abs_if)
- then show thesis
- proof
- assume "\<bar>x - a\<bar> < 1 / 2"
- then show ?thesis
- by (rule_tac n=n in that) (auto simp: a_def \<open>0 < n\<close>)
- next
- assume "\<bar>x - b\<bar> < 1 / 2"
- then show ?thesis
- by (rule_tac n="Suc n" in that) (auto simp: b_def \<open>0 < n\<close>)
- qed
-qed
-
-
-lemma Schottky_lemma3:
- fixes z::complex
- assumes "z \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)})
- \<union> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
- shows "cos(pi * cos(pi * z)) = 1 \<or> cos(pi * cos(pi * z)) = -1"
-proof -
- have sqrt2 [simp]: "complex_of_real (sqrt x) * complex_of_real (sqrt x) = x" if "x \<ge> 0" for x::real
- by (metis abs_of_nonneg of_real_mult real_sqrt_mult_self that)
- have 1: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) -
- (ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
- inverse
- (exp (\<i> * (of_int m * complex_of_real pi) -
- (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
- if "n > 0" for m n
- proof -
- have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> inverse e^2 - 2 * n*inverse e + 1 = 0" for n e::complex
- by (auto simp: field_simps power2_eq_square)
- have [simp]: "1 \<le> real n * real n"
- by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
- show ?thesis
- apply (simp add: eeq)
- using Schottky_lemma1 [OF that]
- apply (auto simp: eeq exp_diff exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
- apply (rule_tac x="int n" in exI)
- apply (auto simp: power2_eq_square algebra_simps)
- apply (rule_tac x="- int n" in exI)
- apply (auto simp: power2_eq_square algebra_simps)
- done
- qed
- have 2: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) +
- (ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
- inverse
- (exp (\<i> * (of_int m * complex_of_real pi) +
- (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
- if "n > 0" for m n
- proof -
- have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> e^2 - 2 * n*e + 1 = 0" for n e::complex
- by (auto simp: field_simps power2_eq_square)
- have [simp]: "1 \<le> real n * real n"
- by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
- show ?thesis
- apply (simp add: eeq)
- using Schottky_lemma1 [OF that]
- apply (auto simp: exp_add exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
- apply (rule_tac x="int n" in exI)
- apply (auto simp: power2_eq_square algebra_simps)
- apply (rule_tac x="- int n" in exI)
- apply (auto simp: power2_eq_square algebra_simps)
- done
- qed
- have "\<exists>x. cos (complex_of_real pi * z) = of_int x"
- using assms
- apply safe
- apply (auto simp: Ints_def cos_exp_eq exp_minus Complex_eq)
- apply (auto simp: algebra_simps dest: 1 2)
- done
- then have "sin(pi * cos(pi * z)) ^ 2 = 0"
- by (simp add: Complex_Transcendental.sin_eq_0)
- then have "1 - cos(pi * cos(pi * z)) ^ 2 = 0"
- by (simp add: sin_squared_eq)
- then show ?thesis
- using power2_eq_1_iff by auto
-qed
-
-
-theorem Schottky:
- assumes holf: "f holomorphic_on cball 0 1"
- and nof0: "norm(f 0) \<le> r"
- and not01: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not>(f z = 0 \<or> f z = 1)"
- and "0 < t" "t < 1" "norm z \<le> t"
- shows "norm(f z) \<le> exp(pi * exp(pi * (2 + 2 * r + 12 * t / (1 - t))))"
-proof -
- obtain h where holf: "h holomorphic_on cball 0 1"
- and nh0: "norm (h 0) \<le> 1 + norm(2 * f 0 - 1) / 3"
- and h: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 = cos(of_real pi * h z)"
- proof (rule Schottky_lemma0 [of "\<lambda>z. 2 * f z - 1" "cball 0 1" 0])
- show "(\<lambda>z. 2 * f z - 1) holomorphic_on cball 0 1"
- by (intro holomorphic_intros holf)
- show "contractible (cball (0::complex) 1)"
- by (auto simp: convex_imp_contractible)
- show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 \<noteq> 1 \<and> 2 * f z - 1 \<noteq> - 1"
- using not01 by force
- qed auto
- obtain g where holg: "g holomorphic_on cball 0 1"
- and ng0: "norm(g 0) \<le> 1 + norm(h 0) / 3"
- and g: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z = cos(of_real pi * g z)"
- proof (rule Schottky_lemma0 [OF holf convex_imp_contractible, of 0])
- show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z \<noteq> 1 \<and> h z \<noteq> - 1"
- using h not01 by fastforce+
- qed auto
- have g0_2_f0: "norm(g 0) \<le> 2 + norm(f 0)"
- proof -
- have "cmod (2 * f 0 - 1) \<le> cmod (2 * f 0) + 1"
- by (metis norm_one norm_triangle_ineq4)
- also have "... \<le> 2 + cmod (f 0) * 3"
- by simp
- finally have "1 + norm(2 * f 0 - 1) / 3 \<le> (2 + norm(f 0) - 1) * 3"
- apply (simp add: field_split_simps)
- using norm_ge_zero [of "f 0 * 2 - 1"]
- by linarith
- with nh0 have "norm(h 0) \<le> (2 + norm(f 0) - 1) * 3"
- by linarith
- then have "1 + norm(h 0) / 3 \<le> 2 + norm(f 0)"
- by simp
- with ng0 show ?thesis
- by auto
- qed
- have "z \<in> ball 0 1"
- using assms by auto
- have norm_g_12: "norm(g z - g 0) \<le> (12 * t) / (1 - t)"
- proof -
- obtain g' where g': "\<And>x. x \<in> cball 0 1 \<Longrightarrow> (g has_field_derivative g' x) (at x within cball 0 1)"
- using holg [unfolded holomorphic_on_def field_differentiable_def] by metis
- have int_g': "(g' has_contour_integral g z - g 0) (linepath 0 z)"
- using contour_integral_primitive [OF g' valid_path_linepath, of 0 z]
- using \<open>z \<in> ball 0 1\<close> segment_bound1 by fastforce
- have "cmod (g' w) \<le> 12 / (1 - t)" if "w \<in> closed_segment 0 z" for w
- proof -
- have w: "w \<in> ball 0 1"
- using segment_bound [OF that] \<open>z \<in> ball 0 1\<close> by simp
- have ttt: "\<And>z. z \<in> frontier (cball 0 1) \<Longrightarrow> 1 - t \<le> dist w z"
- using \<open>norm z \<le> t\<close> segment_bound1 [OF \<open>w \<in> closed_segment 0 z\<close>]
- apply (simp add: dist_complex_def)
- by (metis diff_left_mono dist_commute dist_complex_def norm_triangle_ineq2 order_trans)
- have *: "\<lbrakk>\<And>b. (\<exists>w \<in> T \<union> U. w \<in> ball b 1); \<And>x. x \<in> D \<Longrightarrow> g x \<notin> T \<union> U\<rbrakk> \<Longrightarrow> \<nexists>b. ball b 1 \<subseteq> g ` D" for T U D
- by force
- have "\<nexists>b. ball b 1 \<subseteq> g ` cball 0 1"
- proof (rule *)
- show "(\<exists>w \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
- (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)}). w \<in> ball b 1)" for b
- proof -
- obtain m where m: "m \<in> \<int>" "\<bar>Re b - m\<bar> \<le> 1/2"
- by (metis Ints_of_int abs_minus_commute of_int_round_abs_le)
- show ?thesis
- proof (cases "0::real" "Im b" rule: le_cases)
- case le
- then obtain n where "0 < n" and n: "\<bar>Im b - ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
- using Schottky_lemma2 [of "Im b"] by blast
- have "dist b (Complex m (Im b)) \<le> 1/2"
- by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
- moreover
- have "dist (Complex m (Im b)) (Complex m (ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1/2"
- using n by (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
- ultimately have "dist b (Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1"
- by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
- with le m \<open>0 < n\<close> show ?thesis
- apply (rule_tac x = "Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)" in bexI)
- apply (simp_all del: Complex_eq greaterThan_0)
- by blast
- next
- case ge
- then obtain n where "0 < n" and n: "\<bar>- Im b - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
- using Schottky_lemma2 [of "- Im b"] by auto
- have "dist b (Complex m (Im b)) \<le> 1/2"
- by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
- moreover
- have "dist (Complex m (- ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi)) (Complex m (Im b)) < 1/2"
- using n
- apply (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
- by (metis add.commute add_uminus_conv_diff)
- ultimately have "dist b (Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1"
- by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
- with ge m \<open>0 < n\<close> show ?thesis
- apply (rule_tac x = "Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)" in bexI)
- apply (simp_all del: Complex_eq greaterThan_0)
- by blast
- qed
- qed
- show "g v \<notin> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
- (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
- if "v \<in> cball 0 1" for v
- using not01 [OF that]
- by (force simp: g [OF that, symmetric] h [OF that, symmetric] dest!: Schottky_lemma3 [of "g v"])
- qed
- then have 12: "(1 - t) * cmod (deriv g w) / 12 < 1"
- using Bloch_general [OF holg _ ttt, of 1] w by force
- have "g field_differentiable at w within cball 0 1"
- using holg w by (simp add: holomorphic_on_def)
- then have "g field_differentiable at w within ball 0 1"
- using ball_subset_cball field_differentiable_within_subset by blast
- with w have der_gw: "(g has_field_derivative deriv g w) (at w)"
- by (simp add: field_differentiable_within_open [of _ "ball 0 1"] field_differentiable_derivI)
- with DERIV_unique [OF der_gw] g' w have "deriv g w = g' w"
- by (metis open_ball at_within_open ball_subset_cball has_field_derivative_subset subsetCE)
- then show "cmod (g' w) \<le> 12 / (1 - t)"
- using g' 12 \<open>t < 1\<close> by (simp add: field_simps)
- qed
- then have "cmod (g z - g 0) \<le> 12 / (1 - t) * cmod z"
- using has_contour_integral_bound_linepath [OF int_g', of "12/(1 - t)"] assms
- by simp
- with \<open>cmod z \<le> t\<close> \<open>t < 1\<close> show ?thesis
- by (simp add: field_split_simps)
- qed
- have fz: "f z = (1 + cos(of_real pi * h z)) / 2"
- using h \<open>z \<in> ball 0 1\<close> by (auto simp: field_simps)
- have "cmod (f z) \<le> exp (cmod (complex_of_real pi * h z))"
- by (simp add: fz mult.commute norm_cos_plus1_le)
- also have "... \<le> exp (pi * exp (pi * (2 + 2 * r + 12 * t / (1 - t))))"
- proof (simp add: norm_mult)
- have "cmod (g z - g 0) \<le> 12 * t / (1 - t)"
- using norm_g_12 \<open>t < 1\<close> by (simp add: norm_mult)
- then have "cmod (g z) - cmod (g 0) \<le> 12 * t / (1 - t)"
- using norm_triangle_ineq2 order_trans by blast
- then have *: "cmod (g z) \<le> 2 + 2 * r + 12 * t / (1 - t)"
- using g0_2_f0 norm_ge_zero [of "f 0"] nof0
- by linarith
- have "cmod (h z) \<le> exp (cmod (complex_of_real pi * g z))"
- using \<open>z \<in> ball 0 1\<close> by (simp add: g norm_cos_le)
- also have "... \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))"
- using \<open>t < 1\<close> nof0 * by (simp add: norm_mult)
- finally show "cmod (h z) \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))" .
- qed
- finally show ?thesis .
-qed
-
-
-subsection\<open>The Little Picard Theorem\<close>
-
-theorem Landau_Picard:
- obtains R
- where "\<And>z. 0 < R z"
- "\<And>f. \<lbrakk>f holomorphic_on cball 0 (R(f 0));
- \<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1\<rbrakk> \<Longrightarrow> norm(deriv f 0) < 1"
-proof -
- define R where "R \<equiv> \<lambda>z. 3 * exp(pi * exp(pi*(2 + 2 * cmod z + 12)))"
- show ?thesis
- proof
- show Rpos: "\<And>z. 0 < R z"
- by (auto simp: R_def)
- show "norm(deriv f 0) < 1"
- if holf: "f holomorphic_on cball 0 (R(f 0))"
- and Rf: "\<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1" for f
- proof -
- let ?r = "R(f 0)"
- define g where "g \<equiv> f \<circ> (\<lambda>z. of_real ?r * z)"
- have "0 < ?r"
- using Rpos by blast
- have holg: "g holomorphic_on cball 0 1"
- unfolding g_def
- apply (intro holomorphic_intros holomorphic_on_compose holomorphic_on_subset [OF holf])
- using Rpos by (auto simp: less_imp_le norm_mult)
- have *: "norm(g z) \<le> exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12 * t / (1 - t))))"
- if "0 < t" "t < 1" "norm z \<le> t" for t z
- proof (rule Schottky [OF holg])
- show "cmod (g 0) \<le> cmod (f 0)"
- by (simp add: g_def)
- show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not> (g z = 0 \<or> g z = 1)"
- using Rpos by (simp add: g_def less_imp_le norm_mult Rf)
- qed (auto simp: that)
- have C1: "g holomorphic_on ball 0 (1 / 2)"
- by (rule holomorphic_on_subset [OF holg]) auto
- have C2: "continuous_on (cball 0 (1 / 2)) g"
- by (meson cball_divide_subset_numeral holg holomorphic_on_imp_continuous_on holomorphic_on_subset)
- have C3: "cmod (g z) \<le> R (f 0) / 3" if "cmod (0 - z) = 1/2" for z
- proof -
- have "norm(g z) \<le> exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12)))"
- using * [of "1/2"] that by simp
- also have "... = ?r / 3"
- by (simp add: R_def)
- finally show ?thesis .
- qed
- then have cmod_g'_le: "cmod (deriv g 0) * 3 \<le> R (f 0) * 2"
- using Cauchy_inequality [OF C1 C2 _ C3, of 1] by simp
- have holf': "f holomorphic_on ball 0 (R(f 0))"
- by (rule holomorphic_on_subset [OF holf]) auto
- then have fd0: "f field_differentiable at 0"
- by (rule holomorphic_on_imp_differentiable_at [OF _ open_ball])
- (auto simp: Rpos [of "f 0"])
- have g_eq: "deriv g 0 = of_real ?r * deriv f 0"
- apply (rule DERIV_imp_deriv)
- apply (simp add: g_def)
- by (metis DERIV_chain DERIV_cmult_Id fd0 field_differentiable_derivI mult.commute mult_zero_right)
- show ?thesis
- using cmod_g'_le Rpos [of "f 0"] by (simp add: g_eq norm_mult)
- qed
- qed
-qed
-
-lemma little_Picard_01:
- assumes holf: "f holomorphic_on UNIV" and f01: "\<And>z. f z \<noteq> 0 \<and> f z \<noteq> 1"
- obtains c where "f = (\<lambda>x. c)"
-proof -
- obtain R
- where Rpos: "\<And>z. 0 < R z"
- and R: "\<And>h. \<lbrakk>h holomorphic_on cball 0 (R(h 0));
- \<And>z. norm z \<le> R(h 0) \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1\<rbrakk> \<Longrightarrow> norm(deriv h 0) < 1"
- using Landau_Picard by metis
- have contf: "continuous_on UNIV f"
- by (simp add: holf holomorphic_on_imp_continuous_on)
- show ?thesis
- proof (cases "\<forall>x. deriv f x = 0")
- case True
- obtain c where "\<And>x. f(x) = c"
- apply (rule DERIV_zero_connected_constant [OF connected_UNIV open_UNIV finite.emptyI contf])
- apply (metis True DiffE holf holomorphic_derivI open_UNIV, auto)
- done
- then show ?thesis
- using that by auto
- next
- case False
- then obtain w where w: "deriv f w \<noteq> 0" by auto
- define fw where "fw \<equiv> (f \<circ> (\<lambda>z. w + z / deriv f w))"
- have norm_let1: "norm(deriv fw 0) < 1"
- proof (rule R)
- show "fw holomorphic_on cball 0 (R (fw 0))"
- unfolding fw_def
- by (intro holomorphic_intros holomorphic_on_compose w holomorphic_on_subset [OF holf] subset_UNIV)
- show "fw z \<noteq> 0 \<and> fw z \<noteq> 1" if "cmod z \<le> R (fw 0)" for z
- using f01 by (simp add: fw_def)
- qed
- have "(fw has_field_derivative deriv f w * inverse (deriv f w)) (at 0)"
- apply (simp add: fw_def)
- apply (rule DERIV_chain)
- using holf holomorphic_derivI apply force
- apply (intro derivative_eq_intros w)
- apply (auto simp: field_simps)
- done
- then show ?thesis
- using norm_let1 w by (simp add: DERIV_imp_deriv)
- qed
-qed
-
-
-theorem little_Picard:
- assumes holf: "f holomorphic_on UNIV"
- and "a \<noteq> b" "range f \<inter> {a,b} = {}"
- obtains c where "f = (\<lambda>x. c)"
-proof -
- let ?g = "\<lambda>x. 1/(b - a)*(f x - b) + 1"
- obtain c where "?g = (\<lambda>x. c)"
- proof (rule little_Picard_01)
- show "?g holomorphic_on UNIV"
- by (intro holomorphic_intros holf)
- show "\<And>z. ?g z \<noteq> 0 \<and> ?g z \<noteq> 1"
- using assms by (auto simp: field_simps)
- qed auto
- then have "?g x = c" for x
- by meson
- then have "f x = c * (b-a) + a" for x
- using assms by (auto simp: field_simps)
- then show ?thesis
- using that by blast
-qed
-
-
-text\<open>A couple of little applications of Little Picard\<close>
-
-lemma holomorphic_periodic_fixpoint:
- assumes holf: "f holomorphic_on UNIV"
- and "p \<noteq> 0" and per: "\<And>z. f(z + p) = f z"
- obtains x where "f x = x"
-proof -
- have False if non: "\<And>x. f x \<noteq> x"
- proof -
- obtain c where "(\<lambda>z. f z - z) = (\<lambda>z. c)"
- proof (rule little_Picard)
- show "(\<lambda>z. f z - z) holomorphic_on UNIV"
- by (simp add: holf holomorphic_on_diff)
- show "range (\<lambda>z. f z - z) \<inter> {p,0} = {}"
- using assms non by auto (metis add.commute diff_eq_eq)
- qed (auto simp: assms)
- with per show False
- by (metis add.commute add_cancel_left_left \<open>p \<noteq> 0\<close> diff_add_cancel)
- qed
- then show ?thesis
- using that by blast
-qed
-
-
-lemma holomorphic_involution_point:
- assumes holfU: "f holomorphic_on UNIV" and non: "\<And>a. f \<noteq> (\<lambda>x. a + x)"
- obtains x where "f(f x) = x"
-proof -
- { assume non_ff [simp]: "\<And>x. f(f x) \<noteq> x"
- then have non_fp [simp]: "f z \<noteq> z" for z
- by metis
- have holf: "f holomorphic_on X" for X
- using assms holomorphic_on_subset by blast
- obtain c where c: "(\<lambda>x. (f(f x) - x)/(f x - x)) = (\<lambda>x. c)"
- proof (rule little_Picard_01)
- show "(\<lambda>x. (f(f x) - x)/(f x - x)) holomorphic_on UNIV"
- apply (intro holomorphic_intros holf holomorphic_on_compose [unfolded o_def, OF holf])
- using non_fp by auto
- qed auto
- then obtain "c \<noteq> 0" "c \<noteq> 1"
- by (metis (no_types, hide_lams) non_ff diff_zero divide_eq_0_iff right_inverse_eq)
- have eq: "f(f x) - c * f x = x*(1 - c)" for x
- using fun_cong [OF c, of x] by (simp add: field_simps)
- have df_times_dff: "deriv f z * (deriv f (f z) - c) = 1 * (1 - c)" for z
- proof (rule DERIV_unique)
- show "((\<lambda>x. f (f x) - c * f x) has_field_derivative
- deriv f z * (deriv f (f z) - c)) (at z)"
- apply (intro derivative_eq_intros)
- apply (rule DERIV_chain [unfolded o_def, of f])
- apply (auto simp: algebra_simps intro!: holomorphic_derivI [OF holfU])
- done
- show "((\<lambda>x. f (f x) - c * f x) has_field_derivative 1 * (1 - c)) (at z)"
- by (simp add: eq mult_commute_abs)
- qed
- { fix z::complex
- obtain k where k: "deriv f \<circ> f = (\<lambda>x. k)"
- proof (rule little_Picard)
- show "(deriv f \<circ> f) holomorphic_on UNIV"
- by (meson holfU holomorphic_deriv holomorphic_on_compose holomorphic_on_subset open_UNIV subset_UNIV)
- obtain "deriv f (f x) \<noteq> 0" "deriv f (f x) \<noteq> c" for x
- using df_times_dff \<open>c \<noteq> 1\<close> eq_iff_diff_eq_0
- by (metis lambda_one mult_zero_left mult_zero_right)
- then show "range (deriv f \<circ> f) \<inter> {0,c} = {}"
- by force
- qed (use \<open>c \<noteq> 0\<close> in auto)
- have "\<not> f constant_on UNIV"
- by (meson UNIV_I non_ff constant_on_def)
- with holf open_mapping_thm have "open(range f)"
- by blast
- obtain l where l: "\<And>x. f x - k * x = l"
- proof (rule DERIV_zero_connected_constant [of UNIV "{}" "\<lambda>x. f x - k * x"], simp_all)
- have "deriv f w - k = 0" for w
- proof (rule analytic_continuation [OF _ open_UNIV connected_UNIV subset_UNIV, of "\<lambda>z. deriv f z - k" "f z" "range f" w])
- show "(\<lambda>z. deriv f z - k) holomorphic_on UNIV"
- by (intro holomorphic_intros holf open_UNIV)
- show "f z islimpt range f"
- by (metis (no_types, lifting) IntI UNIV_I \<open>open (range f)\<close> image_eqI inf.absorb_iff2 inf_aci(1) islimpt_UNIV islimpt_eq_acc_point open_Int top_greatest)
- show "\<And>z. z \<in> range f \<Longrightarrow> deriv f z - k = 0"
- by (metis comp_def diff_self image_iff k)
- qed auto
- moreover
- have "((\<lambda>x. f x - k * x) has_field_derivative deriv f x - k) (at x)" for x
- by (metis DERIV_cmult_Id Deriv.field_differentiable_diff UNIV_I field_differentiable_derivI holf holomorphic_on_def)
- ultimately
- show "\<forall>x. ((\<lambda>x. f x - k * x) has_field_derivative 0) (at x)"
- by auto
- show "continuous_on UNIV (\<lambda>x. f x - k * x)"
- by (simp add: continuous_on_diff holf holomorphic_on_imp_continuous_on)
- qed (auto simp: connected_UNIV)
- have False
- proof (cases "k=1")
- case True
- then have "\<exists>x. k * x + l \<noteq> a + x" for a
- using l non [of a] ext [of f "(+) a"]
- by (metis add.commute diff_eq_eq)
- with True show ?thesis by auto
- next
- case False
- have "\<And>x. (1 - k) * x \<noteq> f 0"
- using l [of 0] apply (simp add: algebra_simps)
- by (metis diff_add_cancel l mult.commute non_fp)
- then show False
- by (metis False eq_iff_diff_eq_0 mult.commute nonzero_mult_div_cancel_right times_divide_eq_right)
- qed
- }
- }
- then show thesis
- using that by blast
-qed
-
-
-subsection\<open>The Arzelà --Ascoli theorem\<close>
-
-lemma subsequence_diagonalization_lemma:
- fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
- assumes sub: "\<And>i r. \<exists>k. strict_mono (k :: nat \<Rightarrow> nat) \<and> P i (r \<circ> k)"
- and P_P: "\<And>i r::nat \<Rightarrow> 'a. \<And>k1 k2 N.
- \<lbrakk>P i (r \<circ> k1); \<And>j. N \<le> j \<Longrightarrow> \<exists>j'. j \<le> j' \<and> k2 j = k1 j'\<rbrakk> \<Longrightarrow> P i (r \<circ> k2)"
- obtains k where "strict_mono (k :: nat \<Rightarrow> nat)" "\<And>i. P i (r \<circ> k)"
-proof -
- obtain kk where "\<And>i r. strict_mono (kk i r :: nat \<Rightarrow> nat) \<and> P i (r \<circ> (kk i r))"
- using sub by metis
- then have sub_kk: "\<And>i r. strict_mono (kk i r)" and P_kk: "\<And>i r. P i (r \<circ> (kk i r))"
- by auto
- define rr where "rr \<equiv> rec_nat (kk 0 r) (\<lambda>n x. x \<circ> kk (Suc n) (r \<circ> x))"
- then have [simp]: "rr 0 = kk 0 r" "\<And>n. rr(Suc n) = rr n \<circ> kk (Suc n) (r \<circ> rr n)"
- by auto
- show thesis
- proof
- have sub_rr: "strict_mono (rr i)" for i
- using sub_kk by (induction i) (auto simp: strict_mono_def o_def)
- have P_rr: "P i (r \<circ> rr i)" for i
- using P_kk by (induction i) (auto simp: o_def)
- have "i \<le> i+d \<Longrightarrow> rr i n \<le> rr (i+d) n" for d i n
- proof (induction d)
- case 0 then show ?case
- by simp
- next
- case (Suc d) then show ?case
- apply simp
- using seq_suble [OF sub_kk] order_trans strict_mono_less_eq [OF sub_rr] by blast
- qed
- then have "\<And>i j n. i \<le> j \<Longrightarrow> rr i n \<le> rr j n"
- by (metis le_iff_add)
- show "strict_mono (\<lambda>n. rr n n)"
- apply (simp add: strict_mono_Suc_iff)
- by (meson lessI less_le_trans seq_suble strict_monoD sub_kk sub_rr)
- have "\<exists>j. i \<le> j \<and> rr (n+d) i = rr n j" for d n i
- apply (induction d arbitrary: i, auto)
- by (meson order_trans seq_suble sub_kk)
- then have "\<And>m n i. n \<le> m \<Longrightarrow> \<exists>j. i \<le> j \<and> rr m i = rr n j"
- by (metis le_iff_add)
- then show "P i (r \<circ> (\<lambda>n. rr n n))" for i
- by (meson P_rr P_P)
- qed
-qed
-
-lemma function_convergent_subsequence:
- fixes f :: "[nat,'a] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
- assumes "countable S" and M: "\<And>n::nat. \<And>x. x \<in> S \<Longrightarrow> norm(f n x) \<le> M"
- obtains k where "strict_mono (k::nat\<Rightarrow>nat)" "\<And>x. x \<in> S \<Longrightarrow> \<exists>l. (\<lambda>n. f (k n) x) \<longlonglongrightarrow> l"
-proof (cases "S = {}")
- case True
- then show ?thesis
- using strict_mono_id that by fastforce
-next
- case False
- with \<open>countable S\<close> obtain \<sigma> :: "nat \<Rightarrow> 'a" where \<sigma>: "S = range \<sigma>"
- using uncountable_def by blast
- obtain k where "strict_mono k" and k: "\<And>i. \<exists>l. (\<lambda>n. (f \<circ> k) n (\<sigma> i)) \<longlonglongrightarrow> l"
- proof (rule subsequence_diagonalization_lemma
- [of "\<lambda>i r. \<exists>l. ((\<lambda>n. (f \<circ> r) n (\<sigma> i)) \<longlongrightarrow> l) sequentially" id])
- show "\<exists>k::nat\<Rightarrow>nat. strict_mono k \<and> (\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k)) n (\<sigma> i)) \<longlonglongrightarrow> l)" for i r
- proof -
- have "f (r n) (\<sigma> i) \<in> cball 0 M" for n
- by (simp add: \<sigma> M)
- then show ?thesis
- using compact_def [of "cball (0::'b) M"] apply simp
- apply (drule_tac x="(\<lambda>n. f (r n) (\<sigma> i))" in spec)
- apply (force simp: o_def)
- done
- qed
- show "\<And>i r k1 k2 N.
- \<lbrakk>\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k1)) n (\<sigma> i)) \<longlonglongrightarrow> l; \<And>j. N \<le> j \<Longrightarrow> \<exists>j'\<ge>j. k2 j = k1 j'\<rbrakk>
- \<Longrightarrow> \<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k2)) n (\<sigma> i)) \<longlonglongrightarrow> l"
- apply (simp add: lim_sequentially)
- apply (erule ex_forward all_forward imp_forward)+
- apply auto
- by (metis (no_types, hide_lams) le_cases order_trans)
- qed auto
- with \<sigma> that show ?thesis
- by force
-qed
-
-
-theorem Arzela_Ascoli:
- fixes \<F> :: "[nat,'a::euclidean_space] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
- assumes "compact S"
- and M: "\<And>n x. x \<in> S \<Longrightarrow> norm(\<F> n x) \<le> M"
- and equicont:
- "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
- \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n y. y \<in> S \<and> norm(x - y) < d \<longrightarrow> norm(\<F> n x - \<F> n y) < e)"
- obtains g k where "continuous_on S g" "strict_mono (k :: nat \<Rightarrow> nat)"
- "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(k n) x - g x) < e"
-proof -
- have UEQ: "\<And>e. 0 < e \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e)"
- apply (rule compact_uniformly_equicontinuous [OF \<open>compact S\<close>, of "range \<F>"])
- using equicont by (force simp: dist_commute dist_norm)+
- have "continuous_on S g"
- if "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(r n) x - g x) < e"
- for g:: "'a \<Rightarrow> 'b" and r :: "nat \<Rightarrow> nat"
- proof (rule uniform_limit_theorem [of _ "\<F> \<circ> r"])
- show "\<forall>\<^sub>F n in sequentially. continuous_on S ((\<F> \<circ> r) n)"
- apply (simp add: eventually_sequentially)
- apply (rule_tac x=0 in exI)
- using UEQ apply (force simp: continuous_on_iff)
- done
- show "uniform_limit S (\<F> \<circ> r) g sequentially"
- apply (simp add: uniform_limit_iff eventually_sequentially)
- by (metis dist_norm that)
- qed auto
- moreover
- obtain R where "countable R" "R \<subseteq> S" and SR: "S \<subseteq> closure R"
- by (metis separable that)
- obtain k where "strict_mono k" and k: "\<And>x. x \<in> R \<Longrightarrow> \<exists>l. (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> l"
- apply (rule function_convergent_subsequence [OF \<open>countable R\<close> M])
- using \<open>R \<subseteq> S\<close> apply force+
- done
- then have Cauchy: "Cauchy ((\<lambda>n. \<F> (k n) x))" if "x \<in> R" for x
- using convergent_eq_Cauchy that by blast
- have "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> x \<in> S \<longrightarrow> dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
- if "0 < e" for e
- proof -
- obtain d where "0 < d"
- and d: "\<And>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e/3"
- by (metis UEQ \<open>0 < e\<close> divide_pos_pos zero_less_numeral)
- obtain T where "T \<subseteq> R" and "finite T" and T: "S \<subseteq> (\<Union>c\<in>T. ball c d)"
- proof (rule compactE_image [OF \<open>compact S\<close>, of R "(\<lambda>x. ball x d)"])
- have "closure R \<subseteq> (\<Union>c\<in>R. ball c d)"
- apply clarsimp
- using \<open>0 < d\<close> closure_approachable by blast
- with SR show "S \<subseteq> (\<Union>c\<in>R. ball c d)"
- by auto
- qed auto
- have "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (\<F> (k m) x) (\<F> (k n) x) < e/3" if "x \<in> R" for x
- using Cauchy \<open>0 < e\<close> that unfolding Cauchy_def
- by (metis less_divide_eq_numeral1(1) mult_zero_left)
- then obtain MF where MF: "\<And>x m n. \<lbrakk>x \<in> R; m \<ge> MF x; n \<ge> MF x\<rbrakk> \<Longrightarrow> norm (\<F> (k m) x - \<F> (k n) x) < e/3"
- using dist_norm by metis
- have "dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
- if m: "Max (MF ` T) \<le> m" and n: "Max (MF ` T) \<le> n" "x \<in> S" for m n x
- proof -
- obtain t where "t \<in> T" and t: "x \<in> ball t d"
- using \<open>x \<in> S\<close> T by auto
- have "norm(\<F> (k m) t - \<F> (k m) x) < e / 3"
- by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> d dist_norm mem_ball subset_iff t \<open>x \<in> S\<close>)
- moreover
- have "norm(\<F> (k n) t - \<F> (k n) x) < e / 3"
- by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> subsetD d dist_norm mem_ball t \<open>x \<in> S\<close>)
- moreover
- have "norm(\<F> (k m) t - \<F> (k n) t) < e / 3"
- proof (rule MF)
- show "t \<in> R"
- using \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> by blast
- show "MF t \<le> m" "MF t \<le> n"
- by (meson Max_ge \<open>finite T\<close> \<open>t \<in> T\<close> finite_imageI imageI le_trans m n)+
- qed
- ultimately
- show ?thesis
- unfolding dist_norm [symmetric] o_def
- by (metis dist_triangle_third dist_commute)
- qed
- then show ?thesis
- by force
- qed
- then have "\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> S. norm(\<F>(k n) x - g x) < e"
- using uniformly_convergent_eq_cauchy [of "\<lambda>x. x \<in> S" "\<F> \<circ> k"]
- apply (simp add: o_def dist_norm)
- by meson
- ultimately show thesis
- by (metis that \<open>strict_mono k\<close>)
-qed
-
-
-
-subsubsection\<^marker>\<open>tag important\<close>\<open>Montel's theorem\<close>
-
-text\<open>a sequence of holomorphic functions uniformly bounded
-on compact subsets of an open set S has a subsequence that converges to a
-holomorphic function, and converges \emph{uniformly} on compact subsets of S.\<close>
-
-
-theorem Montel:
- fixes \<F> :: "[nat,complex] \<Rightarrow> complex"
- assumes "open S"
- and \<H>: "\<And>h. h \<in> \<H> \<Longrightarrow> h holomorphic_on S"
- and bounded: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K. norm(h z) \<le> B"
- and rng_f: "range \<F> \<subseteq> \<H>"
- obtains g r
- where "g holomorphic_on S" "strict_mono (r :: nat \<Rightarrow> nat)"
- "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>n. \<F> (r n) x) \<longlongrightarrow> g x) sequentially"
- "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) g sequentially"
-proof -
- obtain K where comK: "\<And>n. compact(K n)" and KS: "\<And>n::nat. K n \<subseteq> S"
- and subK: "\<And>X. \<lbrakk>compact X; X \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X \<subseteq> K n"
- using open_Union_compact_subsets [OF \<open>open S\<close>] by metis
- then have "\<And>i. \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K i. norm(h z) \<le> B"
- by (simp add: bounded)
- then obtain B where B: "\<And>i h z. \<lbrakk>h \<in> \<H>; z \<in> K i\<rbrakk> \<Longrightarrow> norm(h z) \<le> B i"
- by metis
- have *: "\<exists>r g. strict_mono (r::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r) n x - g x) < e)"
- if "\<And>n. \<F> n \<in> \<H>" for \<F> i
- proof -
- obtain g k where "continuous_on (K i) g" "strict_mono (k::nat\<Rightarrow>nat)"
- "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm(\<F>(k n) x - g x) < e"
- proof (rule Arzela_Ascoli [of "K i" "\<F>" "B i"])
- show "\<exists>d>0. \<forall>n y. y \<in> K i \<and> cmod (z - y) < d \<longrightarrow> cmod (\<F> n z - \<F> n y) < e"
- if z: "z \<in> K i" and "0 < e" for z e
- proof -
- obtain r where "0 < r" and r: "cball z r \<subseteq> S"
- using z KS [of i] \<open>open S\<close> by (force simp: open_contains_cball)
- have "cball z (2 / 3 * r) \<subseteq> cball z r"
- using \<open>0 < r\<close> by (simp add: cball_subset_cball_iff)
- then have z23S: "cball z (2 / 3 * r) \<subseteq> S"
- using r by blast
- obtain M where "0 < M" and M: "\<And>n w. dist z w \<le> 2/3 * r \<Longrightarrow> norm(\<F> n w) \<le> M"
- proof -
- obtain N where N: "\<forall>n\<ge>N. cball z (2/3 * r) \<subseteq> K n"
- using subK compact_cball [of z "(2 / 3 * r)"] z23S by force
- have "cmod (\<F> n w) \<le> \<bar>B N\<bar> + 1" if "dist z w \<le> 2 / 3 * r" for n w
- proof -
- have "w \<in> K N"
- using N mem_cball that by blast
- then have "cmod (\<F> n w) \<le> B N"
- using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
- also have "... \<le> \<bar>B N\<bar> + 1"
- by simp
- finally show ?thesis .
- qed
- then show ?thesis
- by (rule_tac M="\<bar>B N\<bar> + 1" in that) auto
- qed
- have "cmod (\<F> n z - \<F> n y) < e"
- if "y \<in> K i" and y_near_z: "cmod (z - y) < r/3" "cmod (z - y) < e * r / (6 * M)"
- for n y
- proof -
- have "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral
- (2 * pi) * \<i> * winding_number (circlepath z (2 / 3 * r)) \<xi> * \<F> n \<xi>)
- (circlepath z (2 / 3 * r))"
- if "dist \<xi> z < (2 / 3 * r)" for \<xi>
- proof (rule Cauchy_integral_formula_convex_simple)
- have "\<F> n holomorphic_on S"
- by (simp add: \<H> \<open>\<And>n. \<F> n \<in> \<H>\<close>)
- with z23S show "\<F> n holomorphic_on cball z (2 / 3 * r)"
- using holomorphic_on_subset by blast
- qed (use that \<open>0 < r\<close> in \<open>auto simp: dist_commute\<close>)
- then have *: "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral (2 * pi) * \<i> * \<F> n \<xi>)
- (circlepath z (2 / 3 * r))"
- if "dist \<xi> z < (2 / 3 * r)" for \<xi>
- using that by (simp add: winding_number_circlepath dist_norm)
- have y: "((\<lambda>w. \<F> n w / (w - y)) has_contour_integral (2 * pi) * \<i> * \<F> n y)
- (circlepath z (2 / 3 * r))"
- apply (rule *)
- using that \<open>0 < r\<close> by (simp only: dist_norm norm_minus_commute)
- have z: "((\<lambda>w. \<F> n w / (w - z)) has_contour_integral (2 * pi) * \<i> * \<F> n z)
- (circlepath z (2 / 3 * r))"
- apply (rule *)
- using \<open>0 < r\<close> by simp
- have le_er: "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) \<le> e / r"
- if "cmod (x - z) = r/3 + r/3" for x
- proof -
- have "\<not> (cmod (x - y) < r/3)"
- using y_near_z(1) that \<open>M > 0\<close> \<open>r > 0\<close>
- by (metis (full_types) norm_diff_triangle_less norm_minus_commute order_less_irrefl)
- then have r4_le_xy: "r/4 \<le> cmod (x - y)"
- using \<open>r > 0\<close> by simp
- then have neq: "x \<noteq> y" "x \<noteq> z"
- using that \<open>r > 0\<close> by (auto simp: field_split_simps norm_minus_commute)
- have leM: "cmod (\<F> n x) \<le> M"
- by (simp add: M dist_commute dist_norm that)
- have "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) = cmod (\<F> n x) * cmod (1 / (x - y) - 1 / (x - z))"
- by (metis (no_types, lifting) divide_inverse mult.left_neutral norm_mult right_diff_distrib')
- also have "... = cmod (\<F> n x) * cmod ((y - z) / ((x - y) * (x - z)))"
- using neq by (simp add: field_split_simps)
- also have "... = cmod (\<F> n x) * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
- by (simp add: norm_mult norm_divide that)
- also have "... \<le> M * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
- apply (rule mult_mono)
- apply (rule leM)
- using \<open>r > 0\<close> \<open>M > 0\<close> neq by auto
- also have "... < M * ((e * r / (6 * M)) / (cmod(x - y) * (2/3 * r)))"
- unfolding mult_less_cancel_left
- using y_near_z(2) \<open>M > 0\<close> \<open>r > 0\<close> neq
- apply (simp add: field_simps mult_less_0_iff norm_minus_commute)
- done
- also have "... \<le> e/r"
- using \<open>e > 0\<close> \<open>r > 0\<close> r4_le_xy by (simp add: field_split_simps)
- finally show ?thesis by simp
- qed
- have "(2 * pi) * cmod (\<F> n y - \<F> n z) = cmod ((2 * pi) * \<i> * \<F> n y - (2 * pi) * \<i> * \<F> n z)"
- by (simp add: right_diff_distrib [symmetric] norm_mult)
- also have "cmod ((2 * pi) * \<i> * \<F> n y - (2 * pi) * \<i> * \<F> n z) \<le> e / r * (2 * pi * (2 / 3 * r))"
- apply (rule has_contour_integral_bound_circlepath [OF has_contour_integral_diff [OF y z], of "e/r"])
- using \<open>e > 0\<close> \<open>r > 0\<close> le_er by auto
- also have "... = (2 * pi) * e * ((2 / 3))"
- using \<open>r > 0\<close> by (simp add: field_split_simps)
- finally have "cmod (\<F> n y - \<F> n z) \<le> e * (2 / 3)"
- by simp
- also have "... < e"
- using \<open>e > 0\<close> by simp
- finally show ?thesis by (simp add: norm_minus_commute)
- qed
- then show ?thesis
- apply (rule_tac x="min (r/3) ((e * r)/(6 * M))" in exI)
- using \<open>0 < e\<close> \<open>0 < r\<close> \<open>0 < M\<close> by simp
- qed
- show "\<And>n x. x \<in> K i \<Longrightarrow> cmod (\<F> n x) \<le> B i"
- using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
- qed (use comK in \<open>fastforce+\<close>)
- then show ?thesis
- by fastforce
- qed
- have "\<exists>k g. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r \<circ> k) n x - g x) < e)"
- for i r
- apply (rule *)
- using rng_f by auto
- then have **: "\<And>i r. \<exists>k. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> (r \<circ> k)) n x - g x) < e)"
- by (force simp: o_assoc)
- obtain k :: "nat \<Rightarrow> nat" where "strict_mono k"
- and "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> (id \<circ> k)) n x - g x) < e"
- apply (rule subsequence_diagonalization_lemma [OF **, of id])
- apply (erule ex_forward all_forward imp_forward)+
- apply auto
- apply (rule_tac x="max N Na" in exI, fastforce+)
- done
- then have lt_e: "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - g x) < e"
- by simp
- have "\<exists>l. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - l) < e" if "z \<in> S" for z
- proof -
- obtain G where G: "\<And>i e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - G i x) < e"
- using lt_e by metis
- obtain N where "\<And>n. n \<ge> N \<Longrightarrow> z \<in> K n"
- using subK [of "{z}"] that \<open>z \<in> S\<close> by auto
- moreover have "\<And>e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> k) n x - G N x) < e"
- using G by auto
- ultimately show ?thesis
- by (metis comp_apply order_refl)
- qed
- then obtain g where g: "\<And>z e. \<lbrakk>z \<in> S; e > 0\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - g z) < e"
- by metis
- show ?thesis
- proof
- show g_lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> g x"
- by (simp add: lim_sequentially g dist_norm)
- have dg_le_e: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>T. cmod (\<F> (k n) x - g x) < e"
- if T: "compact T" "T \<subseteq> S" and "0 < e" for T e
- proof -
- obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> T \<subseteq> K n"
- using subK [OF T] by blast
- obtain h where h: "\<And>e. e>0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> k) n x - h x) < e"
- using lt_e by blast
- have geq: "g w = h w" if "w \<in> T" for w
- apply (rule LIMSEQ_unique [of "\<lambda>n. \<F>(k n) w"])
- using \<open>T \<subseteq> S\<close> g_lim that apply blast
- using h N that by (force simp: lim_sequentially dist_norm)
- show ?thesis
- using T h N \<open>0 < e\<close> by (fastforce simp add: geq)
- qed
- then show "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk>
- \<Longrightarrow> uniform_limit K (\<F> \<circ> k) g sequentially"
- by (simp add: uniform_limit_iff dist_norm eventually_sequentially)
- show "g holomorphic_on S"
- proof (rule holomorphic_uniform_sequence [OF \<open>open S\<close> \<H>])
- show "\<And>n. (\<F> \<circ> k) n \<in> \<H>"
- by (simp add: range_subsetD rng_f)
- show "\<exists>d>0. cball z d \<subseteq> S \<and> uniform_limit (cball z d) (\<lambda>n. (\<F> \<circ> k) n) g sequentially"
- if "z \<in> S" for z
- proof -
- obtain d where d: "d>0" "cball z d \<subseteq> S"
- using \<open>open S\<close> \<open>z \<in> S\<close> open_contains_cball by blast
- then have "uniform_limit (cball z d) (\<F> \<circ> k) g sequentially"
- using dg_le_e compact_cball by (auto simp: uniform_limit_iff eventually_sequentially dist_norm)
- with d show ?thesis by blast
- qed
- qed
- qed (auto simp: \<open>strict_mono k\<close>)
-qed
-
-
-
-subsection\<open>Some simple but useful cases of Hurwitz's theorem\<close>
-
-proposition Hurwitz_no_zeros:
- assumes S: "open S" "connected S"
- and holf: "\<And>n::nat. \<F> n holomorphic_on S"
- and holg: "g holomorphic_on S"
- and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
- and nonconst: "\<not> g constant_on S"
- and nz: "\<And>n z. z \<in> S \<Longrightarrow> \<F> n z \<noteq> 0"
- and "z0 \<in> S"
- shows "g z0 \<noteq> 0"
-proof
- assume g0: "g z0 = 0"
- obtain h r m
- where "0 < m" "0 < r" and subS: "ball z0 r \<subseteq> S"
- and holh: "h holomorphic_on ball z0 r"
- and geq: "\<And>w. w \<in> ball z0 r \<Longrightarrow> g w = (w - z0)^m * h w"
- and hnz: "\<And>w. w \<in> ball z0 r \<Longrightarrow> h w \<noteq> 0"
- by (blast intro: holomorphic_factor_zero_nonconstant [OF holg S \<open>z0 \<in> S\<close> g0 nonconst])
- then have holf0: "\<F> n holomorphic_on ball z0 r" for n
- by (meson holf holomorphic_on_subset)
- have *: "((\<lambda>z. deriv (\<F> n) z / \<F> n z) has_contour_integral 0) (circlepath z0 (r/2))" for n
- proof (rule Cauchy_theorem_disc_simple [of _ z0 r])
- show "(\<lambda>z. deriv (\<F> n) z / \<F> n z) holomorphic_on ball z0 r"
- apply (intro holomorphic_intros holomorphic_deriv holf holf0 open_ball nz)
- using \<open>ball z0 r \<subseteq> S\<close> by blast
- qed (use \<open>0 < r\<close> in auto)
- have hol_dg: "deriv g holomorphic_on S"
- by (simp add: \<open>open S\<close> holg holomorphic_deriv)
- have "continuous_on (sphere z0 (r/2)) (deriv g)"
- apply (intro holomorphic_on_imp_continuous_on holomorphic_on_subset [OF hol_dg])
- using \<open>0 < r\<close> subS by auto
- then have "compact (deriv g ` (sphere z0 (r/2)))"
- by (rule compact_continuous_image [OF _ compact_sphere])
- then have bo_dg: "bounded (deriv g ` (sphere z0 (r/2)))"
- using compact_imp_bounded by blast
- have "continuous_on (sphere z0 (r/2)) (cmod \<circ> g)"
- apply (intro continuous_intros holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holg])
- using \<open>0 < r\<close> subS by auto
- then have "compact ((cmod \<circ> g) ` sphere z0 (r/2))"
- by (rule compact_continuous_image [OF _ compact_sphere])
- moreover have "(cmod \<circ> g) ` sphere z0 (r/2) \<noteq> {}"
- using \<open>0 < r\<close> by auto
- ultimately obtain b where b: "b \<in> (cmod \<circ> g) ` sphere z0 (r/2)"
- "\<And>t. t \<in> (cmod \<circ> g) ` sphere z0 (r/2) \<Longrightarrow> b \<le> t"
- using compact_attains_inf [of "(norm \<circ> g) ` (sphere z0 (r/2))"] by blast
- have "(\<lambda>n. contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv (\<F> n) z / \<F> n z)) \<longlonglongrightarrow>
- contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
- proof (rule contour_integral_uniform_limit_circlepath)
- show "\<forall>\<^sub>F n in sequentially. (\<lambda>z. deriv (\<F> n) z / \<F> n z) contour_integrable_on circlepath z0 (r/2)"
- using * contour_integrable_on_def eventually_sequentiallyI by meson
- show "uniform_limit (sphere z0 (r/2)) (\<lambda>n z. deriv (\<F> n) z / \<F> n z) (\<lambda>z. deriv g z / g z) sequentially"
- proof (rule uniform_lim_divide [OF _ _ bo_dg])
- show "uniform_limit (sphere z0 (r/2)) (\<lambda>a. deriv (\<F> a)) (deriv g) sequentially"
- proof (rule uniform_limitI)
- fix e::real
- assume "0 < e"
- have *: "dist (deriv (\<F> n) w) (deriv g w) < e"
- if e8: "\<And>x. dist z0 x \<le> 3 * r / 4 \<Longrightarrow> dist (\<F> n x) (g x) * 8 < r * e"
- and w: "dist w z0 = r/2" for n w
- proof -
- have "ball w (r/4) \<subseteq> ball z0 r" "cball w (r/4) \<subseteq> ball z0 r"
- using \<open>0 < r\<close> by (simp_all add: ball_subset_ball_iff cball_subset_ball_iff w)
- with subS have wr4_sub: "ball w (r/4) \<subseteq> S" "cball w (r/4) \<subseteq> S" by force+
- moreover
- have "(\<lambda>z. \<F> n z - g z) holomorphic_on S"
- by (intro holomorphic_intros holf holg)
- ultimately have hol: "(\<lambda>z. \<F> n z - g z) holomorphic_on ball w (r/4)"
- and cont: "continuous_on (cball w (r / 4)) (\<lambda>z. \<F> n z - g z)"
- using holomorphic_on_subset by (blast intro: holomorphic_on_imp_continuous_on)+
- have "w \<in> S"
- using \<open>0 < r\<close> wr4_sub by auto
- have "\<And>y. dist w y < r / 4 \<Longrightarrow> dist z0 y \<le> 3 * r / 4"
- apply (rule dist_triangle_le [where z=w])
- using w by (simp add: dist_commute)
- with e8 have in_ball: "\<And>y. y \<in> ball w (r/4) \<Longrightarrow> \<F> n y - g y \<in> ball 0 (r/4 * e/2)"
- by (simp add: dist_norm [symmetric])
- have "\<F> n field_differentiable at w"
- by (metis holomorphic_on_imp_differentiable_at \<open>w \<in> S\<close> holf \<open>open S\<close>)
- moreover
- have "g field_differentiable at w"
- using \<open>w \<in> S\<close> \<open>open S\<close> holg holomorphic_on_imp_differentiable_at by auto
- moreover
- have "cmod (deriv (\<lambda>w. \<F> n w - g w) w) * 2 \<le> e"
- apply (rule Cauchy_higher_deriv_bound [OF hol cont in_ball, of 1, simplified])
- using \<open>r > 0\<close> by auto
- ultimately have "dist (deriv (\<F> n) w) (deriv g w) \<le> e/2"
- by (simp add: dist_norm)
- then show ?thesis
- using \<open>e > 0\<close> by auto
- qed
- have "cball z0 (3 * r / 4) \<subseteq> ball z0 r"
- by (simp add: cball_subset_ball_iff \<open>0 < r\<close>)
- with subS have "uniform_limit (cball z0 (3 * r/4)) \<F> g sequentially"
- by (force intro: ul_g)
- then have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>cball z0 (3 * r / 4). dist (\<F> n x) (g x) < r / 4 * e / 2"
- using \<open>0 < e\<close> \<open>0 < r\<close> by (force simp: intro!: uniform_limitD)
- then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> sphere z0 (r/2). dist (deriv (\<F> n) x) (deriv g x) < e"
- apply (simp add: eventually_sequentially)
- apply (elim ex_forward all_forward imp_forward asm_rl)
- using * apply (force simp: dist_commute)
- done
- qed
- show "uniform_limit (sphere z0 (r/2)) \<F> g sequentially"
- proof (rule uniform_limitI)
- fix e::real
- assume "0 < e"
- have "sphere z0 (r/2) \<subseteq> ball z0 r"
- using \<open>0 < r\<close> by auto
- with subS have "uniform_limit (sphere z0 (r/2)) \<F> g sequentially"
- by (force intro: ul_g)
- then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> sphere z0 (r/2). dist (\<F> n x) (g x) < e"
- apply (rule uniform_limitD)
- using \<open>0 < e\<close> by force
- qed
- show "b > 0" "\<And>x. x \<in> sphere z0 (r/2) \<Longrightarrow> b \<le> cmod (g x)"
- using b \<open>0 < r\<close> by (fastforce simp: geq hnz)+
- qed
- qed (use \<open>0 < r\<close> in auto)
- then have "(\<lambda>n. 0) \<longlonglongrightarrow> contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
- by (simp add: contour_integral_unique [OF *])
- then have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) = 0"
- by (simp add: LIMSEQ_const_iff)
- moreover
- have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) =
- contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z)"
- proof (rule contour_integral_eq, use \<open>0 < r\<close> in simp)
- fix w
- assume w: "dist z0 w * 2 = r"
- then have w_inb: "w \<in> ball z0 r"
- using \<open>0 < r\<close> by auto
- have h_der: "(h has_field_derivative deriv h w) (at w)"
- using holh holomorphic_derivI w_inb by blast
- have "deriv g w = ((of_nat m * h w + deriv h w * (w - z0)) * (w - z0) ^ m) / (w - z0)"
- if "r = dist z0 w * 2" "w \<noteq> z0"
- proof -
- have "((\<lambda>w. (w - z0) ^ m * h w) has_field_derivative
- (m * h w + deriv h w * (w - z0)) * (w - z0) ^ m / (w - z0)) (at w)"
- apply (rule derivative_eq_intros h_der refl)+
- using that \<open>m > 0\<close> \<open>0 < r\<close> apply (simp add: divide_simps distrib_right)
- apply (metis Suc_pred mult.commute power_Suc)
- done
- then show ?thesis
- apply (rule DERIV_imp_deriv [OF has_field_derivative_transform_within_open [where S = "ball z0 r"]])
- using that \<open>m > 0\<close> \<open>0 < r\<close>
- apply (simp_all add: hnz geq)
- done
- qed
- with \<open>0 < r\<close> \<open>0 < m\<close> w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
- by (auto simp: geq field_split_simps hnz)
- qed
- moreover
- have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z) =
- 2 * of_real pi * \<i> * m + 0"
- proof (rule contour_integral_unique [OF has_contour_integral_add])
- show "((\<lambda>x. m / (x - z0)) has_contour_integral 2 * of_real pi * \<i> * m) (circlepath z0 (r/2))"
- by (force simp: \<open>0 < r\<close> intro: Cauchy_integral_circlepath_simple)
- show "((\<lambda>x. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
- apply (rule Cauchy_theorem_disc_simple [of _ z0 r])
- using hnz holh holomorphic_deriv holomorphic_on_divide \<open>0 < r\<close>
- apply force+
- done
- qed
- ultimately show False using \<open>0 < m\<close> by auto
-qed
-
-corollary Hurwitz_injective:
- assumes S: "open S" "connected S"
- and holf: "\<And>n::nat. \<F> n holomorphic_on S"
- and holg: "g holomorphic_on S"
- and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
- and nonconst: "\<not> g constant_on S"
- and inj: "\<And>n. inj_on (\<F> n) S"
- shows "inj_on g S"
-proof -
- have False if z12: "z1 \<in> S" "z2 \<in> S" "z1 \<noteq> z2" "g z2 = g z1" for z1 z2
- proof -
- obtain z0 where "z0 \<in> S" and z0: "g z0 \<noteq> g z2"
- using constant_on_def nonconst by blast
- have "(\<lambda>z. g z - g z1) holomorphic_on S"
- by (intro holomorphic_intros holg)
- then obtain r where "0 < r" "ball z2 r \<subseteq> S" "\<And>z. dist z2 z < r \<and> z \<noteq> z2 \<Longrightarrow> g z \<noteq> g z1"
- apply (rule isolated_zeros [of "\<lambda>z. g z - g z1" S z2 z0])
- using S \<open>z0 \<in> S\<close> z0 z12 by auto
- have "g z2 - g z1 \<noteq> 0"
- proof (rule Hurwitz_no_zeros [of "S - {z1}" "\<lambda>n z. \<F> n z - \<F> n z1" "\<lambda>z. g z - g z1"])
- show "open (S - {z1})"
- by (simp add: S open_delete)
- show "connected (S - {z1})"
- by (simp add: connected_open_delete [OF S])
- show "\<And>n. (\<lambda>z. \<F> n z - \<F> n z1) holomorphic_on S - {z1}"
- by (intro holomorphic_intros holomorphic_on_subset [OF holf]) blast
- show "(\<lambda>z. g z - g z1) holomorphic_on S - {z1}"
- by (intro holomorphic_intros holomorphic_on_subset [OF holg]) blast
- show "uniform_limit K (\<lambda>n z. \<F> n z - \<F> n z1) (\<lambda>z. g z - g z1) sequentially"
- if "compact K" "K \<subseteq> S - {z1}" for K
- proof (rule uniform_limitI)
- fix e::real
- assume "e > 0"
- have "uniform_limit K \<F> g sequentially"
- using that ul_g by fastforce
- then have K: "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> K. dist (\<F> n x) (g x) < e/2"
- using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
- have "uniform_limit {z1} \<F> g sequentially"
- by (simp add: ul_g z12)
- then have "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> {z1}. dist (\<F> n x) (g x) < e/2"
- using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
- then have z1: "\<forall>\<^sub>F n in sequentially. dist (\<F> n z1) (g z1) < e/2"
- by simp
- have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e/2 + e/2"
- apply (rule eventually_mono [OF eventually_conj [OF K z1]])
- apply (simp add: dist_norm algebra_simps del: divide_const_simps)
- by (metis add.commute dist_commute dist_norm dist_triangle_add_half)
- have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e/2 + e/2"
- using eventually_conj [OF K z1]
- apply (rule eventually_mono)
- by (metis (no_types, hide_lams) diff_add_eq diff_diff_eq2 dist_commute dist_norm dist_triangle_add_half field_sum_of_halves)
- then
- show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e"
- by simp
- qed
- show "\<not> (\<lambda>z. g z - g z1) constant_on S - {z1}"
- unfolding constant_on_def
- by (metis Diff_iff \<open>z0 \<in> S\<close> empty_iff insert_iff right_minus_eq z0 z12)
- show "\<And>n z. z \<in> S - {z1} \<Longrightarrow> \<F> n z - \<F> n z1 \<noteq> 0"
- by (metis DiffD1 DiffD2 eq_iff_diff_eq_0 inj inj_onD insertI1 \<open>z1 \<in> S\<close>)
- show "z2 \<in> S - {z1}"
- using \<open>z2 \<in> S\<close> \<open>z1 \<noteq> z2\<close> by auto
- qed
- with z12 show False by auto
- qed
- then show ?thesis by (auto simp: inj_on_def)
-qed
-
-
-
-subsection\<open>The Great Picard theorem\<close>
-
-lemma GPicard1:
- assumes S: "open S" "connected S" and "w \<in> S" "0 < r" "Y \<subseteq> X"
- and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
- and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
- and r: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> r"
- obtains B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S" "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
-proof -
- obtain e where "e > 0" and e: "cball w e \<subseteq> S"
- using assms open_contains_cball_eq by blast
- show ?thesis
- proof
- show "0 < exp(pi * exp(pi * (2 + 2 * r + 12)))"
- by simp
- show "ball w (e / 2) \<subseteq> S"
- using e ball_divide_subset_numeral ball_subset_cball by blast
- show "cmod (h z) \<le> exp (pi * exp (pi * (2 + 2 * r + 12)))"
- if "h \<in> Y" "z \<in> ball w (e / 2)" for h z
- proof -
- have "h \<in> X"
- using \<open>Y \<subseteq> X\<close> \<open>h \<in> Y\<close> by blast
- with holX have "h holomorphic_on S"
- by auto
- then have "h holomorphic_on cball w e"
- by (metis e holomorphic_on_subset)
- then have hol_h_o: "(h \<circ> (\<lambda>z. (w + of_real e * z))) holomorphic_on cball 0 1"
- apply (intro holomorphic_intros holomorphic_on_compose)
- apply (erule holomorphic_on_subset)
- using that \<open>e > 0\<close> by (auto simp: dist_norm norm_mult)
- have norm_le_r: "cmod ((h \<circ> (\<lambda>z. w + complex_of_real e * z)) 0) \<le> r"
- by (auto simp: r \<open>h \<in> Y\<close>)
- have le12: "norm (of_real(inverse e) * (z - w)) \<le> 1/2"
- using that \<open>e > 0\<close> by (simp add: inverse_eq_divide dist_norm norm_minus_commute norm_divide)
- have non01: "\<And>z::complex. cmod z \<le> 1 \<Longrightarrow> h (w + e * z) \<noteq> 0 \<and> h (w + e * z) \<noteq> 1"
- apply (rule X01 [OF \<open>h \<in> X\<close>])
- apply (rule subsetD [OF e])
- using \<open>0 < e\<close> by (auto simp: dist_norm norm_mult)
- have "cmod (h z) \<le> cmod (h (w + of_real e * (inverse e * (z - w))))"
- using \<open>0 < e\<close> by (simp add: field_split_simps)
- also have "... \<le> exp (pi * exp (pi * (14 + 2 * r)))"
- using r [OF \<open>h \<in> Y\<close>] Schottky [OF hol_h_o norm_le_r _ _ _ le12] non01 by auto
- finally
- show ?thesis by simp
- qed
- qed (use \<open>e > 0\<close> in auto)
-qed
-
-lemma GPicard2:
- assumes "S \<subseteq> T" "connected T" "S \<noteq> {}" "open S" "\<And>x. \<lbrakk>x islimpt S; x \<in> T\<rbrakk> \<Longrightarrow> x \<in> S"
- shows "S = T"
- by (metis assms open_subset connected_clopen closedin_limpt)
-
-
-lemma GPicard3:
- assumes S: "open S" "connected S" "w \<in> S" and "Y \<subseteq> X"
- and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
- and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
- and no_hw_le1: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> 1"
- and "compact K" "K \<subseteq> S"
- obtains B where "\<And>h z. \<lbrakk>h \<in> Y; z \<in> K\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
-proof -
- define U where "U \<equiv> {z \<in> S. \<exists>B Z. 0 < B \<and> open Z \<and> z \<in> Z \<and> Z \<subseteq> S \<and>
- (\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B)}"
- then have "U \<subseteq> S" by blast
- have "U = S"
- proof (rule GPicard2 [OF \<open>U \<subseteq> S\<close> \<open>connected S\<close>])
- show "U \<noteq> {}"
- proof -
- obtain B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S"
- and "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
- apply (rule GPicard1 [OF S zero_less_one \<open>Y \<subseteq> X\<close> holX])
- using no_hw_le1 X01 by force+
- then show ?thesis
- unfolding U_def using \<open>w \<in> S\<close> by blast
- qed
- show "open U"
- unfolding open_subopen [of U] by (auto simp: U_def)
- fix v
- assume v: "v islimpt U" "v \<in> S"
- have "\<not> (\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v))"
- proof
- assume "\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v)"
- then have "\<forall>n. \<exists>h\<in>Y. Suc n < cmod (h v)"
- by simp
- then obtain \<F> where FY: "\<And>n. \<F> n \<in> Y" and ltF: "\<And>n. Suc n < cmod (\<F> n v)"
- by metis
- define \<G> where "\<G> \<equiv> \<lambda>n z. inverse(\<F> n z)"
- have hol\<G>: "\<G> n holomorphic_on S" for n
- apply (simp add: \<G>_def)
- using FY X01 \<open>Y \<subseteq> X\<close> holX apply (blast intro: holomorphic_on_inverse)
- done
- have \<G>not0: "\<G> n z \<noteq> 0" and \<G>not1: "\<G> n z \<noteq> 1" if "z \<in> S" for n z
- using FY X01 \<open>Y \<subseteq> X\<close> that by (force simp: \<G>_def)+
- have \<G>_le1: "cmod (\<G> n v) \<le> 1" for n
- using less_le_trans linear ltF
- by (fastforce simp add: \<G>_def norm_inverse inverse_le_1_iff)
- define W where "W \<equiv> {h. h holomorphic_on S \<and> (\<forall>z \<in> S. h z \<noteq> 0 \<and> h z \<noteq> 1)}"
- obtain B Z where "0 < B" "open Z" "v \<in> Z" "Z \<subseteq> S"
- and B: "\<And>h z. \<lbrakk>h \<in> range \<G>; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
- apply (rule GPicard1 [OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> zero_less_one, of "range \<G>" W])
- using hol\<G> \<G>not0 \<G>not1 \<G>_le1 by (force simp: W_def)+
- then obtain e where "e > 0" and e: "ball v e \<subseteq> Z"
- by (meson open_contains_ball)
- obtain h j where holh: "h holomorphic_on ball v e" and "strict_mono j"
- and lim: "\<And>x. x \<in> ball v e \<Longrightarrow> (\<lambda>n. \<G> (j n) x) \<longlonglongrightarrow> h x"
- and ulim: "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk>
- \<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
- proof (rule Montel)
- show "\<And>h. h \<in> range \<G> \<Longrightarrow> h holomorphic_on ball v e"
- by (metis \<open>Z \<subseteq> S\<close> e hol\<G> holomorphic_on_subset imageE)
- show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>range \<G>. \<forall>z\<in>K. cmod (h z) \<le> B"
- using B e by blast
- qed auto
- have "h v = 0"
- proof (rule LIMSEQ_unique)
- show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> h v"
- using \<open>e > 0\<close> lim by simp
- have lt_Fj: "real x \<le> cmod (\<F> (j x) v)" for x
- by (metis of_nat_Suc ltF \<open>strict_mono j\<close> add.commute less_eq_real_def less_le_trans nat_le_real_less seq_suble)
- show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> 0"
- proof (rule Lim_null_comparison [OF eventually_sequentiallyI lim_inverse_n])
- show "cmod (\<G> (j x) v) \<le> inverse (real x)" if "1 \<le> x" for x
- using that by (simp add: \<G>_def norm_inverse_le_norm [OF lt_Fj])
- qed
- qed
- have "h v \<noteq> 0"
- proof (rule Hurwitz_no_zeros [of "ball v e" "\<G> \<circ> j" h])
- show "\<And>n. (\<G> \<circ> j) n holomorphic_on ball v e"
- using \<open>Z \<subseteq> S\<close> e hol\<G> by force
- show "\<And>n z. z \<in> ball v e \<Longrightarrow> (\<G> \<circ> j) n z \<noteq> 0"
- using \<G>not0 \<open>Z \<subseteq> S\<close> e by fastforce
- show "\<not> h constant_on ball v e"
- proof (clarsimp simp: constant_on_def)
- fix c
- have False if "\<And>z. dist v z < e \<Longrightarrow> h z = c"
- proof -
- have "h v = c"
- by (simp add: \<open>0 < e\<close> that)
- obtain y where "y \<in> U" "y \<noteq> v" and y: "dist y v < e"
- using v \<open>e > 0\<close> by (auto simp: islimpt_approachable)
- then obtain C T where "y \<in> S" "C > 0" "open T" "y \<in> T" "T \<subseteq> S"
- and "\<And>h z'. \<lbrakk>h \<in> Y; z' \<in> T\<rbrakk> \<Longrightarrow> cmod (h z') \<le> C"
- using \<open>y \<in> U\<close> by (auto simp: U_def)
- then have le_C: "\<And>n. cmod (\<F> n y) \<le> C"
- using FY by blast
- have "\<forall>\<^sub>F n in sequentially. dist (\<G> (j n) y) (h y) < inverse C"
- using uniform_limitD [OF ulim [of "{y}"], of "inverse C"] \<open>C > 0\<close> y
- by (simp add: dist_commute)
- then obtain n where "dist (\<G> (j n) y) (h y) < inverse C"
- by (meson eventually_at_top_linorder order_refl)
- moreover
- have "h y = h v"
- by (metis \<open>h v = c\<close> dist_commute that y)
- ultimately have "norm (\<G> (j n) y) < inverse C"
- by (simp add: \<open>h v = 0\<close>)
- then have "C < norm (\<F> (j n) y)"
- apply (simp add: \<G>_def)
- by (metis FY X01 \<open>0 < C\<close> \<open>y \<in> S\<close> \<open>Y \<subseteq> X\<close> inverse_less_iff_less norm_inverse subsetD zero_less_norm_iff)
- show False
- using \<open>C < cmod (\<F> (j n) y)\<close> le_C not_less by blast
- qed
- then show "\<exists>x\<in>ball v e. h x \<noteq> c" by force
- qed
- show "h holomorphic_on ball v e"
- by (simp add: holh)
- show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
- by (simp add: ulim)
- qed (use \<open>e > 0\<close> in auto)
- with \<open>h v = 0\<close> show False by blast
- qed
- then show "v \<in> U"
- apply (clarsimp simp add: U_def v)
- apply (rule GPicard1[OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> _ \<open>Y \<subseteq> X\<close> holX])
- using X01 no_hw_le1 apply (meson | force simp: not_less)+
- done
- qed
- have "\<And>x. x \<in> K \<longrightarrow> x \<in> U"
- using \<open>U = S\<close> \<open>K \<subseteq> S\<close> by blast
- then have "\<And>x. x \<in> K \<longrightarrow> (\<exists>B Z. 0 < B \<and> open Z \<and> x \<in> Z \<and>
- (\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B))"
- unfolding U_def by blast
- then obtain F Z where F: "\<And>x. x \<in> K \<Longrightarrow> open (Z x) \<and> x \<in> Z x \<and>
- (\<forall>h z'. h \<in> Y \<and> z' \<in> Z x \<longrightarrow> norm(h z') \<le> F x)"
- by metis
- then obtain L where "L \<subseteq> K" "finite L" and L: "K \<subseteq> (\<Union>c \<in> L. Z c)"
- by (auto intro: compactE_image [OF \<open>compact K\<close>, of K Z])
- then have *: "\<And>x h z'. \<lbrakk>x \<in> L; h \<in> Y \<and> z' \<in> Z x\<rbrakk> \<Longrightarrow> cmod (h z') \<le> F x"
- using F by blast
- have "\<exists>B. \<forall>h z. h \<in> Y \<and> z \<in> K \<longrightarrow> norm(h z) \<le> B"
- proof (cases "L = {}")
- case True with L show ?thesis by simp
- next
- case False
- with \<open>finite L\<close> show ?thesis
- apply (rule_tac x = "Max (F ` L)" in exI)
- apply (simp add: linorder_class.Max_ge_iff)
- using * F by (metis L UN_E subsetD)
- qed
- with that show ?thesis by metis
-qed
-
-
-lemma GPicard4:
- assumes "0 < k" and holf: "f holomorphic_on (ball 0 k - {0})"
- and AE: "\<And>e. \<lbrakk>0 < e; e < k\<rbrakk> \<Longrightarrow> \<exists>d. 0 < d \<and> d < e \<and> (\<forall>z \<in> sphere 0 d. norm(f z) \<le> B)"
- obtains \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> norm(f z) \<le> B"
-proof -
- obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k/2" and \<epsilon>: "\<And>z. norm z = \<epsilon> \<Longrightarrow> norm(f z) \<le> B"
- using AE [of "k/2"] \<open>0 < k\<close> by auto
- show ?thesis
- proof
- show "\<epsilon> < k"
- using \<open>0 < k\<close> \<open>\<epsilon> < k/2\<close> by auto
- show "cmod (f \<xi>) \<le> B" if \<xi>: "\<xi> \<in> ball 0 \<epsilon> - {0}" for \<xi>
- proof -
- obtain d where "0 < d" "d < norm \<xi>" and d: "\<And>z. norm z = d \<Longrightarrow> norm(f z) \<le> B"
- using AE [of "norm \<xi>"] \<open>\<epsilon> < k\<close> \<xi> by auto
- have [simp]: "closure (cball 0 \<epsilon> - ball 0 d) = cball 0 \<epsilon> - ball 0 d"
- by (blast intro!: closure_closed)
- have [simp]: "interior (cball 0 \<epsilon> - ball 0 d) = ball 0 \<epsilon> - cball (0::complex) d"
- using \<open>0 < \<epsilon>\<close> \<open>0 < d\<close> by (simp add: interior_diff)
- have *: "norm(f w) \<le> B" if "w \<in> cball 0 \<epsilon> - ball 0 d" for w
- proof (rule maximum_modulus_frontier [of f "cball 0 \<epsilon> - ball 0 d"])
- show "f holomorphic_on interior (cball 0 \<epsilon> - ball 0 d)"
- apply (rule holomorphic_on_subset [OF holf])
- using \<open>\<epsilon> < k\<close> \<open>0 < d\<close> that by auto
- show "continuous_on (closure (cball 0 \<epsilon> - ball 0 d)) f"
- apply (rule holomorphic_on_imp_continuous_on)
- apply (rule holomorphic_on_subset [OF holf])
- using \<open>0 < d\<close> \<open>\<epsilon> < k\<close> by auto
- show "\<And>z. z \<in> frontier (cball 0 \<epsilon> - ball 0 d) \<Longrightarrow> cmod (f z) \<le> B"
- apply (simp add: frontier_def)
- using \<epsilon> d less_eq_real_def by blast
- qed (use that in auto)
- show ?thesis
- using * \<open>d < cmod \<xi>\<close> that by auto
- qed
- qed (use \<open>0 < \<epsilon>\<close> in auto)
-qed
-
-
-lemma GPicard5:
- assumes holf: "f holomorphic_on (ball 0 1 - {0})"
- and f01: "\<And>z. z \<in> ball 0 1 - {0} \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1"
- obtains e B where "0 < e" "e < 1" "0 < B"
- "(\<forall>z \<in> ball 0 e - {0}. norm(f z) \<le> B) \<or>
- (\<forall>z \<in> ball 0 e - {0}. norm(f z) \<ge> B)"
-proof -
- have [simp]: "1 + of_nat n \<noteq> (0::complex)" for n
- using of_nat_eq_0_iff by fastforce
- have [simp]: "cmod (1 + of_nat n) = 1 + of_nat n" for n
- by (metis norm_of_nat of_nat_Suc)
- have *: "(\<lambda>x::complex. x / of_nat (Suc n)) ` (ball 0 1 - {0}) \<subseteq> ball 0 1 - {0}" for n
- by (auto simp: norm_divide field_split_simps split: if_split_asm)
- define h where "h \<equiv> \<lambda>n z::complex. f (z / (Suc n))"
- have holh: "(h n) holomorphic_on ball 0 1 - {0}" for n
- unfolding h_def
- proof (rule holomorphic_on_compose_gen [unfolded o_def, OF _ holf *])
- show "(\<lambda>x. x / of_nat (Suc n)) holomorphic_on ball 0 1 - {0}"
- by (intro holomorphic_intros) auto
- qed
- have h01: "\<And>n z. z \<in> ball 0 1 - {0} \<Longrightarrow> h n z \<noteq> 0 \<and> h n z \<noteq> 1"
- unfolding h_def
- apply (rule f01)
- using * by force
- obtain w where w: "w \<in> ball 0 1 - {0::complex}"
- by (rule_tac w = "1/2" in that) auto
- consider "infinite {n. norm(h n w) \<le> 1}" | "infinite {n. 1 \<le> norm(h n w)}"
- by (metis (mono_tags, lifting) infinite_nat_iff_unbounded_le le_cases mem_Collect_eq)
- then show ?thesis
- proof cases
- case 1
- with infinite_enumerate obtain r :: "nat \<Rightarrow> nat"
- where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<le> 1}"
- by blast
- obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (h \<circ> r)\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
- proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
- show "range (h \<circ> r) \<subseteq>
- {g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 1)}"
- apply clarsimp
- apply (intro conjI holomorphic_intros holomorphic_on_compose holh)
- using h01 apply auto
- done
- show "connected (ball 0 1 - {0::complex})"
- by (simp add: connected_open_delete)
- qed (use r in auto)
- have normf_le_B: "cmod(f z) \<le> B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
- proof -
- have *: "\<And>w. norm w = 1/2 \<Longrightarrow> cmod((f (w / (1 + of_nat (r n))))) \<le> B"
- using B by (auto simp: h_def o_def)
- have half: "norm (z * (1 + of_nat (r n))) = 1/2"
- by (simp add: norm_mult divide_simps that)
- show ?thesis
- using * [OF half] by simp
- qed
- obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod(f z) \<le> B"
- proof (rule GPicard4 [OF zero_less_one holf, of B])
- fix e::real
- assume "0 < e" "e < 1"
- obtain n where "(1/e - 2) / 2 < real n"
- using reals_Archimedean2 by blast
- also have "... \<le> r n"
- using \<open>strict_mono r\<close> by (simp add: seq_suble)
- finally have "(1/e - 2) / 2 < real (r n)" .
- with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
- by (simp add: field_simps)
- show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod (f z) \<le> B)"
- apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
- using normf_le_B by (simp add: e)
- qed blast
- then have \<epsilon>: "cmod (f z) \<le> \<bar>B\<bar> + 1" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
- using that by fastforce
- have "0 < \<bar>B\<bar> + 1"
- by simp
- then show ?thesis
- apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
- using \<epsilon> by auto
- next
- case 2
- with infinite_enumerate obtain r :: "nat \<Rightarrow> nat"
- where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<ge> 1}"
- by blast
- obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (\<lambda>n. inverse \<circ> h (r n))\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
- proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
- show "range (\<lambda>n. inverse \<circ> h (r n)) \<subseteq>
- {g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 1)}"
- apply clarsimp
- apply (intro conjI holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holh] holomorphic_on_compose)
- using h01 apply auto
- done
- show "connected (ball 0 1 - {0::complex})"
- by (simp add: connected_open_delete)
- show "\<And>j. j \<in> range (\<lambda>n. inverse \<circ> h (r n)) \<Longrightarrow> cmod (j w) \<le> 1"
- using r norm_inverse_le_norm by fastforce
- qed (use r in auto)
- have norm_if_le_B: "cmod(inverse (f z)) \<le> B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
- proof -
- have *: "inverse (cmod((f (z / (1 + of_nat (r n)))))) \<le> B" if "norm z = 1/2" for z
- using B [OF that] by (force simp: norm_inverse h_def)
- have half: "norm (z * (1 + of_nat (r n))) = 1/2"
- by (simp add: norm_mult divide_simps that)
- show ?thesis
- using * [OF half] by (simp add: norm_inverse)
- qed
- have hol_if: "(inverse \<circ> f) holomorphic_on (ball 0 1 - {0})"
- by (metis (no_types, lifting) holf comp_apply f01 holomorphic_on_inverse holomorphic_transform)
- obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" and leB: "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod((inverse \<circ> f) z) \<le> B"
- proof (rule GPicard4 [OF zero_less_one hol_if, of B])
- fix e::real
- assume "0 < e" "e < 1"
- obtain n where "(1/e - 2) / 2 < real n"
- using reals_Archimedean2 by blast
- also have "... \<le> r n"
- using \<open>strict_mono r\<close> by (simp add: seq_suble)
- finally have "(1/e - 2) / 2 < real (r n)" .
- with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
- by (simp add: field_simps)
- show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod ((inverse \<circ> f) z) \<le> B)"
- apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
- using norm_if_le_B by (simp add: e)
- qed blast
- have \<epsilon>: "cmod (f z) \<ge> inverse B" and "B > 0" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
- proof -
- have "inverse (cmod (f z)) \<le> B"
- using leB that by (simp add: norm_inverse)
- moreover
- have "f z \<noteq> 0"
- using \<open>\<epsilon> < 1\<close> f01 that by auto
- ultimately show "cmod (f z) \<ge> inverse B"
- by (simp add: norm_inverse inverse_le_imp_le)
- show "B > 0"
- using \<open>f z \<noteq> 0\<close> \<open>inverse (cmod (f z)) \<le> B\<close> not_le order.trans by fastforce
- qed
- then have "B > 0"
- by (metis \<open>0 < \<epsilon>\<close> dense leI order.asym vector_choose_size)
- then have "inverse B > 0"
- by (simp add: field_split_simps)
- then show ?thesis
- apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
- using \<epsilon> by auto
- qed
-qed
-
-
-lemma GPicard6:
- assumes "open M" "z \<in> M" "a \<noteq> 0" and holf: "f holomorphic_on (M - {z})"
- and f0a: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> 0 \<and> f w \<noteq> a"
- obtains r where "0 < r" "ball z r \<subseteq> M"
- "bounded(f ` (ball z r - {z})) \<or>
- bounded((inverse \<circ> f) ` (ball z r - {z}))"
-proof -
- obtain r where "0 < r" and r: "ball z r \<subseteq> M"
- using assms openE by blast
- let ?g = "\<lambda>w. f (z + of_real r * w) / a"
- obtain e B where "0 < e" "e < 1" "0 < B"
- and B: "(\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<le> B) \<or> (\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<ge> B)"
- proof (rule GPicard5)
- show "?g holomorphic_on ball 0 1 - {0}"
- apply (intro holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holf])
- using \<open>0 < r\<close> \<open>a \<noteq> 0\<close> r
- by (auto simp: dist_norm norm_mult subset_eq)
- show "\<And>w. w \<in> ball 0 1 - {0} \<Longrightarrow> f (z + of_real r * w) / a \<noteq> 0 \<and> f (z + of_real r * w) / a \<noteq> 1"
- apply (simp add: field_split_simps \<open>a \<noteq> 0\<close>)
- apply (rule f0a)
- using \<open>0 < r\<close> r by (auto simp: dist_norm norm_mult subset_eq)
- qed
- show ?thesis
- proof
- show "0 < e*r"
- by (simp add: \<open>0 < e\<close> \<open>0 < r\<close>)
- have "ball z (e * r) \<subseteq> ball z r"
- by (simp add: \<open>0 < r\<close> \<open>e < 1\<close> order.strict_implies_order subset_ball)
- then show "ball z (e * r) \<subseteq> M"
- using r by blast
- consider "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<le> B" | "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<ge> B"
- using B by blast
- then show "bounded (f ` (ball z (e * r) - {z})) \<or>
- bounded ((inverse \<circ> f) ` (ball z (e * r) - {z}))"
- proof cases
- case 1
- have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<le> B * norm a" for w
- using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 1 [of "(w - z) / r"]
- by (simp add: norm_divide dist_norm field_split_simps)
- then show ?thesis
- by (force simp: intro!: boundedI)
- next
- case 2
- have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<ge> B * norm a" for w
- using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 2 [of "(w - z) / r"]
- by (simp add: norm_divide dist_norm field_split_simps)
- then have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> inverse (cmod (f w)) \<le> inverse (B * norm a)" for w
- by (metis \<open>0 < B\<close> \<open>a \<noteq> 0\<close> mult_pos_pos norm_inverse norm_inverse_le_norm zero_less_norm_iff)
- then show ?thesis
- by (force simp: norm_inverse intro!: boundedI)
- qed
- qed
-qed
-
-
-theorem great_Picard:
- assumes "open M" "z \<in> M" "a \<noteq> b" and holf: "f holomorphic_on (M - {z})"
- and fab: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> a \<and> f w \<noteq> b"
- obtains l where "(f \<longlongrightarrow> l) (at z) \<or> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
-proof -
- obtain r where "0 < r" and zrM: "ball z r \<subseteq> M"
- and r: "bounded((\<lambda>z. f z - a) ` (ball z r - {z})) \<or>
- bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
- proof (rule GPicard6 [OF \<open>open M\<close> \<open>z \<in> M\<close>])
- show "b - a \<noteq> 0"
- using assms by auto
- show "(\<lambda>z. f z - a) holomorphic_on M - {z}"
- by (intro holomorphic_intros holf)
- qed (use fab in auto)
- have holfb: "f holomorphic_on ball z r - {z}"
- apply (rule holomorphic_on_subset [OF holf])
- using zrM by auto
- have holfb_i: "(\<lambda>z. inverse(f z - a)) holomorphic_on ball z r - {z}"
- apply (intro holomorphic_intros holfb)
- using fab zrM by fastforce
- show ?thesis
- using r
- proof
- assume "bounded ((\<lambda>z. f z - a) ` (ball z r - {z}))"
- then obtain B where B: "\<And>w. w \<in> (\<lambda>z. f z - a) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
- by (force simp: bounded_iff)
- have "\<forall>\<^sub>F w in at z. cmod (f w - a) \<le> B"
- apply (simp add: eventually_at)
- apply (rule_tac x=r in exI)
- using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
- then have "\<exists>B. \<forall>\<^sub>F w in at z. cmod (f w) \<le> B"
- apply (rule_tac x="B + norm a" in exI)
- apply (erule eventually_mono)
- by (metis add.commute add_le_cancel_right norm_triangle_sub order.trans)
- then obtain g where holg: "g holomorphic_on ball z r" and gf: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = f w"
- using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb] by auto
- then have "g \<midarrow>z\<rightarrow> g z"
- apply (simp add: continuous_at [symmetric])
- using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
- then have "(f \<longlongrightarrow> g z) (at z)"
- apply (rule Lim_transform_within_open [of g "g z" z UNIV "ball z r"])
- using \<open>0 < r\<close> by (auto simp: gf)
- then show ?thesis
- using that by blast
- next
- assume "bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
- then obtain B where B: "\<And>w. w \<in> (inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
- by (force simp: bounded_iff)
- have "\<forall>\<^sub>F w in at z. cmod (inverse (f w - a)) \<le> B"
- apply (simp add: eventually_at)
- apply (rule_tac x=r in exI)
- using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
- then have "\<exists>B. \<forall>\<^sub>F z in at z. cmod (inverse (f z - a)) \<le> B"
- by blast
- then obtain g where holg: "g holomorphic_on ball z r" and gf: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = inverse (f w - a)"
- using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb_i] by auto
- then have gz: "g \<midarrow>z\<rightarrow> g z"
- apply (simp add: continuous_at [symmetric])
- using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
- have gnz: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w \<noteq> 0"
- using gf fab zrM by fastforce
- show ?thesis
- proof (cases "g z = 0")
- case True
- have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
- by (auto simp: field_simps)
- have "(inverse \<circ> f) \<midarrow>z\<rightarrow> 0"
- proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
- show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> 0"
- using True by (auto simp: intro!: tendsto_eq_intros gz)
- show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
- using * gf gnz by simp
- qed (use \<open>0 < r\<close> in auto)
- with that show ?thesis by blast
- next
- case False
- show ?thesis
- proof (cases "1 + a * g z = 0")
- case True
- have "(f \<longlongrightarrow> 0) (at z)"
- proof (rule Lim_transform_within_open [of "\<lambda>w. (1 + a * g w) / g w" _ _ _ "ball z r"])
- show "(\<lambda>w. (1 + a * g w) / g w) \<midarrow>z\<rightarrow> 0"
- apply (rule tendsto_eq_intros refl gz \<open>g z \<noteq> 0\<close>)+
- by (simp add: True)
- show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> (1 + a * g x) / g x = f x"
- using fab fab zrM by (fastforce simp add: gf field_split_simps)
- qed (use \<open>0 < r\<close> in auto)
- then show ?thesis
- using that by blast
- next
- case False
- have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
- by (auto simp: field_simps)
- have "(inverse \<circ> f) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
- proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
- show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
- using False by (auto simp: False intro!: tendsto_eq_intros gz)
- show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
- using * gf gnz by simp
- qed (use \<open>0 < r\<close> in auto)
- with that show ?thesis by blast
- qed
- qed
- qed
-qed
-
-
-corollary great_Picard_alt:
- assumes M: "open M" "z \<in> M" and holf: "f holomorphic_on (M - {z})"
- and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
- obtains a where "- {a} \<subseteq> f ` (M - {z})"
- apply (simp add: subset_iff image_iff)
- by (metis great_Picard [OF M _ holf] non)
-
-
-corollary great_Picard_infinite:
- assumes M: "open M" "z \<in> M" and holf: "f holomorphic_on (M - {z})"
- and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
- obtains a where "\<And>w. w \<noteq> a \<Longrightarrow> infinite {x. x \<in> M - {z} \<and> f x = w}"
-proof -
- have False if "a \<noteq> b" and ab: "finite {x. x \<in> M - {z} \<and> f x = a}" "finite {x. x \<in> M - {z} \<and> f x = b}" for a b
- proof -
- have finab: "finite {x. x \<in> M - {z} \<and> f x \<in> {a,b}}"
- using finite_UnI [OF ab] unfolding mem_Collect_eq insert_iff empty_iff
- by (simp add: conj_disj_distribL)
- obtain r where "0 < r" and zrM: "ball z r \<subseteq> M" and r: "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a,b}\<rbrakk> \<Longrightarrow> x \<notin> ball z r"
- proof -
- obtain e where "e > 0" and e: "ball z e \<subseteq> M"
- using assms openE by blast
- show ?thesis
- proof (cases "{x \<in> M - {z}. f x \<in> {a, b}} = {}")
- case True
- then show ?thesis
- apply (rule_tac r=e in that)
- using e \<open>e > 0\<close> by auto
- next
- case False
- let ?r = "min e (Min (dist z ` {x \<in> M - {z}. f x \<in> {a,b}}))"
- show ?thesis
- proof
- show "0 < ?r"
- using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> by auto
- have "ball z ?r \<subseteq> ball z e"
- by (simp add: subset_ball)
- with e show "ball z ?r \<subseteq> M" by blast
- show "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a, b}\<rbrakk> \<Longrightarrow> x \<notin> ball z ?r"
- using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> by auto
- qed
- qed
- qed
- have holfb: "f holomorphic_on (ball z r - {z})"
- apply (rule holomorphic_on_subset [OF holf])
- using zrM by auto
- show ?thesis
- apply (rule great_Picard [OF open_ball _ \<open>a \<noteq> b\<close> holfb])
- using non \<open>0 < r\<close> r zrM by auto
- qed
- with that show thesis
- by meson
-qed
-
-theorem Casorati_Weierstrass:
- assumes "open M" "z \<in> M" "f holomorphic_on (M - {z})"
- and "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
- shows "closure(f ` (M - {z})) = UNIV"
-proof -
- obtain a where a: "- {a} \<subseteq> f ` (M - {z})"
- using great_Picard_alt [OF assms] .
- have "UNIV = closure(- {a})"
- by (simp add: closure_interior)
- also have "... \<subseteq> closure(f ` (M - {z}))"
- by (simp add: a closure_mono)
- finally show ?thesis
- by blast
-qed
-
-end
--- a/src/HOL/Analysis/Line_Segment.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Line_Segment.thy Sat Nov 30 13:47:33 2019 +0100
@@ -576,6 +576,42 @@
fixes u::real shows "closed_segment u v = (\<lambda>x. (v - u) * x + u) ` {0..1}"
by (simp add: add.commute [of u] image_affinity_atLeastAtMost [where c=u] closed_segment_eq_real_ivl)
+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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def 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 (force simp: u closed_segment_def algebra_simps scaleR_conv_of_real complex_eq_iff)
+qed
+
lemma dist_in_closed_segment:
fixes a :: "'a :: euclidean_space"
assumes "x \<in> closed_segment a b"
--- a/src/HOL/Analysis/Path_Connected.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Path_Connected.thy Sat Nov 30 13:47:33 2019 +0100
@@ -1063,6 +1063,9 @@
definition\<^marker>\<open>tag important\<close> shiftpath :: "real \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> real \<Rightarrow> 'a"
where "shiftpath a f = (\<lambda>x. if (a + x) \<le> 1 then f (a + x) else f (a + x - 1))"
+lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
+ by (auto simp: shiftpath_def)
+
lemma pathstart_shiftpath: "a \<le> 1 \<Longrightarrow> pathstart (shiftpath a g) = g a"
unfolding pathstart_def shiftpath_def by auto
@@ -1273,6 +1276,55 @@
fixes a::"'a::linordered_idom" shows "\<lbrakk>a \<le> 1; b \<le> 1; 0 \<le> u; u \<le> 1\<rbrakk> \<Longrightarrow> (1 - u) * a + u * b \<le> 1"
using mult_left_le [of a "1-u"] mult_left_le [of b u] by auto
+lemma linepath_in_path:
+ shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
+ by (auto simp: segment linepath_def)
+
+lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
+ by (auto simp: segment linepath_def)
+
+lemma linepath_in_convex_hull:
+ fixes x::real
+ assumes a: "a \<in> convex hull s"
+ and b: "b \<in> convex hull s"
+ and x: "0\<le>x" "x\<le>1"
+ shows "linepath a b x \<in> convex hull s"
+ apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
+ using x
+ apply (auto simp: linepath_image_01 [symmetric])
+ done
+
+lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
+ by (simp add: linepath_def)
+
+lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
+ by (simp add: linepath_def)
+
+lemma bounded_linear_linepath:
+ assumes "bounded_linear f"
+ shows "f (linepath a b x) = linepath (f a) (f b) x"
+proof -
+ interpret f: bounded_linear f by fact
+ show ?thesis by (simp add: linepath_def f.add f.scale)
+qed
+
+lemma bounded_linear_linepath':
+ assumes "bounded_linear f"
+ shows "f \<circ> linepath a b = linepath (f a) (f b)"
+ using bounded_linear_linepath[OF assms] by (simp add: fun_eq_iff)
+
+lemma linepath_cnj': "cnj \<circ> linepath a b = linepath (cnj a) (cnj b)"
+ by (simp add: linepath_def fun_eq_iff)
+
+lemma differentiable_linepath [intro]: "linepath a b differentiable at x within A"
+ by (auto simp: linepath_def)
+
+lemma has_vector_derivative_linepath_within:
+ "(linepath a b has_vector_derivative (b - a)) (at x within s)"
+apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
+apply (rule derivative_eq_intros | simp)+
+done
+
subsection\<^marker>\<open>tag unimportant\<close>\<open>Segments via convex hulls\<close>
@@ -4003,4 +4055,60 @@
shows "\<exists>g. homeomorphism S T f g"
using assms injective_into_1d_eq_homeomorphism by blast
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Rectangular paths\<close>
+
+definition\<^marker>\<open>tag unimportant\<close> rectpath where
+ "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
+ in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
+
+lemma path_rectpath [simp, intro]: "path (rectpath a b)"
+ by (simp add: Let_def rectpath_def)
+
+lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
+ by (simp add: rectpath_def Let_def)
+
+lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
+ by (simp add: rectpath_def Let_def)
+
+lemma simple_path_rectpath [simp, intro]:
+ assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3"
+ shows "simple_path (rectpath a1 a3)"
+ unfolding rectpath_def Let_def using assms
+ by (intro simple_path_join_loop arc_join arc_linepath)
+ (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)
+
+lemma path_image_rectpath:
+ assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
+ shows "path_image (rectpath a1 a3) =
+ {z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union>
+ {z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs")
+proof -
+ define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
+ have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union>
+ closed_segment a4 a3 \<union> closed_segment a1 a4"
+ by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
+ a2_def a4_def Un_assoc)
+ also have "\<dots> = ?rhs" using assms
+ by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
+ closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
+ finally show ?thesis .
+qed
+
+lemma path_image_rectpath_subset_cbox:
+ assumes "Re a \<le> Re b" "Im a \<le> Im b"
+ shows "path_image (rectpath a b) \<subseteq> cbox a b"
+ using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)
+
+lemma path_image_rectpath_inter_box:
+ assumes "Re a \<le> Re b" "Im a \<le> Im b"
+ shows "path_image (rectpath a b) \<inter> box a b = {}"
+ using assms by (auto simp: path_image_rectpath in_box_complex_iff)
+
+lemma path_image_rectpath_cbox_minus_box:
+ assumes "Re a \<le> Re b" "Im a \<le> Im b"
+ shows "path_image (rectpath a b) = cbox a b - box a b"
+ using assms by (auto simp: path_image_rectpath in_cbox_complex_iff
+ in_box_complex_iff)
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Smooth_Paths.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,490 @@
+(*
+ Material originally from HOL Light, ported by Larry Paulson, moved here by Manuel Eberl
+*)
+section\<^marker>\<open>tag unimportant\<close> \<open>Smooth paths\<close>
+theory Smooth_Paths
+ imports
+ Retracts
+begin
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Homeomorphisms of arc images\<close>
+
+lemma homeomorphism_arc:
+ fixes g :: "real \<Rightarrow> 'a::t2_space"
+ assumes "arc g"
+ obtains h where "homeomorphism {0..1} (path_image g) g h"
+using assms by (force simp: arc_def homeomorphism_compact path_def path_image_def)
+
+lemma homeomorphic_arc_image_interval:
+ fixes g :: "real \<Rightarrow> 'a::t2_space" and a::real
+ assumes "arc g" "a < b"
+ shows "(path_image g) homeomorphic {a..b}"
+proof -
+ have "(path_image g) homeomorphic {0..1::real}"
+ by (meson assms(1) homeomorphic_def homeomorphic_sym homeomorphism_arc)
+ also have "\<dots> homeomorphic {a..b}"
+ using assms by (force intro: homeomorphic_closed_intervals_real)
+ finally show ?thesis .
+qed
+
+lemma homeomorphic_arc_images:
+ fixes g :: "real \<Rightarrow> 'a::t2_space" and h :: "real \<Rightarrow> 'b::t2_space"
+ assumes "arc g" "arc h"
+ shows "(path_image g) homeomorphic (path_image h)"
+proof -
+ have "(path_image g) homeomorphic {0..1::real}"
+ by (meson assms homeomorphic_def homeomorphic_sym homeomorphism_arc)
+ also have "\<dots> homeomorphic (path_image h)"
+ by (meson assms homeomorphic_def homeomorphism_arc)
+ finally show ?thesis .
+qed
+
+lemma path_connected_arc_complement:
+ fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "arc \<gamma>" "2 \<le> DIM('a)"
+ shows "path_connected(- path_image \<gamma>)"
+proof -
+ have "path_image \<gamma> homeomorphic {0..1::real}"
+ by (simp add: assms homeomorphic_arc_image_interval)
+ then
+ show ?thesis
+ apply (rule path_connected_complement_homeomorphic_convex_compact)
+ apply (auto simp: assms)
+ done
+qed
+
+lemma connected_arc_complement:
+ fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "arc \<gamma>" "2 \<le> DIM('a)"
+ shows "connected(- path_image \<gamma>)"
+ by (simp add: assms path_connected_arc_complement path_connected_imp_connected)
+
+lemma inside_arc_empty:
+ fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
+ assumes "arc \<gamma>"
+ shows "inside(path_image \<gamma>) = {}"
+proof (cases "DIM('a) = 1")
+ case True
+ then show ?thesis
+ using assms connected_arc_image connected_convex_1_gen inside_convex by blast
+next
+ case False
+ show ?thesis
+ proof (rule inside_bounded_complement_connected_empty)
+ show "connected (- path_image \<gamma>)"
+ apply (rule connected_arc_complement [OF assms])
+ using False
+ by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym)
+ show "bounded (path_image \<gamma>)"
+ by (simp add: assms bounded_arc_image)
+ qed
+qed
+
+lemma inside_simple_curve_imp_closed:
+ fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "\<lbrakk>simple_path \<gamma>; x \<in> inside(path_image \<gamma>)\<rbrakk> \<Longrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
+ using arc_simple_path inside_arc_empty by blast
+
+
+subsection \<open>Piecewise differentiability of paths\<close>
+
+lemma continuous_on_joinpaths_D1:
+ "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
+ apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> ((*)(inverse 2))"])
+ apply (rule continuous_intros | simp)+
+ apply (auto elim!: continuous_on_subset simp: joinpaths_def)
+ done
+
+lemma continuous_on_joinpaths_D2:
+ "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
+ apply (rule continuous_on_eq [of _ "(g1 +++ g2) \<circ> (\<lambda>x. inverse 2*x + 1/2)"])
+ apply (rule continuous_intros | simp)+
+ apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
+ done
+
+lemma piecewise_differentiable_D1:
+ assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}"
+ shows "g1 piecewise_differentiable_on {0..1}"
+proof -
+ obtain S where cont: "continuous_on {0..1} g1" and "finite S"
+ and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
+ using assms unfolding piecewise_differentiable_on_def
+ by (blast dest!: continuous_on_joinpaths_D1)
+ show ?thesis
+ unfolding piecewise_differentiable_on_def
+ proof (intro exI conjI ballI cont)
+ show "finite (insert 1 (((*)2) ` S))"
+ by (simp add: \<open>finite S\<close>)
+ show "g1 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
+ proof (rule_tac d="dist (x/2) (1/2)" in differentiable_transform_within)
+ have "g1 +++ g2 differentiable at (x / 2) within {0..1/2}"
+ by (rule differentiable_subset [OF S [of "x/2"]] | use that in force)+
+ then show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x within {0..1}"
+ using image_affinity_atLeastAtMost_div [of 2 0 "0::real" 1]
+ by (auto intro: differentiable_chain_within)
+ qed (use that in \<open>auto simp: joinpaths_def\<close>)
+ qed
+qed
+
+lemma piecewise_differentiable_D2:
+ assumes "(g1 +++ g2) piecewise_differentiable_on {0..1}" and eq: "pathfinish g1 = pathstart g2"
+ shows "g2 piecewise_differentiable_on {0..1}"
+proof -
+ have [simp]: "g1 1 = g2 0"
+ using eq by (simp add: pathfinish_def pathstart_def)
+ obtain S where cont: "continuous_on {0..1} g2" and "finite S"
+ and S: "\<And>x. x \<in> {0..1} - S \<Longrightarrow> g1 +++ g2 differentiable at x within {0..1}"
+ using assms unfolding piecewise_differentiable_on_def
+ by (blast dest!: continuous_on_joinpaths_D2)
+ show ?thesis
+ unfolding piecewise_differentiable_on_def
+ proof (intro exI conjI ballI cont)
+ show "finite (insert 0 ((\<lambda>x. 2*x-1)`S))"
+ by (simp add: \<open>finite S\<close>)
+ show "g2 differentiable at x within {0..1}" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1)`S)" for x
+ proof (rule_tac d="dist ((x+1)/2) (1/2)" in differentiable_transform_within)
+ have x2: "(x + 1) / 2 \<notin> S"
+ using that
+ apply (clarsimp simp: image_iff)
+ by (metis add.commute add_diff_cancel_left' mult_2 field_sum_of_halves)
+ have "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
+ by (rule differentiable_chain_within differentiable_subset [OF S [of "(x+1)/2"]] | use x2 that in force)+
+ then show "g1 +++ g2 \<circ> (\<lambda>x. (x+1) / 2) differentiable at x within {0..1}"
+ by (auto intro: differentiable_chain_within)
+ show "(g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'" if "x' \<in> {0..1}" "dist x' x < dist ((x + 1) / 2) (1/2)" for x'
+ proof -
+ have [simp]: "(2*x'+2)/2 = x'+1"
+ by (simp add: field_split_simps)
+ show ?thesis
+ using that by (auto simp: joinpaths_def)
+ qed
+ qed (use that in \<open>auto simp: joinpaths_def\<close>)
+ qed
+qed
+
+lemma piecewise_C1_differentiable_D1:
+ fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
+ assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
+ shows "g1 piecewise_C1_differentiable_on {0..1}"
+proof -
+ obtain S where "finite S"
+ and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
+ and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
+ using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
+ proof (rule differentiable_transform_within)
+ show "g1 +++ g2 \<circ> (*) (inverse 2) differentiable at x"
+ using that g12D
+ apply (simp only: joinpaths_def)
+ by (rule differentiable_chain_at derivative_intros | force)+
+ show "\<And>x'. \<lbrakk>dist x' x < dist (x/2) (1/2)\<rbrakk>
+ \<Longrightarrow> (g1 +++ g2 \<circ> (*) (inverse 2)) x' = g1 x'"
+ using that by (auto simp: dist_real_def joinpaths_def)
+ qed (use that in \<open>auto simp: dist_real_def\<close>)
+ have [simp]: "vector_derivative (g1 \<circ> (*) 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
+ if "x \<in> {0..1} - insert 1 ((*) 2 ` S)" for x
+ apply (subst vector_derivative_chain_at)
+ using that
+ apply (rule derivative_eq_intros g1D | simp)+
+ done
+ have "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
+ using co12 by (rule continuous_on_subset) force
+ then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 \<circ> (*)2) (at x))"
+ proof (rule continuous_on_eq [OF _ vector_derivative_at])
+ show "(g1 +++ g2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)"
+ if "x \<in> {0..1/2} - insert (1/2) S" for x
+ proof (rule has_vector_derivative_transform_within)
+ show "(g1 \<circ> (*) 2 has_vector_derivative vector_derivative (g1 \<circ> (*) 2) (at x)) (at x)"
+ using that
+ by (force intro: g1D differentiable_chain_at simp: vector_derivative_works [symmetric])
+ show "\<And>x'. \<lbrakk>dist x' x < dist x (1/2)\<rbrakk> \<Longrightarrow> (g1 \<circ> (*) 2) x' = (g1 +++ g2) x'"
+ using that by (auto simp: dist_norm joinpaths_def)
+ qed (use that in \<open>auto simp: dist_norm\<close>)
+ qed
+ have "continuous_on ({0..1} - insert 1 ((*) 2 ` S))
+ ((\<lambda>x. 1/2 * vector_derivative (g1 \<circ> (*)2) (at x)) \<circ> (*)(1/2))"
+ apply (rule continuous_intros)+
+ using coDhalf
+ apply (simp add: scaleR_conv_of_real image_set_diff image_image)
+ done
+ then have con_g1: "continuous_on ({0..1} - insert 1 ((*) 2 ` S)) (\<lambda>x. vector_derivative g1 (at x))"
+ by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
+ have "continuous_on {0..1} g1"
+ using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
+ with \<open>finite S\<close> show ?thesis
+ apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ apply (rule_tac x="insert 1 (((*)2)`S)" in exI)
+ apply (simp add: g1D con_g1)
+ done
+qed
+
+lemma piecewise_C1_differentiable_D2:
+ fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
+ assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
+ shows "g2 piecewise_C1_differentiable_on {0..1}"
+proof -
+ obtain S where "finite S"
+ and co12: "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
+ and g12D: "\<forall>x\<in>{0..1} - S. g1 +++ g2 differentiable at x"
+ using assms by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
+ proof (rule differentiable_transform_within)
+ show "g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2) differentiable at x"
+ using g12D that
+ apply (simp only: joinpaths_def)
+ apply (drule_tac x= "(x+1) / 2" in bspec, force simp: field_split_simps)
+ apply (rule differentiable_chain_at derivative_intros | force)+
+ done
+ show "\<And>x'. dist x' x < dist ((x + 1) / 2) (1/2) \<Longrightarrow> (g1 +++ g2 \<circ> (\<lambda>x. (x + 1) / 2)) x' = g2 x'"
+ using that by (auto simp: dist_real_def joinpaths_def field_simps)
+ qed (use that in \<open>auto simp: dist_norm\<close>)
+ have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
+ if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)" for x
+ using that by (auto simp: vector_derivative_chain_at field_split_simps g2D)
+ have "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
+ using co12 by (rule continuous_on_subset) force
+ then have coDhalf: "continuous_on ({1/2..1} - insert (1/2) S) (\<lambda>x. vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x))"
+ proof (rule continuous_on_eq [OF _ vector_derivative_at])
+ show "(g1 +++ g2 has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
+ (at x)"
+ if "x \<in> {1 / 2..1} - insert (1 / 2) S" for x
+ proof (rule_tac f="g2 \<circ> (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
+ show "(g2 \<circ> (\<lambda>x. 2 * x - 1) has_vector_derivative vector_derivative (g2 \<circ> (\<lambda>x. 2 * x - 1)) (at x))
+ (at x)"
+ using that by (force intro: g2D differentiable_chain_at simp: vector_derivative_works [symmetric])
+ show "\<And>x'. \<lbrakk>dist x' x < dist (3 / 4) ((x + 1) / 2)\<rbrakk> \<Longrightarrow> (g2 \<circ> (\<lambda>x. 2 * x - 1)) x' = (g1 +++ g2) x'"
+ using that by (auto simp: dist_norm joinpaths_def add_divide_distrib)
+ qed (use that in \<open>auto simp: dist_norm\<close>)
+ qed
+ have [simp]: "((\<lambda>x. (x+1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` S))) = ({1/2..1} - insert (1/2) S)"
+ apply (simp add: image_set_diff inj_on_def image_image)
+ apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
+ done
+ have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S))
+ ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) \<circ> (\<lambda>x. (x+1)/2))"
+ by (rule continuous_intros | simp add: coDhalf)+
+ then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` S)) (\<lambda>x. vector_derivative g2 (at x))"
+ by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
+ have "continuous_on {0..1} g2"
+ using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
+ with \<open>finite S\<close> show ?thesis
+ apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` S)" in exI)
+ apply (simp add: g2D con_g2)
+ done
+qed
+
+
+subsection \<open>Valid paths, and their start and finish\<close>
+
+definition\<^marker>\<open>tag important\<close> valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
+ where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
+
+definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
+ where "closed_path g \<equiv> g 0 = g 1"
+
+text\<open>In particular, all results for paths apply\<close>
+
+lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
+ by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
+
+lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
+ by (metis connected_path_image valid_path_imp_path)
+
+lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
+ by (metis compact_path_image valid_path_imp_path)
+
+lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
+ by (metis bounded_path_image valid_path_imp_path)
+
+lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
+ by (metis closed_path_image valid_path_imp_path)
+
+lemma valid_path_compose:
+ assumes "valid_path g"
+ and der: "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable (at x)"
+ and con: "continuous_on (path_image g) (deriv f)"
+ shows "valid_path (f \<circ> g)"
+proof -
+ obtain S where "finite S" and g_diff: "g C1_differentiable_on {0..1} - S"
+ using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
+ have "f \<circ> g differentiable at t" when "t\<in>{0..1} - S" for t
+ proof (rule differentiable_chain_at)
+ show "g differentiable at t" using \<open>valid_path g\<close>
+ by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - S\<close> that)
+ next
+ have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
+ then show "f differentiable at (g t)"
+ using der[THEN field_differentiable_imp_differentiable] by auto
+ qed
+ moreover have "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
+ proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
+ rule continuous_intros)
+ show "continuous_on ({0..1} - S) (\<lambda>x. vector_derivative g (at x))"
+ using g_diff C1_differentiable_on_eq by auto
+ next
+ have "continuous_on {0..1} (\<lambda>x. deriv f (g x))"
+ using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def]
+ \<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def
+ by blast
+ then show "continuous_on ({0..1} - S) (\<lambda>x. deriv f (g x))"
+ using continuous_on_subset by blast
+ next
+ show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
+ when "t \<in> {0..1} - S" for t
+ proof (rule vector_derivative_chain_at_general[symmetric])
+ show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
+ next
+ have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
+ then show "f field_differentiable at (g t)" using der by auto
+ qed
+ qed
+ ultimately have "f \<circ> g C1_differentiable_on {0..1} - S"
+ using C1_differentiable_on_eq by blast
+ moreover have "path (f \<circ> g)"
+ apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]])
+ using der
+ by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at)
+ ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
+ using \<open>finite S\<close> by auto
+qed
+
+lemma valid_path_uminus_comp[simp]:
+ fixes g::"real \<Rightarrow> 'a ::real_normed_field"
+ shows "valid_path (uminus \<circ> g) \<longleftrightarrow> valid_path g"
+proof
+ show "valid_path g \<Longrightarrow> valid_path (uminus \<circ> g)" for g::"real \<Rightarrow> 'a"
+ by (auto intro!: valid_path_compose derivative_intros)
+ then show "valid_path g" when "valid_path (uminus \<circ> g)"
+ by (metis fun.map_comp group_add_class.minus_comp_minus id_comp that)
+qed
+
+lemma valid_path_offset[simp]:
+ shows "valid_path (\<lambda>t. g t - z) \<longleftrightarrow> valid_path g"
+proof
+ show *: "valid_path (g::real\<Rightarrow>'a) \<Longrightarrow> valid_path (\<lambda>t. g t - z)" for g z
+ unfolding valid_path_def
+ by (fastforce intro:derivative_intros C1_differentiable_imp_piecewise piecewise_C1_differentiable_diff)
+ show "valid_path (\<lambda>t. g t - z) \<Longrightarrow> valid_path g"
+ using *[of "\<lambda>t. g t - z" "-z",simplified] .
+qed
+
+lemma valid_path_imp_reverse:
+ assumes "valid_path g"
+ shows "valid_path(reversepath g)"
+proof -
+ obtain S where "finite S" and S: "g C1_differentiable_on ({0..1} - S)"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ then have "finite ((-) 1 ` S)"
+ by auto
+ moreover have "(reversepath g C1_differentiable_on ({0..1} - (-) 1 ` S))"
+ unfolding reversepath_def
+ apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
+ using S
+ by (force simp: finite_vimageI inj_on_def C1_differentiable_on_eq elim!: continuous_on_subset)+
+ ultimately show ?thesis using assms
+ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
+qed
+
+lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
+ using valid_path_imp_reverse by force
+
+lemma valid_path_join:
+ assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
+ shows "valid_path(g1 +++ g2)"
+proof -
+ have "g1 1 = g2 0"
+ using assms by (auto simp: pathfinish_def pathstart_def)
+ moreover have "(g1 \<circ> (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
+ apply (rule piecewise_C1_differentiable_compose)
+ using assms
+ apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
+ apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
+ done
+ moreover have "(g2 \<circ> (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
+ apply (rule piecewise_C1_differentiable_compose)
+ using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
+ by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
+ simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
+ ultimately show ?thesis
+ apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
+ apply (rule piecewise_C1_differentiable_cases)
+ apply (auto simp: o_def)
+ done
+qed
+
+lemma valid_path_join_D1:
+ fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
+ shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
+ unfolding valid_path_def
+ by (rule piecewise_C1_differentiable_D1)
+
+lemma valid_path_join_D2:
+ fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
+ shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
+ unfolding valid_path_def
+ by (rule piecewise_C1_differentiable_D2)
+
+lemma valid_path_join_eq [simp]:
+ fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
+ shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
+ using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
+
+lemma valid_path_shiftpath [intro]:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "valid_path(shiftpath a g)"
+ using assms
+ apply (auto simp: valid_path_def shiftpath_alt_def)
+ apply (rule piecewise_C1_differentiable_cases)
+ apply (auto simp: algebra_simps)
+ apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
+ apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
+ apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
+ apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
+ done
+
+lemma vector_derivative_linepath_within:
+ "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
+ apply (rule vector_derivative_within_cbox [of 0 "1::real", simplified])
+ apply (auto simp: has_vector_derivative_linepath_within)
+ done
+
+lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
+ by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
+
+lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
+ apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
+ apply (rule_tac x="{}" in exI)
+ apply (simp add: differentiable_on_def differentiable_def)
+ using has_vector_derivative_def has_vector_derivative_linepath_within
+ apply (fastforce simp add: continuous_on_eq_continuous_within)
+ done
+
+lemma valid_path_subpath:
+ fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
+ assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
+ shows "valid_path(subpath u v g)"
+proof (cases "v=u")
+ case True
+ then show ?thesis
+ unfolding valid_path_def subpath_def
+ by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
+next
+ case False
+ have "(g \<circ> (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
+ apply (rule piecewise_C1_differentiable_compose)
+ apply (simp add: C1_differentiable_imp_piecewise)
+ apply (simp add: image_affinity_atLeastAtMost)
+ using assms False
+ apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
+ apply (subst Int_commute)
+ apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
+ done
+ then show ?thesis
+ by (auto simp: o_def valid_path_def subpath_def)
+qed
+
+lemma valid_path_rectpath [simp, intro]: "valid_path (rectpath a b)"
+ by (simp add: Let_def rectpath_def)
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Vitali_Covering_Theorem.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Vitali_Covering_Theorem.thy Sat Nov 30 13:47:33 2019 +0100
@@ -584,7 +584,7 @@
have "\<exists>U. case p of (x,d) \<Rightarrow> S \<inter> ball x d \<subseteq> U \<and>
U \<in> lmeasurable \<and> ?\<mu> U < e / ?\<mu> (ball z 1) * ?\<mu> (ball x d)"
if "p \<in> C" for p
- using that Csub by auto
+ using that Csub unfolding case_prod_unfold by blast
then obtain U where U:
"\<And>p. p \<in> C \<Longrightarrow>
case p of (x,d) \<Rightarrow> S \<inter> ball x d \<subseteq> U p \<and>
--- a/src/HOL/Analysis/Winding_Numbers.thy Wed Nov 27 16:54:33 2019 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1211 +0,0 @@
-section \<open>Winding Numbers\<close>
-
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2017)\<close>
-
-theory Winding_Numbers
-imports
- Polytope
- Jordan_Curve
- Riemann_Mapping
-begin
-
-lemma simply_connected_inside_simple_path:
- fixes p :: "real \<Rightarrow> complex"
- shows "simple_path p \<Longrightarrow> simply_connected(inside(path_image p))"
- using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties
- by fastforce
-
-lemma simply_connected_Int:
- fixes S :: "complex set"
- assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S \<inter> T)"
- shows "simply_connected (S \<inter> T)"
- using assms by (force simp: simply_connected_eq_winding_number_zero open_Int)
-
-subsection\<open>Winding number for a triangle\<close>
-
-lemma wn_triangle1:
- assumes "0 \<in> interior(convex hull {a,b,c})"
- shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))"
-proof -
- { assume 0: "Im(a/b) \<le> 0" "0 \<le> Im(b/c)"
- have "0 \<notin> interior (convex hull {a,b,c})"
- proof (cases "a=0 \<or> b=0 \<or> c=0")
- case True then show ?thesis
- by (auto simp: not_in_interior_convex_hull_3)
- next
- case False
- then have "b \<noteq> 0" by blast
- { fix x y::complex and u::real
- assume eq_f': "Im x * Re b \<le> Im b * Re x" "Im y * Re b \<le> Im b * Re y" "0 \<le> u" "u \<le> 1"
- then have "((1 - u) * Im x) * Re b \<le> Im b * ((1 - u) * Re x)"
- by (simp add: mult_left_mono mult.assoc mult.left_commute [of "Im b"])
- then have "((1 - u) * Im x + u * Im y) * Re b \<le> Im b * ((1 - u) * Re x + u * Re y)"
- using eq_f' ordered_comm_semiring_class.comm_mult_left_mono
- by (fastforce simp add: algebra_simps)
- }
- with False 0 have "convex hull {a,b,c} \<le> {z. Im z * Re b \<le> Im b * Re z}"
- apply (simp add: Complex.Im_divide divide_simps complex_neq_0 [symmetric])
- apply (simp add: algebra_simps)
- apply (rule hull_minimal)
- apply (auto simp: algebra_simps convex_alt)
- done
- moreover have "0 \<notin> interior({z. Im z * Re b \<le> Im b * Re z})"
- proof
- assume "0 \<in> interior {z. Im z * Re b \<le> Im b * Re z}"
- then obtain e where "e>0" and e: "ball 0 e \<subseteq> {z. Im z * Re b \<le> Im b * Re z}"
- by (meson mem_interior)
- define z where "z \<equiv> - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * \<i>"
- have "z \<in> ball 0 e"
- using \<open>e>0\<close>
- apply (simp add: z_def dist_norm)
- apply (rule le_less_trans [OF norm_triangle_ineq4])
- apply (simp add: norm_mult abs_sgn_eq)
- done
- then have "z \<in> {z. Im z * Re b \<le> Im b * Re z}"
- using e by blast
- then show False
- using \<open>e>0\<close> \<open>b \<noteq> 0\<close>
- apply (simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm)
- apply (auto simp: algebra_simps)
- apply (meson less_asym less_trans mult_pos_pos neg_less_0_iff_less)
- by (metis less_asym mult_pos_pos neg_less_0_iff_less)
- qed
- ultimately show ?thesis
- using interior_mono by blast
- qed
- } with assms show ?thesis by blast
-qed
-
-lemma wn_triangle2_0:
- assumes "0 \<in> interior(convex hull {a,b,c})"
- shows
- "0 < Im((b - a) * cnj (b)) \<and>
- 0 < Im((c - b) * cnj (c)) \<and>
- 0 < Im((a - c) * cnj (a))
- \<or>
- Im((b - a) * cnj (b)) < 0 \<and>
- 0 < Im((b - c) * cnj (b)) \<and>
- 0 < Im((a - b) * cnj (a)) \<and>
- 0 < Im((c - a) * cnj (c))"
-proof -
- have [simp]: "{b,c,a} = {a,b,c}" "{c,a,b} = {a,b,c}" by auto
- show ?thesis
- using wn_triangle1 [OF assms] wn_triangle1 [of b c a] wn_triangle1 [of c a b] assms
- by (auto simp: algebra_simps Im_complex_div_gt_0 Im_complex_div_lt_0 not_le not_less)
-qed
-
-lemma wn_triangle2:
- assumes "z \<in> interior(convex hull {a,b,c})"
- shows "0 < Im((b - a) * cnj (b - z)) \<and>
- 0 < Im((c - b) * cnj (c - z)) \<and>
- 0 < Im((a - c) * cnj (a - z))
- \<or>
- Im((b - a) * cnj (b - z)) < 0 \<and>
- 0 < Im((b - c) * cnj (b - z)) \<and>
- 0 < Im((a - b) * cnj (a - z)) \<and>
- 0 < Im((c - a) * cnj (c - z))"
-proof -
- have 0: "0 \<in> interior(convex hull {a-z, b-z, c-z})"
- using assms convex_hull_translation [of "-z" "{a,b,c}"]
- interior_translation [of "-z"]
- by (simp cong: image_cong_simp)
- show ?thesis using wn_triangle2_0 [OF 0]
- by simp
-qed
-
-lemma wn_triangle3:
- assumes z: "z \<in> interior(convex hull {a,b,c})"
- and "0 < Im((b-a) * cnj (b-z))"
- "0 < Im((c-b) * cnj (c-z))"
- "0 < Im((a-c) * cnj (a-z))"
- shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1"
-proof -
- have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
- using z interior_of_triangle [of a b c]
- by (auto simp: closed_segment_def)
- have gt0: "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)"
- using assms
- by (simp add: winding_number_linepath_pos_lt path_image_join winding_number_join_pos_combined)
- have lt2: "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2"
- using winding_number_lt_half_linepath [of _ a b]
- using winding_number_lt_half_linepath [of _ b c]
- using winding_number_lt_half_linepath [of _ c a] znot
- apply (fastforce simp add: winding_number_join path_image_join)
- done
- show ?thesis
- by (rule winding_number_eq_1) (simp_all add: path_image_join gt0 lt2)
-qed
-
-proposition winding_number_triangle:
- assumes z: "z \<in> interior(convex hull {a,b,c})"
- shows "winding_number(linepath a b +++ linepath b c +++ linepath c a) z =
- (if 0 < Im((b - a) * cnj (b - z)) then 1 else -1)"
-proof -
- have [simp]: "{a,c,b} = {a,b,c}" by auto
- have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
- using z interior_of_triangle [of a b c]
- by (auto simp: closed_segment_def)
- then have [simp]: "z \<notin> closed_segment b a" "z \<notin> closed_segment c b" "z \<notin> closed_segment a c"
- using closed_segment_commute by blast+
- have *: "winding_number (linepath a b +++ linepath b c +++ linepath c a) z =
- winding_number (reversepath (linepath a c +++ linepath c b +++ linepath b a)) z"
- by (simp add: reversepath_joinpaths winding_number_join not_in_path_image_join)
- show ?thesis
- using wn_triangle2 [OF z] apply (rule disjE)
- apply (simp add: wn_triangle3 z)
- apply (simp add: path_image_join winding_number_reversepath * wn_triangle3 z)
- done
-qed
-
-subsection\<open>Winding numbers for simple closed paths\<close>
-
-lemma winding_number_from_innerpath:
- assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b"
- and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b"
- and "simple_path c" and c: "pathstart c = a" "pathfinish c = b"
- and c1c2: "path_image c1 \<inter> path_image c2 = {a,b}"
- and c1c: "path_image c1 \<inter> path_image c = {a,b}"
- and c2c: "path_image c2 \<inter> path_image c = {a,b}"
- and ne_12: "path_image c \<inter> inside(path_image c1 \<union> path_image c2) \<noteq> {}"
- and z: "z \<in> inside(path_image c1 \<union> path_image c)"
- and wn_d: "winding_number (c1 +++ reversepath c) z = d"
- and "a \<noteq> b" "d \<noteq> 0"
- obtains "z \<in> inside(path_image c1 \<union> path_image c2)" "winding_number (c1 +++ reversepath c2) z = d"
-proof -
- obtain 0: "inside(path_image c1 \<union> path_image c) \<inter> inside(path_image c2 \<union> path_image c) = {}"
- and 1: "inside(path_image c1 \<union> path_image c) \<union> inside(path_image c2 \<union> path_image c) \<union>
- (path_image c - {a,b}) = inside(path_image c1 \<union> path_image c2)"
- by (rule split_inside_simple_closed_curve
- [OF \<open>simple_path c1\<close> c1 \<open>simple_path c2\<close> c2 \<open>simple_path c\<close> c \<open>a \<noteq> b\<close> c1c2 c1c c2c ne_12])
- have znot: "z \<notin> path_image c" "z \<notin> path_image c1" "z \<notin> path_image c2"
- using union_with_outside z 1 by auto
- have wn_cc2: "winding_number (c +++ reversepath c2) z = 0"
- apply (rule winding_number_zero_in_outside)
- apply (simp_all add: \<open>simple_path c2\<close> c c2 \<open>simple_path c\<close> simple_path_imp_path path_image_join)
- by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z znot)
- show ?thesis
- proof
- show "z \<in> inside (path_image c1 \<union> path_image c2)"
- using "1" z by blast
- have "winding_number c1 z - winding_number c z = d "
- using assms znot
- by (metis wn_d winding_number_join simple_path_imp_path winding_number_reversepath add.commute path_image_reversepath path_reversepath pathstart_reversepath uminus_add_conv_diff)
- then show "winding_number (c1 +++ reversepath c2) z = d"
- using wn_cc2 by (simp add: winding_number_join simple_path_imp_path assms znot winding_number_reversepath)
- qed
-qed
-
-lemma simple_closed_path_wn1:
- fixes a::complex and e::real
- assumes "0 < e"
- and sp_pl: "simple_path(p +++ linepath (a - e) (a + e))"
- and psp: "pathstart p = a + e"
- and pfp: "pathfinish p = a - e"
- and disj: "ball a e \<inter> path_image p = {}"
-obtains z where "z \<in> inside (path_image (p +++ linepath (a - e) (a + e)))"
- "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1"
-proof -
- have "arc p" and arc_lp: "arc (linepath (a - e) (a + e))"
- and pap: "path_image p \<inter> path_image (linepath (a - e) (a + e)) \<subseteq> {pathstart p, a-e}"
- using simple_path_join_loop_eq [of "linepath (a - e) (a + e)" p] assms by auto
- have mid_eq_a: "midpoint (a - e) (a + e) = a"
- by (simp add: midpoint_def)
- then have "a \<in> path_image(p +++ linepath (a - e) (a + e))"
- apply (simp add: assms path_image_join)
- by (metis midpoint_in_closed_segment)
- have "a \<in> frontier(inside (path_image(p +++ linepath (a - e) (a + e))))"
- apply (simp add: assms Jordan_inside_outside)
- apply (simp_all add: assms path_image_join)
- by (metis mid_eq_a midpoint_in_closed_segment)
- with \<open>0 < e\<close> obtain c where c: "c \<in> inside (path_image(p +++ linepath (a - e) (a + e)))"
- and dac: "dist a c < e"
- by (auto simp: frontier_straddle)
- then have "c \<notin> path_image(p +++ linepath (a - e) (a + e))"
- using inside_no_overlap by blast
- then have "c \<notin> path_image p"
- "c \<notin> closed_segment (a - of_real e) (a + of_real e)"
- by (simp_all add: assms path_image_join)
- with \<open>0 < e\<close> dac have "c \<notin> affine hull {a - of_real e, a + of_real e}"
- by (simp add: segment_as_ball not_le)
- with \<open>0 < e\<close> have *: "\<not> collinear {a - e, c,a + e}"
- using collinear_3_affine_hull [of "a-e" "a+e"] by (auto simp: insert_commute)
- have 13: "1/3 + 1/3 + 1/3 = (1::real)" by simp
- have "(1/3) *\<^sub>R (a - of_real e) + (1/3) *\<^sub>R c + (1/3) *\<^sub>R (a + of_real e) \<in> interior(convex hull {a - e, c, a + e})"
- using interior_convex_hull_3_minimal [OF * DIM_complex]
- by clarsimp (metis 13 zero_less_divide_1_iff zero_less_numeral)
- then obtain z where z: "z \<in> interior(convex hull {a - e, c, a + e})" by force
- have [simp]: "z \<notin> closed_segment (a - e) c"
- by (metis DIM_complex Diff_iff IntD2 inf_sup_absorb interior_of_triangle z)
- have [simp]: "z \<notin> closed_segment (a + e) (a - e)"
- by (metis DIM_complex DiffD2 Un_iff interior_of_triangle z)
- have [simp]: "z \<notin> closed_segment c (a + e)"
- by (metis (no_types, lifting) DIM_complex DiffD2 Un_insert_right inf_sup_aci(5) insertCI interior_of_triangle mk_disjoint_insert z)
- show thesis
- proof
- have "norm (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z) = 1"
- using winding_number_triangle [OF z] by simp
- have zin: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union> path_image p)"
- and zeq: "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
- winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
- proof (rule winding_number_from_innerpath
- [of "linepath (a + e) (a - e)" "a+e" "a-e" p
- "linepath (a + e) c +++ linepath c (a - e)" z
- "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"])
- show sp_aec: "simple_path (linepath (a + e) c +++ linepath c (a - e))"
- proof (rule arc_imp_simple_path [OF arc_join])
- show "arc (linepath (a + e) c)"
- by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathstart_in_path_image psp)
- show "arc (linepath c (a - e))"
- by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathfinish_in_path_image pfp)
- show "path_image (linepath (a + e) c) \<inter> path_image (linepath c (a - e)) \<subseteq> {pathstart (linepath c (a - e))}"
- by clarsimp (metis "*" IntI Int_closed_segment closed_segment_commute singleton_iff)
- qed auto
- show "simple_path p"
- using \<open>arc p\<close> arc_simple_path by blast
- show sp_ae2: "simple_path (linepath (a + e) (a - e))"
- using \<open>arc p\<close> arc_distinct_ends pfp psp by fastforce
- show pa: "pathfinish (linepath (a + e) (a - e)) = a - e"
- "pathstart (linepath (a + e) c +++ linepath c (a - e)) = a + e"
- "pathfinish (linepath (a + e) c +++ linepath c (a - e)) = a - e"
- "pathstart p = a + e" "pathfinish p = a - e"
- "pathstart (linepath (a + e) (a - e)) = a + e"
- by (simp_all add: assms)
- show 1: "path_image (linepath (a + e) (a - e)) \<inter> path_image p = {a + e, a - e}"
- proof
- show "path_image (linepath (a + e) (a - e)) \<inter> path_image p \<subseteq> {a + e, a - e}"
- using pap closed_segment_commute psp segment_convex_hull by fastforce
- show "{a + e, a - e} \<subseteq> path_image (linepath (a + e) (a - e)) \<inter> path_image p"
- using pap pathfinish_in_path_image pathstart_in_path_image pfp psp by fastforce
- qed
- show 2: "path_image (linepath (a + e) (a - e)) \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) =
- {a + e, a - e}" (is "?lhs = ?rhs")
- proof
- have "\<not> collinear {c, a + e, a - e}"
- using * by (simp add: insert_commute)
- then have "convex hull {a + e, a - e} \<inter> convex hull {a + e, c} = {a + e}"
- "convex hull {a + e, a - e} \<inter> convex hull {c, a - e} = {a - e}"
- by (metis (full_types) Int_closed_segment insert_commute segment_convex_hull)+
- then show "?lhs \<subseteq> ?rhs"
- by (metis Int_Un_distrib equalityD1 insert_is_Un path_image_join path_image_linepath path_join_eq path_linepath segment_convex_hull simple_path_def sp_aec)
- show "?rhs \<subseteq> ?lhs"
- using segment_convex_hull by (simp add: path_image_join)
- qed
- have "path_image p \<inter> path_image (linepath (a + e) c) \<subseteq> {a + e}"
- proof (clarsimp simp: path_image_join)
- fix x
- assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment (a + e) c"
- then have "dist x a \<ge> e"
- by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
- with x_ac dac \<open>e > 0\<close> show "x = a + e"
- by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
- qed
- moreover
- have "path_image p \<inter> path_image (linepath c (a - e)) \<subseteq> {a - e}"
- proof (clarsimp simp: path_image_join)
- fix x
- assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment c (a - e)"
- then have "dist x a \<ge> e"
- by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
- with x_ac dac \<open>e > 0\<close> show "x = a - e"
- by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
- qed
- ultimately
- have "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) \<subseteq> {a + e, a - e}"
- by (force simp: path_image_join)
- then show 3: "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}"
- apply (rule equalityI)
- apply (clarsimp simp: path_image_join)
- apply (metis pathstart_in_path_image psp pathfinish_in_path_image pfp)
- done
- show 4: "path_image (linepath (a + e) c +++ linepath c (a - e)) \<inter>
- inside (path_image (linepath (a + e) (a - e)) \<union> path_image p) \<noteq> {}"
- apply (clarsimp simp: path_image_join segment_convex_hull disjoint_iff_not_equal)
- by (metis (no_types, hide_lams) UnI1 Un_commute c closed_segment_commute ends_in_segment(1) path_image_join
- path_image_linepath pathstart_linepath pfp segment_convex_hull)
- show zin_inside: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union>
- path_image (linepath (a + e) c +++ linepath c (a - e)))"
- apply (simp add: path_image_join)
- by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute)
- show 5: "winding_number
- (linepath (a + e) (a - e) +++ reversepath (linepath (a + e) c +++ linepath c (a - e))) z =
- winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
- by (simp add: reversepath_joinpaths path_image_join winding_number_join)
- show 6: "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z \<noteq> 0"
- by (simp add: winding_number_triangle z)
- show "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
- winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
- by (metis 1 2 3 4 5 6 pa sp_aec sp_ae2 \<open>arc p\<close> \<open>simple_path p\<close> arc_distinct_ends winding_number_from_innerpath zin_inside)
- qed (use assms \<open>e > 0\<close> in auto)
- show "z \<in> inside (path_image (p +++ linepath (a - e) (a + e)))"
- using zin by (simp add: assms path_image_join Un_commute closed_segment_commute)
- then have "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) =
- cmod ((winding_number(reversepath (p +++ linepath (a - e) (a + e))) z))"
- apply (subst winding_number_reversepath)
- using simple_path_imp_path sp_pl apply blast
- apply (metis IntI emptyE inside_no_overlap)
- by (simp add: inside_def)
- also have "... = cmod (winding_number(linepath (a + e) (a - e) +++ reversepath p) z)"
- by (simp add: pfp reversepath_joinpaths)
- also have "... = cmod (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z)"
- by (simp add: zeq)
- also have "... = 1"
- using z by (simp add: interior_of_triangle winding_number_triangle)
- finally show "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1" .
- qed
-qed
-
-lemma simple_closed_path_wn2:
- fixes a::complex and d e::real
- assumes "0 < d" "0 < e"
- and sp_pl: "simple_path(p +++ linepath (a - d) (a + e))"
- and psp: "pathstart p = a + e"
- and pfp: "pathfinish p = a - d"
-obtains z where "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
- "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
-proof -
- have [simp]: "a + of_real x \<in> closed_segment (a - \<alpha>) (a - \<beta>) \<longleftrightarrow> x \<in> closed_segment (-\<alpha>) (-\<beta>)" for x \<alpha> \<beta>::real
- using closed_segment_translation_eq [of a]
- by (metis (no_types, hide_lams) add_uminus_conv_diff of_real_minus of_real_closed_segment)
- have [simp]: "a - of_real x \<in> closed_segment (a + \<alpha>) (a + \<beta>) \<longleftrightarrow> -x \<in> closed_segment \<alpha> \<beta>" for x \<alpha> \<beta>::real
- by (metis closed_segment_translation_eq diff_conv_add_uminus of_real_closed_segment of_real_minus)
- have "arc p" and arc_lp: "arc (linepath (a - d) (a + e))" and "path p"
- and pap: "path_image p \<inter> closed_segment (a - d) (a + e) \<subseteq> {a+e, a-d}"
- using simple_path_join_loop_eq [of "linepath (a - d) (a + e)" p] assms arc_imp_path by auto
- have "0 \<in> closed_segment (-d) e"
- using \<open>0 < d\<close> \<open>0 < e\<close> closed_segment_eq_real_ivl by auto
- then have "a \<in> path_image (linepath (a - d) (a + e))"
- using of_real_closed_segment [THEN iffD2]
- by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
- then have "a \<notin> path_image p"
- using \<open>0 < d\<close> \<open>0 < e\<close> pap by auto
- then obtain k where "0 < k" and k: "ball a k \<inter> (path_image p) = {}"
- using \<open>0 < e\<close> \<open>path p\<close> not_on_path_ball by blast
- define kde where "kde \<equiv> (min k (min d e)) / 2"
- have "0 < kde" "kde < k" "kde < d" "kde < e"
- using \<open>0 < k\<close> \<open>0 < d\<close> \<open>0 < e\<close> by (auto simp: kde_def)
- let ?q = "linepath (a + kde) (a + e) +++ p +++ linepath (a - d) (a - kde)"
- have "- kde \<in> closed_segment (-d) e"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
- then have a_diff_kde: "a - kde \<in> closed_segment (a - d) (a + e)"
- using of_real_closed_segment [THEN iffD2]
- by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
- then have clsub2: "closed_segment (a - d) (a - kde) \<subseteq> closed_segment (a - d) (a + e)"
- by (simp add: subset_closed_segment)
- then have "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a + e, a - d}"
- using pap by force
- moreover
- have "a + e \<notin> path_image p \<inter> closed_segment (a - d) (a - kde)"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
- ultimately have sub_a_diff_d: "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a - d}"
- by blast
- have "kde \<in> closed_segment (-d) e"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
- then have a_diff_kde: "a + kde \<in> closed_segment (a - d) (a + e)"
- using of_real_closed_segment [THEN iffD2]
- by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
- then have clsub1: "closed_segment (a + kde) (a + e) \<subseteq> closed_segment (a - d) (a + e)"
- by (simp add: subset_closed_segment)
- then have "closed_segment (a + kde) (a + e) \<inter> path_image p \<subseteq> {a + e, a - d}"
- using pap by force
- moreover
- have "closed_segment (a + kde) (a + e) \<inter> closed_segment (a - d) (a - kde) = {}"
- proof (clarsimp intro!: equals0I)
- fix y
- assume y1: "y \<in> closed_segment (a + kde) (a + e)"
- and y2: "y \<in> closed_segment (a - d) (a - kde)"
- obtain u where u: "y = a + of_real u" and "0 < u"
- using y1 \<open>0 < kde\<close> \<open>kde < e\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
- apply (rule_tac u = "(1 - u)*kde + u*e" in that)
- apply (auto simp: scaleR_conv_of_real algebra_simps)
- by (meson le_less_trans less_add_same_cancel2 less_eq_real_def mult_left_mono)
- moreover
- obtain v where v: "y = a + of_real v" and "v \<le> 0"
- using y2 \<open>0 < kde\<close> \<open>0 < d\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
- apply (rule_tac v = "- ((1 - u)*d + u*kde)" in that)
- apply (force simp: scaleR_conv_of_real algebra_simps)
- by (meson less_eq_real_def neg_le_0_iff_le segment_bound_lemma)
- ultimately show False
- by auto
- qed
- moreover have "a - d \<notin> closed_segment (a + kde) (a + e)"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
- ultimately have sub_a_plus_e:
- "closed_segment (a + kde) (a + e) \<inter> (path_image p \<union> closed_segment (a - d) (a - kde))
- \<subseteq> {a + e}"
- by auto
- have "kde \<in> closed_segment (-kde) e"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
- then have a_add_kde: "a + kde \<in> closed_segment (a - kde) (a + e)"
- using of_real_closed_segment [THEN iffD2]
- by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
- have "closed_segment (a - kde) (a + kde) \<inter> closed_segment (a + kde) (a + e) = {a + kde}"
- by (metis a_add_kde Int_closed_segment)
- moreover
- have "path_image p \<inter> closed_segment (a - kde) (a + kde) = {}"
- proof (rule equals0I, clarify)
- fix y assume "y \<in> path_image p" "y \<in> closed_segment (a - kde) (a + kde)"
- with equals0D [OF k, of y] \<open>0 < kde\<close> \<open>kde < k\<close> show False
- by (auto simp: dist_norm dest: dist_decreases_closed_segment [where c=a])
- qed
- moreover
- have "- kde \<in> closed_segment (-d) kde"
- using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
- then have a_diff_kde': "a - kde \<in> closed_segment (a - d) (a + kde)"
- using of_real_closed_segment [THEN iffD2]
- by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
- then have "closed_segment (a - d) (a - kde) \<inter> closed_segment (a - kde) (a + kde) = {a - kde}"
- by (metis Int_closed_segment)
- ultimately
- have pa_subset_pm_kde: "path_image ?q \<inter> closed_segment (a - kde) (a + kde) \<subseteq> {a - kde, a + kde}"
- by (auto simp: path_image_join assms)
- have ge_kde1: "\<exists>y. x = a + y \<and> y \<ge> kde" if "x \<in> closed_segment (a + kde) (a + e)" for x
- using that \<open>kde < e\<close> mult_le_cancel_left
- apply (auto simp: in_segment)
- apply (rule_tac x="(1-u)*kde + u*e" in exI)
- apply (fastforce simp: algebra_simps scaleR_conv_of_real)
- done
- have ge_kde2: "\<exists>y. x = a + y \<and> y \<le> -kde" if "x \<in> closed_segment (a - d) (a - kde)" for x
- using that \<open>kde < d\<close> affine_ineq
- apply (auto simp: in_segment)
- apply (rule_tac x="- ((1-u)*d + u*kde)" in exI)
- apply (fastforce simp: algebra_simps scaleR_conv_of_real)
- done
- have notin_paq: "x \<notin> path_image ?q" if "dist a x < kde" for x
- using that using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < k\<close>
- apply (auto simp: path_image_join assms dist_norm dest!: ge_kde1 ge_kde2)
- by (meson k disjoint_iff_not_equal le_less_trans less_eq_real_def mem_ball that)
- obtain z where zin: "z \<in> inside (path_image (?q +++ linepath (a - kde) (a + kde)))"
- and z1: "cmod (winding_number (?q +++ linepath (a - kde) (a + kde)) z) = 1"
- proof (rule simple_closed_path_wn1 [of kde ?q a])
- show "simple_path (?q +++ linepath (a - kde) (a + kde))"
- proof (intro simple_path_join_loop conjI)
- show "arc ?q"
- proof (rule arc_join)
- show "arc (linepath (a + kde) (a + e))"
- using \<open>kde < e\<close> \<open>arc p\<close> by (force simp: pfp)
- show "arc (p +++ linepath (a - d) (a - kde))"
- using \<open>kde < d\<close> \<open>kde < e\<close> \<open>arc p\<close> sub_a_diff_d by (force simp: pfp intro: arc_join)
- qed (auto simp: psp pfp path_image_join sub_a_plus_e)
- show "arc (linepath (a - kde) (a + kde))"
- using \<open>0 < kde\<close> by auto
- qed (use pa_subset_pm_kde in auto)
- qed (use \<open>0 < kde\<close> notin_paq in auto)
- have eq: "path_image (?q +++ linepath (a - kde) (a + kde)) = path_image (p +++ linepath (a - d) (a + e))"
- (is "?lhs = ?rhs")
- proof
- show "?lhs \<subseteq> ?rhs"
- using clsub1 clsub2 apply (auto simp: path_image_join assms)
- by (meson subsetCE subset_closed_segment)
- show "?rhs \<subseteq> ?lhs"
- apply (simp add: path_image_join assms Un_ac)
- by (metis Un_closed_segment Un_assoc a_diff_kde a_diff_kde' le_supI2 subset_refl)
- qed
- show thesis
- proof
- show zzin: "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
- by (metis eq zin)
- then have znotin: "z \<notin> path_image p"
- by (metis ComplD Un_iff inside_Un_outside path_image_join pathfinish_linepath pathstart_reversepath pfp reversepath_linepath)
- have znotin_de: "z \<notin> closed_segment (a - d) (a + kde)"
- by (metis ComplD Un_iff Un_closed_segment a_diff_kde inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
- have "winding_number (linepath (a - d) (a + e)) z =
- winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z"
- apply (rule winding_number_split_linepath)
- apply (simp add: a_diff_kde)
- by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
- also have "... = winding_number (linepath (a + kde) (a + e)) z +
- (winding_number (linepath (a - d) (a - kde)) z +
- winding_number (linepath (a - kde) (a + kde)) z)"
- by (simp add: winding_number_split_linepath [of "a-kde", symmetric] znotin_de a_diff_kde')
- finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
- winding_number p z + winding_number (linepath (a + kde) (a + e)) z +
- (winding_number (linepath (a - d) (a - kde)) z +
- winding_number (linepath (a - kde) (a + kde)) z)"
- by (metis (no_types, lifting) ComplD Un_iff \<open>arc p\<close> add.assoc arc_imp_path eq path_image_join path_join_path_ends path_linepath simple_path_imp_path sp_pl union_with_outside winding_number_join zin)
- also have "... = winding_number ?q z + winding_number (linepath (a - kde) (a + kde)) z"
- using \<open>path p\<close> znotin assms zzin clsub1
- apply (subst winding_number_join, auto)
- apply (metis (no_types, hide_lams) ComplD Un_iff contra_subsetD inside_Un_outside path_image_join path_image_linepath pathstart_linepath)
- apply (metis Un_iff Un_closed_segment a_diff_kde' not_in_path_image_join path_image_linepath znotin_de)
- by (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_de)
- also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z"
- using \<open>path p\<close> assms zin
- apply (subst winding_number_join [symmetric], auto)
- apply (metis ComplD Un_iff path_image_join pathfinish_join pathfinish_linepath pathstart_linepath union_with_outside)
- by (metis Un_iff Un_closed_segment a_diff_kde' znotin_de)
- finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
- winding_number (?q +++ linepath (a - kde) (a + kde)) z" .
- then show "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
- by (simp add: z1)
- qed
-qed
-
-lemma simple_closed_path_wn3:
- fixes p :: "real \<Rightarrow> complex"
- assumes "simple_path p" and loop: "pathfinish p = pathstart p"
- obtains z where "z \<in> inside (path_image p)" "cmod (winding_number p z) = 1"
-proof -
- have ins: "inside(path_image p) \<noteq> {}" "open(inside(path_image p))"
- "connected(inside(path_image p))"
- and out: "outside(path_image p) \<noteq> {}" "open(outside(path_image p))"
- "connected(outside(path_image p))"
- and bo: "bounded(inside(path_image p))" "\<not> bounded(outside(path_image p))"
- and ins_out: "inside(path_image p) \<inter> outside(path_image p) = {}"
- "inside(path_image p) \<union> outside(path_image p) = - path_image p"
- and fro: "frontier(inside(path_image p)) = path_image p"
- "frontier(outside(path_image p)) = path_image p"
- using Jordan_inside_outside [OF assms] by auto
- obtain a where a: "a \<in> inside(path_image p)"
- using \<open>inside (path_image p) \<noteq> {}\<close> by blast
- obtain d::real where "0 < d" and d_fro: "a - d \<in> frontier (inside (path_image p))"
- and d_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < d\<rbrakk> \<Longrightarrow> (a - \<epsilon>) \<in> inside (path_image p)"
- apply (rule ray_to_frontier [of "inside (path_image p)" a "-1"])
- using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
- apply (auto simp: of_real_def)
- done
- obtain e::real where "0 < e" and e_fro: "a + e \<in> frontier (inside (path_image p))"
- and e_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < e\<rbrakk> \<Longrightarrow> (a + \<epsilon>) \<in> inside (path_image p)"
- apply (rule ray_to_frontier [of "inside (path_image p)" a 1])
- using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
- apply (auto simp: of_real_def)
- done
- obtain t0 where "0 \<le> t0" "t0 \<le> 1" and pt: "p t0 = a - d"
- using a d_fro fro by (auto simp: path_image_def)
- obtain q where "simple_path q" and q_ends: "pathstart q = a - d" "pathfinish q = a - d"
- and q_eq_p: "path_image q = path_image p"
- and wn_q_eq_wn_p: "\<And>z. z \<in> inside(path_image p) \<Longrightarrow> winding_number q z = winding_number p z"
- proof
- show "simple_path (shiftpath t0 p)"
- by (simp add: pathstart_shiftpath pathfinish_shiftpath
- simple_path_shiftpath path_image_shiftpath \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> assms)
- show "pathstart (shiftpath t0 p) = a - d"
- using pt by (simp add: \<open>t0 \<le> 1\<close> pathstart_shiftpath)
- show "pathfinish (shiftpath t0 p) = a - d"
- by (simp add: \<open>0 \<le> t0\<close> loop pathfinish_shiftpath pt)
- show "path_image (shiftpath t0 p) = path_image p"
- by (simp add: \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> loop path_image_shiftpath)
- show "winding_number (shiftpath t0 p) z = winding_number p z"
- if "z \<in> inside (path_image p)" for z
- by (metis ComplD Un_iff \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> \<open>simple_path p\<close> atLeastAtMost_iff inside_Un_outside
- loop simple_path_imp_path that winding_number_shiftpath)
- qed
- have ad_not_ae: "a - d \<noteq> a + e"
- by (metis \<open>0 < d\<close> \<open>0 < e\<close> add.left_inverse add_left_cancel add_uminus_conv_diff
- le_add_same_cancel2 less_eq_real_def not_less of_real_add of_real_def of_real_eq_0_iff pt)
- have ad_ae_q: "{a - d, a + e} \<subseteq> path_image q"
- using \<open>path_image q = path_image p\<close> d_fro e_fro fro(1) by auto
- have ada: "open_segment (a - d) a \<subseteq> inside (path_image p)"
- proof (clarsimp simp: in_segment)
- fix u::real assume "0 < u" "u < 1"
- with d_int have "a - (1 - u) * d \<in> inside (path_image p)"
- by (metis \<open>0 < d\<close> add.commute diff_add_cancel left_diff_distrib' less_add_same_cancel2 less_eq_real_def mult.left_neutral zero_less_mult_iff)
- then show "(1 - u) *\<^sub>R (a - d) + u *\<^sub>R a \<in> inside (path_image p)"
- by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
- qed
- have aae: "open_segment a (a + e) \<subseteq> inside (path_image p)"
- proof (clarsimp simp: in_segment)
- fix u::real assume "0 < u" "u < 1"
- with e_int have "a + u * e \<in> inside (path_image p)"
- by (meson \<open>0 < e\<close> less_eq_real_def mult_less_cancel_right2 not_less zero_less_mult_iff)
- then show "(1 - u) *\<^sub>R a + u *\<^sub>R (a + e) \<in> inside (path_image p)"
- apply (simp add: algebra_simps)
- by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
- qed
- have "complex_of_real (d * d + (e * e + d * (e + e))) \<noteq> 0"
- using ad_not_ae
- by (metis \<open>0 < d\<close> \<open>0 < e\<close> add_strict_left_mono less_add_same_cancel1 not_sum_squares_lt_zero
- of_real_eq_0_iff zero_less_double_add_iff_zero_less_single_add zero_less_mult_iff)
- then have a_in_de: "a \<in> open_segment (a - d) (a + e)"
- using ad_not_ae \<open>0 < d\<close> \<open>0 < e\<close>
- apply (auto simp: in_segment algebra_simps scaleR_conv_of_real)
- apply (rule_tac x="d / (d+e)" in exI)
- apply (auto simp: field_simps)
- done
- then have "open_segment (a - d) (a + e) \<subseteq> inside (path_image p)"
- using ada a aae Un_open_segment [of a "a-d" "a+e"] by blast
- then have "path_image q \<inter> open_segment (a - d) (a + e) = {}"
- using inside_no_overlap by (fastforce simp: q_eq_p)
- with ad_ae_q have paq_Int_cs: "path_image q \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
- by (simp add: closed_segment_eq_open)
- obtain t where "0 \<le> t" "t \<le> 1" and qt: "q t = a + e"
- using a e_fro fro ad_ae_q by (auto simp: path_defs)
- then have "t \<noteq> 0"
- by (metis ad_not_ae pathstart_def q_ends(1))
- then have "t \<noteq> 1"
- by (metis ad_not_ae pathfinish_def q_ends(2) qt)
- have q01: "q 0 = a - d" "q 1 = a - d"
- using q_ends by (auto simp: pathstart_def pathfinish_def)
- obtain z where zin: "z \<in> inside (path_image (subpath t 0 q +++ linepath (a - d) (a + e)))"
- and z1: "cmod (winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z) = 1"
- proof (rule simple_closed_path_wn2 [of d e "subpath t 0 q" a], simp_all add: q01)
- show "simple_path (subpath t 0 q +++ linepath (a - d) (a + e))"
- proof (rule simple_path_join_loop, simp_all add: qt q01)
- have "inj_on q (closed_segment t 0)"
- using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close>
- by (fastforce simp: simple_path_def inj_on_def closed_segment_eq_real_ivl)
- then show "arc (subpath t 0 q)"
- using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close>
- by (simp add: arc_subpath_eq simple_path_imp_path)
- show "arc (linepath (a - d) (a + e))"
- by (simp add: ad_not_ae)
- show "path_image (subpath t 0 q) \<inter> closed_segment (a - d) (a + e) \<subseteq> {a + e, a - d}"
- using qt paq_Int_cs \<open>simple_path q\<close> \<open>0 \<le> t\<close> \<open>t \<le> 1\<close>
- by (force simp: dest: rev_subsetD [OF _ path_image_subpath_subset] intro: simple_path_imp_path)
- qed
- qed (auto simp: \<open>0 < d\<close> \<open>0 < e\<close> qt)
- have pa01_Un: "path_image (subpath 0 t q) \<union> path_image (subpath 1 t q) = path_image q"
- unfolding path_image_subpath
- using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> by (force simp: path_image_def image_iff)
- with paq_Int_cs have pa_01q:
- "(path_image (subpath 0 t q) \<union> path_image (subpath 1 t q)) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
- by metis
- have z_notin_ed: "z \<notin> closed_segment (a + e) (a - d)"
- using zin q01 by (simp add: path_image_join closed_segment_commute inside_def)
- have z_notin_0t: "z \<notin> path_image (subpath 0 t q)"
- by (metis (no_types, hide_lams) IntI Un_upper1 subsetD empty_iff inside_no_overlap path_image_join
- path_image_subpath_commute pathfinish_subpath pathstart_def pathstart_linepath q_ends(1) qt subpath_trivial zin)
- have [simp]: "- winding_number (subpath t 0 q) z = winding_number (subpath 0 t q) z"
- by (metis \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> atLeastAtMost_iff zero_le_one
- path_image_subpath_commute path_subpath real_eq_0_iff_le_ge_0
- reversepath_subpath simple_path_imp_path winding_number_reversepath z_notin_0t)
- obtain z_in_q: "z \<in> inside(path_image q)"
- and wn_q: "winding_number (subpath 0 t q +++ subpath t 1 q) z = - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
- proof (rule winding_number_from_innerpath
- [of "subpath 0 t q" "a-d" "a+e" "subpath 1 t q" "linepath (a - d) (a + e)"
- z "- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"],
- simp_all add: q01 qt pa01_Un reversepath_subpath)
- show "simple_path (subpath 0 t q)" "simple_path (subpath 1 t q)"
- by (simp_all add: \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close> simple_path_subpath)
- show "simple_path (linepath (a - d) (a + e))"
- using ad_not_ae by blast
- show "path_image (subpath 0 t q) \<inter> path_image (subpath 1 t q) = {a - d, a + e}" (is "?lhs = ?rhs")
- proof
- show "?lhs \<subseteq> ?rhs"
- using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 1\<close> q_ends qt q01
- by (force simp: pathfinish_def qt simple_path_def path_image_subpath)
- show "?rhs \<subseteq> ?lhs"
- using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
- qed
- show "path_image (subpath 0 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
- proof
- show "?lhs \<subseteq> ?rhs" using paq_Int_cs pa01_Un by fastforce
- show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
- qed
- show "path_image (subpath 1 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
- proof
- show "?lhs \<subseteq> ?rhs" by (auto simp: pa_01q [symmetric])
- show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
- qed
- show "closed_segment (a - d) (a + e) \<inter> inside (path_image q) \<noteq> {}"
- using a a_in_de open_closed_segment pa01_Un q_eq_p by fastforce
- show "z \<in> inside (path_image (subpath 0 t q) \<union> closed_segment (a - d) (a + e))"
- by (metis path_image_join path_image_linepath path_image_subpath_commute pathfinish_subpath pathstart_linepath q01(1) zin)
- show "winding_number (subpath 0 t q +++ linepath (a + e) (a - d)) z =
- - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
- using z_notin_ed z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
- by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric])
- show "- d \<noteq> e"
- using ad_not_ae by auto
- show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z \<noteq> 0"
- using z1 by auto
- qed
- show ?thesis
- proof
- show "z \<in> inside (path_image p)"
- using q_eq_p z_in_q by auto
- then have [simp]: "z \<notin> path_image q"
- by (metis disjoint_iff_not_equal inside_no_overlap q_eq_p)
- have [simp]: "z \<notin> path_image (subpath 1 t q)"
- using inside_def pa01_Un z_in_q by fastforce
- have "winding_number(subpath 0 t q +++ subpath t 1 q) z = winding_number(subpath 0 1 q) z"
- using z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
- by (simp add: simple_path_imp_path qt path_image_subpath_commute winding_number_join winding_number_subpath_combine)
- with wn_q have "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z = - winding_number q z"
- by auto
- with z1 have "cmod (winding_number q z) = 1"
- by simp
- with z1 wn_q_eq_wn_p show "cmod (winding_number p z) = 1"
- using z1 wn_q_eq_wn_p by (simp add: \<open>z \<in> inside (path_image p)\<close>)
- qed
-qed
-
-proposition simple_closed_path_winding_number_inside:
- assumes "simple_path \<gamma>"
- obtains "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1"
- | "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = -1"
-proof (cases "pathfinish \<gamma> = pathstart \<gamma>")
- case True
- have "path \<gamma>"
- by (simp add: assms simple_path_imp_path)
- then have const: "winding_number \<gamma> constant_on inside(path_image \<gamma>)"
- proof (rule winding_number_constant)
- show "connected (inside(path_image \<gamma>))"
- by (simp add: Jordan_inside_outside True assms)
- qed (use inside_no_overlap True in auto)
- obtain z where zin: "z \<in> inside (path_image \<gamma>)" and z1: "cmod (winding_number \<gamma> z) = 1"
- using simple_closed_path_wn3 [of \<gamma>] True assms by blast
- have "winding_number \<gamma> z \<in> \<int>"
- using zin integer_winding_number [OF \<open>path \<gamma>\<close> True] inside_def by blast
- with z1 consider "winding_number \<gamma> z = 1" | "winding_number \<gamma> z = -1"
- apply (auto simp: Ints_def abs_if split: if_split_asm)
- by (metis of_int_1 of_int_eq_iff of_int_minus)
- with that const zin show ?thesis
- unfolding constant_on_def by metis
-next
- case False
- then show ?thesis
- using inside_simple_curve_imp_closed assms that(2) by blast
-qed
-
-lemma simple_closed_path_abs_winding_number_inside:
- assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
- shows "\<bar>Re (winding_number \<gamma> z)\<bar> = 1"
- by (metis assms norm_minus_cancel norm_one one_complex.simps(1) real_norm_def simple_closed_path_winding_number_inside uminus_complex.simps(1))
-
-lemma simple_closed_path_norm_winding_number_inside:
- assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
- shows "norm (winding_number \<gamma> z) = 1"
-proof -
- have "pathfinish \<gamma> = pathstart \<gamma>"
- using assms inside_simple_curve_imp_closed by blast
- with assms integer_winding_number have "winding_number \<gamma> z \<in> \<int>"
- by (simp add: inside_def simple_path_def)
- then show ?thesis
- by (metis assms norm_minus_cancel norm_one simple_closed_path_winding_number_inside)
-qed
-
-lemma simple_closed_path_winding_number_cases:
- "\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> {-1,0,1}"
-apply (simp add: inside_Un_outside [of "path_image \<gamma>", symmetric, unfolded set_eq_iff Set.Compl_iff] del: inside_Un_outside)
- apply (rule simple_closed_path_winding_number_inside)
- using simple_path_def winding_number_zero_in_outside by blast+
-
-lemma simple_closed_path_winding_number_pos:
- "\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>; 0 < Re(winding_number \<gamma> z)\<rbrakk>
- \<Longrightarrow> winding_number \<gamma> z = 1"
-using simple_closed_path_winding_number_cases
- by fastforce
-
-subsection \<open>Winding number for rectangular paths\<close>
-
-definition\<^marker>\<open>tag important\<close> rectpath where
- "rectpath a1 a3 = (let a2 = Complex (Re a3) (Im a1); a4 = Complex (Re a1) (Im a3)
- in linepath a1 a2 +++ linepath a2 a3 +++ linepath a3 a4 +++ linepath a4 a1)"
-
-lemma path_rectpath [simp, intro]: "path (rectpath a b)"
- by (simp add: Let_def rectpath_def)
-
-lemma valid_path_rectpath [simp, intro]: "valid_path (rectpath a b)"
- by (simp add: Let_def rectpath_def)
-
-lemma pathstart_rectpath [simp]: "pathstart (rectpath a1 a3) = a1"
- by (simp add: rectpath_def Let_def)
-
-lemma pathfinish_rectpath [simp]: "pathfinish (rectpath a1 a3) = a1"
- by (simp add: rectpath_def Let_def)
-
-lemma simple_path_rectpath [simp, intro]:
- assumes "Re a1 \<noteq> Re a3" "Im a1 \<noteq> Im a3"
- shows "simple_path (rectpath a1 a3)"
- unfolding rectpath_def Let_def using assms
- by (intro simple_path_join_loop arc_join arc_linepath)
- (auto simp: complex_eq_iff path_image_join closed_segment_same_Re closed_segment_same_Im)
-
-lemma path_image_rectpath:
- assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
- shows "path_image (rectpath a1 a3) =
- {z. Re z \<in> {Re a1, Re a3} \<and> Im z \<in> {Im a1..Im a3}} \<union>
- {z. Im z \<in> {Im a1, Im a3} \<and> Re z \<in> {Re a1..Re a3}}" (is "?lhs = ?rhs")
-proof -
- define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
- have "?lhs = closed_segment a1 a2 \<union> closed_segment a2 a3 \<union>
- closed_segment a4 a3 \<union> closed_segment a1 a4"
- by (simp_all add: rectpath_def Let_def path_image_join closed_segment_commute
- a2_def a4_def Un_assoc)
- also have "\<dots> = ?rhs" using assms
- by (auto simp: rectpath_def Let_def path_image_join a2_def a4_def
- closed_segment_same_Re closed_segment_same_Im closed_segment_eq_real_ivl)
- finally show ?thesis .
-qed
-
-lemma path_image_rectpath_subset_cbox:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) \<subseteq> cbox a b"
- using assms by (auto simp: path_image_rectpath in_cbox_complex_iff)
-
-lemma path_image_rectpath_inter_box:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) \<inter> box a b = {}"
- using assms by (auto simp: path_image_rectpath in_box_complex_iff)
-
-lemma path_image_rectpath_cbox_minus_box:
- assumes "Re a \<le> Re b" "Im a \<le> Im b"
- shows "path_image (rectpath a b) = cbox a b - box a b"
- using assms by (auto simp: path_image_rectpath in_cbox_complex_iff
- in_box_complex_iff)
-
-proposition winding_number_rectpath:
- assumes "z \<in> box a1 a3"
- shows "winding_number (rectpath a1 a3) z = 1"
-proof -
- from assms have less: "Re a1 < Re a3" "Im a1 < Im a3"
- by (auto simp: in_box_complex_iff)
- define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
- let ?l1 = "linepath a1 a2" and ?l2 = "linepath a2 a3"
- and ?l3 = "linepath a3 a4" and ?l4 = "linepath a4 a1"
- from assms and less have "z \<notin> path_image (rectpath a1 a3)"
- by (auto simp: path_image_rectpath_cbox_minus_box)
- also have "path_image (rectpath a1 a3) =
- path_image ?l1 \<union> path_image ?l2 \<union> path_image ?l3 \<union> path_image ?l4"
- by (simp add: rectpath_def Let_def path_image_join Un_assoc a2_def a4_def)
- finally have "z \<notin> \<dots>" .
- moreover have "\<forall>l\<in>{?l1,?l2,?l3,?l4}. Re (winding_number l z) > 0"
- unfolding ball_simps HOL.simp_thms a2_def a4_def
- by (intro conjI; (rule winding_number_linepath_pos_lt;
- (insert assms, auto simp: a2_def a4_def in_box_complex_iff mult_neg_neg))+)
- ultimately have "Re (winding_number (rectpath a1 a3) z) > 0"
- by (simp add: winding_number_join path_image_join rectpath_def Let_def a2_def a4_def)
- thus "winding_number (rectpath a1 a3) z = 1" using assms less
- by (intro simple_closed_path_winding_number_pos simple_path_rectpath)
- (auto simp: path_image_rectpath_cbox_minus_box)
-qed
-
-proposition winding_number_rectpath_outside:
- assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
- assumes "z \<notin> cbox a1 a3"
- shows "winding_number (rectpath a1 a3) z = 0"
- using assms by (intro winding_number_zero_outside[OF _ _ _ assms(3)]
- path_image_rectpath_subset_cbox) simp_all
-
-text\<open>A per-function version for continuous logs, a kind of monodromy\<close>
-proposition\<^marker>\<open>tag unimportant\<close> winding_number_compose_exp:
- assumes "path p"
- shows "winding_number (exp \<circ> p) 0 = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
-proof -
- obtain e where "0 < e" and e: "\<And>t. t \<in> {0..1} \<Longrightarrow> e \<le> norm(exp(p t))"
- proof
- have "closed (path_image (exp \<circ> p))"
- by (simp add: assms closed_path_image holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image)
- then show "0 < setdist {0} (path_image (exp \<circ> p))"
- by (metis exp_not_eq_zero imageE image_comp infdist_eq_setdist infdist_pos_not_in_closed path_defs(4) path_image_nonempty)
- next
- fix t::real
- assume "t \<in> {0..1}"
- have "setdist {0} (path_image (exp \<circ> p)) \<le> dist 0 (exp (p t))"
- apply (rule setdist_le_dist)
- using \<open>t \<in> {0..1}\<close> path_image_def by fastforce+
- then show "setdist {0} (path_image (exp \<circ> p)) \<le> cmod (exp (p t))"
- by simp
- qed
- have "bounded (path_image p)"
- by (simp add: assms bounded_path_image)
- then obtain B where "0 < B" and B: "path_image p \<subseteq> cball 0 B"
- by (meson bounded_pos mem_cball_0 subsetI)
- let ?B = "cball (0::complex) (B+1)"
- have "uniformly_continuous_on ?B exp"
- using holomorphic_on_exp holomorphic_on_imp_continuous_on
- by (force intro: compact_uniformly_continuous)
- then obtain d where "d > 0"
- and d: "\<And>x x'. \<lbrakk>x\<in>?B; x'\<in>?B; dist x' x < d\<rbrakk> \<Longrightarrow> norm (exp x' - exp x) < e"
- using \<open>e > 0\<close> by (auto simp: uniformly_continuous_on_def dist_norm)
- then have "min 1 d > 0"
- by force
- then obtain g where pfg: "polynomial_function g" and "g 0 = p 0" "g 1 = p 1"
- and gless: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm(g t - p t) < min 1 d"
- using path_approx_polynomial_function [OF \<open>path p\<close>] \<open>d > 0\<close> \<open>0 < e\<close>
- unfolding pathfinish_def pathstart_def by meson
- have "winding_number (exp \<circ> p) 0 = winding_number (exp \<circ> g) 0"
- proof (rule winding_number_nearby_paths_eq [symmetric])
- show "path (exp \<circ> p)" "path (exp \<circ> g)"
- by (simp_all add: pfg assms holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image path_polynomial_function)
- next
- fix t :: "real"
- assume t: "t \<in> {0..1}"
- with gless have "norm(g t - p t) < 1"
- using min_less_iff_conj by blast
- moreover have ptB: "norm (p t) \<le> B"
- using B t by (force simp: path_image_def)
- ultimately have "cmod (g t) \<le> B + 1"
- by (meson add_mono_thms_linordered_field(4) le_less_trans less_imp_le norm_triangle_sub)
- with ptB gless t have "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < e"
- by (auto simp: dist_norm d)
- with e t show "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < cmod ((exp \<circ> p) t - 0)"
- by fastforce
- qed (use \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> in \<open>auto simp: pathfinish_def pathstart_def\<close>)
- also have "... = 1 / (of_real (2 * pi) * \<i>) * contour_integral (exp \<circ> g) (\<lambda>w. 1 / (w - 0))"
- proof (rule winding_number_valid_path)
- have "continuous_on (path_image g) (deriv exp)"
- by (metis DERIV_exp DERIV_imp_deriv continuous_on_cong holomorphic_on_exp holomorphic_on_imp_continuous_on)
- then show "valid_path (exp \<circ> g)"
- by (simp add: field_differentiable_within_exp pfg valid_path_compose valid_path_polynomial_function)
- show "0 \<notin> path_image (exp \<circ> g)"
- by (auto simp: path_image_def)
- qed
- also have "... = 1 / (of_real (2 * pi) * \<i>) * integral {0..1} (\<lambda>x. vector_derivative g (at x))"
- proof (simp add: contour_integral_integral, rule integral_cong)
- fix t :: "real"
- assume t: "t \<in> {0..1}"
- show "vector_derivative (exp \<circ> g) (at t) / exp (g t) = vector_derivative g (at t)"
- proof -
- have "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)"
- by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def
- has_vector_derivative_polynomial_function pfg vector_derivative_works)
- moreover have "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)"
- apply (rule field_vector_diff_chain_at)
- apply (metis has_vector_derivative_polynomial_function pfg vector_derivative_at)
- using DERIV_exp has_field_derivative_def apply blast
- done
- ultimately show ?thesis
- by (simp add: divide_simps, rule vector_derivative_unique_at)
- qed
- qed
- also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
- proof -
- have "((\<lambda>x. vector_derivative g (at x)) has_integral g 1 - g 0) {0..1}"
- apply (rule fundamental_theorem_of_calculus [OF zero_le_one])
- by (metis has_vector_derivative_at_within has_vector_derivative_polynomial_function pfg vector_derivative_at)
- then show ?thesis
- apply (simp add: pathfinish_def pathstart_def)
- using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto
- qed
- finally show ?thesis .
-qed
-
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
-
-lemma winding_number_as_continuous_log:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- obtains q where "path q"
- "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
-proof -
- let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
- show ?thesis
- proof
- have *: "continuous (at t within {0..1}) (\<lambda>x. winding_number (subpath 0 x p) \<zeta>)"
- if t: "t \<in> {0..1}" for t
- proof -
- let ?B = "ball (p t) (norm(p t - \<zeta>))"
- have "p t \<noteq> \<zeta>"
- using path_image_def that \<zeta> by blast
- then have "simply_connected ?B"
- by (simp add: convex_imp_simply_connected)
- then have "\<And>f::complex\<Rightarrow>complex. continuous_on ?B f \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> \<noteq> 0)
- \<longrightarrow> (\<exists>g. continuous_on ?B g \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> = exp (g \<zeta>)))"
- by (simp add: simply_connected_eq_continuous_log)
- moreover have "continuous_on ?B (\<lambda>w. w - \<zeta>)"
- by (intro continuous_intros)
- moreover have "(\<forall>z \<in> ?B. z - \<zeta> \<noteq> 0)"
- by (auto simp: dist_norm)
- ultimately obtain g where contg: "continuous_on ?B g"
- and geq: "\<And>z. z \<in> ?B \<Longrightarrow> z - \<zeta> = exp (g z)" by blast
- obtain d where "0 < d" and d:
- "\<And>x. \<lbrakk>x \<in> {0..1}; dist x t < d\<rbrakk> \<Longrightarrow> dist (p x) (p t) < cmod (p t - \<zeta>)"
- using \<open>path p\<close> t unfolding path_def continuous_on_iff
- by (metis \<open>p t \<noteq> \<zeta>\<close> right_minus_eq zero_less_norm_iff)
- have "((\<lambda>x. winding_number (\<lambda>w. subpath 0 x p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0) \<longlongrightarrow> 0)
- (at t within {0..1})"
- proof (rule Lim_transform_within [OF _ \<open>d > 0\<close>])
- have "continuous (at t within {0..1}) (g o p)"
- proof (rule continuous_within_compose)
- show "continuous (at t within {0..1}) p"
- using \<open>path p\<close> continuous_on_eq_continuous_within path_def that by blast
- show "continuous (at (p t) within p ` {0..1}) g"
- by (metis (no_types, lifting) open_ball UNIV_I \<open>p t \<noteq> \<zeta>\<close> centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff)
- qed
- with LIM_zero have "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0))) \<longlongrightarrow> 0) (at t within {0..1})"
- by (auto simp: subpath_def continuous_within o_def)
- then show "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)) \<longlongrightarrow> 0)
- (at t within {0..1})"
- by (simp add: tendsto_divide_zero)
- show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>) =
- winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 - winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- if "u \<in> {0..1}" "0 < dist u t" "dist u t < d" for u
- proof -
- have "closed_segment t u \<subseteq> {0..1}"
- using closed_segment_eq_real_ivl t that by auto
- then have piB: "path_image(subpath t u p) \<subseteq> ?B"
- apply (clarsimp simp add: path_image_subpath_gen)
- by (metis subsetD le_less_trans \<open>dist u t < d\<close> d dist_commute dist_in_closed_segment)
- have *: "path (g \<circ> subpath t u p)"
- apply (rule path_continuous_image)
- using \<open>path p\<close> t that apply auto[1]
- using piB contg continuous_on_subset by blast
- have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)
- = winding_number (exp \<circ> g \<circ> subpath t u p) 0"
- using winding_number_compose_exp [OF *]
- by (simp add: pathfinish_def pathstart_def o_assoc)
- also have "... = winding_number (\<lambda>w. subpath t u p w - \<zeta>) 0"
- proof (rule winding_number_cong)
- have "exp(g y) = y - \<zeta>" if "y \<in> (subpath t u p) ` {0..1}" for y
- by (metis that geq path_image_def piB subset_eq)
- then show "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> (exp \<circ> g \<circ> subpath t u p) x = subpath t u p x - \<zeta>"
- by auto
- qed
- also have "... = winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 -
- winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
- apply (simp add: winding_number_offset [symmetric])
- using winding_number_subpath_combine [OF \<open>path p\<close> \<zeta>, of 0 t u] \<open>t \<in> {0..1}\<close> \<open>u \<in> {0..1}\<close>
- by (simp add: add.commute eq_diff_eq)
- finally show ?thesis .
- qed
- qed
- then show ?thesis
- by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff)
- qed
- show "path ?q"
- unfolding path_def
- by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
-
- have "\<zeta> \<noteq> p 0"
- by (metis \<zeta> pathstart_def pathstart_in_path_image)
- then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- by (simp add: pathfinish_def pathstart_def)
- show "p t = \<zeta> + exp (?q t)" if "t \<in> {0..1}" for t
- proof -
- have "path (subpath 0 t p)"
- using \<open>path p\<close> that by auto
- moreover
- have "\<zeta> \<notin> path_image (subpath 0 t p)"
- using \<zeta> [unfolded path_image_def] that by (auto simp: path_image_subpath)
- ultimately show ?thesis
- using winding_number_exp_2pi [of "subpath 0 t p" \<zeta>] \<open>\<zeta> \<noteq> p 0\<close>
- by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def)
- qed
- qed
-qed
-
-subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
-
-lemma winding_number_homotopic_loops_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_loops (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume [simp]: ?lhs
- obtain q where "path q"
- and qeq: "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
- and peq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
- using winding_number_as_continuous_log [OF assms] by blast
- have *: "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r)
- {0..1} (-{\<zeta>}) ((\<lambda>w. \<zeta> + exp w) \<circ> q) ((\<lambda>w. \<zeta> + exp w) \<circ> (\<lambda>t. 0))"
- proof (rule homotopic_with_compose_continuous_left)
- show "homotopic_with_canon (\<lambda>f. pathfinish ((\<lambda>w. \<zeta> + exp w) \<circ> f) = pathstart ((\<lambda>w. \<zeta> + exp w) \<circ> f))
- {0..1} UNIV q (\<lambda>t. 0)"
- proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def)
- have "homotopic_loops UNIV q (\<lambda>t. 0)"
- by (rule homotopic_loops_linear) (use qeq \<open>path q\<close> in \<open>auto simp: path_defs\<close>)
- then have "homotopic_with (\<lambda>r. r 1 = r 0) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (simp add: homotopic_loops_def pathfinish_def pathstart_def)
- then show "homotopic_with (\<lambda>h. exp (h 1) = exp (h 0)) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
- by (rule homotopic_with_mono) simp
- qed
- show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
- by (rule continuous_intros)+
- show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
- by auto
- qed
- then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
- by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def)
- then have "homotopic_loops (-{\<zeta>}) p (\<lambda>t. \<zeta> + 1)"
- by (simp add: homotopic_loops_def)
- then show ?rhs ..
-next
- assume ?rhs
- then obtain a where "homotopic_loops (-{\<zeta>}) p (\<lambda>t. a)" ..
- then have "winding_number p \<zeta> = winding_number (\<lambda>t. a) \<zeta>" "a \<noteq> \<zeta>"
- using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+
- moreover have "winding_number (\<lambda>t. a) \<zeta> = 0"
- by (metis winding_number_zero_const \<open>a \<noteq> \<zeta>\<close>)
- ultimately show ?lhs by metis
-qed
-
-lemma winding_number_homotopic_paths_null_explicit_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p (linepath (pathstart p) (pathstart p))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms])
- apply (rule homotopic_loops_imp_homotopic_paths_null)
- apply (simp add: linepath_refl)
- done
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial)
-qed
-
-lemma winding_number_homotopic_paths_null_eq:
- assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
- shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_paths (-{\<zeta>}) p (\<lambda>t. a))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)
-next
- assume ?rhs
- then show ?lhs
- by (metis \<zeta> homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)
-qed
-
-proposition winding_number_homotopic_paths_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have "winding_number (p +++ reversepath q) \<zeta> = 0"
- using assms by (simp add: winding_number_join winding_number_reversepath)
- moreover
- have "path (p +++ reversepath q)" "\<zeta> \<notin> path_image (p +++ reversepath q)"
- using assms by (auto simp: not_in_path_image_join)
- ultimately obtain a where "homotopic_paths (- {\<zeta>}) (p +++ reversepath q) (linepath a a)"
- using winding_number_homotopic_paths_null_explicit_eq by blast
- then show ?rhs
- using homotopic_paths_imp_pathstart assms
- by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts)
-next
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_paths)
-qed
-
-lemma winding_number_homotopic_loops_eq:
- assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
- and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
- and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q"
- shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_loops (-{\<zeta>}) p q"
- (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- have "pathstart p \<noteq> \<zeta>" "pathstart q \<noteq> \<zeta>"
- using \<zeta>p \<zeta>q by blast+
- moreover have "path_connected (-{\<zeta>})"
- by (simp add: path_connected_punctured_universe)
- ultimately obtain r where "path r" and rim: "path_image r \<subseteq> -{\<zeta>}"
- and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q"
- by (auto simp: path_connected_def)
- then have "pathstart r \<noteq> \<zeta>" by blast
- have "homotopic_loops (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- proof (rule homotopic_paths_imp_homotopic_loops)
- show "homotopic_paths (- {\<zeta>}) p (r +++ q +++ reversepath r)"
- by (metis (mono_tags, hide_lams) \<open>path r\<close> L \<zeta>p \<zeta>q \<open>path p\<close> \<open>path q\<close> homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq)
- qed (use loops pas in auto)
- moreover have "homotopic_loops (- {\<zeta>}) (r +++ q +++ reversepath r) q"
- using rim \<zeta>q by (auto simp: homotopic_loops_conjugate paf \<open>path q\<close> \<open>path r\<close> loops)
- ultimately show ?rhs
- using homotopic_loops_trans by metis
-next
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_loops)
-qed
-
-end
-
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Theorem.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,7159 @@
+section \<open>Complex Path Integrals and Cauchy's Integral Theorem\<close>
+
+text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2015)\<close>
+
+theory Cauchy_Integral_Theorem
+imports
+ "HOL-Analysis.Analysis"
+begin
+
+lemma leibniz_rule_holomorphic:
+ fixes f::"complex \<Rightarrow> 'b::euclidean_space \<Rightarrow> complex"
+ assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
+ assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
+ assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
+ assumes "convex U"
+ shows "(\<lambda>x. integral (cbox a b) (f x)) holomorphic_on U"
+ using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
+ by (auto simp: holomorphic_on_def)
+
+lemma Ln_measurable [measurable]: "Ln \<in> measurable borel borel"
+proof -
+ have *: "Ln (-of_real x) = of_real (ln x) + \<i> * pi" if "x > 0" for x
+ using that by (subst Ln_minus) (auto simp: Ln_of_real)
+ have **: "Ln (of_real x) = of_real (ln (-x)) + \<i> * pi" if "x < 0" for x
+ using *[of "-x"] that by simp
+ have cont: "(\<lambda>x. indicat_real (- \<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel_measurable borel"
+ by (intro borel_measurable_continuous_on_indicator continuous_intros) auto
+ have "(\<lambda>x. if x \<in> \<real>\<^sub>\<le>\<^sub>0 then ln (-Re x) + \<i> * pi else indicator (-\<real>\<^sub>\<le>\<^sub>0) x *\<^sub>R Ln x) \<in> borel \<rightarrow>\<^sub>M borel"
+ (is "?f \<in> _") by (rule measurable_If_set[OF _ cont]) auto
+ hence "(\<lambda>x. if x = 0 then Ln 0 else ?f x) \<in> borel \<rightarrow>\<^sub>M borel" by measurable
+ also have "(\<lambda>x. if x = 0 then Ln 0 else ?f x) = Ln"
+ by (auto simp: fun_eq_iff ** nonpos_Reals_def)
+ finally show ?thesis .
+qed
+
+lemma powr_complex_measurable [measurable]:
+ assumes [measurable]: "f \<in> measurable M borel" "g \<in> measurable M borel"
+ shows "(\<lambda>x. f x powr g x :: complex) \<in> measurable M borel"
+ using assms by (simp add: powr_def)
+
+subsection\<open>Contour Integrals along a path\<close>
+
+text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
+
+text\<open>piecewise differentiable function on [0,1]\<close>
+
+definition\<^marker>\<open>tag important\<close> has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
+ (infixr "has'_contour'_integral" 50)
+ where "(f has_contour_integral i) g \<equiv>
+ ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
+ has_integral i) {0..1}"
+
+definition\<^marker>\<open>tag important\<close> contour_integrable_on
+ (infixr "contour'_integrable'_on" 50)
+ where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
+
+definition\<^marker>\<open>tag important\<close> contour_integral
+ where "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i=0"
+
+lemma not_integrable_contour_integral: "\<not> f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
+ unfolding contour_integrable_on_def contour_integral_def by blast
+
+lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
+ apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
+ using has_integral_unique by blast
+
+lemma has_contour_integral_eqpath:
+ "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
+ contour_integral p f = contour_integral \<gamma> f\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral y) \<gamma>"
+using contour_integrable_on_def contour_integral_unique by auto
+
+lemma has_contour_integral_integral:
+ "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
+ by (metis contour_integral_unique contour_integrable_on_def)
+
+lemma has_contour_integral_unique:
+ "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
+ using has_integral_unique
+ by (auto simp: has_contour_integral_def)
+
+lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
+ using contour_integrable_on_def by blast
+
+text\<open>Show that we can forget about the localized derivative.\<close>
+
+lemma has_integral_localized_vector_derivative:
+ "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
+ ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
+proof -
+ have *: "{a..b} - {a,b} = interior {a..b}"
+ by (simp add: atLeastAtMost_diff_ends)
+ show ?thesis
+ apply (rule has_integral_spike_eq [of "{a,b}"])
+ apply (auto simp: at_within_interior [of _ "{a..b}"])
+ done
+qed
+
+lemma integrable_on_localized_vector_derivative:
+ "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
+ (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
+ by (simp add: integrable_on_def has_integral_localized_vector_derivative)
+
+lemma has_contour_integral:
+ "(f has_contour_integral i) g \<longleftrightarrow>
+ ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
+ by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
+
+lemma contour_integrable_on:
+ "f contour_integrable_on g \<longleftrightarrow>
+ (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
+ by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path\<close>
+
+
+
+lemma has_contour_integral_reversepath:
+ assumes "valid_path g" and f: "(f has_contour_integral i) g"
+ shows "(f has_contour_integral (-i)) (reversepath g)"
+proof -
+ { fix S x
+ assume xs: "g C1_differentiable_on ({0..1} - S)" "x \<notin> (-) 1 ` S" "0 \<le> x" "x \<le> 1"
+ have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
+ - vector_derivative g (at (1 - x) within {0..1})"
+ proof -
+ obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
+ using xs
+ by (force simp: has_vector_derivative_def C1_differentiable_on_def)
+ have "(g \<circ> (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
+ by (intro vector_diff_chain_within has_vector_derivative_at_within [OF f'] derivative_eq_intros | simp)+
+ then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
+ by (simp add: o_def)
+ show ?thesis
+ using xs
+ by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
+ qed
+ } note * = this
+ obtain S where S: "continuous_on {0..1} g" "finite S" "g C1_differentiable_on {0..1} - S"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have "((\<lambda>x. - (f (g (1 - x)) * vector_derivative g (at (1 - x) within {0..1}))) has_integral -i)
+ {0..1}"
+ using has_integral_affinity01 [where m= "-1" and c=1, OF f [unfolded has_contour_integral_def]]
+ by (simp add: has_integral_neg)
+ then show ?thesis
+ using S
+ apply (clarsimp simp: reversepath_def has_contour_integral_def)
+ apply (rule_tac S = "(\<lambda>x. 1 - x) ` S" in has_integral_spike_finite)
+ apply (auto simp: *)
+ done
+qed
+
+lemma contour_integrable_reversepath:
+ "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
+ using has_contour_integral_reversepath contour_integrable_on_def by blast
+
+lemma contour_integrable_reversepath_eq:
+ "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
+ using contour_integrable_reversepath valid_path_reversepath by fastforce
+
+lemma contour_integral_reversepath:
+ assumes "valid_path g"
+ shows "contour_integral (reversepath g) f = - (contour_integral g f)"
+proof (cases "f contour_integrable_on g")
+ case True then show ?thesis
+ by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
+next
+ case False then have "\<not> f contour_integrable_on (reversepath g)"
+ by (simp add: assms contour_integrable_reversepath_eq)
+ with False show ?thesis by (simp add: not_integrable_contour_integral)
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Joining two paths together\<close>
+
+lemma has_contour_integral_join:
+ assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
+ "valid_path g1" "valid_path g2"
+ shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
+proof -
+ obtain s1 s2
+ where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
+ and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
+ using assms
+ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
+ and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
+ using assms
+ by (auto simp: has_contour_integral)
+ have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
+ and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
+ using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
+ has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
+ by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
+ have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
+ 2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
+ apply (simp_all add: dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
+ apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ using s1
+ apply (auto simp: algebra_simps vector_derivative_works)
+ done
+ have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
+ 2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
+ apply (simp_all add: dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
+ apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ using s2
+ apply (auto simp: algebra_simps vector_derivative_works)
+ done
+ have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
+ apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) ((*)2 -` s1)"])
+ using s1
+ apply (force intro: finite_vimageI [where h = "(*)2"] inj_onI)
+ apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
+ done
+ moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
+ apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
+ using s2
+ apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
+ apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
+ done
+ ultimately
+ show ?thesis
+ apply (simp add: has_contour_integral)
+ apply (rule has_integral_combine [where c = "1/2"], auto)
+ done
+qed
+
+lemma contour_integrable_joinI:
+ assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
+ "valid_path g1" "valid_path g2"
+ shows "f contour_integrable_on (g1 +++ g2)"
+ using assms
+ by (meson has_contour_integral_join contour_integrable_on_def)
+
+lemma contour_integrable_joinD1:
+ assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
+ shows "f contour_integrable_on g1"
+proof -
+ obtain s1
+ where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
+ using assms
+ apply (auto simp: contour_integrable_on)
+ apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
+ apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
+ done
+ then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
+ by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
+ have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
+ 2 *\<^sub>R vector_derivative g1 (at z)" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
+ apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
+ using s1
+ apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
+ done
+ show ?thesis
+ using s1
+ apply (auto simp: contour_integrable_on)
+ apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
+ apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
+ done
+qed
+
+lemma contour_integrable_joinD2:
+ assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
+ shows "f contour_integrable_on g2"
+proof -
+ obtain s2
+ where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
+ using assms
+ apply (auto simp: contour_integrable_on)
+ apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
+ apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
+ apply (simp add: image_affinity_atLeastAtMost_diff)
+ done
+ then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
+ integrable_on {0..1}"
+ by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
+ have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
+ vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
+ 2 *\<^sub>R vector_derivative g2 (at z)" for z
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
+ apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
+ using s2
+ apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
+ vector_derivative_works add_divide_distrib)
+ done
+ show ?thesis
+ using s2
+ apply (auto simp: contour_integrable_on)
+ apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
+ apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
+ done
+qed
+
+lemma contour_integrable_join [simp]:
+ shows
+ "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
+ \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
+using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
+
+lemma contour_integral_join [simp]:
+ shows
+ "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
+ \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
+ by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Shifting the starting point of a (closed) path\<close>
+
+lemma has_contour_integral_shiftpath:
+ assumes f: "(f has_contour_integral i) g" "valid_path g"
+ and a: "a \<in> {0..1}"
+ shows "(f has_contour_integral i) (shiftpath a g)"
+proof -
+ obtain s
+ where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
+ using assms by (auto simp: has_contour_integral)
+ then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
+ integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
+ apply (rule has_integral_unique)
+ apply (subst add.commute)
+ apply (subst Henstock_Kurzweil_Integration.integral_combine)
+ using assms * integral_unique by auto
+ { fix x
+ have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
+ vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
+ unfolding shiftpath_def
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
+ apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
+ apply (intro derivative_eq_intros | simp)+
+ using g
+ apply (drule_tac x="x+a" in bspec)
+ using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
+ done
+ } note vd1 = this
+ { fix x
+ have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
+ vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
+ unfolding shiftpath_def
+ apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
+ apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
+ apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
+ apply (intro derivative_eq_intros | simp)+
+ using g
+ apply (drule_tac x="x+a-1" in bspec)
+ using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
+ done
+ } note vd2 = this
+ have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
+ using * a by (fastforce intro: integrable_subinterval_real)
+ have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
+ apply (rule integrable_subinterval_real)
+ using * a by auto
+ have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
+ has_integral integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x))) {0..1 - a}"
+ apply (rule has_integral_spike_finite
+ [where S = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
+ using s apply blast
+ using a apply (auto simp: algebra_simps vd1)
+ apply (force simp: shiftpath_def add.commute)
+ using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
+ apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
+ done
+ moreover
+ have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
+ has_integral integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))) {1 - a..1}"
+ apply (rule has_integral_spike_finite
+ [where S = "{1-a} \<union> (\<lambda>x. x-a+1) ` s" and f = "\<lambda>x. f(g(a+x-1)) * vector_derivative g (at(a+x-1))"])
+ using s apply blast
+ using a apply (auto simp: algebra_simps vd2)
+ apply (force simp: shiftpath_def add.commute)
+ using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
+ apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
+ apply (simp add: algebra_simps)
+ done
+ ultimately show ?thesis
+ using a
+ by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
+qed
+
+lemma has_contour_integral_shiftpath_D:
+ assumes "(f has_contour_integral i) (shiftpath a g)"
+ "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "(f has_contour_integral i) g"
+proof -
+ obtain s
+ where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ { fix x
+ assume x: "0 < x" "x < 1" "x \<notin> s"
+ then have gx: "g differentiable at x"
+ using g by auto
+ have "vector_derivative g (at x within {0..1}) =
+ vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
+ apply (rule vector_derivative_at_within_ivl
+ [OF has_vector_derivative_transform_within_open
+ [where f = "(shiftpath (1 - a) (shiftpath a g))" and S = "{0<..<1}-s"]])
+ using s g assms x
+ apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
+ at_within_interior [of _ "{0..1}"] vector_derivative_works [symmetric])
+ apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
+ apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
+ done
+ } note vd = this
+ have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
+ using assms by (auto intro!: has_contour_integral_shiftpath)
+ show ?thesis
+ apply (simp add: has_contour_integral_def)
+ apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _ fi [unfolded has_contour_integral_def]])
+ using s assms vd
+ apply (auto simp: Path_Connected.shiftpath_shiftpath)
+ done
+qed
+
+lemma has_contour_integral_shiftpath_eq:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
+ using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
+
+lemma contour_integrable_on_shiftpath_eq:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
+using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
+
+lemma contour_integral_shiftpath:
+ assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
+ shows "contour_integral (shiftpath a g) f = contour_integral g f"
+ using assms
+ by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More about straight-line paths\<close>
+
+lemma has_contour_integral_linepath:
+ shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
+ ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
+ by (simp add: has_contour_integral)
+
+lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
+ by (simp add: has_contour_integral_linepath)
+
+lemma has_contour_integral_trivial_iff [simp]: "(f has_contour_integral i) (linepath a a) \<longleftrightarrow> i=0"
+ using has_contour_integral_unique by blast
+
+lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
+ using has_contour_integral_trivial contour_integral_unique by blast
+
+
+subsection\<open>Relation to subpath construction\<close>
+
+lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
+ by (simp add: has_contour_integral subpath_def)
+
+lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
+ using has_contour_integral_subpath_refl contour_integrable_on_def by blast
+
+lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
+ by (simp add: contour_integral_unique)
+
+lemma has_contour_integral_subpath:
+ assumes f: "f contour_integrable_on g" and g: "valid_path g"
+ and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "(f has_contour_integral integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
+ (subpath u v g)"
+proof (cases "v=u")
+ case True
+ then show ?thesis
+ using f by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
+next
+ case False
+ obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
+ using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
+ have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
+ has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
+ {0..1}"
+ using f uv
+ apply (simp add: contour_integrable_on subpath_def has_contour_integral)
+ apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
+ apply (simp_all add: has_integral_integral)
+ apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
+ apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
+ apply (simp add: divide_simps False)
+ done
+ { fix x
+ have "x \<in> {0..1} \<Longrightarrow>
+ x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
+ vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
+ apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
+ apply (intro derivative_eq_intros | simp)+
+ apply (cut_tac s [of "(v - u) * x + u"])
+ using uv mult_left_le [of x "v-u"]
+ apply (auto simp: vector_derivative_works)
+ done
+ } note vd = this
+ show ?thesis
+ apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
+ using fs assms
+ apply (simp add: False subpath_def has_contour_integral)
+ apply (rule_tac S = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
+ apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
+ done
+qed
+
+lemma contour_integrable_subpath:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
+ shows "f contour_integrable_on (subpath u v g)"
+ apply (cases u v rule: linorder_class.le_cases)
+ apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
+ apply (subst reversepath_subpath [symmetric])
+ apply (rule contour_integrable_reversepath)
+ using assms apply (blast intro: valid_path_subpath)
+ apply (simp add: contour_integrable_on_def)
+ using assms apply (blast intro: has_contour_integral_subpath)
+ done
+
+lemma has_integral_contour_integral_subpath:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
+ has_integral contour_integral (subpath u v g) f) {u..v}"
+ using assms
+ apply (auto simp: has_integral_integrable_integral)
+ apply (rule integrable_on_subcbox [where a=u and b=v and S = "{0..1}", simplified])
+ apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
+ done
+
+lemma contour_integral_subcontour_integral:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
+ shows "contour_integral (subpath u v g) f =
+ integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
+ using assms has_contour_integral_subpath contour_integral_unique by blast
+
+lemma contour_integral_subpath_combine_less:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
+ "u<v" "v<w"
+ shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
+ contour_integral (subpath u w g) f"
+ using assms apply (auto simp: contour_integral_subcontour_integral)
+ apply (rule Henstock_Kurzweil_Integration.integral_combine, auto)
+ apply (rule integrable_on_subcbox [where a=u and b=w and S = "{0..1}", simplified])
+ apply (auto simp: contour_integrable_on)
+ done
+
+lemma contour_integral_subpath_combine:
+ assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
+ shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
+ contour_integral (subpath u w g) f"
+proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
+ case True
+ have *: "subpath v u g = reversepath(subpath u v g) \<and>
+ subpath w u g = reversepath(subpath u w g) \<and>
+ subpath w v g = reversepath(subpath v w g)"
+ by (auto simp: reversepath_subpath)
+ have "u < v \<and> v < w \<or>
+ u < w \<and> w < v \<or>
+ v < u \<and> u < w \<or>
+ v < w \<and> w < u \<or>
+ w < u \<and> u < v \<or>
+ w < v \<and> v < u"
+ using True assms by linarith
+ with assms show ?thesis
+ using contour_integral_subpath_combine_less [of f g u v w]
+ contour_integral_subpath_combine_less [of f g u w v]
+ contour_integral_subpath_combine_less [of f g v u w]
+ contour_integral_subpath_combine_less [of f g v w u]
+ contour_integral_subpath_combine_less [of f g w u v]
+ contour_integral_subpath_combine_less [of f g w v u]
+ apply simp
+ apply (elim disjE)
+ apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
+ valid_path_subpath algebra_simps)
+ done
+next
+ case False
+ then show ?thesis
+ apply (auto)
+ using assms
+ by (metis eq_neg_iff_add_eq_0 contour_integral_reversepath reversepath_subpath valid_path_subpath)
+qed
+
+lemma contour_integral_integral:
+ "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
+ by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
+
+lemma contour_integral_cong:
+ assumes "g = g'" "\<And>x. x \<in> path_image g \<Longrightarrow> f x = f' x"
+ shows "contour_integral g f = contour_integral g' f'"
+ unfolding contour_integral_integral using assms
+ by (intro integral_cong) (auto simp: path_image_def)
+
+
+text \<open>Contour integral along a segment on the real axis\<close>
+
+lemma has_contour_integral_linepath_Reals_iff:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "(f has_contour_integral I) (linepath a b) \<longleftrightarrow>
+ ((\<lambda>x. f (of_real x)) has_integral I) {Re a..Re b}"
+proof -
+ from assms have [simp]: "of_real (Re a) = a" "of_real (Re b) = b"
+ by (simp_all add: complex_eq_iff)
+ from assms have "a \<noteq> b" by auto
+ have "((\<lambda>x. f (of_real x)) has_integral I) (cbox (Re a) (Re b)) \<longleftrightarrow>
+ ((\<lambda>x. f (a + b * of_real x - a * of_real x)) has_integral I /\<^sub>R (Re b - Re a)) {0..1}"
+ by (subst has_integral_affinity_iff [of "Re b - Re a" _ "Re a", symmetric])
+ (insert assms, simp_all add: field_simps scaleR_conv_of_real)
+ also have "(\<lambda>x. f (a + b * of_real x - a * of_real x)) =
+ (\<lambda>x. (f (a + b * of_real x - a * of_real x) * (b - a)) /\<^sub>R (Re b - Re a))"
+ using \<open>a \<noteq> b\<close> by (auto simp: field_simps fun_eq_iff scaleR_conv_of_real)
+ also have "(\<dots> has_integral I /\<^sub>R (Re b - Re a)) {0..1} \<longleftrightarrow>
+ ((\<lambda>x. f (linepath a b x) * (b - a)) has_integral I) {0..1}" using assms
+ by (subst has_integral_cmul_iff) (auto simp: linepath_def scaleR_conv_of_real algebra_simps)
+ also have "\<dots> \<longleftrightarrow> (f has_contour_integral I) (linepath a b)" unfolding has_contour_integral_def
+ by (intro has_integral_cong) (simp add: vector_derivative_linepath_within)
+ finally show ?thesis by simp
+qed
+
+lemma contour_integrable_linepath_Reals_iff:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "(f contour_integrable_on linepath a b) \<longleftrightarrow>
+ (\<lambda>x. f (of_real x)) integrable_on {Re a..Re b}"
+ using has_contour_integral_linepath_Reals_iff[OF assms, of f]
+ by (auto simp: contour_integrable_on_def integrable_on_def)
+
+lemma contour_integral_linepath_Reals_eq:
+ fixes a b :: complex and f :: "complex \<Rightarrow> complex"
+ assumes "a \<in> Reals" "b \<in> Reals" "Re a < Re b"
+ shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (of_real x))"
+proof (cases "f contour_integrable_on linepath a b")
+ case True
+ thus ?thesis using has_contour_integral_linepath_Reals_iff[OF assms, of f]
+ using has_contour_integral_integral has_contour_integral_unique by blast
+next
+ case False
+ thus ?thesis using contour_integrable_linepath_Reals_iff[OF assms, of f]
+ by (simp add: not_integrable_contour_integral not_integrable_integral)
+qed
+
+
+
+text\<open>Cauchy's theorem where there's a primitive\<close>
+
+lemma contour_integral_primitive_lemma:
+ fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
+ assumes "a \<le> b"
+ and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "g piecewise_differentiable_on {a..b}" "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
+ shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
+ has_integral (f(g b) - f(g a))) {a..b}"
+proof -
+ obtain k where k: "finite k" "\<forall>x\<in>{a..b} - k. g differentiable (at x within {a..b})" and cg: "continuous_on {a..b} g"
+ using assms by (auto simp: piecewise_differentiable_on_def)
+ have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
+ apply (rule continuous_on_compose [OF cg, unfolded o_def])
+ using assms
+ apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
+ done
+ { fix x::real
+ assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
+ then have "g differentiable at x within {a..b}"
+ using k by (simp add: differentiable_at_withinI)
+ then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
+ by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
+ then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
+ by (simp add: has_vector_derivative_def scaleR_conv_of_real)
+ have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
+ using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
+ then have fdiff: "(f has_derivative (*) (f' (g x))) (at (g x) within g ` {a..b})"
+ by (simp add: has_field_derivative_def)
+ have "((\<lambda>x. f (g x)) has_vector_derivative f' (g x) * vector_derivative g (at x within {a..b})) (at x within {a..b})"
+ using diff_chain_within [OF gdiff fdiff]
+ by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
+ } note * = this
+ show ?thesis
+ apply (rule fundamental_theorem_of_calculus_interior_strong)
+ using k assms cfg *
+ apply (auto simp: at_within_Icc_at)
+ done
+qed
+
+lemma contour_integral_primitive:
+ assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "valid_path g" "path_image g \<subseteq> s"
+ shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
+ using assms
+ apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
+ apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
+ done
+
+corollary Cauchy_theorem_primitive:
+ assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
+ and "valid_path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g"
+ shows "(f' has_contour_integral 0) g"
+ using assms
+ by (metis diff_self contour_integral_primitive)
+
+text\<open>Existence of path integral for continuous function\<close>
+lemma contour_integrable_continuous_linepath:
+ assumes "continuous_on (closed_segment a b) f"
+ shows "f contour_integrable_on (linepath a b)"
+proof -
+ have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) \<circ> linepath a b)"
+ apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
+ apply (rule continuous_intros | simp add: assms)+
+ done
+ then show ?thesis
+ apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
+ apply (rule integrable_continuous [of 0 "1::real", simplified])
+ apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
+ apply (auto simp: vector_derivative_linepath_within)
+ done
+qed
+
+lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
+ by (rule has_derivative_imp_has_field_derivative)
+ (rule derivative_intros | simp)+
+
+lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
+ apply (rule contour_integral_unique)
+ using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
+ apply (auto simp: field_simps has_field_der_id)
+ done
+
+lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
+ by (simp add: contour_integrable_continuous_linepath)
+
+lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
+ by (simp add: contour_integrable_continuous_linepath)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetical combining theorems\<close>
+
+lemma has_contour_integral_neg:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
+ by (simp add: has_integral_neg has_contour_integral_def)
+
+lemma has_contour_integral_add:
+ "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
+ by (simp add: has_integral_add has_contour_integral_def algebra_simps)
+
+lemma has_contour_integral_diff:
+ "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
+ by (simp add: has_integral_diff has_contour_integral_def algebra_simps)
+
+lemma has_contour_integral_lmul:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
+apply (simp add: has_contour_integral_def)
+apply (drule has_integral_mult_right)
+apply (simp add: algebra_simps)
+done
+
+lemma has_contour_integral_rmul:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
+apply (drule has_contour_integral_lmul)
+apply (simp add: mult.commute)
+done
+
+lemma has_contour_integral_div:
+ "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
+ by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
+
+lemma has_contour_integral_eq:
+ "\<lbrakk>(f has_contour_integral y) p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> (g has_contour_integral y) p"
+apply (simp add: path_image_def has_contour_integral_def)
+by (metis (no_types, lifting) image_eqI has_integral_eq)
+
+lemma has_contour_integral_bound_linepath:
+ assumes "(f has_contour_integral i) (linepath a b)"
+ "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm i \<le> B * norm(b - a)"
+proof -
+ { fix x::real
+ assume x: "0 \<le> x" "x \<le> 1"
+ have "norm (f (linepath a b x)) *
+ norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
+ by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
+ } note * = this
+ have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
+ apply (rule has_integral_bound
+ [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
+ using assms * unfolding has_contour_integral_def
+ apply (auto simp: norm_mult)
+ done
+ then show ?thesis
+ by (auto simp: content_real)
+qed
+
+(*UNUSED
+lemma has_contour_integral_bound_linepath_strong:
+ fixes a :: real and f :: "complex \<Rightarrow> real"
+ assumes "(f has_contour_integral i) (linepath a b)"
+ "finite k"
+ "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm i \<le> B*norm(b - a)"
+*)
+
+lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
+ unfolding has_contour_integral_linepath
+ by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
+
+lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
+ by (simp add: has_contour_integral_def)
+
+lemma has_contour_integral_is_0:
+ "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
+ by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
+
+lemma has_contour_integral_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
+ by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Operations on path integrals\<close>
+
+lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
+ by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
+
+lemma contour_integral_neg:
+ "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
+
+lemma contour_integral_add:
+ "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
+ contour_integral g f1 + contour_integral g f2"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
+
+lemma contour_integral_diff:
+ "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
+ contour_integral g f1 - contour_integral g f2"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
+
+lemma contour_integral_lmul:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
+
+lemma contour_integral_rmul:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
+
+lemma contour_integral_div:
+ shows "f contour_integrable_on g
+ \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
+ by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
+
+lemma contour_integral_eq:
+ "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
+ apply (simp add: contour_integral_def)
+ using has_contour_integral_eq
+ by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
+
+lemma contour_integral_eq_0:
+ "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
+ by (simp add: has_contour_integral_is_0 contour_integral_unique)
+
+lemma contour_integral_bound_linepath:
+ shows
+ "\<lbrakk>f contour_integrable_on (linepath a b);
+ 0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
+ apply (rule has_contour_integral_bound_linepath [of f])
+ apply (auto simp: has_contour_integral_integral)
+ done
+
+lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
+ by (simp add: contour_integral_unique has_contour_integral_0)
+
+lemma contour_integral_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
+ \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
+ by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
+
+lemma contour_integrable_eq:
+ "\<lbrakk>f contour_integrable_on p; \<And>x. x \<in> path_image p \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g contour_integrable_on p"
+ unfolding contour_integrable_on_def
+ by (metis has_contour_integral_eq)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic theorems for path integrability\<close>
+
+lemma contour_integrable_neg:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
+ using has_contour_integral_neg contour_integrable_on_def by blast
+
+lemma contour_integrable_add:
+ "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
+ using has_contour_integral_add contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_diff:
+ "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
+ using has_contour_integral_diff contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_lmul:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
+ using has_contour_integral_lmul contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_rmul:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
+ using has_contour_integral_rmul contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_div:
+ "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
+ using has_contour_integral_div contour_integrable_on_def
+ by fastforce
+
+lemma contour_integrable_sum:
+ "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
+ \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
+ unfolding contour_integrable_on_def
+ by (metis has_contour_integral_sum)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Reversing a path integral\<close>
+
+lemma has_contour_integral_reverse_linepath:
+ "(f has_contour_integral i) (linepath a b)
+ \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
+ using has_contour_integral_reversepath valid_path_linepath by fastforce
+
+lemma contour_integral_reverse_linepath:
+ "continuous_on (closed_segment a b) f
+ \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
+apply (rule contour_integral_unique)
+apply (rule has_contour_integral_reverse_linepath)
+by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
+
+
+(* Splitting a path integral in a flat way.*)
+
+lemma has_contour_integral_split:
+ assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "(f has_contour_integral (i + j)) (linepath a b)"
+proof (cases "k = 0 \<or> k = 1")
+ case True
+ then show ?thesis
+ using assms by auto
+next
+ case False
+ then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
+ using assms by auto
+ have c': "c = k *\<^sub>R (b - a) + a"
+ by (metis diff_add_cancel c)
+ have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
+ by (simp add: algebra_simps c')
+ { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
+ have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
+ using False apply (simp add: c' algebra_simps)
+ apply (simp add: real_vector.scale_left_distrib [symmetric] field_split_simps)
+ done
+ have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
+ using k has_integral_affinity01 [OF *, of "inverse k" "0"]
+ apply (simp add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
+ apply (auto dest: has_integral_cmul [where c = "inverse k"])
+ done
+ } note fi = this
+ { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
+ have **: "\<And>x. (((1 - x) / (1 - k)) *\<^sub>R c + ((x - k) / (1 - k)) *\<^sub>R b) = ((1 - x) *\<^sub>R a + x *\<^sub>R b)"
+ using k
+ apply (simp add: c' field_simps)
+ apply (simp add: scaleR_conv_of_real divide_simps)
+ apply (simp add: field_simps)
+ done
+ have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
+ using k has_integral_affinity01 [OF *, of "inverse(1 - k)" "-(k/(1 - k))"]
+ apply (simp add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
+ apply (auto dest: has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
+ done
+ } note fj = this
+ show ?thesis
+ using f k
+ apply (simp add: has_contour_integral_linepath)
+ apply (simp add: linepath_def)
+ apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
+ done
+qed
+
+lemma continuous_on_closed_segment_transform:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "continuous_on (closed_segment a c) f"
+proof -
+ have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
+ using c by (simp add: algebra_simps)
+ have "closed_segment a c \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
+ then show "continuous_on (closed_segment a c) f"
+ by (rule continuous_on_subset [OF f])
+qed
+
+lemma contour_integral_split:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and k: "0 \<le> k" "k \<le> 1"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
+proof -
+ have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
+ using c by (simp add: algebra_simps)
+ have "closed_segment a c \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(1) in_segment(1) k subset_closed_segment)
+ moreover have "closed_segment c b \<subseteq> closed_segment a b"
+ by (metis c' ends_in_segment(2) in_segment(1) k subset_closed_segment)
+ ultimately
+ have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
+ by (auto intro: continuous_on_subset [OF f])
+ show ?thesis
+ by (rule contour_integral_unique) (meson "*" c contour_integrable_continuous_linepath has_contour_integral_integral has_contour_integral_split k)
+qed
+
+lemma contour_integral_split_linepath:
+ assumes f: "continuous_on (closed_segment a b) f"
+ and c: "c \<in> closed_segment a b"
+ shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
+ using c by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
+
+text\<open>The special case of midpoints used in the main quadrisection\<close>
+
+lemma has_contour_integral_midpoint:
+ assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
+ "(f has_contour_integral j) (linepath (midpoint a b) b)"
+ shows "(f has_contour_integral (i + j)) (linepath a b)"
+ apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
+ using assms
+ apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
+ done
+
+lemma contour_integral_midpoint:
+ "continuous_on (closed_segment a b) f
+ \<Longrightarrow> contour_integral (linepath a b) f =
+ contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
+ apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
+ apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
+ done
+
+
+text\<open>A couple of special case lemmas that are useful below\<close>
+
+lemma triangle_linear_has_chain_integral:
+ "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+ apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
+ apply (auto intro!: derivative_eq_intros)
+ done
+
+lemma has_chain_integral_chain_integral3:
+ "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
+ \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
+ apply (subst contour_integral_unique [symmetric], assumption)
+ apply (drule has_contour_integral_integrable)
+ apply (simp add: valid_path_join)
+ done
+
+lemma has_chain_integral_chain_integral4:
+ "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
+ \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i"
+ apply (subst contour_integral_unique [symmetric], assumption)
+ apply (drule has_contour_integral_integrable)
+ apply (simp add: valid_path_join)
+ done
+
+subsection\<open>Reversing the order in a double path integral\<close>
+
+text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
+
+lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
+ by (auto simp: cbox_Pair_eq)
+
+lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
+ by (auto simp: cbox_Pair_eq)
+
+proposition contour_integral_swap:
+ assumes fcon: "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
+ and vp: "valid_path g" "valid_path h"
+ and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
+ and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
+ shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
+ contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
+proof -
+ have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
+ using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g x, h t))"
+ by (rule ext) simp
+ have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>t. (g t, h x))"
+ by (rule ext) simp
+ have fcon_im1: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
+ by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
+ have fcon_im2: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> continuous_on ((\<lambda>t. (g t, h x)) ` {0..1}) (\<lambda>(x, y). f x y)"
+ by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
+ have "\<And>y. y \<in> {0..1} \<Longrightarrow> continuous_on {0..1} (\<lambda>x. f (g x) (h y))"
+ by (subst fgh2) (rule fcon_im2 gcon continuous_intros | simp)+
+ then have vdg: "\<And>y. y \<in> {0..1} \<Longrightarrow> (\<lambda>x. f (g x) (h y) * vector_derivative g (at x)) integrable_on {0..1}"
+ using continuous_on_mult gvcon integrable_continuous_real by blast
+ have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) \<circ> fst"
+ by auto
+ then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
+ apply (rule ssubst)
+ apply (rule continuous_intros | simp add: gvcon)+
+ done
+ have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) \<circ> snd"
+ by auto
+ then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
+ apply (rule ssubst)
+ apply (rule continuous_intros | simp add: hvcon)+
+ done
+ have "(\<lambda>x. f (g (fst x)) (h (snd x))) = (\<lambda>(y1,y2). f y1 y2) \<circ> (\<lambda>w. ((g \<circ> fst) w, (h \<circ> snd) w))"
+ by auto
+ then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
+ apply (rule ssubst)
+ apply (rule gcon hcon continuous_intros | simp)+
+ apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
+ done
+ have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
+ integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
+ proof (rule integral_cong [OF contour_integral_rmul [symmetric]])
+ show "\<And>x. x \<in> {0..1} \<Longrightarrow> f (g x) contour_integrable_on h"
+ unfolding contour_integrable_on
+ apply (rule integrable_continuous_real)
+ apply (rule continuous_on_mult [OF _ hvcon])
+ apply (subst fgh1)
+ apply (rule fcon_im1 hcon continuous_intros | simp)+
+ done
+ qed
+ also have "\<dots> = integral {0..1}
+ (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
+ unfolding contour_integral_integral
+ apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
+ apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
+ unfolding integral_mult_left [symmetric]
+ apply (simp only: mult_ac)
+ done
+ also have "\<dots> = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
+ unfolding contour_integral_integral
+ apply (rule integral_cong)
+ unfolding integral_mult_left [symmetric]
+ apply (simp add: algebra_simps)
+ done
+ finally show ?thesis
+ by (simp add: contour_integral_integral)
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The key quadrisection step\<close>
+
+lemma norm_sum_half:
+ assumes "norm(a + b) \<ge> e"
+ shows "norm a \<ge> e/2 \<or> norm b \<ge> e/2"
+proof -
+ have "e \<le> norm (- a - b)"
+ by (simp add: add.commute assms norm_minus_commute)
+ thus ?thesis
+ using norm_triangle_ineq4 order_trans by fastforce
+qed
+
+lemma norm_sum_lemma:
+ assumes "e \<le> norm (a + b + c + d)"
+ shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
+proof -
+ have "e \<le> norm ((a + b) + (c + d))" using assms
+ by (simp add: algebra_simps)
+ then show ?thesis
+ by (auto dest!: norm_sum_half)
+qed
+
+lemma Cauchy_theorem_quadrisection:
+ assumes f: "continuous_on (convex hull {a,b,c}) f"
+ and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
+ and e: "e * K^2 \<le>
+ norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
+ shows "\<exists>a' b' c'.
+ a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
+ dist a' b' \<le> K/2 \<and> dist b' c' \<le> K/2 \<and> dist c' a' \<le> K/2 \<and>
+ e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
+ (is "\<exists>x y z. ?\<Phi> x y z")
+proof -
+ note divide_le_eq_numeral1 [simp del]
+ define a' where "a' = midpoint b c"
+ define b' where "b' = midpoint c a"
+ define c' where "c' = midpoint a b"
+ have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
+ using f continuous_on_subset segments_subset_convex_hull by metis+
+ have fcont': "continuous_on (closed_segment c' b') f"
+ "continuous_on (closed_segment a' c') f"
+ "continuous_on (closed_segment b' a') f"
+ unfolding a'_def b'_def c'_def
+ by (rule continuous_on_subset [OF f],
+ metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
+ let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
+ have *: "?pathint a b + ?pathint b c + ?pathint c a =
+ (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
+ (?pathint a' c' + ?pathint c' b + ?pathint b a') +
+ (?pathint a' c + ?pathint c b' + ?pathint b' a') +
+ (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
+ by (simp add: fcont' contour_integral_reverse_linepath) (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
+ have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
+ by (metis left_diff_distrib mult.commute norm_mult_numeral1)
+ have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
+ by (simp add: norm_minus_commute)
+ consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
+ "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
+ "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
+ "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
+ using assms unfolding * by (blast intro: that dest!: norm_sum_lemma)
+ then show ?thesis
+ proof cases
+ case 1 then have "?\<Phi> a c' b'"
+ using assms
+ apply (clarsimp simp: c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
+ done
+ then show ?thesis by blast
+ next
+ case 2 then have "?\<Phi> a' c' b"
+ using assms
+ apply (clarsimp simp: a'_def c'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
+ done
+ then show ?thesis by blast
+ next
+ case 3 then have "?\<Phi> a' c b'"
+ using assms
+ apply (clarsimp simp: a'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
+ done
+ then show ?thesis by blast
+ next
+ case 4 then have "?\<Phi> a' b' c'"
+ using assms
+ apply (clarsimp simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
+ apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real field_split_simps)
+ done
+ then show ?thesis by blast
+ qed
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for triangles\<close>
+
+lemma triangle_points_closer:
+ fixes a::complex
+ shows "\<lbrakk>x \<in> convex hull {a,b,c}; y \<in> convex hull {a,b,c}\<rbrakk>
+ \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
+ norm(x - y) \<le> norm(b - c) \<or>
+ norm(x - y) \<le> norm(c - a)"
+ using simplex_extremal_le [of "{a,b,c}"]
+ by (auto simp: norm_minus_commute)
+
+lemma holomorphic_point_small_triangle:
+ assumes x: "x \<in> S"
+ and f: "continuous_on S f"
+ and cd: "f field_differentiable (at x within S)"
+ and e: "0 < e"
+ shows "\<exists>k>0. \<forall>a b c. dist a b \<le> k \<and> dist b c \<le> k \<and> dist c a \<le> k \<and>
+ x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> S
+ \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
+ contour_integral(linepath c a) f)
+ \<le> e*(dist a b + dist b c + dist c a)^2"
+ (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
+proof -
+ have le_of_3: "\<And>a x y z. \<lbrakk>0 \<le> x*y; 0 \<le> x*z; 0 \<le> y*z; a \<le> (e*(x + y + z))*x + (e*(x + y + z))*y + (e*(x + y + z))*z\<rbrakk>
+ \<Longrightarrow> a \<le> e*(x + y + z)^2"
+ by (simp add: algebra_simps power2_eq_square)
+ have disj_le: "\<lbrakk>x \<le> a \<or> x \<le> b \<or> x \<le> c; 0 \<le> a; 0 \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> x \<le> a + b + c"
+ for x::real and a b c
+ by linarith
+ have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
+ if "convex hull {a, b, c} \<subseteq> S" for a b c
+ using segments_subset_convex_hull that
+ by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
+ note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
+ { fix f' a b c d
+ assume d: "0 < d"
+ and f': "\<And>y. \<lbrakk>cmod (y - x) \<le> d; y \<in> S\<rbrakk> \<Longrightarrow> cmod (f y - f x - f' * (y - x)) \<le> e * cmod (y - x)"
+ and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
+ and xc: "x \<in> convex hull {a, b, c}"
+ and S: "convex hull {a, b, c} \<subseteq> S"
+ have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
+ contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
+ contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
+ contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
+ apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF S])
+ apply (simp add: field_simps)
+ done
+ { fix y
+ assume yc: "y \<in> convex hull {a,b,c}"
+ have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
+ proof (rule f')
+ show "cmod (y - x) \<le> d"
+ by (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
+ qed (use S yc in blast)
+ also have "\<dots> \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
+ by (simp add: yc e xc disj_le [OF triangle_points_closer])
+ finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
+ } note cm_le = this
+ have "?normle a b c"
+ unfolding dist_norm pa
+ apply (rule le_of_3)
+ using f' xc S e
+ apply simp_all
+ apply (intro norm_triangle_le add_mono path_bound)
+ apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
+ apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
+ done
+ } note * = this
+ show ?thesis
+ using cd e
+ apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
+ apply (clarify dest!: spec mp)
+ using * unfolding dist_norm
+ apply blast
+ done
+qed
+
+
+text\<open>Hence the most basic theorem for a triangle.\<close>
+
+locale Chain =
+ fixes x0 At Follows
+ assumes At0: "At x0 0"
+ and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
+begin
+ primrec f where
+ "f 0 = x0"
+ | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
+
+ lemma At: "At (f n) n"
+ proof (induct n)
+ case 0 show ?case
+ by (simp add: At0)
+ next
+ case (Suc n) show ?case
+ by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
+ qed
+
+ lemma Follows: "Follows (f(Suc n)) (f n)"
+ by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
+
+ declare f.simps(2) [simp del]
+end
+
+lemma Chain3:
+ assumes At0: "At x0 y0 z0 0"
+ and AtSuc: "\<And>x y z n. At x y z n \<Longrightarrow> \<exists>x' y' z'. At x' y' z' (Suc n) \<and> Follows x' y' z' x y z"
+ obtains f g h where
+ "f 0 = x0" "g 0 = y0" "h 0 = z0"
+ "\<And>n. At (f n) (g n) (h n) n"
+ "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
+proof -
+ interpret three: Chain "(x0,y0,z0)" "\<lambda>(x,y,z). At x y z" "\<lambda>(x',y',z'). \<lambda>(x,y,z). Follows x' y' z' x y z"
+ apply unfold_locales
+ using At0 AtSuc by auto
+ show ?thesis
+ apply (rule that [of "\<lambda>n. fst (three.f n)" "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
+ using three.At three.Follows
+ apply simp_all
+ apply (simp_all add: split_beta')
+ done
+qed
+
+proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle:
+ assumes "f holomorphic_on (convex hull {a,b,c})"
+ shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+proof -
+ have contf: "continuous_on (convex hull {a,b,c}) f"
+ by (metis assms holomorphic_on_imp_continuous_on)
+ let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
+ { fix y::complex
+ assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
+ and ynz: "y \<noteq> 0"
+ define K where "K = 1 + max (dist a b) (max (dist b c) (dist c a))"
+ define e where "e = norm y / K^2"
+ have K1: "K \<ge> 1" by (simp add: K_def max.coboundedI1)
+ then have K: "K > 0" by linarith
+ have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
+ by (simp_all add: K_def)
+ have e: "e > 0"
+ unfolding e_def using ynz K1 by simp
+ define At where "At x y z n \<longleftrightarrow>
+ convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
+ dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
+ norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
+ for x y z n
+ have At0: "At a b c 0"
+ using fy
+ by (simp add: At_def e_def has_chain_integral_chain_integral3)
+ { fix x y z n
+ assume At: "At x y z n"
+ then have contf': "continuous_on (convex hull {x,y,z}) f"
+ using contf At_def continuous_on_subset by metis
+ have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
+ using At Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
+ apply (simp add: At_def algebra_simps)
+ apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
+ done
+ } note AtSuc = this
+ obtain fa fb fc
+ where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
+ and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
+ and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
+ "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
+ "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
+ and no: "\<And>n. norm(?pathint (fa n) (fb n) +
+ ?pathint (fb n) (fc n) +
+ ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
+ and conv_le: "\<And>n. convex hull {fa(Suc n), fb(Suc n), fc(Suc n)} \<subseteq> convex hull {fa n, fb n, fc n}"
+ apply (rule Chain3 [of At, OF At0 AtSuc])
+ apply (auto simp: At_def)
+ done
+ obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}"
+ proof (rule bounded_closed_nest)
+ show "\<And>n. closed (convex hull {fa n, fb n, fc n})"
+ by (simp add: compact_imp_closed finite_imp_compact_convex_hull)
+ show "\<And>m n. m \<le> n \<Longrightarrow> convex hull {fa n, fb n, fc n} \<subseteq> convex hull {fa m, fb m, fc m}"
+ by (erule transitive_stepwise_le) (auto simp: conv_le)
+ qed (fastforce intro: finite_imp_bounded_convex_hull)+
+ then have xin: "x \<in> convex hull {a,b,c}"
+ using assms f0 by blast
+ then have fx: "f field_differentiable at x within (convex hull {a,b,c})"
+ using assms holomorphic_on_def by blast
+ { fix k n
+ assume k: "0 < k"
+ and le:
+ "\<And>x' y' z'.
+ \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
+ x \<in> convex hull {x',y',z'};
+ convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
+ \<Longrightarrow>
+ cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
+ \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
+ and Kk: "K / k < 2 ^ n"
+ have "K / 2 ^ n < k" using Kk k
+ by (auto simp: field_simps)
+ then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
+ using dist [of n] k
+ by linarith+
+ have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
+ \<le> (3 * K / 2 ^ n)\<^sup>2"
+ using dist [of n] e K
+ by (simp add: abs_le_square_iff [symmetric])
+ have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
+ by linarith
+ have "e * (dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2 \<le> e * (3 * K / 2 ^ n)\<^sup>2"
+ using ynz dle e mult_le_cancel_left_pos by blast
+ also have "\<dots> <
+ cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
+ using no [of n] e K
+ apply (simp add: e_def field_simps)
+ apply (simp only: zero_less_norm_iff [symmetric])
+ done
+ finally have False
+ using le [OF DD x cosb] by auto
+ } then
+ have ?thesis
+ using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
+ apply clarsimp
+ apply (rule_tac y1="K/k" in exE [OF real_arch_pow[of 2]], force+)
+ done
+ }
+ moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
+ by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1)
+ segments_subset_convex_hull(3) segments_subset_convex_hull(5))
+ ultimately show ?thesis
+ using has_contour_integral_integral by fastforce
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Version needing function holomorphic in interior only\<close>
+
+lemma Cauchy_theorem_flat_lemma:
+ assumes f: "continuous_on (convex hull {a,b,c}) f"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ and k: "0 \<le> k"
+ shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof -
+ have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
+ using f continuous_on_subset segments_subset_convex_hull by metis+
+ show ?thesis
+ proof (cases "k \<le> 1")
+ case True show ?thesis
+ by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc)
+ next
+ case False then show ?thesis
+ using fabc c
+ apply (subst contour_integral_split [of a c f "1/k" b, symmetric])
+ apply (metis closed_segment_commute fabc(3))
+ apply (auto simp: k contour_integral_reverse_linepath)
+ done
+ qed
+qed
+
+lemma Cauchy_theorem_flat:
+ assumes f: "continuous_on (convex hull {a,b,c}) f"
+ and c: "c - a = k *\<^sub>R (b - a)"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "0 \<le> k")
+ case True with assms show ?thesis
+ by (blast intro: Cauchy_theorem_flat_lemma)
+next
+ case False
+ have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
+ using f continuous_on_subset segments_subset_convex_hull by metis+
+ moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f +
+ contour_integral (linepath c b) f = 0"
+ apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
+ using False c
+ apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
+ done
+ ultimately show ?thesis
+ apply (auto simp: contour_integral_reverse_linepath)
+ using add_eq_0_iff by force
+qed
+
+lemma Cauchy_theorem_triangle_interior:
+ assumes contf: "continuous_on (convex hull {a,b,c}) f"
+ and holf: "f holomorphic_on interior (convex hull {a,b,c})"
+ shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+proof -
+ have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
+ using contf continuous_on_subset segments_subset_convex_hull by metis+
+ have "bounded (f ` (convex hull {a,b,c}))"
+ by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
+ then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
+ by (auto simp: dest!: bounded_pos [THEN iffD1])
+ have "bounded (convex hull {a,b,c})"
+ by (simp add: bounded_convex_hull)
+ then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
+ using bounded_pos_less by blast
+ then have diff_2C: "norm(x - y) \<le> 2*C"
+ if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
+ proof -
+ have "cmod x \<le> C"
+ using x by (meson Cno not_le not_less_iff_gr_or_eq)
+ hence "cmod (x - y) \<le> C + C"
+ using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
+ thus "cmod (x - y) \<le> 2 * C"
+ by (metis mult_2)
+ qed
+ have contf': "continuous_on (convex hull {b,a,c}) f"
+ using contf by (simp add: insert_commute)
+ { fix y::complex
+ assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
+ and ynz: "y \<noteq> 0"
+ have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y"
+ by (rule has_chain_integral_chain_integral3 [OF fy])
+ have ?thesis
+ proof (cases "c=a \<or> a=b \<or> b=c")
+ case True then show ?thesis
+ using Cauchy_theorem_flat [OF contf, of 0]
+ using has_chain_integral_chain_integral3 [OF fy] ynz
+ by (force simp: fabc contour_integral_reverse_linepath)
+ next
+ case False
+ then have car3: "card {a, b, c} = Suc (DIM(complex))"
+ by auto
+ { assume "interior(convex hull {a,b,c}) = {}"
+ then have "collinear{a,b,c}"
+ using interior_convex_hull_eq_empty [OF car3]
+ by (simp add: collinear_3_eq_affine_dependent)
+ with False obtain d where "c \<noteq> a" "a \<noteq> b" "b \<noteq> c" "c - b = d *\<^sub>R (a - b)"
+ by (auto simp: collinear_3 collinear_lemma)
+ then have "False"
+ using False Cauchy_theorem_flat [OF contf'] pi_eq_y ynz
+ by (simp add: fabc add_eq_0_iff contour_integral_reverse_linepath)
+ }
+ then obtain d where d: "d \<in> interior (convex hull {a, b, c})"
+ by blast
+ { fix d1
+ assume d1_pos: "0 < d1"
+ and d1: "\<And>x x'. \<lbrakk>x\<in>convex hull {a, b, c}; x'\<in>convex hull {a, b, c}; cmod (x' - x) < d1\<rbrakk>
+ \<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)"
+ define e where "e = min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))"
+ define shrink where "shrink x = x - e *\<^sub>R (x - d)" for x
+ let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
+ have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B"
+ using d1_pos \<open>C>0\<close> \<open>B>0\<close> ynz by (simp_all add: e_def)
+ then have eCB: "24 * e * C * B \<le> cmod y"
+ using \<open>C>0\<close> \<open>B>0\<close> by (simp add: field_simps)
+ have e_le_d1: "e * (4 * C) \<le> d1"
+ using e \<open>C>0\<close> by (simp add: field_simps)
+ have "shrink a \<in> interior(convex hull {a,b,c})"
+ "shrink b \<in> interior(convex hull {a,b,c})"
+ "shrink c \<in> interior(convex hull {a,b,c})"
+ using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
+ then have fhp0: "(f has_contour_integral 0)
+ (linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))"
+ by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal)
+ then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0"
+ by (simp add: has_chain_integral_chain_integral3)
+ have fpi_abc: "f contour_integrable_on linepath (shrink a) (shrink b)"
+ "f contour_integrable_on linepath (shrink b) (shrink c)"
+ "f contour_integrable_on linepath (shrink c) (shrink a)"
+ using fhp0 by (auto simp: valid_path_join dest: has_contour_integral_integrable)
+ have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)"
+ using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric])
+ have sh_eq: "\<And>a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)"
+ by (simp add: algebra_simps)
+ have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
+ using False \<open>C>0\<close> diff_2C [of b a] ynz
+ by (auto simp: field_split_simps hull_inc)
+ have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u
+ apply (cases "x=0", simp add: \<open>0<C\<close>)
+ using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
+ { fix u v
+ assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v"
+ and fpi_uv: "f contour_integrable_on linepath (shrink u) (shrink v)"
+ have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})"
+ "shrink v \<in> interior(convex hull {a,b,c})"
+ using d e uv
+ by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
+ have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B"
+ using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD])
+ have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y"
+ apply (rule order_trans [OF _ eCB])
+ using e \<open>B>0\<close> diff_2C [of u v] uv
+ by (auto simp: field_simps)
+ { fix x::real assume x: "0\<le>x" "x\<le>1"
+ have cmod_less_4C: "cmod ((1 - x) *\<^sub>R u - (1 - x) *\<^sub>R d) + cmod (x *\<^sub>R v - x *\<^sub>R d) < (C+C) + (C+C)"
+ apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0])
+ using uv x d interior_subset
+ apply (auto simp: hull_inc intro!: less_C)
+ done
+ have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))"
+ by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real)
+ have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1"
+ apply (simp only: ll norm_mult scaleR_diff_right)
+ using \<open>e>0\<close> cmod_less_4C apply (force intro: norm_triangle_lt less_le_trans [OF _ e_le_d1])
+ done
+ have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)"
+ using x uv shr_uv cmod_less_dt
+ by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
+ also have "\<dots> \<le> cmod y / cmod (v - u) / 12"
+ using False uv \<open>C>0\<close> diff_2C [of v u] ynz
+ by (auto simp: field_split_simps hull_inc)
+ finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
+ by simp
+ then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y"
+ using uv False by (auto simp: field_simps)
+ have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
+ cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
+ \<le> B * (cmod y / 24 / C / B * 2 * C) + 2 * C * (cmod y / 24 / C)"
+ apply (rule add_mono [OF mult_mono])
+ using By_uv e \<open>0 < B\<close> \<open>0 < C\<close> x apply (simp_all add: cmod_fuv cmod_shr cmod_12_le)
+ apply (simp add: field_simps)
+ done
+ also have "\<dots> \<le> cmod y / 6"
+ by simp
+ finally have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
+ cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
+ \<le> cmod y / 6" .
+ } note cmod_diff_le = this
+ have f_uv: "continuous_on (closed_segment u v) f"
+ by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull])
+ have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)"
+ by (simp add: algebra_simps)
+ have "norm (?pathint (shrink u) (shrink v) - ?pathint u v)
+ \<le> (B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)) * content (cbox 0 (1::real))"
+ apply (rule has_integral_bound
+ [of _ "\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)"
+ _ 0 1])
+ using ynz \<open>0 < B\<close> \<open>0 < C\<close>
+ apply (simp_all del: le_divide_eq_numeral1)
+ apply (simp add: has_integral_diff has_contour_integral_linepath [symmetric] has_contour_integral_integral
+ fpi_uv f_uv contour_integrable_continuous_linepath)
+ apply (auto simp: ** norm_triangle_le norm_mult cmod_diff_le simp del: le_divide_eq_numeral1)
+ done
+ also have "\<dots> \<le> norm y / 6"
+ by simp
+ finally have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6" .
+ } note * = this
+ have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6"
+ using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
+ moreover
+ have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6"
+ using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
+ moreover
+ have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6"
+ using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
+ ultimately
+ have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) +
+ (?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a))
+ \<le> norm y / 6 + norm y / 6 + norm y / 6"
+ by (metis norm_triangle_le add_mono)
+ also have "\<dots> = norm y / 2"
+ by simp
+ finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) -
+ (?pathint a b + ?pathint b c + ?pathint c a))
+ \<le> norm y / 2"
+ by (simp add: algebra_simps)
+ then
+ have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2"
+ by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff)
+ then have "False"
+ using pi_eq_y ynz by auto
+ }
+ note * = this
+ have "uniformly_continuous_on (convex hull {a,b,c}) f"
+ by (simp add: contf compact_convex_hull compact_uniformly_continuous)
+ moreover have "norm y / (24 * C) > 0"
+ using ynz \<open>C > 0\<close> by auto
+ ultimately obtain \<delta> where "\<delta> > 0" and
+ "\<forall>x\<in>convex hull {a, b, c}. \<forall>x'\<in>convex hull {a, b, c}.
+ dist x' x < \<delta> \<longrightarrow> dist (f x') (f x) < cmod y / (24 * C)"
+ using \<open>C > 0\<close> ynz unfolding uniformly_continuous_on_def dist_norm by blast
+ hence False using *[of \<delta>] by (auto simp: dist_norm)
+ then show ?thesis ..
+ qed
+ }
+ moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
+ using fabc contour_integrable_continuous_linepath by auto
+ ultimately show ?thesis
+ using has_contour_integral_integral by fastforce
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Version allowing finite number of exceptional points\<close>
+
+proposition\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_triangle_cofinite:
+ assumes "continuous_on (convex hull {a,b,c}) f"
+ and "finite S"
+ and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - S \<Longrightarrow> f field_differentiable (at x))"
+ shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+using assms
+proof (induction "card S" arbitrary: a b c S rule: less_induct)
+ case (less S a b c)
+ show ?case
+ proof (cases "S={}")
+ case True with less show ?thesis
+ by (fastforce simp: holomorphic_on_def field_differentiable_at_within Cauchy_theorem_triangle_interior)
+ next
+ case False
+ then obtain d S' where d: "S = insert d S'" "d \<notin> S'"
+ by (meson Set.set_insert all_not_in_conv)
+ then show ?thesis
+ proof (cases "d \<in> convex hull {a,b,c}")
+ case False
+ show "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+ proof (rule less.hyps)
+ show "\<And>x. x \<in> interior (convex hull {a, b, c}) - S' \<Longrightarrow> f field_differentiable at x"
+ using False d interior_subset by (auto intro!: less.prems)
+ qed (use d less.prems in auto)
+ next
+ case True
+ have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}"
+ by (meson True hull_subset insert_subset convex_hull_subset)
+ have abd: "(f has_contour_integral 0) (linepath a b +++ linepath b d +++ linepath d a)"
+ proof (rule less.hyps)
+ show "\<And>x. x \<in> interior (convex hull {a, b, d}) - S' \<Longrightarrow> f field_differentiable at x"
+ using d not_in_interior_convex_hull_3
+ by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
+ qed (use d continuous_on_subset [OF _ *] less.prems in auto)
+ have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}"
+ by (meson True hull_subset insert_subset convex_hull_subset)
+ have bcd: "(f has_contour_integral 0) (linepath b c +++ linepath c d +++ linepath d b)"
+ proof (rule less.hyps)
+ show "\<And>x. x \<in> interior (convex hull {b, c, d}) - S' \<Longrightarrow> f field_differentiable at x"
+ using d not_in_interior_convex_hull_3
+ by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
+ qed (use d continuous_on_subset [OF _ *] less.prems in auto)
+ have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}"
+ by (meson True hull_subset insert_subset convex_hull_subset)
+ have cad: "(f has_contour_integral 0) (linepath c a +++ linepath a d +++ linepath d c)"
+ proof (rule less.hyps)
+ show "\<And>x. x \<in> interior (convex hull {c, a, d}) - S' \<Longrightarrow> f field_differentiable at x"
+ using d not_in_interior_convex_hull_3
+ by (clarsimp intro!: less.prems) (metis * insert_absorb insert_subset interior_mono)
+ qed (use d continuous_on_subset [OF _ *] less.prems in auto)
+ have "f contour_integrable_on linepath a b"
+ using less.prems abd contour_integrable_joinD1 contour_integrable_on_def by blast
+ moreover have "f contour_integrable_on linepath b c"
+ using less.prems bcd contour_integrable_joinD1 contour_integrable_on_def by blast
+ moreover have "f contour_integrable_on linepath c a"
+ using less.prems cad contour_integrable_joinD1 contour_integrable_on_def by blast
+ ultimately have fpi: "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
+ by auto
+ { fix y::complex
+ assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
+ and ynz: "y \<noteq> 0"
+ have cont_ad: "continuous_on (closed_segment a d) f"
+ by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3))
+ have cont_bd: "continuous_on (closed_segment b d) f"
+ by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1))
+ have cont_cd: "continuous_on (closed_segment c d) f"
+ by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2))
+ have "contour_integral (linepath a b) f = - (contour_integral (linepath b d) f + (contour_integral (linepath d a) f))"
+ "contour_integral (linepath b c) f = - (contour_integral (linepath c d) f + (contour_integral (linepath d b) f))"
+ "contour_integral (linepath c a) f = - (contour_integral (linepath a d) f + contour_integral (linepath d c) f)"
+ using has_chain_integral_chain_integral3 [OF abd]
+ has_chain_integral_chain_integral3 [OF bcd]
+ has_chain_integral_chain_integral3 [OF cad]
+ by (simp_all add: algebra_simps add_eq_0_iff)
+ then have ?thesis
+ using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 contour_integral_reverse_linepath by fastforce
+ }
+ then show ?thesis
+ using fpi contour_integrable_on_def by blast
+ qed
+ qed
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Cauchy's theorem for an open starlike set\<close>
+
+lemma starlike_convex_subset:
+ assumes S: "a \<in> S" "closed_segment b c \<subseteq> S" and subs: "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S"
+ shows "convex hull {a,b,c} \<subseteq> S"
+ using S
+ apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull)
+ apply (meson subs convexD convex_closed_segment ends_in_segment(1) ends_in_segment(2) subsetCE)
+ done
+
+lemma triangle_contour_integrals_starlike_primitive:
+ assumes contf: "continuous_on S f"
+ and S: "a \<in> S" "open S"
+ and x: "x \<in> S"
+ and subs: "\<And>y. y \<in> S \<Longrightarrow> closed_segment a y \<subseteq> S"
+ and zer: "\<And>b c. closed_segment b c \<subseteq> S
+ \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x)"
+proof -
+ let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
+ { fix e y
+ assume e: "0 < e" and bxe: "ball x e \<subseteq> S" and close: "cmod (y - x) < e"
+ have y: "y \<in> S"
+ using bxe close by (force simp: dist_norm norm_minus_commute)
+ have cont_ayf: "continuous_on (closed_segment a y) f"
+ using contf continuous_on_subset subs y by blast
+ have xys: "closed_segment x y \<subseteq> S"
+ apply (rule order_trans [OF _ bxe])
+ using close
+ by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound)
+ have "?pathint a y - ?pathint a x = ?pathint x y"
+ using zer [OF xys] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
+ } note [simp] = this
+ { fix e::real
+ assume e: "0 < e"
+ have cont_atx: "continuous (at x) f"
+ using x S contf continuous_on_eq_continuous_at by blast
+ then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2"
+ unfolding continuous_at Lim_at dist_norm using e
+ by (drule_tac x="e/2" in spec) force
+ obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> S" using \<open>open S\<close> x
+ by (auto simp: open_contains_ball)
+ have dpos: "min d1 d2 > 0" using d1 d2 by simp
+ { fix y
+ assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2"
+ have y: "y \<in> S"
+ using d2 close by (force simp: dist_norm norm_minus_commute)
+ have "closed_segment x y \<subseteq> S"
+ using close d2 by (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1))
+ then have fxy: "f contour_integrable_on linepath x y"
+ by (metis contour_integrable_continuous_linepath continuous_on_subset [OF contf])
+ then obtain i where i: "(f has_contour_integral i) (linepath x y)"
+ by (auto simp: contour_integrable_on_def)
+ then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
+ by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
+ then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
+ proof (rule has_contour_integral_bound_linepath)
+ show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2"
+ by (meson close d1_less le_less_trans less_imp_le min.strict_boundedE segment_bound1)
+ qed (use e in simp)
+ also have "\<dots> < e * cmod (y - x)"
+ by (simp add: e yx)
+ finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
+ using i yx by (simp add: contour_integral_unique divide_less_eq)
+ }
+ then have "\<exists>d>0. \<forall>y. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
+ using dpos by blast
+ }
+ then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) \<midarrow>x\<rightarrow> 0"
+ by (simp add: Lim_at dist_norm inverse_eq_divide)
+ show ?thesis
+ apply (simp add: has_field_derivative_def has_derivative_at2 bounded_linear_mult_right)
+ apply (rule Lim_transform [OF * tendsto_eventually])
+ using \<open>open S\<close> x apply (force simp: dist_norm open_contains_ball inverse_eq_divide [symmetric] eventually_at)
+ done
+qed
+
+(** Existence of a primitive.*)
+lemma holomorphic_starlike_primitive:
+ fixes f :: "complex \<Rightarrow> complex"
+ assumes contf: "continuous_on S f"
+ and S: "starlike S" and os: "open S"
+ and k: "finite k"
+ and fcd: "\<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x"
+ shows "\<exists>g. \<forall>x \<in> S. (g has_field_derivative f x) (at x)"
+proof -
+ obtain a where a: "a\<in>S" and a_cs: "\<And>x. x\<in>S \<Longrightarrow> closed_segment a x \<subseteq> S"
+ using S by (auto simp: starlike_def)
+ { fix x b c
+ assume "x \<in> S" "closed_segment b c \<subseteq> S"
+ then have abcs: "convex hull {a, b, c} \<subseteq> S"
+ by (simp add: a a_cs starlike_convex_subset)
+ then have "continuous_on (convex hull {a, b, c}) f"
+ by (simp add: continuous_on_subset [OF contf])
+ then have "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+ using abcs interior_subset by (force intro: fcd Cauchy_theorem_triangle_cofinite [OF _ k])
+ } note 0 = this
+ show ?thesis
+ apply (intro exI ballI)
+ apply (rule triangle_contour_integrals_starlike_primitive [OF contf a os], assumption)
+ apply (metis a_cs)
+ apply (metis has_chain_integral_chain_integral3 0)
+ done
+qed
+
+lemma Cauchy_theorem_starlike:
+ "\<lbrakk>open S; starlike S; finite k; continuous_on S f;
+ \<And>x. x \<in> S - k \<Longrightarrow> f field_differentiable at x;
+ valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral 0) g"
+ by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open)
+
+lemma Cauchy_theorem_starlike_simple:
+ "\<lbrakk>open S; starlike S; f holomorphic_on S; valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral 0) g"
+apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI])
+apply (simp_all add: holomorphic_on_imp_continuous_on)
+apply (metis at_within_open holomorphic_on_def)
+done
+
+subsection\<open>Cauchy's theorem for a convex set\<close>
+
+text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close>
+
+lemma triangle_contour_integrals_convex_primitive:
+ assumes contf: "continuous_on S f"
+ and S: "a \<in> S" "convex S"
+ and x: "x \<in> S"
+ and zer: "\<And>b c. \<lbrakk>b \<in> S; c \<in> S\<rbrakk>
+ \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x within S)"
+proof -
+ let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
+ { fix y
+ assume y: "y \<in> S"
+ have cont_ayf: "continuous_on (closed_segment a y) f"
+ using S y by (meson contf continuous_on_subset convex_contains_segment)
+ have xys: "closed_segment x y \<subseteq> S" (*?*)
+ using convex_contains_segment S x y by auto
+ have "?pathint a y - ?pathint a x = ?pathint x y"
+ using zer [OF x y] contour_integral_reverse_linepath [OF cont_ayf] add_eq_0_iff by force
+ } note [simp] = this
+ { fix e::real
+ assume e: "0 < e"
+ have cont_atx: "continuous (at x within S) f"
+ using x S contf by (simp add: continuous_on_eq_continuous_within)
+ then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. \<lbrakk>y \<in> S; cmod (y - x) < d1\<rbrakk> \<Longrightarrow> cmod (f y - f x) < e/2"
+ unfolding continuous_within Lim_within dist_norm using e
+ by (drule_tac x="e/2" in spec) force
+ { fix y
+ assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> S"
+ have fxy: "f contour_integrable_on linepath x y"
+ using convex_contains_segment S x y
+ by (blast intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
+ then obtain i where i: "(f has_contour_integral i) (linepath x y)"
+ by (auto simp: contour_integrable_on_def)
+ then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
+ by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
+ then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
+ proof (rule has_contour_integral_bound_linepath)
+ show "\<And>u. u \<in> closed_segment x y \<Longrightarrow> cmod (f u - f x) \<le> e / 2"
+ by (meson assms(3) close convex_contains_segment d1_less le_less_trans less_imp_le segment_bound1 subset_iff x y)
+ qed (use e in simp)
+ also have "\<dots> < e * cmod (y - x)"
+ by (simp add: e yx)
+ finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
+ using i yx by (simp add: contour_integral_unique divide_less_eq)
+ }
+ then have "\<exists>d>0. \<forall>y\<in>S. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
+ using d1 by blast
+ }
+ then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) \<longlongrightarrow> 0) (at x within S)"
+ by (simp add: Lim_within dist_norm inverse_eq_divide)
+ show ?thesis
+ apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
+ apply (rule Lim_transform [OF * tendsto_eventually])
+ using linordered_field_no_ub
+ apply (force simp: inverse_eq_divide [symmetric] eventually_at)
+ done
+qed
+
+lemma contour_integral_convex_primitive:
+ assumes "convex S" "continuous_on S f"
+ "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
+ obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
+proof (cases "S={}")
+ case False
+ with assms that show ?thesis
+ by (blast intro: triangle_contour_integrals_convex_primitive has_chain_integral_chain_integral3)
+qed auto
+
+lemma holomorphic_convex_primitive:
+ fixes f :: "complex \<Rightarrow> complex"
+ assumes "convex S" "finite K" and contf: "continuous_on S f"
+ and fd: "\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x"
+ obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
+proof (rule contour_integral_convex_primitive [OF \<open>convex S\<close> contf Cauchy_theorem_triangle_cofinite])
+ have *: "convex hull {a, b, c} \<subseteq> S" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c
+ by (simp add: \<open>convex S\<close> hull_minimal that)
+ show "continuous_on (convex hull {a, b, c}) f" if "a \<in> S" "b \<in> S" "c \<in> S" for a b c
+ by (meson "*" contf continuous_on_subset that)
+ show "f field_differentiable at x" if "a \<in> S" "b \<in> S" "c \<in> S" "x \<in> interior (convex hull {a, b, c}) - K" for a b c x
+ by (metis "*" DiffD1 DiffD2 DiffI fd interior_mono subsetCE that)
+qed (use assms in \<open>force+\<close>)
+
+lemma holomorphic_convex_primitive':
+ fixes f :: "complex \<Rightarrow> complex"
+ assumes "convex S" and "open S" and "f holomorphic_on S"
+ obtains g where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
+proof (rule holomorphic_convex_primitive)
+ fix x assume "x \<in> interior S - {}"
+ with assms show "f field_differentiable at x"
+ by (auto intro!: holomorphic_on_imp_differentiable_at simp: interior_open)
+qed (use assms in \<open>auto intro: holomorphic_on_imp_continuous_on\<close>)
+
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_convex:
+ "\<lbrakk>continuous_on S f; convex S; finite K;
+ \<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x;
+ valid_path g; path_image g \<subseteq> S; pathfinish g = pathstart g\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral 0) g"
+ by (metis holomorphic_convex_primitive Cauchy_theorem_primitive)
+
+corollary Cauchy_theorem_convex_simple:
+ "\<lbrakk>f holomorphic_on S; convex S;
+ valid_path g; path_image g \<subseteq> S;
+ pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
+ apply (rule Cauchy_theorem_convex [where K = "{}"])
+ apply (simp_all add: holomorphic_on_imp_continuous_on)
+ using at_within_interior holomorphic_on_def interior_subset by fastforce
+
+text\<open>In particular for a disc\<close>
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc:
+ "\<lbrakk>finite K; continuous_on (cball a e) f;
+ \<And>x. x \<in> ball a e - K \<Longrightarrow> f field_differentiable at x;
+ valid_path g; path_image g \<subseteq> cball a e;
+ pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
+ by (auto intro: Cauchy_theorem_convex)
+
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_theorem_disc_simple:
+ "\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e;
+ pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
+by (simp add: Cauchy_theorem_convex_simple)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Generalize integrability to local primitives\<close>
+
+lemma contour_integral_local_primitive_lemma:
+ fixes f :: "complex\<Rightarrow>complex"
+ shows
+ "\<lbrakk>g piecewise_differentiable_on {a..b};
+ \<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s);
+ \<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk>
+ \<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b}))
+ integrable_on {a..b}"
+ apply (cases "cbox a b = {}", force)
+ apply (simp add: integrable_on_def)
+ apply (rule exI)
+ apply (rule contour_integral_primitive_lemma, assumption+)
+ using atLeastAtMost_iff by blast
+
+lemma contour_integral_local_primitive_any:
+ fixes f :: "complex \<Rightarrow> complex"
+ assumes gpd: "g piecewise_differentiable_on {a..b}"
+ and dh: "\<And>x. x \<in> s
+ \<Longrightarrow> \<exists>d h. 0 < d \<and>
+ (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
+ and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
+ shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}"
+proof -
+ { fix x
+ assume x: "a \<le> x" "x \<le> b"
+ obtain d h where d: "0 < d"
+ and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))"
+ using x gs dh by (metis atLeastAtMost_iff)
+ have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast
+ then obtain e where e: "e>0" and lessd: "\<And>x'. x' \<in> {a..b} \<Longrightarrow> \<bar>x' - x\<bar> < e \<Longrightarrow> cmod (g x' - g x) < d"
+ using x d
+ apply (auto simp: dist_norm continuous_on_iff)
+ apply (drule_tac x=x in bspec)
+ using x apply simp
+ apply (drule_tac x=d in spec, auto)
+ done
+ have "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> {u..v} \<subseteq> ball x d \<and> (u \<le> v \<longrightarrow> a \<le> u \<and> v \<le> b) \<longrightarrow>
+ (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}"
+ apply (rule_tac x=e in exI)
+ using e
+ apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify)
+ apply (rule_tac f = h and s = "g ` {u..v}" in contour_integral_local_primitive_lemma)
+ apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset)
+ apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force)
+ done
+ } then
+ show ?thesis
+ by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified])
+qed
+
+lemma contour_integral_local_primitive:
+ fixes f :: "complex \<Rightarrow> complex"
+ assumes g: "valid_path g" "path_image g \<subseteq> s"
+ and dh: "\<And>x. x \<in> s
+ \<Longrightarrow> \<exists>d h. 0 < d \<and>
+ (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
+ shows "f contour_integrable_on g"
+ using g
+ apply (simp add: valid_path_def path_image_def contour_integrable_on_def has_contour_integral_def
+ has_integral_localized_vector_derivative integrable_on_def [symmetric])
+ using contour_integral_local_primitive_any [OF _ dh]
+ by (meson image_subset_iff piecewise_C1_imp_differentiable)
+
+
+text\<open>In particular if a function is holomorphic\<close>
+
+lemma contour_integrable_holomorphic:
+ assumes contf: "continuous_on s f"
+ and os: "open s"
+ and k: "finite k"
+ and g: "valid_path g" "path_image g \<subseteq> s"
+ and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x"
+ shows "f contour_integrable_on g"
+proof -
+ { fix z
+ assume z: "z \<in> s"
+ obtain d where "d>0" and d: "ball z d \<subseteq> s" using \<open>open s\<close> z
+ by (auto simp: open_contains_ball)
+ then have contfb: "continuous_on (ball z d) f"
+ using contf continuous_on_subset by blast
+ obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)"
+ by (metis holomorphic_convex_primitive [OF convex_ball k contfb fcd] d interior_subset Diff_iff subsetD)
+ then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)"
+ by (metis open_ball at_within_open d os subsetCE)
+ then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
+ by (force simp: dist_norm norm_minus_commute)
+ then have "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
+ using \<open>0 < d\<close> by blast
+ }
+ then show ?thesis
+ by (rule contour_integral_local_primitive [OF g])
+qed
+
+lemma contour_integrable_holomorphic_simple:
+ assumes fh: "f holomorphic_on S"
+ and os: "open S"
+ and g: "valid_path g" "path_image g \<subseteq> S"
+ shows "f contour_integrable_on g"
+ apply (rule contour_integrable_holomorphic [OF _ os Finite_Set.finite.emptyI g])
+ apply (simp add: fh holomorphic_on_imp_continuous_on)
+ using fh by (simp add: field_differentiable_def holomorphic_on_open os)
+
+lemma continuous_on_inversediff:
+ fixes z:: "'a::real_normed_field" shows "z \<notin> S \<Longrightarrow> continuous_on S (\<lambda>w. 1 / (w - z))"
+ by (rule continuous_intros | force)+
+
+lemma contour_integrable_inversediff:
+ "\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) contour_integrable_on g"
+apply (rule contour_integrable_holomorphic_simple [of _ "UNIV-{z}"])
+apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros)
+done
+
+text\<open>Key fact that path integral is the same for a "nearby" path. This is the
+ main lemma for the homotopy form of Cauchy's theorem and is also useful
+ if we want "without loss of generality" to assume some nice properties of a
+ path (e.g. smoothness). It can also be used to define the integrals of
+ analytic functions over arbitrary continuous paths. This is just done for
+ winding numbers now.
+\<close>
+
+text\<open>A technical definition to avoid duplication of similar proofs,
+ for paths joined at the ends versus looping paths\<close>
+definition linked_paths :: "bool \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
+ where "linked_paths atends g h ==
+ (if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
+ else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)"
+
+text\<open>This formulation covers two cases: \<^term>\<open>g\<close> and \<^term>\<open>h\<close> share their
+ start and end points; \<^term>\<open>g\<close> and \<^term>\<open>h\<close> both loop upon themselves.\<close>
+lemma contour_integral_nearby:
+ assumes os: "open S" and p: "path p" "path_image p \<subseteq> S"
+ shows "\<exists>d. 0 < d \<and>
+ (\<forall>g h. valid_path g \<and> valid_path h \<and>
+ (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
+ linked_paths atends g h
+ \<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and>
+ (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f))"
+proof -
+ have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> S"
+ using open_contains_ball os p(2) by blast
+ then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> S"
+ by metis
+ define cover where "cover = (\<lambda>z. ball z (ee z/3)) ` (path_image p)"
+ have "compact (path_image p)"
+ by (metis p(1) compact_path_image)
+ moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))"
+ using ee by auto
+ ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D"
+ by (simp add: compact_eq_Heine_Borel cover_def)
+ then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D"
+ by blast
+ then obtain k where k: "k \<subseteq> {0..1}" "finite k" and D_eq: "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k"
+ apply (simp add: cover_def path_image_def image_comp)
+ apply (blast dest!: finite_subset_image [OF \<open>finite D\<close>])
+ done
+ then have kne: "k \<noteq> {}"
+ using D by auto
+ have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p"
+ using k by (auto simp: path_image_def)
+ then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))"
+ by (metis ee)
+ define e where "e = Min((ee \<circ> p) ` k)"
+ have fin_eep: "finite ((ee \<circ> p) ` k)"
+ using k by blast
+ have "0 < e"
+ using ee k by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi)
+ have "uniformly_continuous_on {0..1} p"
+ using p by (simp add: path_def compact_uniformly_continuous)
+ then obtain d::real where d: "d>0"
+ and de: "\<And>x x'. \<bar>x' - x\<bar> < d \<Longrightarrow> x\<in>{0..1} \<Longrightarrow> x'\<in>{0..1} \<Longrightarrow> cmod (p x' - p x) < e/3"
+ unfolding uniformly_continuous_on_def dist_norm real_norm_def
+ by (metis divide_pos_pos \<open>0 < e\<close> zero_less_numeral)
+ then obtain N::nat where N: "N>0" "inverse N < d"
+ using real_arch_inverse [of d] by auto
+ show ?thesis
+ proof (intro exI conjI allI; clarify?)
+ show "e/3 > 0"
+ using \<open>0 < e\<close> by simp
+ fix g h
+ assume g: "valid_path g" and ghp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3 \<and> cmod (h t - p t) < e / 3"
+ and h: "valid_path h"
+ and joins: "linked_paths atends g h"
+ { fix t::real
+ assume t: "0 \<le> t" "t \<le> 1"
+ then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)"
+ using \<open>path_image p \<subseteq> \<Union>D\<close> D_eq by (force simp: path_image_def)
+ then have ele: "e \<le> ee (p u)" using fin_eep
+ by (simp add: e_def)
+ have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3"
+ using ghp t by auto
+ with ele have "cmod (g t - p t) < ee (p u) / 3"
+ "cmod (h t - p t) < ee (p u) / 3"
+ by linarith+
+ then have "g t \<in> ball(p u) (ee(p u))" "h t \<in> ball(p u) (ee(p u))"
+ using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"]
+ norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u
+ by (force simp: dist_norm ball_def norm_minus_commute)+
+ then have "g t \<in> S" "h t \<in> S" using ee u k
+ by (auto simp: path_image_def ball_def)
+ }
+ then have ghs: "path_image g \<subseteq> S" "path_image h \<subseteq> S"
+ by (auto simp: path_image_def)
+ moreover
+ { fix f
+ assume fhols: "f holomorphic_on S"
+ then have fpa: "f contour_integrable_on g" "f contour_integrable_on h"
+ using g ghs h holomorphic_on_imp_continuous_on os contour_integrable_holomorphic_simple
+ by blast+
+ have contf: "continuous_on S f"
+ by (simp add: fhols holomorphic_on_imp_continuous_on)
+ { fix z
+ assume z: "z \<in> path_image p"
+ have "f holomorphic_on ball z (ee z)"
+ using fhols ee z holomorphic_on_subset by blast
+ then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))"
+ using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified]
+ by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball)
+ }
+ then obtain ff where ff:
+ "\<And>z w. \<lbrakk>z \<in> path_image p; w \<in> ball z (ee z)\<rbrakk> \<Longrightarrow> (ff z has_field_derivative f w) (at w)"
+ by metis
+ { fix n
+ assume n: "n \<le> N"
+ then have "contour_integral(subpath 0 (n/N) h) f - contour_integral(subpath 0 (n/N) g) f =
+ contour_integral(linepath (g(n/N)) (h(n/N))) f - contour_integral(linepath (g 0) (h 0)) f"
+ proof (induct n)
+ case 0 show ?case by simp
+ next
+ case (Suc n)
+ obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)"
+ using \<open>path_image p \<subseteq> \<Union>D\<close> [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems
+ by (force simp: path_image_def)
+ then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3"
+ by (simp add: dist_norm)
+ have e3le: "e/3 \<le> ee (p t) / 3" using fin_eep t
+ by (simp add: e_def)
+ { fix x
+ assume x: "n/N \<le> x" "x \<le> (1 + n)/N"
+ then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1"
+ using Suc.prems by auto
+ then have x01: "0 \<le> x" "x \<le> 1"
+ using x by linarith+
+ have "cmod (p t - p x) < ee (p t) / 3 + e/3"
+ proof (rule norm_diff_triangle_less [OF ptu de])
+ show "\<bar>real n / real N - x\<bar> < d"
+ using x N by (auto simp: field_simps)
+ qed (use x01 Suc.prems in auto)
+ then have ptx: "cmod (p t - p x) < 2*ee (p t)/3"
+ using e3le eepi [OF t] by simp
+ have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 "
+ apply (rule norm_diff_triangle_less [OF ptx])
+ using ghp x01 by (simp add: norm_minus_commute)
+ also have "\<dots> \<le> ee (p t)"
+ using e3le eepi [OF t] by simp
+ finally have gg: "cmod (p t - g x) < ee (p t)" .
+ have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 "
+ apply (rule norm_diff_triangle_less [OF ptx])
+ using ghp x01 by (simp add: norm_minus_commute)
+ also have "\<dots> \<le> ee (p t)"
+ using e3le eepi [OF t] by simp
+ finally have "cmod (p t - g x) < ee (p t)"
+ "cmod (p t - h x) < ee (p t)"
+ using gg by auto
+ } note ptgh_ee = this
+ have "closed_segment (g (real n / real N)) (h (real n / real N)) = path_image (linepath (h (n/N)) (g (n/N)))"
+ by (simp add: closed_segment_commute)
+ also have pi_hgn: "\<dots> \<subseteq> ball (p t) (ee (p t))"
+ using ptgh_ee [of "n/N"] Suc.prems
+ by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
+ finally have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> S"
+ using ee pi t by blast
+ have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N))) \<subseteq> ball (p t) (ee (p t))"
+ using ptgh_ee [of "(1+n)/N"] Suc.prems
+ by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
+ then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> S"
+ using \<open>N>0\<close> Suc.prems ee pi t
+ by (auto simp: Path_Connected.path_image_join field_simps)
+ have pi_subset_ball:
+ "path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++
+ subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N)))
+ \<subseteq> ball (p t) (ee (p t))"
+ apply (intro subset_path_image_join pi_hgn pi_ghn')
+ using \<open>N>0\<close> Suc.prems
+ apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
+ done
+ have pi0: "(f has_contour_integral 0)
+ (subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
+ subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))"
+ apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"])
+ apply (metis ff open_ball at_within_open pi t)
+ using Suc.prems pi_subset_ball apply (simp_all add: valid_path_join valid_path_subpath g h)
+ done
+ have fpa1: "f contour_integrable_on subpath (real n / real N) (real (Suc n) / real N) g"
+ using Suc.prems by (simp add: contour_integrable_subpath g fpa)
+ have fpa2: "f contour_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))"
+ using gh_n's
+ by (auto intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
+ have fpa3: "f contour_integrable_on linepath (h (real n / real N)) (g (real n / real N))"
+ using gh_ns
+ by (auto simp: closed_segment_commute intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
+ have eq0: "contour_integral (subpath (n/N) ((Suc n) / real N) g) f +
+ contour_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f +
+ contour_integral (subpath ((Suc n) / N) (n/N) h) f +
+ contour_integral (linepath (h (n/N)) (g (n/N))) f = 0"
+ using contour_integral_unique [OF pi0] Suc.prems
+ by (simp add: g h fpa valid_path_subpath contour_integrable_subpath
+ fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc)
+ have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
+ \<lbrakk>hn - gn = ghn - gh0;
+ gd + ghn' + he + hgn = (0::complex);
+ hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0"
+ by (auto simp: algebra_simps)
+ have "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
+ contour_integral (subpath 0 (n/N) h) f + contour_integral (subpath (n/N) ((Suc n) / N) h) f"
+ unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"]
+ using Suc.prems by (simp add: h fpa contour_integral_reversepath valid_path_subpath contour_integrable_subpath)
+ also have "\<dots> = contour_integral (subpath 0 ((Suc n) / N) h) f"
+ using Suc.prems by (simp add: contour_integral_subpath_combine h fpa)
+ finally have pi0_eq:
+ "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
+ contour_integral (subpath 0 ((Suc n) / N) h) f" .
+ show ?case
+ apply (rule * [OF Suc.hyps eq0 pi0_eq])
+ using Suc.prems
+ apply (simp_all add: g h fpa contour_integral_subpath_combine
+ contour_integral_reversepath [symmetric] contour_integrable_continuous_linepath
+ continuous_on_subset [OF contf gh_ns])
+ done
+ qed
+ } note ind = this
+ have "contour_integral h f = contour_integral g f"
+ using ind [OF order_refl] N joins
+ by (simp add: linked_paths_def pathstart_def pathfinish_def split: if_split_asm)
+ }
+ ultimately
+ show "path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)"
+ by metis
+ qed
+qed
+
+
+lemma
+ assumes "open S" "path p" "path_image p \<subseteq> S"
+ shows contour_integral_nearby_ends:
+ "\<exists>d. 0 < d \<and>
+ (\<forall>g h. valid_path g \<and> valid_path h \<and>
+ (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
+ pathstart h = pathstart g \<and> pathfinish h = pathfinish g
+ \<longrightarrow> path_image g \<subseteq> S \<and>
+ path_image h \<subseteq> S \<and>
+ (\<forall>f. f holomorphic_on S
+ \<longrightarrow> contour_integral h f = contour_integral g f))"
+ and contour_integral_nearby_loops:
+ "\<exists>d. 0 < d \<and>
+ (\<forall>g h. valid_path g \<and> valid_path h \<and>
+ (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
+ pathfinish g = pathstart g \<and> pathfinish h = pathstart h
+ \<longrightarrow> path_image g \<subseteq> S \<and>
+ path_image h \<subseteq> S \<and>
+ (\<forall>f. f holomorphic_on S
+ \<longrightarrow> contour_integral h f = contour_integral g f))"
+ using contour_integral_nearby [OF assms, where atends=True]
+ using contour_integral_nearby [OF assms, where atends=False]
+ unfolding linked_paths_def by simp_all
+
+lemma C1_differentiable_polynomial_function:
+ fixes p :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on S"
+ by (metis continuous_on_polymonial_function C1_differentiable_on_def has_vector_derivative_polynomial_function)
+
+lemma valid_path_polynomial_function:
+ fixes p :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "polynomial_function p \<Longrightarrow> valid_path p"
+by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
+
+lemma valid_path_subpath_trivial [simp]:
+ fixes g :: "real \<Rightarrow> 'a::euclidean_space"
+ shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
+ by (simp add: subpath_def valid_path_polynomial_function)
+
+lemma contour_integral_bound_exists:
+assumes S: "open S"
+ and g: "valid_path g"
+ and pag: "path_image g \<subseteq> S"
+ shows "\<exists>L. 0 < L \<and>
+ (\<forall>f B. f holomorphic_on S \<and> (\<forall>z \<in> S. norm(f z) \<le> B)
+ \<longrightarrow> norm(contour_integral g f) \<le> L*B)"
+proof -
+ have "path g" using g
+ by (simp add: valid_path_imp_path)
+ then obtain d::real and p
+ where d: "0 < d"
+ and p: "polynomial_function p" "path_image p \<subseteq> S"
+ and pi: "\<And>f. f holomorphic_on S \<Longrightarrow> contour_integral g f = contour_integral p f"
+ using contour_integral_nearby_ends [OF S \<open>path g\<close> pag]
+ apply clarify
+ apply (drule_tac x=g in spec)
+ apply (simp only: assms)
+ apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function)
+ done
+ then obtain p' where p': "polynomial_function p'"
+ "\<And>x. (p has_vector_derivative (p' x)) (at x)"
+ by (blast intro: has_vector_derivative_polynomial_function that)
+ then have "bounded(p' ` {0..1})"
+ using continuous_on_polymonial_function
+ by (force simp: intro!: compact_imp_bounded compact_continuous_image)
+ then obtain L where L: "L>0" and nop': "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> norm (p' x) \<le> L"
+ by (force simp: bounded_pos)
+ { fix f B
+ assume f: "f holomorphic_on S" and B: "\<And>z. z\<in>S \<Longrightarrow> cmod (f z) \<le> B"
+ then have "f contour_integrable_on p \<and> valid_path p"
+ using p S
+ by (blast intro: valid_path_polynomial_function contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on)
+ moreover have "cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B" if "0 \<le> x" "x \<le> 1" for x
+ proof (rule mult_mono)
+ show "cmod (vector_derivative p (at x)) \<le> L"
+ by (metis nop' p'(2) that vector_derivative_at)
+ show "cmod (f (p x)) \<le> B"
+ by (metis B atLeastAtMost_iff imageI p(2) path_defs(4) subset_eq that)
+ qed (use \<open>L>0\<close> in auto)
+ ultimately have "cmod (contour_integral g f) \<le> L * B"
+ apply (simp only: pi [OF f])
+ apply (simp only: contour_integral_integral)
+ apply (rule order_trans [OF integral_norm_bound_integral])
+ apply (auto simp: mult.commute integral_norm_bound_integral contour_integrable_on [symmetric] norm_mult)
+ done
+ } then
+ show ?thesis using \<open>L > 0\<close>
+ by (intro exI[of _ L]) auto
+qed
+
+text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
+
+subsection \<open>Winding Numbers\<close>
+
+definition\<^marker>\<open>tag important\<close> winding_number_prop :: "[real \<Rightarrow> complex, complex, real, real \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "winding_number_prop \<gamma> z e p n \<equiv>
+ valid_path p \<and> z \<notin> path_image p \<and>
+ pathstart p = pathstart \<gamma> \<and>
+ pathfinish p = pathfinish \<gamma> \<and>
+ (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+
+definition\<^marker>\<open>tag important\<close> winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
+ "winding_number \<gamma> z \<equiv> SOME n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
+
+
+lemma winding_number:
+ assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
+ shows "\<exists>p. winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain d
+ where d: "d>0"
+ and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
+ pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
+ path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
+ (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
+ using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
+ (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
+ define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
+ have "\<exists>n. \<forall>e > 0. \<exists>p. winding_number_prop \<gamma> z e p n"
+ proof (rule_tac x=nn in exI, clarify)
+ fix e::real
+ assume e: "e>0"
+ obtain p where p: "polynomial_function p \<and>
+ pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d/2))"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
+ have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto simp: intro!: holomorphic_intros)
+ then show "\<exists>p. winding_number_prop \<gamma> z e p nn"
+ apply (rule_tac x=p in exI)
+ using pi_eq [of h p] h p d
+ apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def winding_number_prop_def)
+ done
+ qed
+ then show ?thesis
+ unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
+qed
+
+lemma winding_number_unique:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and pi: "\<And>e. e>0 \<Longrightarrow> \<exists>p. winding_number_prop \<gamma> z e p n"
+ shows "winding_number \<gamma> z = n"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain e
+ where e: "e>0"
+ and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
+ pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
+ contour_integral h2 f = contour_integral h1 f"
+ using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ obtain p where p: "winding_number_prop \<gamma> z e p n"
+ using pi [OF e] by blast
+ obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
+ using winding_number [OF \<gamma> e] by blast
+ have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
+ using p by (auto simp: winding_number_prop_def)
+ also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
+ proof (rule pi_eq)
+ show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto intro!: holomorphic_intros)
+ qed (use p q in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
+ also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
+ using q by (auto simp: winding_number_prop_def)
+ finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
+ then show ?thesis
+ by simp
+qed
+
+(*NB not winding_number_prop here due to the loop in p*)
+lemma winding_number_unique_loop:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and pi:
+ "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
+ pathfinish p = pathstart p \<and>
+ (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+ shows "winding_number \<gamma> z = n"
+proof -
+ have "path_image \<gamma> \<subseteq> UNIV - {z}"
+ using assms by blast
+ then obtain e
+ where e: "e>0"
+ and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
+ pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
+ contour_integral h2 f = contour_integral h1 f"
+ using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
+ obtain p where p:
+ "valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
+ (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
+ using pi [OF e] by blast
+ obtain q where q: "winding_number_prop \<gamma> z e q (winding_number \<gamma> z)"
+ using winding_number [OF \<gamma> e] by blast
+ have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
+ using p by auto
+ also have "\<dots> = contour_integral q (\<lambda>w. 1 / (w - z))"
+ proof (rule pi_eq)
+ show "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
+ by (auto intro!: holomorphic_intros)
+ qed (use p q loop in \<open>auto simp: winding_number_prop_def norm_minus_commute\<close>)
+ also have "\<dots> = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
+ using q by (auto simp: winding_number_prop_def)
+ finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
+ then show ?thesis
+ by simp
+qed
+
+proposition winding_number_valid_path:
+ assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
+ by (rule winding_number_unique)
+ (use assms in \<open>auto simp: valid_path_imp_path winding_number_prop_def\<close>)
+
+proposition has_contour_integral_winding_number:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
+by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
+
+lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
+ by (simp add: winding_number_valid_path)
+
+lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
+ by (simp add: path_image_subpath winding_number_valid_path)
+
+lemma winding_number_join:
+ assumes \<gamma>1: "path \<gamma>1" "z \<notin> path_image \<gamma>1"
+ and \<gamma>2: "path \<gamma>2" "z \<notin> path_image \<gamma>2"
+ and "pathfinish \<gamma>1 = pathstart \<gamma>2"
+ shows "winding_number(\<gamma>1 +++ \<gamma>2) z = winding_number \<gamma>1 z + winding_number \<gamma>2 z"
+proof (rule winding_number_unique)
+ show "\<exists>p. winding_number_prop (\<gamma>1 +++ \<gamma>2) z e p
+ (winding_number \<gamma>1 z + winding_number \<gamma>2 z)" if "e > 0" for e
+ proof -
+ obtain p1 where "winding_number_prop \<gamma>1 z e p1 (winding_number \<gamma>1 z)"
+ using \<open>0 < e\<close> \<gamma>1 winding_number by blast
+ moreover
+ obtain p2 where "winding_number_prop \<gamma>2 z e p2 (winding_number \<gamma>2 z)"
+ using \<open>0 < e\<close> \<gamma>2 winding_number by blast
+ ultimately
+ have "winding_number_prop (\<gamma>1+++\<gamma>2) z e (p1+++p2) (winding_number \<gamma>1 z + winding_number \<gamma>2 z)"
+ using assms
+ apply (simp add: winding_number_prop_def not_in_path_image_join contour_integrable_inversediff algebra_simps)
+ apply (auto simp: joinpaths_def)
+ done
+ then show ?thesis
+ by blast
+ qed
+qed (use assms in \<open>auto simp: not_in_path_image_join\<close>)
+
+lemma winding_number_reversepath:
+ assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
+ shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
+proof (rule winding_number_unique)
+ show "\<exists>p. winding_number_prop (reversepath \<gamma>) z e p (- winding_number \<gamma> z)" if "e > 0" for e
+ proof -
+ obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+ using \<open>0 < e\<close> assms winding_number by blast
+ then have "winding_number_prop (reversepath \<gamma>) z e (reversepath p) (- winding_number \<gamma> z)"
+ using assms
+ apply (simp add: winding_number_prop_def contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
+ apply (auto simp: reversepath_def)
+ done
+ then show ?thesis
+ by blast
+ qed
+qed (use assms in auto)
+
+lemma winding_number_shiftpath:
+ assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
+ and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
+ shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
+proof (rule winding_number_unique_loop)
+ show "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and> pathfinish p = pathstart p \<and>
+ (\<forall>t\<in>{0..1}. cmod (shiftpath a \<gamma> t - p t) < e) \<and>
+ contour_integral p (\<lambda>w. 1 / (w - z)) =
+ complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ if "e > 0" for e
+ proof -
+ obtain p where "winding_number_prop \<gamma> z e p (winding_number \<gamma> z)"
+ using \<open>0 < e\<close> assms winding_number by blast
+ then show ?thesis
+ apply (rule_tac x="shiftpath a p" in exI)
+ using assms that
+ apply (auto simp: winding_number_prop_def path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath contour_integral_shiftpath)
+ apply (simp add: shiftpath_def)
+ done
+ qed
+qed (use assms in \<open>auto simp: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath\<close>)
+
+lemma winding_number_split_linepath:
+ assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
+ shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
+proof -
+ have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
+ using assms by (meson convex_contains_segment convex_segment ends_in_segment subsetCE)+
+ then show ?thesis
+ using assms
+ by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
+qed
+
+lemma winding_number_cong:
+ "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
+ by (simp add: winding_number_def winding_number_prop_def pathstart_def pathfinish_def)
+
+lemma winding_number_constI:
+ assumes "c\<noteq>z" "\<And>t. \<lbrakk>0\<le>t; t\<le>1\<rbrakk> \<Longrightarrow> g t = c"
+ shows "winding_number g z = 0"
+proof -
+ have "winding_number g z = winding_number (linepath c c) z"
+ apply (rule winding_number_cong)
+ using assms unfolding linepath_def by auto
+ moreover have "winding_number (linepath c c) z =0"
+ apply (rule winding_number_trivial)
+ using assms by auto
+ ultimately show ?thesis by auto
+qed
+
+lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
+ unfolding winding_number_def
+proof (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
+ fix n e g
+ assume "0 < e" and g: "winding_number_prop p z e g n"
+ then show "\<exists>r. winding_number_prop (\<lambda>w. p w - z) 0 e r n"
+ by (rule_tac x="\<lambda>t. g t - z" in exI)
+ (force simp: winding_number_prop_def contour_integral_integral valid_path_def path_defs
+ vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
+next
+ fix n e g
+ assume "0 < e" and g: "winding_number_prop (\<lambda>w. p w - z) 0 e g n"
+ then show "\<exists>r. winding_number_prop p z e r n"
+ apply (rule_tac x="\<lambda>t. g t + z" in exI)
+ apply (simp add: winding_number_prop_def contour_integral_integral valid_path_def path_defs
+ piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
+ apply (force simp: algebra_simps)
+ done
+qed
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Some lemmas about negating a path\<close>
+
+lemma valid_path_negatepath: "valid_path \<gamma> \<Longrightarrow> valid_path (uminus \<circ> \<gamma>)"
+ unfolding o_def using piecewise_C1_differentiable_neg valid_path_def by blast
+
+lemma has_contour_integral_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and cint: "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>"
+ shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
+proof -
+ obtain S where cont: "continuous_on {0..1} \<gamma>" and "finite S" and diff: "\<gamma> C1_differentiable_on {0..1} - S"
+ using \<gamma> by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
+ have "((\<lambda>x. - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))) has_integral i) {0..1}"
+ using cint by (auto simp: has_contour_integral_def dest: has_integral_neg)
+ then
+ have "((\<lambda>x. f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1})) has_integral i) {0..1}"
+ proof (rule rev_iffD1 [OF _ has_integral_spike_eq])
+ show "negligible S"
+ by (simp add: \<open>finite S\<close> negligible_finite)
+ show "f (- \<gamma> x) * vector_derivative (uminus \<circ> \<gamma>) (at x within {0..1}) =
+ - (f (- \<gamma> x) * vector_derivative \<gamma> (at x within {0..1}))"
+ if "x \<in> {0..1} - S" for x
+ proof -
+ have "vector_derivative (uminus \<circ> \<gamma>) (at x within cbox 0 1) = - vector_derivative \<gamma> (at x within cbox 0 1)"
+ proof (rule vector_derivative_within_cbox)
+ show "(uminus \<circ> \<gamma> has_vector_derivative - vector_derivative \<gamma> (at x within cbox 0 1)) (at x within cbox 0 1)"
+ using that unfolding o_def
+ by (metis C1_differentiable_on_eq UNIV_I diff differentiable_subset has_vector_derivative_minus subsetI that vector_derivative_works)
+ qed (use that in auto)
+ then show ?thesis
+ by simp
+ qed
+ qed
+ then show ?thesis by (simp add: has_contour_integral_def)
+qed
+
+lemma winding_number_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and 0: "0 \<notin> path_image \<gamma>"
+ shows "winding_number(uminus \<circ> \<gamma>) 0 = winding_number \<gamma> 0"
+proof -
+ have "(/) 1 contour_integrable_on \<gamma>"
+ using "0" \<gamma> contour_integrable_inversediff by fastforce
+ then have "((\<lambda>z. 1/z) has_contour_integral contour_integral \<gamma> ((/) 1)) \<gamma>"
+ by (rule has_contour_integral_integral)
+ then have "((\<lambda>z. 1 / - z) has_contour_integral - contour_integral \<gamma> ((/) 1)) \<gamma>"
+ using has_contour_integral_neg by auto
+ then show ?thesis
+ using assms
+ apply (simp add: winding_number_valid_path valid_path_negatepath image_def path_defs)
+ apply (simp add: contour_integral_unique has_contour_integral_negatepath)
+ done
+qed
+
+lemma contour_integrable_negatepath:
+ assumes \<gamma>: "valid_path \<gamma>" and pi: "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
+ shows "f contour_integrable_on (uminus \<circ> \<gamma>)"
+ by (metis \<gamma> add.inverse_inverse contour_integrable_on_def has_contour_integral_negatepath pi)
+
+(* A combined theorem deducing several things piecewise.*)
+lemma winding_number_join_pos_combined:
+ "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
+ valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
+ \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
+ by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
+
+
+subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Useful sufficient conditions for the winding number to be positive\<close>
+
+lemma Re_winding_number:
+ "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
+ \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
+by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
+
+lemma winding_number_pos_le:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
+ shows "0 \<le> Re(winding_number \<gamma> z)"
+proof -
+ have ge0: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
+ using ge by (simp add: Complex.Im_divide algebra_simps x)
+ let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
+ let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
+ have hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ have "0 \<le> Im (?int z)"
+ proof (rule has_integral_component_nonneg [of \<i>, simplified])
+ show "\<And>x. x \<in> cbox 0 1 \<Longrightarrow> 0 \<le> Im (if 0 < x \<and> x < 1 then ?vd x else 0)"
+ by (force simp: ge0)
+ show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else 0) has_integral ?int z) (cbox 0 1)"
+ by (rule has_integral_spike_interior [OF hi]) simp
+ qed
+ then show ?thesis
+ by (simp add: Re_winding_number [OF \<gamma>] field_simps)
+qed
+
+lemma winding_number_pos_lt_lemma:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and e: "0 < e"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ shows "0 < Re(winding_number \<gamma> z)"
+proof -
+ let ?vd = "\<lambda>x. 1 / (\<gamma> x - z) * vector_derivative \<gamma> (at x)"
+ let ?int = "\<lambda>z. contour_integral \<gamma> (\<lambda>w. 1 / (w - z))"
+ have hi: "(?vd has_integral ?int z) (cbox 0 1)"
+ unfolding box_real
+ apply (subst has_contour_integral [symmetric])
+ using \<gamma> by (simp add: contour_integrable_inversediff has_contour_integral_integral)
+ have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
+ proof (rule has_integral_component_le [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}", simplified])
+ show "((\<lambda>x. if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e) has_integral ?int z) {0..1}"
+ by (rule has_integral_spike_interior [OF hi, simplified box_real]) (use e in simp)
+ show "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow>
+ e \<le> Im (if 0 < x \<and> x < 1 then ?vd x else \<i> * complex_of_real e)"
+ by (simp add: ge)
+ qed (use has_integral_const_real [of _ 0 1] in auto)
+ with e show ?thesis
+ by (simp add: Re_winding_number [OF \<gamma>] field_simps)
+qed
+
+lemma winding_number_pos_lt:
+ assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
+ and e: "0 < e"
+ and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
+ shows "0 < Re (winding_number \<gamma> z)"
+proof -
+ have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
+ using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
+ then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
+ using bounded_pos [THEN iffD1, OF bm] by blast
+ { fix x::real assume x: "0 < x" "x < 1"
+ then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
+ by (simp add: path_image_def power2_eq_square mult_mono')
+ with x have "\<gamma> x \<noteq> z" using \<gamma>
+ using path_image_def by fastforce
+ then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
+ using B ge [OF x] B2 e
+ apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
+ apply (auto simp: divide_left_mono divide_right_mono)
+ done
+ then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ by (simp add: complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
+ } note * = this
+ show ?thesis
+ using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
+qed
+
+subsection\<open>The winding number is an integer\<close>
+
+text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
+ Also on page 134 of Serge Lang's book with the name title, etc.\<close>
+
+lemma exp_fg:
+ fixes z::complex
+ assumes g: "(g has_vector_derivative g') (at x within s)"
+ and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
+ and z: "g x \<noteq> z"
+ shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
+proof -
+ have *: "(exp \<circ> (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
+ using assms unfolding has_vector_derivative_def scaleR_conv_of_real
+ by (auto intro!: derivative_eq_intros)
+ show ?thesis
+ apply (rule has_vector_derivative_eq_rhs)
+ using z
+ apply (auto intro!: derivative_eq_intros * [unfolded o_def] g)
+ done
+qed
+
+lemma winding_number_exp_integral:
+ fixes z::complex
+ assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
+ and ab: "a \<le> b"
+ and z: "z \<notin> \<gamma> ` {a..b}"
+ shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
+ (is "?thesis1")
+ "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
+ (is "?thesis2")
+proof -
+ let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
+ have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
+ using z by force
+ have cong: "continuous_on {a..b} \<gamma>"
+ using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
+ obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
+ using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
+ have \<circ>: "open ({a<..<b} - k)"
+ using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
+ moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
+ by force
+ ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
+ by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
+ { fix w
+ assume "w \<noteq> z"
+ have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
+ by (auto simp: dist_norm intro!: continuous_intros)
+ moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
+ by (auto simp: intro!: derivative_eq_intros)
+ ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
+ using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
+ by (force simp: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
+ }
+ then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
+ by meson
+ have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
+ unfolding integrable_on_def [symmetric]
+ proof (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>]])
+ show "\<exists>d h. 0 < d \<and>
+ (\<forall>y. cmod (y - w) < d \<longrightarrow> (h has_field_derivative inverse (y - z))(at y within - {z}))"
+ if "w \<in> - {z}" for w
+ apply (rule_tac x="norm(w - z)" in exI)
+ using that inverse_eq_divide has_field_derivative_at_within h
+ by (metis Compl_insert DiffD2 insertCI right_minus_eq zero_less_norm_iff)
+ qed simp
+ have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
+ unfolding box_real [symmetric] divide_inverse_commute
+ by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
+ with ab show ?thesis1
+ by (simp add: divide_inverse_commute integral_def integrable_on_def)
+ { fix t
+ assume t: "t \<in> {a..b}"
+ have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
+ using z by (auto intro!: continuous_intros simp: dist_norm)
+ have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
+ unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
+ obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
+ (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
+ using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
+ by simp (auto simp: ball_def dist_norm that)
+ { fix x D
+ assume x: "x \<notin> k" "a < x" "x < b"
+ then have "x \<in> interior ({a..b} - k)"
+ using open_subset_interior [OF \<circ>] by fastforce
+ then have con: "isCont ?D\<gamma> x"
+ using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
+ then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
+ by (rule continuous_at_imp_continuous_within)
+ have gdx: "\<gamma> differentiable at x"
+ using x by (simp add: g_diff_at)
+ have "\<And>d. \<lbrakk>x \<notin> k; a < x; x < b;
+ (\<gamma> has_vector_derivative d) (at x); a \<le> t; t \<le> b\<rbrakk>
+ \<Longrightarrow> ((\<lambda>x. integral {a..x}
+ (\<lambda>x. ?D\<gamma> x /
+ (\<gamma> x - z))) has_vector_derivative
+ d / (\<gamma> x - z))
+ (at x within {a..b})"
+ apply (rule has_vector_derivative_eq_rhs)
+ apply (rule integral_has_vector_derivative_continuous_at [where S = "{}", simplified])
+ apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
+ done
+ then have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
+ (at x within {a..b})"
+ using x gdx t
+ apply (clarsimp simp add: differentiable_iff_scaleR)
+ apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_withinI)
+ apply (simp_all add: has_vector_derivative_def [symmetric])
+ done
+ } note * = this
+ have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
+ apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
+ using t
+ apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous_1 [OF vg_int] simp add: ab)+
+ done
+ }
+ with ab show ?thesis2
+ by (simp add: divide_inverse_commute integral_def)
+qed
+
+lemma winding_number_exp_2pi:
+ "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
+ \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
+using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def winding_number_prop_def
+ by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
+
+lemma integer_winding_number_eq:
+ assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
+proof -
+ obtain p where p: "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
+ and eq: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using winding_number [OF assms, of 1] unfolding winding_number_prop_def by auto
+ then have wneq: "winding_number \<gamma> z = winding_number p z"
+ using eq winding_number_valid_path by force
+ have iff: "(winding_number \<gamma> z \<in> \<int>) \<longleftrightarrow> (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
+ using eq by (simp add: exp_eq_1 complex_is_Int_iff)
+ have "exp (contour_integral p (\<lambda>w. 1 / (w - z))) = (\<gamma> 1 - z) / (\<gamma> 0 - z)"
+ using p winding_number_exp_integral(2) [of p 0 1 z]
+ apply (simp add: valid_path_def path_defs contour_integral_integral exp_minus field_split_simps)
+ by (metis path_image_def pathstart_def pathstart_in_path_image)
+ then have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
+ using p wneq iff by (auto simp: path_defs)
+ then show ?thesis using p eq
+ by (auto simp: winding_number_valid_path)
+qed
+
+theorem integer_winding_number:
+ "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
+by (metis integer_winding_number_eq)
+
+
+text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
+ We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
+
+lemma winding_number_pos_meets:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
+ and w: "w \<noteq> z"
+ shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
+proof -
+ have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
+ using z by (auto simp: path_image_def)
+ have [simp]: "z \<notin> \<gamma> ` {0..1}"
+ using path_image_def z by auto
+ have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
+ using \<gamma> valid_path_def by blast
+ define r where "r = (w - z) / (\<gamma> 0 - z)"
+ have [simp]: "r \<noteq> 0"
+ using w z by (auto simp: r_def)
+ have cont: "continuous_on {0..1}
+ (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
+ by (intro continuous_intros indefinite_integral_continuous_1 winding_number_exp_integral [OF gpd]; simp)
+ have "Arg2pi r \<le> 2*pi"
+ by (simp add: Arg2pi less_eq_real_def)
+ also have "\<dots> \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
+ using 1
+ apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
+ apply (simp add: Complex.Re_divide field_simps power2_eq_square)
+ done
+ finally have "Arg2pi r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
+ then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
+ by (simp add: Arg2pi_ge_0 cont IVT')
+ then obtain t where t: "t \<in> {0..1}"
+ and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg2pi r"
+ by blast
+ define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
+ have iArg: "Arg2pi r = Im i"
+ using eqArg by (simp add: i_def)
+ have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
+ by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
+ have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
+ unfolding i_def
+ apply (rule winding_number_exp_integral [OF gpdt])
+ using t z unfolding path_image_def by force+
+ then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
+ by (simp add: exp_minus field_simps)
+ then have "(w - z) = r * (\<gamma> 0 - z)"
+ by (simp add: r_def)
+ then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
+ apply simp
+ apply (subst Complex_Transcendental.Arg2pi_eq [of r])
+ apply (simp add: iArg)
+ using * apply (simp add: exp_eq_polar field_simps)
+ done
+ with t show ?thesis
+ by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
+qed
+
+lemma winding_number_big_meets:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
+ and w: "w \<noteq> z"
+ shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
+proof -
+ { assume "Re (winding_number \<gamma> z) \<le> - 1"
+ then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
+ by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
+ moreover have "valid_path (reversepath \<gamma>)"
+ using \<gamma> valid_path_imp_reverse by auto
+ moreover have "z \<notin> path_image (reversepath \<gamma>)"
+ by (simp add: z)
+ ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
+ using winding_number_pos_meets w by blast
+ then have ?thesis
+ by simp
+ }
+ then show ?thesis
+ using assms
+ by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
+qed
+
+lemma winding_number_less_1:
+ fixes z::complex
+ shows
+ "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
+ \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
+ \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
+ by (auto simp: not_less dest: winding_number_big_meets)
+
+text\<open>One way of proving that WN=1 for a loop.\<close>
+lemma winding_number_eq_1:
+ fixes z::complex
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
+ shows "winding_number \<gamma> z = 1"
+proof -
+ have "winding_number \<gamma> z \<in> Ints"
+ by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
+ then show ?thesis
+ using 0 2 by (auto simp: Ints_def)
+qed
+
+subsection\<open>Continuity of winding number and invariance on connected sets\<close>
+
+lemma continuous_at_winding_number:
+ fixes z::complex
+ assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "continuous (at z) (winding_number \<gamma>)"
+proof -
+ obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
+ using open_contains_cball [of "- path_image \<gamma>"] z
+ by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
+ then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
+ by (force simp: cball_def dist_norm)
+ have oc: "open (- cball z (e / 2))"
+ by (simp add: closed_def [symmetric])
+ obtain d where "d>0" and pi_eq:
+ "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
+ (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
+ pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
+ \<Longrightarrow>
+ path_image h1 \<subseteq> - cball z (e / 2) \<and>
+ path_image h2 \<subseteq> - cball z (e / 2) \<and>
+ (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
+ using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
+ obtain p where p: "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
+ and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
+ and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by (auto simp: winding_number_prop_def)
+ { fix w
+ assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
+ then have wnotp: "w \<notin> path_image p"
+ using cbg \<open>d>0\<close> \<open>e>0\<close>
+ apply (simp add: path_image_def cball_def dist_norm, clarify)
+ apply (frule pg)
+ apply (drule_tac c="\<gamma> x" in subsetD)
+ apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
+ done
+ have wnotg: "w \<notin> path_image \<gamma>"
+ using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
+ { fix k::real
+ assume k: "k>0"
+ then obtain q where q: "valid_path q" "w \<notin> path_image q"
+ "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
+ and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
+ and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
+ using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
+ by (force simp: min_divide_distrib_right winding_number_prop_def)
+ have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
+ apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
+ apply (frule pg)
+ apply (frule qg)
+ using p q \<open>d>0\<close> e2
+ apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ done
+ then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
+ by (simp add: pi qi)
+ } note pip = this
+ have "path p"
+ using p by (simp add: valid_path_imp_path)
+ then have "winding_number p w = winding_number \<gamma> w"
+ apply (rule winding_number_unique [OF _ wnotp])
+ apply (rule_tac x=p in exI)
+ apply (simp add: p wnotp min_divide_distrib_right pip winding_number_prop_def)
+ done
+ } note wnwn = this
+ obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
+ using p open_contains_cball [of "- path_image p"]
+ by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
+ obtain L
+ where "L>0"
+ and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
+ \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
+ cmod (contour_integral p f) \<le> L * B"
+ using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by blast
+ { fix e::real and w::complex
+ assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
+ then have [simp]: "w \<notin> path_image p"
+ using cbp p(2) \<open>0 < pe\<close>
+ by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
+ have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
+ contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
+ by (simp add: p contour_integrable_inversediff contour_integral_diff)
+ { fix x
+ assume pe: "3/4 * pe < cmod (z - x)"
+ have "cmod (w - x) < pe/4 + cmod (z - x)"
+ by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
+ then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
+ have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
+ using norm_diff_triangle_le by blast
+ also have "\<dots> < pe/4 + cmod (w - x)"
+ using w by (simp add: norm_minus_commute)
+ finally have "pe/2 < cmod (w - x)"
+ using pe by auto
+ then have "(pe/2)^2 < cmod (w - x) ^ 2"
+ apply (rule power_strict_mono)
+ using \<open>pe>0\<close> by auto
+ then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
+ by (simp add: power_divide)
+ have "8 * L * cmod (w - z) < e * pe\<^sup>2"
+ using w \<open>L>0\<close> by (simp add: field_simps)
+ also have "\<dots> < e * 4 * cmod (w - x) * cmod (w - x)"
+ using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
+ also have "\<dots> < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
+ using wx
+ apply (rule mult_strict_left_mono)
+ using pe2 e not_less_iff_gr_or_eq by fastforce
+ finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
+ by simp
+ also have "\<dots> \<le> e * cmod (w - x) * cmod (z - x)"
+ using e by simp
+ finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
+ have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
+ apply (cases "x=z \<or> x=w")
+ using pe \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (force simp: norm_minus_commute)
+ using wx w(2) \<open>L>0\<close> pe pe2 Lwz
+ apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
+ done
+ } note L_cmod_le = this
+ have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
+ apply (rule L)
+ using \<open>pe>0\<close> w
+ apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
+ using \<open>pe>0\<close> w \<open>L>0\<close>
+ apply (auto simp: cball_def dist_norm field_simps L_cmod_le simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
+ done
+ have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
+ apply simp
+ apply (rule le_less_trans [OF *])
+ using \<open>L>0\<close> e
+ apply (force simp: field_simps)
+ done
+ then have "cmod (winding_number p w - winding_number p z) < e"
+ using pi_ge_two e
+ by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
+ } note cmod_wn_diff = this
+ then have "isCont (winding_number p) z"
+ apply (simp add: continuous_at_eps_delta, clarify)
+ apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
+ using \<open>pe>0\<close> \<open>L>0\<close>
+ apply (simp add: dist_norm cmod_wn_diff)
+ done
+ then show ?thesis
+ apply (rule continuous_transform_within [where d = "min d e / 2"])
+ apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
+ done
+qed
+
+corollary continuous_on_winding_number:
+ "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
+ by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
+
+lemma winding_number_constant:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected S" and sg: "S \<inter> path_image \<gamma> = {}"
+ shows "winding_number \<gamma> constant_on S"
+proof -
+ have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
+ if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> S" "z \<in> S" for y z
+ proof -
+ have "winding_number \<gamma> y \<in> \<int>" "winding_number \<gamma> z \<in> \<int>"
+ using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> S\<close> by auto
+ with ne show ?thesis
+ by (auto simp: Ints_def simp flip: of_int_diff)
+ qed
+ have cont: "continuous_on S (\<lambda>w. winding_number \<gamma> w)"
+ using continuous_on_winding_number [OF \<gamma>] sg
+ by (meson continuous_on_subset disjoint_eq_subset_Compl)
+ show ?thesis
+ using "*" zero_less_one
+ by (blast intro: continuous_discrete_range_constant [OF cs cont])
+qed
+
+lemma winding_number_eq:
+ "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> S; z \<in> S; connected S; S \<inter> path_image \<gamma> = {}\<rbrakk>
+ \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
+ using winding_number_constant by (metis constant_on_def)
+
+lemma open_winding_number_levelsets:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
+proof -
+ have opn: "open (- path_image \<gamma>)"
+ by (simp add: closed_path_image \<gamma> open_Compl)
+ { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
+ obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
+ using open_contains_ball [of "- path_image \<gamma>"] opn z
+ by blast
+ have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
+ apply (rule_tac x=e in exI)
+ using e apply (simp add: dist_norm ball_def norm_minus_commute)
+ apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where S = "ball z e"])
+ done
+ } then
+ show ?thesis
+ by (auto simp: open_dist)
+qed
+
+subsection\<open>Winding number is zero "outside" a curve\<close>
+
+proposition winding_number_zero_in_outside:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
+ shows "winding_number \<gamma> z = 0"
+proof -
+ obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
+ using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
+ obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
+ by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
+ have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
+ apply (rule outside_subset_convex)
+ using B subset_ball by auto
+ then have wout: "w \<in> outside (path_image \<gamma>)"
+ using w by blast
+ moreover have "winding_number \<gamma> constant_on outside (path_image \<gamma>)"
+ using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
+ by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
+ ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
+ by (metis (no_types, hide_lams) constant_on_def z)
+ also have "\<dots> = 0"
+ proof -
+ have wnot: "w \<notin> path_image \<gamma>" using wout by (simp add: outside_def)
+ { fix e::real assume "0<e"
+ obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
+ and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
+ and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
+ using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
+ have pip: "path_image p \<subseteq> ball 0 (B + 1)"
+ using B
+ apply (clarsimp simp add: path_image_def dist_norm ball_def)
+ apply (frule (1) pg1)
+ apply (fastforce dest: norm_add_less)
+ done
+ then have "w \<notin> path_image p" using w by blast
+ then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
+ pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
+ (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
+ apply (rule_tac x=p in exI)
+ apply (simp add: p valid_path_polynomial_function)
+ apply (intro conjI)
+ using pge apply (simp add: norm_minus_commute)
+ apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
+ apply (rule holomorphic_intros | simp add: dist_norm)+
+ using mem_ball_0 w apply blast
+ using p apply (simp_all add: valid_path_polynomial_function loop pip)
+ done
+ }
+ then show ?thesis
+ by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
+ qed
+ finally show ?thesis .
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_const: "a \<noteq> z \<Longrightarrow> winding_number (\<lambda>t. a) z = 0"
+ by (rule winding_number_zero_in_outside)
+ (auto simp: pathfinish_def pathstart_def path_polynomial_function)
+
+corollary\<^marker>\<open>tag unimportant\<close> winding_number_zero_outside:
+ "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
+ by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
+
+lemma winding_number_zero_at_infinity:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
+proof -
+ obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
+ using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
+ then show ?thesis
+ apply (rule_tac x="B+1" in exI, clarify)
+ apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
+ apply (meson less_add_one mem_cball_0 not_le order_trans)
+ using ball_subset_cball by blast
+qed
+
+lemma winding_number_zero_point:
+ "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
+ \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
+ using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
+ by (fastforce simp add: compact_path_image)
+
+
+text\<open>If a path winds round a set, it winds rounds its inside.\<close>
+lemma winding_number_around_inside:
+ assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
+ and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
+ shows "winding_number \<gamma> w = winding_number \<gamma> z"
+proof -
+ have ssb: "s \<subseteq> inside(path_image \<gamma>)"
+ proof
+ fix x :: complex
+ assume "x \<in> s"
+ hence "x \<notin> path_image \<gamma>"
+ by (meson disjoint_iff_not_equal s_disj)
+ thus "x \<in> inside (path_image \<gamma>)"
+ using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
+qed
+ show ?thesis
+ apply (rule winding_number_eq [OF \<gamma> loop w])
+ using z apply blast
+ apply (simp add: cls connected_with_inside cos)
+ apply (simp add: Int_Un_distrib2 s_disj, safe)
+ by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
+ qed
+
+
+text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
+lemma winding_number_subpath_continuous:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
+proof -
+ have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ winding_number (subpath 0 x \<gamma>) z"
+ if x: "0 \<le> x" "x \<le> 1" for x
+ proof -
+ have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
+ 1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
+ using assms x
+ apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
+ done
+ also have "\<dots> = winding_number (subpath 0 x \<gamma>) z"
+ apply (subst winding_number_valid_path)
+ using assms x
+ apply (simp_all add: path_image_subpath valid_path_subpath)
+ by (force simp: path_image_def)
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule continuous_on_eq
+ [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
+ integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
+ apply (rule continuous_intros)+
+ apply (rule indefinite_integral_continuous_1)
+ apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
+ using assms
+ apply (simp add: *)
+ done
+qed
+
+lemma winding_number_ivt_pos:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
+ shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
+ apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+lemma winding_number_ivt_neg:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
+ shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
+ apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right], simp)
+ apply (rule winding_number_subpath_continuous [OF \<gamma> z])
+ using assms
+ apply (auto simp: path_image_def image_def)
+ done
+
+lemma winding_number_ivt_abs:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
+ shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
+ using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
+ by force
+
+lemma winding_number_lt_half_lemma:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
+ shows "Re(winding_number \<gamma> z) < 1/2"
+proof -
+ { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
+ then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
+ using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
+ have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
+ using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
+ apply (simp add: t \<gamma> valid_path_imp_path)
+ using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
+ have "b < a \<bullet> \<gamma> 0"
+ proof -
+ have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
+ by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
+ thus ?thesis
+ by blast
+ qed
+ moreover have "b < a \<bullet> \<gamma> t"
+ proof -
+ have "\<gamma> t \<in> {c. b < a \<bullet> c}"
+ by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
+ thus ?thesis
+ by blast
+ qed
+ ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
+ by (simp add: inner_diff_right)+
+ then have False
+ by (simp add: gt inner_mult_right mult_less_0_iff)
+ }
+ then show ?thesis by force
+qed
+
+lemma winding_number_lt_half:
+ assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
+ shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
+proof -
+ have "z \<notin> path_image \<gamma>" using assms by auto
+ with assms show ?thesis
+ apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
+ apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
+ winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
+ done
+qed
+
+lemma winding_number_le_half:
+ assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
+ and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
+ shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
+proof -
+ { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
+ have "isCont (winding_number \<gamma>) z"
+ by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
+ then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
+ using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
+ define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
+ have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
+ unfolding z'_def inner_mult_right' divide_inverse
+ apply (simp add: field_split_simps algebra_simps dot_square_norm power2_eq_square anz)
+ apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
+ done
+ have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
+ using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
+ then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
+ by simp
+ then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
+ using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
+ then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
+ by linarith
+ moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
+ apply (rule winding_number_lt_half [OF \<gamma> *])
+ using azb \<open>d>0\<close> pag
+ apply (auto simp: add_strict_increasing anz field_split_simps dest!: subsetD)
+ done
+ ultimately have False
+ by simp
+ }
+ then show ?thesis by force
+qed
+
+lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
+ using separating_hyperplane_closed_point [of "closed_segment a b" z]
+ apply auto
+ apply (simp add: closed_segment_def)
+ apply (drule less_imp_le)
+ apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
+ apply (auto simp: segment)
+ done
+
+
+text\<open> Positivity of WN for a linepath.\<close>
+lemma winding_number_linepath_pos_lt:
+ assumes "0 < Im ((b - a) * cnj (b - z))"
+ shows "0 < Re(winding_number(linepath a b) z)"
+proof -
+ have z: "z \<notin> path_image (linepath a b)"
+ using assms
+ by (simp add: closed_segment_def) (force simp: algebra_simps)
+ show ?thesis
+ apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
+ apply (simp add: linepath_def algebra_simps)
+ done
+qed
+
+
+subsection\<open>Cauchy's integral formula, again for a convex enclosing set\<close>
+
+lemma Cauchy_integral_formula_weak:
+ assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
+ and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
+ and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ obtain f' where f': "(f has_field_derivative f') (at z)"
+ using fcd [OF z] by (auto simp: field_differentiable_def)
+ have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
+ have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
+ proof (cases "x = z")
+ case True then show ?thesis
+ apply (simp add: continuous_within)
+ apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
+ using has_field_derivative_at_within has_field_derivative_iff f'
+ apply (fastforce simp add:)+
+ done
+ next
+ case False
+ then have dxz: "dist x z > 0" by auto
+ have cf: "continuous (at x within s) f"
+ using conf continuous_on_eq_continuous_within that by blast
+ have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
+ by (rule cf continuous_intros | simp add: False)+
+ then show ?thesis
+ apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
+ apply (force simp: dist_commute)
+ done
+ qed
+ have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
+ have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+ apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
+ using c apply (force simp: continuous_on_eq_continuous_within)
+ apply (rename_tac w)
+ apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
+ apply (simp_all add: dist_pos_lt dist_commute)
+ apply (metis less_irrefl)
+ apply (rule derivative_intros fcd | simp)+
+ done
+ show ?thesis
+ apply (rule has_contour_integral_eq)
+ using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
+ apply (auto simp: ac_simps divide_simps)
+ done
+qed
+
+theorem Cauchy_integral_formula_convex_simple:
+ "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
+ pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
+ \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+ apply (rule Cauchy_integral_formula_weak [where k = "{}"])
+ using holomorphic_on_imp_continuous_on
+ by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
+
+subsection\<open>Homotopy forms of Cauchy's theorem\<close>
+
+lemma Cauchy_theorem_homotopic:
+ assumes hom: "if atends then homotopic_paths s g h else homotopic_loops s g h"
+ and "open s" and f: "f holomorphic_on s"
+ and vpg: "valid_path g" and vph: "valid_path h"
+ shows "contour_integral g f = contour_integral h f"
+proof -
+ have pathsf: "linked_paths atends g h"
+ using hom by (auto simp: linked_paths_def homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish homotopic_loops_imp_loop)
+ obtain k :: "real \<times> real \<Rightarrow> complex"
+ where contk: "continuous_on ({0..1} \<times> {0..1}) k"
+ and ks: "k ` ({0..1} \<times> {0..1}) \<subseteq> s"
+ and k [simp]: "\<forall>x. k (0, x) = g x" "\<forall>x. k (1, x) = h x"
+ and ksf: "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))"
+ using hom pathsf by (auto simp: linked_paths_def homotopic_paths_def homotopic_loops_def homotopic_with_def split: if_split_asm)
+ have ucontk: "uniformly_continuous_on ({0..1} \<times> {0..1}) k"
+ by (blast intro: compact_Times compact_uniformly_continuous [OF contk])
+ { fix t::real assume t: "t \<in> {0..1}"
+ have pak: "path (k \<circ> (\<lambda>u. (t, u)))"
+ unfolding path_def
+ apply (rule continuous_intros continuous_on_subset [OF contk])+
+ using t by force
+ have pik: "path_image (k \<circ> Pair t) \<subseteq> s"
+ using ks t by (auto simp: path_image_def)
+ obtain e where "e>0" and e:
+ "\<And>g h. \<lbrakk>valid_path g; valid_path h;
+ \<forall>u\<in>{0..1}. cmod (g u - (k \<circ> Pair t) u) < e \<and> cmod (h u - (k \<circ> Pair t) u) < e;
+ linked_paths atends g h\<rbrakk>
+ \<Longrightarrow> contour_integral h f = contour_integral g f"
+ using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis
+ obtain d where "d>0" and d:
+ "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> {0..1}; x' \<in> {0..1} \<times> {0..1}; norm (x'-x) < d\<rbrakk> \<Longrightarrow> norm (k x' - k x) < e/4"
+ by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>)
+ { fix t1 t2
+ assume t1: "0 \<le> t1" "t1 \<le> 1" and t2: "0 \<le> t2" "t2 \<le> 1" and ltd: "\<bar>t1 - t\<bar> < d" "\<bar>t2 - t\<bar> < d"
+ have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e"
+ using \<open>e > 0\<close>
+ apply (rule_tac y = k1 in norm_triangle_half_l)
+ apply (auto simp: norm_minus_commute intro: order_less_trans)
+ done
+ have "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
+ (\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and>
+ linked_paths atends g1 g2 \<longrightarrow>
+ contour_integral g2 f = contour_integral g1 f"
+ apply (rule_tac x="e/4" in exI)
+ using t t1 t2 ltd \<open>e > 0\<close>
+ apply (auto intro!: e simp: d no2 simp del: less_divide_eq_numeral1)
+ done
+ }
+ then have "\<exists>e. 0 < e \<and>
+ (\<forall>t1 t2. t1 \<in> {0..1} \<and> t2 \<in> {0..1} \<and> \<bar>t1 - t\<bar> < e \<and> \<bar>t2 - t\<bar> < e
+ \<longrightarrow> (\<exists>d. 0 < d \<and>
+ (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
+ (\<forall>u \<in> {0..1}.
+ norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
+ linked_paths atends g1 g2
+ \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
+ by (rule_tac x=d in exI) (simp add: \<open>d > 0\<close>)
+ }
+ then obtain ee where ee:
+ "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0 \<and>
+ (\<forall>t1 t2. t1 \<in> {0..1} \<longrightarrow> t2 \<in> {0..1} \<longrightarrow> \<bar>t1 - t\<bar> < ee t \<longrightarrow> \<bar>t2 - t\<bar> < ee t
+ \<longrightarrow> (\<exists>d. 0 < d \<and>
+ (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
+ (\<forall>u \<in> {0..1}.
+ norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
+ linked_paths atends g1 g2
+ \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
+ by metis
+ note ee_rule = ee [THEN conjunct2, rule_format]
+ define C where "C = (\<lambda>t. ball t (ee t / 3)) ` {0..1}"
+ obtain C' where C': "C' \<subseteq> C" "finite C'" and C'01: "{0..1} \<subseteq> \<Union>C'"
+ proof (rule compactE [OF compact_interval])
+ show "{0..1} \<subseteq> \<Union>C"
+ using ee [THEN conjunct1] by (auto simp: C_def dist_norm)
+ qed (use C_def in auto)
+ define kk where "kk = {t \<in> {0..1}. ball t (ee t / 3) \<in> C'}"
+ have kk01: "kk \<subseteq> {0..1}" by (auto simp: kk_def)
+ define e where "e = Min (ee ` kk)"
+ have C'_eq: "C' = (\<lambda>t. ball t (ee t / 3)) ` kk"
+ using C' by (auto simp: kk_def C_def)
+ have ee_pos[simp]: "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0"
+ by (simp add: kk_def ee)
+ moreover have "finite kk"
+ using \<open>finite C'\<close> kk01 by (force simp: C'_eq inj_on_def ball_eq_ball_iff dest: ee_pos finite_imageD)
+ moreover have "kk \<noteq> {}" using \<open>{0..1} \<subseteq> \<Union>C'\<close> C'_eq by force
+ ultimately have "e > 0"
+ using finite_less_Inf_iff [of "ee ` kk" 0] kk01 by (force simp: e_def)
+ then obtain N::nat where "N > 0" and N: "1/N < e/3"
+ by (meson divide_pos_pos nat_approx_posE zero_less_Suc zero_less_numeral)
+ have e_le_ee: "\<And>i. i \<in> kk \<Longrightarrow> e \<le> ee i"
+ using \<open>finite kk\<close> by (simp add: e_def Min_le_iff [of "ee ` kk"])
+ have plus: "\<exists>t \<in> kk. x \<in> ball t (ee t / 3)" if "x \<in> {0..1}" for x
+ using C' subsetD [OF C'01 that] unfolding C'_eq by blast
+ have [OF order_refl]:
+ "\<exists>d. 0 < d \<and> (\<forall>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (n/N, u)) < d) \<and> linked_paths atends g j
+ \<longrightarrow> contour_integral j f = contour_integral g f)"
+ if "n \<le> N" for n
+ using that
+ proof (induct n)
+ case 0 show ?case using ee_rule [of 0 0 0]
+ apply clarsimp
+ apply (rule_tac x=d in exI, safe)
+ by (metis diff_self vpg norm_zero)
+ next
+ case (Suc n)
+ then have N01: "n/N \<in> {0..1}" "(Suc n)/N \<in> {0..1}" by auto
+ then obtain t where t: "t \<in> kk" "n/N \<in> ball t (ee t / 3)"
+ using plus [of "n/N"] by blast
+ then have nN_less: "\<bar>n/N - t\<bar> < ee t"
+ by (simp add: dist_norm del: less_divide_eq_numeral1)
+ have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t"
+ using t N \<open>N > 0\<close> e_le_ee [of t]
+ by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps)
+ have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast
+ obtain d1 where "d1 > 0" and d1:
+ "\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2;
+ \<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1;
+ linked_paths atends g1 g2\<rbrakk>
+ \<Longrightarrow> contour_integral g2 f = contour_integral g1 f"
+ using ee [THEN conjunct2, rule_format, OF t01 N01 nN_less n'N_less] by fastforce
+ have "n \<le> N" using Suc.prems by auto
+ with Suc.hyps
+ obtain d2 where "d2 > 0"
+ and d2: "\<And>j. \<lbrakk>valid_path j; \<forall>u\<in>{0..1}. cmod (j u - k (n/N, u)) < d2; linked_paths atends g j\<rbrakk>
+ \<Longrightarrow> contour_integral j f = contour_integral g f"
+ by auto
+ have "continuous_on {0..1} (k \<circ> (\<lambda>u. (n/N, u)))"
+ apply (rule continuous_intros continuous_on_subset [OF contk])+
+ using N01 by auto
+ then have pkn: "path (\<lambda>u. k (n/N, u))"
+ by (simp add: path_def)
+ have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
+ obtain p where "polynomial_function p"
+ and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))"
+ "pathfinish p = pathfinish (\<lambda>u. k (n/N, u))"
+ and pk_le: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (p t - k (n/N, t)) < min d1 d2"
+ using path_approx_polynomial_function [OF pkn min12] by blast
+ then have vpp: "valid_path p" using valid_path_polynomial_function by blast
+ have lpa: "linked_paths atends g p"
+ by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf)
+ show ?case
+ proof (intro exI; safe)
+ fix j
+ assume "valid_path j" "linked_paths atends g j"
+ and "\<forall>u\<in>{0..1}. cmod (j u - k (real (Suc n) / real N, u)) < min d1 d2"
+ then have "contour_integral j f = contour_integral p f"
+ using pk_le N01(1) ksf by (force intro!: vpp d1 simp add: linked_paths_def psf)
+ also have "... = contour_integral g f"
+ using pk_le by (force intro!: vpp d2 lpa)
+ finally show "contour_integral j f = contour_integral g f" .
+ qed (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
+ qed
+ then obtain d where "0 < d"
+ "\<And>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (1,u)) < d) \<and> linked_paths atends g j
+ \<Longrightarrow> contour_integral j f = contour_integral g f"
+ using \<open>N>0\<close> by auto
+ then have "linked_paths atends g h \<Longrightarrow> contour_integral h f = contour_integral g f"
+ using \<open>N>0\<close> vph by fastforce
+ then show ?thesis
+ by (simp add: pathsf)
+qed
+
+proposition Cauchy_theorem_homotopic_paths:
+ assumes hom: "homotopic_paths s g h"
+ and "open s" and f: "f holomorphic_on s"
+ and vpg: "valid_path g" and vph: "valid_path h"
+ shows "contour_integral g f = contour_integral h f"
+ using Cauchy_theorem_homotopic [of True s g h] assms by simp
+
+proposition Cauchy_theorem_homotopic_loops:
+ assumes hom: "homotopic_loops s g h"
+ and "open s" and f: "f holomorphic_on s"
+ and vpg: "valid_path g" and vph: "valid_path h"
+ shows "contour_integral g f = contour_integral h f"
+ using Cauchy_theorem_homotopic [of False s g h] assms by simp
+
+lemma has_contour_integral_newpath:
+ "\<lbrakk>(f has_contour_integral y) h; f contour_integrable_on g; contour_integral g f = contour_integral h f\<rbrakk>
+ \<Longrightarrow> (f has_contour_integral y) g"
+ using has_contour_integral_integral contour_integral_unique by auto
+
+lemma Cauchy_theorem_null_homotopic:
+ "\<lbrakk>f holomorphic_on s; open s; valid_path g; homotopic_loops s g (linepath a a)\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
+ apply (rule has_contour_integral_newpath [where h = "linepath a a"], simp)
+ using contour_integrable_holomorphic_simple
+ apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
+ by (simp add: Cauchy_theorem_homotopic_loops)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More winding number properties\<close>
+
+text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
+
+lemma winding_number_homotopic_paths:
+ assumes "homotopic_paths (-{z}) g h"
+ shows "winding_number g z = winding_number h z"
+proof -
+ have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
+ moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
+ using homotopic_paths_imp_subset [OF assms] by auto
+ ultimately obtain d e where "d > 0" "e > 0"
+ and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
+ \<Longrightarrow> homotopic_paths (-{z}) g p"
+ and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
+ \<Longrightarrow> homotopic_paths (-{z}) h q"
+ using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
+ obtain p where p:
+ "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart g" "pathfinish p = pathfinish g"
+ and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
+ and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
+ using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
+ obtain q where q:
+ "valid_path q" "z \<notin> path_image q"
+ "pathstart q = pathstart h" "pathfinish q = pathfinish h"
+ and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
+ and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
+ using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
+ have "homotopic_paths (- {z}) g p"
+ by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
+ moreover have "homotopic_paths (- {z}) h q"
+ by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
+ ultimately have "homotopic_paths (- {z}) p q"
+ by (blast intro: homotopic_paths_trans homotopic_paths_sym assms)
+ then have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
+ by (rule Cauchy_theorem_homotopic_paths) (auto intro!: holomorphic_intros simp: p q)
+ then show ?thesis
+ by (simp add: pap paq)
+qed
+
+lemma winding_number_homotopic_loops:
+ assumes "homotopic_loops (-{z}) g h"
+ shows "winding_number g z = winding_number h z"
+proof -
+ have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
+ moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
+ using homotopic_loops_imp_subset [OF assms] by auto
+ moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
+ using homotopic_loops_imp_loop [OF assms] by auto
+ ultimately obtain d e where "d > 0" "e > 0"
+ and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
+ \<Longrightarrow> homotopic_loops (-{z}) g p"
+ and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
+ \<Longrightarrow> homotopic_loops (-{z}) h q"
+ using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
+ obtain p where p:
+ "valid_path p" "z \<notin> path_image p"
+ "pathstart p = pathstart g" "pathfinish p = pathfinish g"
+ and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
+ and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
+ using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] unfolding winding_number_prop_def by blast
+ obtain q where q:
+ "valid_path q" "z \<notin> path_image q"
+ "pathstart q = pathstart h" "pathfinish q = pathfinish h"
+ and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
+ and paq: "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
+ using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] unfolding winding_number_prop_def by blast
+ have gp: "homotopic_loops (- {z}) g p"
+ by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
+ have hq: "homotopic_loops (- {z}) h q"
+ by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
+ have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
+ proof (rule Cauchy_theorem_homotopic_loops)
+ show "homotopic_loops (- {z}) p q"
+ by (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
+ qed (auto intro!: holomorphic_intros simp: p q)
+ then show ?thesis
+ by (simp add: pap paq)
+qed
+
+lemma winding_number_paths_linear_eq:
+ "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths)
+
+lemma winding_number_loops_linear_eq:
+ "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops)
+
+lemma winding_number_nearby_paths_eq:
+ "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
+
+lemma winding_number_nearby_loops_eq:
+ "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
+ \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
+ \<Longrightarrow> winding_number h z = winding_number g z"
+ by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
+
+
+lemma winding_number_subpath_combine:
+ "\<lbrakk>path g; z \<notin> path_image g;
+ u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
+ \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
+ winding_number (subpath u w g) z"
+apply (rule trans [OF winding_number_join [THEN sym]
+ winding_number_homotopic_paths [OF homotopic_join_subpaths]])
+ using path_image_subpath_subset by auto
+
+subsection\<open>Partial circle path\<close>
+
+definition\<^marker>\<open>tag important\<close> part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
+ where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
+
+lemma pathstart_part_circlepath [simp]:
+ "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
+by (metis part_circlepath_def pathstart_def pathstart_linepath)
+
+lemma pathfinish_part_circlepath [simp]:
+ "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
+by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
+
+lemma reversepath_part_circlepath[simp]:
+ "reversepath (part_circlepath z r s t) = part_circlepath z r t s"
+ unfolding part_circlepath_def reversepath_def linepath_def
+ by (auto simp:algebra_simps)
+
+lemma has_vector_derivative_part_circlepath [derivative_intros]:
+ "((part_circlepath z r s t) has_vector_derivative
+ (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
+ (at x within X)"
+ apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
+ apply (rule has_vector_derivative_real_field)
+ apply (rule derivative_eq_intros | simp)+
+ done
+
+lemma differentiable_part_circlepath:
+ "part_circlepath c r a b differentiable at x within A"
+ using has_vector_derivative_part_circlepath[of c r a b x A] differentiableI_vector by blast
+
+lemma vector_derivative_part_circlepath:
+ "vector_derivative (part_circlepath z r s t) (at x) =
+ \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
+ using has_vector_derivative_part_circlepath vector_derivative_at by blast
+
+lemma vector_derivative_part_circlepath01:
+ "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
+ \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
+ \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
+ using has_vector_derivative_part_circlepath
+ by (auto simp: vector_derivative_at_within_ivl)
+
+lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
+ apply (simp add: valid_path_def)
+ apply (rule C1_differentiable_imp_piecewise)
+ apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
+ intro!: continuous_intros)
+ done
+
+lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
+ by (simp add: valid_path_imp_path)
+
+proposition path_image_part_circlepath:
+ assumes "s \<le> t"
+ shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
+proof -
+ { fix z::real
+ assume "0 \<le> z" "z \<le> 1"
+ with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
+ apply (rule_tac x="(1 - z) * s + z * t" in exI)
+ apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
+ apply (rule conjI)
+ using mult_right_mono apply blast
+ using affine_ineq by (metis "mult.commute")
+ }
+ moreover
+ { fix z
+ assume "s \<le> z" "z \<le> t"
+ then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
+ apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
+ apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
+ apply (auto simp: field_split_simps)
+ done
+ }
+ ultimately show ?thesis
+ by (fastforce simp add: path_image_def part_circlepath_def)
+qed
+
+lemma path_image_part_circlepath':
+ "path_image (part_circlepath z r s t) = (\<lambda>x. z + r * cis x) ` closed_segment s t"
+proof -
+ have "path_image (part_circlepath z r s t) =
+ (\<lambda>x. z + r * exp(\<i> * of_real x)) ` linepath s t ` {0..1}"
+ by (simp add: image_image path_image_def part_circlepath_def)
+ also have "linepath s t ` {0..1} = closed_segment s t"
+ by (rule linepath_image_01)
+ finally show ?thesis by (simp add: cis_conv_exp)
+qed
+
+lemma path_image_part_circlepath_subset:
+ "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
+by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
+
+lemma in_path_image_part_circlepath:
+ assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
+ shows "norm(w - z) = r"
+proof -
+ have "w \<in> {c. dist z c = r}"
+ by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
+ thus ?thesis
+ by (simp add: dist_norm norm_minus_commute)
+qed
+
+lemma path_image_part_circlepath_subset':
+ assumes "r \<ge> 0"
+ shows "path_image (part_circlepath z r s t) \<subseteq> sphere z r"
+proof (cases "s \<le> t")
+ case True
+ thus ?thesis using path_image_part_circlepath_subset[of s t r z] assms by simp
+next
+ case False
+ thus ?thesis using path_image_part_circlepath_subset[of t s r z] assms
+ by (subst reversepath_part_circlepath [symmetric], subst path_image_reversepath) simp_all
+qed
+
+lemma part_circlepath_cnj: "cnj (part_circlepath c r a b x) = part_circlepath (cnj c) r (-a) (-b) x"
+ by (simp add: part_circlepath_def exp_cnj linepath_def algebra_simps)
+
+lemma contour_integral_bound_part_circlepath:
+ assumes "f contour_integrable_on part_circlepath c r a b"
+ assumes "B \<ge> 0" "r \<ge> 0" "\<And>x. x \<in> path_image (part_circlepath c r a b) \<Longrightarrow> norm (f x) \<le> B"
+ shows "norm (contour_integral (part_circlepath c r a b) f) \<le> B * r * \<bar>b - a\<bar>"
+proof -
+ let ?I = "integral {0..1} (\<lambda>x. f (part_circlepath c r a b x) * \<i> * of_real (r * (b - a)) *
+ exp (\<i> * linepath a b x))"
+ have "norm ?I \<le> integral {0..1} (\<lambda>x::real. B * 1 * (r * \<bar>b - a\<bar>) * 1)"
+ proof (rule integral_norm_bound_integral, goal_cases)
+ case 1
+ with assms(1) show ?case
+ by (simp add: contour_integrable_on vector_derivative_part_circlepath mult_ac)
+ next
+ case (3 x)
+ with assms(2-) show ?case unfolding norm_mult norm_of_real abs_mult
+ by (intro mult_mono) (auto simp: path_image_def)
+ qed auto
+ also have "?I = contour_integral (part_circlepath c r a b) f"
+ by (simp add: contour_integral_integral vector_derivative_part_circlepath mult_ac)
+ finally show ?thesis by simp
+qed
+
+lemma has_contour_integral_part_circlepath_iff:
+ assumes "a < b"
+ shows "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
+ ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}"
+proof -
+ have "(f has_contour_integral I) (part_circlepath c r a b) \<longleftrightarrow>
+ ((\<lambda>x. f (part_circlepath c r a b x) * vector_derivative (part_circlepath c r a b)
+ (at x within {0..1})) has_integral I) {0..1}"
+ unfolding has_contour_integral_def ..
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (part_circlepath c r a b x) * r * (b - a) * \<i> *
+ cis (linepath a b x)) has_integral I) {0..1}"
+ by (intro has_integral_cong, subst vector_derivative_part_circlepath01)
+ (simp_all add: cis_conv_exp)
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>x. f (c + r * exp (\<i> * linepath (of_real a) (of_real b) x)) *
+ r * \<i> * exp (\<i> * linepath (of_real a) (of_real b) x) *
+ vector_derivative (linepath (of_real a) (of_real b))
+ (at x within {0..1})) has_integral I) {0..1}"
+ by (intro has_integral_cong, subst vector_derivative_linepath_within)
+ (auto simp: part_circlepath_def cis_conv_exp of_real_linepath [symmetric])
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>z. f (c + r * exp (\<i> * z)) * r * \<i> * exp (\<i> * z)) has_contour_integral I)
+ (linepath (of_real a) (of_real b))"
+ by (simp add: has_contour_integral_def)
+ also have "\<dots> \<longleftrightarrow> ((\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) has_integral I) {a..b}" using assms
+ by (subst has_contour_integral_linepath_Reals_iff) (simp_all add: cis_conv_exp)
+ finally show ?thesis .
+qed
+
+lemma contour_integrable_part_circlepath_iff:
+ assumes "a < b"
+ shows "f contour_integrable_on (part_circlepath c r a b) \<longleftrightarrow>
+ (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (auto simp: contour_integrable_on_def integrable_on_def
+ has_contour_integral_part_circlepath_iff)
+
+lemma contour_integral_part_circlepath_eq:
+ assumes "a < b"
+ shows "contour_integral (part_circlepath c r a b) f =
+ integral {a..b} (\<lambda>t. f (c + r * cis t) * r * \<i> * cis t)"
+proof (cases "f contour_integrable_on part_circlepath c r a b")
+ case True
+ hence "(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (simp add: contour_integrable_part_circlepath_iff)
+ with True show ?thesis
+ using has_contour_integral_part_circlepath_iff[OF assms]
+ contour_integral_unique has_integral_integrable_integral by blast
+next
+ case False
+ hence "\<not>(\<lambda>t. f (c + r * cis t) * r * \<i> * cis t) integrable_on {a..b}"
+ using assms by (simp add: contour_integrable_part_circlepath_iff)
+ with False show ?thesis
+ by (simp add: not_integrable_contour_integral not_integrable_integral)
+qed
+
+lemma contour_integral_part_circlepath_reverse:
+ "contour_integral (part_circlepath c r a b) f = -contour_integral (part_circlepath c r b a) f"
+ by (subst reversepath_part_circlepath [symmetric], subst contour_integral_reversepath) simp_all
+
+lemma contour_integral_part_circlepath_reverse':
+ "b < a \<Longrightarrow> contour_integral (part_circlepath c r a b) f =
+ -contour_integral (part_circlepath c r b a) f"
+ by (rule contour_integral_part_circlepath_reverse)
+
+lemma finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
+proof (cases "w = 0")
+ case True then show ?thesis by auto
+next
+ case False
+ have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
+ apply (simp add: norm_mult finite_int_iff_bounded_le)
+ apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
+ apply (auto simp: field_split_simps le_floor_iff)
+ done
+ have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
+ by blast
+ show ?thesis
+ apply (subst exp_Ln [OF False, symmetric])
+ apply (simp add: exp_eq)
+ using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
+ done
+qed
+
+lemma finite_bounded_log2:
+ fixes a::complex
+ assumes "a \<noteq> 0"
+ shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
+proof -
+ have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
+ by (rule finite_imageI [OF finite_bounded_log])
+ show ?thesis
+ by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
+qed
+
+lemma has_contour_integral_bound_part_circlepath_strong:
+ assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
+ and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
+ and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
+ shows "cmod i \<le> B * r * (t - s)"
+proof -
+ consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
+ then show ?thesis
+ proof cases
+ case 1 with fi [unfolded has_contour_integral]
+ have "i = 0" by (simp add: vector_derivative_part_circlepath)
+ with assms show ?thesis by simp
+ next
+ case 2
+ have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
+ have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
+ by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
+ have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
+ proof -
+ define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
+ have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
+ apply (rule finite_vimageI [OF finite_bounded_log2])
+ using \<open>s < t\<close> apply (auto simp: inj_of_real)
+ done
+ show ?thesis
+ apply (simp add: part_circlepath_def linepath_def vimage_def)
+ apply (rule finite_subset [OF _ fin])
+ using le
+ apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
+ done
+ qed
+ then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
+ by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
+ have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
+ else f(part_circlepath z r s t x) *
+ vector_derivative (part_circlepath z r s t) (at x)) has_integral i) {0..1}"
+ by (rule has_integral_spike [OF negligible_finite [OF fin01]]) (use fi has_contour_integral in auto)
+ have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
+ by (auto intro!: B [unfolded path_image_def image_def, simplified])
+ show ?thesis
+ apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
+ using assms apply force
+ apply (simp add: norm_mult vector_derivative_part_circlepath)
+ using le * "2" \<open>r > 0\<close> by auto
+ qed
+qed
+
+lemma has_contour_integral_bound_part_circlepath:
+ "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
+ 0 \<le> B; 0 < r; s \<le> t;
+ \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*r*(t - s)"
+ by (auto intro: has_contour_integral_bound_part_circlepath_strong)
+
+lemma contour_integrable_continuous_part_circlepath:
+ "continuous_on (path_image (part_circlepath z r s t)) f
+ \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
+ apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
+ apply (rule integrable_continuous_real)
+ apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
+ done
+
+proposition winding_number_part_circlepath_pos_less:
+ assumes "s < t" and no: "norm(w - z) < r"
+ shows "0 < Re (winding_number(part_circlepath z r s t) w)"
+proof -
+ have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
+ note valid_path_part_circlepath
+ moreover have " w \<notin> path_image (part_circlepath z r s t)"
+ using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
+ moreover have "0 < r * (t - s) * (r - cmod (w - z))"
+ using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
+ ultimately show ?thesis
+ apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
+ apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
+ apply (rule mult_left_mono)+
+ using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
+ apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
+ using assms \<open>0 < r\<close> by auto
+qed
+
+lemma simple_path_part_circlepath:
+ "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
+proof (cases "r = 0 \<or> s = t")
+ case True
+ then show ?thesis
+ unfolding part_circlepath_def simple_path_def
+ by (rule disjE) (force intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
+next
+ case False then have "r \<noteq> 0" "s \<noteq> t" by auto
+ have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
+ by (simp add: algebra_simps)
+ have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
+ \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
+ by auto
+ have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
+ (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
+ by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
+ intro: exI [where x = "-n" for n])
+ have 1: "\<bar>s - t\<bar> \<le> 2 * pi"
+ if "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1"
+ proof (rule ccontr)
+ assume "\<not> \<bar>s - t\<bar> \<le> 2 * pi"
+ then have *: "\<And>n. t - s \<noteq> of_int n * \<bar>s - t\<bar>"
+ using False that [of "2*pi / \<bar>t - s\<bar>"]
+ by (simp add: abs_minus_commute divide_simps)
+ show False
+ using * [of 1] * [of "-1"] by auto
+ qed
+ have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
+ proof -
+ have "t-s = 2 * (real_of_int n * pi)/x"
+ using that by (simp add: field_simps)
+ then show ?thesis by (metis abs_minus_commute)
+ qed
+ have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
+ by force
+ show ?thesis using False
+ apply (simp add: simple_path_def)
+ apply (simp add: part_circlepath_def linepath_def exp_eq * ** abs01 del: Set.insert_iff)
+ apply (subst abs_away)
+ apply (auto simp: 1)
+ apply (rule ccontr)
+ apply (auto simp: 2 field_split_simps abs_mult dest: of_int_leD)
+ done
+qed
+
+lemma arc_part_circlepath:
+ assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
+ shows "arc (part_circlepath z r s t)"
+proof -
+ have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
+ and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
+ proof (rule ccontr)
+ assume "x \<noteq> y"
+ have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
+ by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
+ then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
+ by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
+ with \<open>x \<noteq> y\<close> have st: "s-t = (of_int n * (pi * 2) / (y-x))"
+ by (force simp: field_simps)
+ have "\<bar>real_of_int n\<bar> < \<bar>y - x\<bar>"
+ using assms \<open>x \<noteq> y\<close> by (simp add: st abs_mult field_simps)
+ then show False
+ using assms x y st by (auto dest: of_int_lessD)
+ qed
+ show ?thesis
+ using assms
+ apply (simp add: arc_def)
+ apply (simp add: part_circlepath_def inj_on_def exp_eq)
+ apply (blast intro: *)
+ done
+qed
+
+subsection\<open>Special case of one complete circle\<close>
+
+definition\<^marker>\<open>tag important\<close> circlepath :: "[complex, real, real] \<Rightarrow> complex"
+ where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
+
+lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
+ by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
+
+lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
+ by (simp add: circlepath_def)
+
+lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
+ by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
+
+lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
+proof -
+ have "z + of_real r * exp (2 * pi * \<i> * (x + 1/2)) =
+ z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
+ by (simp add: divide_simps) (simp add: algebra_simps)
+ also have "\<dots> = z - r * exp (2 * pi * \<i> * x)"
+ by (simp add: exp_add)
+ finally show ?thesis
+ by (simp add: circlepath path_image_def sphere_def dist_norm)
+qed
+
+lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
+ using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
+ by (simp add: add.commute)
+
+lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
+ using circlepath_add1 [of z r "x-1/2"]
+ by (simp add: add.commute)
+
+lemma path_image_circlepath_minus_subset:
+ "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
+ apply (simp add: path_image_def image_def circlepath_minus, clarify)
+ apply (case_tac "xa \<le> 1/2", force)
+ apply (force simp: circlepath_add_half)+
+ done
+
+lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
+ using path_image_circlepath_minus_subset by fastforce
+
+lemma has_vector_derivative_circlepath [derivative_intros]:
+ "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
+ (at x within X)"
+ apply (simp add: circlepath_def scaleR_conv_of_real)
+ apply (rule derivative_eq_intros)
+ apply (simp add: algebra_simps)
+ done
+
+lemma vector_derivative_circlepath:
+ "vector_derivative (circlepath z r) (at x) =
+ 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
+using has_vector_derivative_circlepath vector_derivative_at by blast
+
+lemma vector_derivative_circlepath01:
+ "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
+ \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
+ 2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
+ using has_vector_derivative_circlepath
+ by (auto simp: vector_derivative_at_within_ivl)
+
+lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
+ by (simp add: circlepath_def)
+
+lemma path_circlepath [simp]: "path (circlepath z r)"
+ by (simp add: valid_path_imp_path)
+
+lemma path_image_circlepath_nonneg:
+ assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
+proof -
+ have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
+ proof (cases "x = z")
+ case True then show ?thesis by force
+ next
+ case False
+ define w where "w = x - z"
+ then have "w \<noteq> 0" by (simp add: False)
+ have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
+ using cis_conv_exp complex_eq_iff by auto
+ show ?thesis
+ apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
+ apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
+ apply (rule_tac x="t / (2*pi)" in image_eqI)
+ apply (simp add: field_simps \<open>w \<noteq> 0\<close>)
+ using False **
+ apply (auto simp: w_def)
+ done
+ qed
+ show ?thesis
+ unfolding circlepath path_image_def sphere_def dist_norm
+ by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
+qed
+
+lemma path_image_circlepath [simp]:
+ "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
+ using path_image_circlepath_minus
+ by (force simp: path_image_circlepath_nonneg abs_if)
+
+lemma has_contour_integral_bound_circlepath_strong:
+ "\<lbrakk>(f has_contour_integral i) (circlepath z r);
+ finite k; 0 \<le> B; 0 < r;
+ \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*(2*pi*r)"
+ unfolding circlepath_def
+ by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
+
+lemma has_contour_integral_bound_circlepath:
+ "\<lbrakk>(f has_contour_integral i) (circlepath z r);
+ 0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
+ \<Longrightarrow> norm i \<le> B*(2*pi*r)"
+ by (auto intro: has_contour_integral_bound_circlepath_strong)
+
+lemma contour_integrable_continuous_circlepath:
+ "continuous_on (path_image (circlepath z r)) f
+ \<Longrightarrow> f contour_integrable_on (circlepath z r)"
+ by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
+
+lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
+ by (simp add: circlepath_def simple_path_part_circlepath)
+
+lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
+ by (simp add: sphere_def dist_norm norm_minus_commute)
+
+lemma contour_integral_circlepath:
+ assumes "r > 0"
+ shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
+proof (rule contour_integral_unique)
+ show "((\<lambda>w. 1 / (w - z)) has_contour_integral 2 * complex_of_real pi * \<i>) (circlepath z r)"
+ unfolding has_contour_integral_def using assms
+ apply (subst has_integral_cong)
+ apply (simp add: vector_derivative_circlepath01)
+ using has_integral_const_real [of _ 0 1] apply (force simp: circlepath)
+ done
+qed
+
+lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
+ apply (rule winding_number_unique_loop)
+ apply (simp_all add: sphere_def valid_path_imp_path)
+ apply (rule_tac x="circlepath z r" in exI)
+ apply (simp add: sphere_def contour_integral_circlepath)
+ done
+
+proposition winding_number_circlepath:
+ assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
+proof (cases "w = z")
+ case True then show ?thesis
+ using assms winding_number_circlepath_centre by auto
+next
+ case False
+ have [simp]: "r > 0"
+ using assms le_less_trans norm_ge_zero by blast
+ define r' where "r' = norm(w - z)"
+ have "r' < r"
+ by (simp add: assms r'_def)
+ have disjo: "cball z r' \<inter> sphere z r = {}"
+ using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
+ have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
+ proof (rule winding_number_around_inside [where s = "cball z r'"])
+ show "winding_number (circlepath z r) z \<noteq> 0"
+ by (simp add: winding_number_circlepath_centre)
+ show "cball z r' \<inter> path_image (circlepath z r) = {}"
+ by (simp add: disjo less_eq_real_def)
+ qed (auto simp: r'_def dist_norm norm_minus_commute)
+ also have "\<dots> = 1"
+ by (simp add: winding_number_circlepath_centre)
+ finally show ?thesis .
+qed
+
+
+text\<open> Hence the Cauchy formula for points inside a circle.\<close>
+
+theorem Cauchy_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f" and holf: "f holomorphic_on (ball z r)" and wz: "norm(w - z) < r"
+ shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+ (circlepath z r)"
+proof -
+ have "r > 0"
+ using assms le_less_trans norm_ge_zero by blast
+ have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
+ (circlepath z r)"
+ proof (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
+ show "\<And>x. x \<in> interior (cball z r) - {} \<Longrightarrow>
+ f field_differentiable at x"
+ using holf holomorphic_on_imp_differentiable_at by auto
+ have "w \<notin> sphere z r"
+ by simp (metis dist_commute dist_norm not_le order_refl wz)
+ then show "path_image (circlepath z r) \<subseteq> cball z r - {w}"
+ using \<open>r > 0\<close> by (auto simp add: cball_def sphere_def)
+ qed (use wz in \<open>simp_all add: dist_norm norm_minus_commute contf\<close>)
+ then show ?thesis
+ by (simp add: winding_number_circlepath assms)
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> Cauchy_integral_circlepath_simple:
+ assumes "f holomorphic_on cball z r" "norm(w - z) < r"
+ shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
+ (circlepath z r)"
+using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
+
+
+lemma no_bounded_connected_component_imp_winding_number_zero:
+ assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
+ and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
+ shows "winding_number g z = 0"
+apply (rule winding_number_zero_in_outside)
+apply (simp_all add: assms)
+by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
+
+lemma no_bounded_path_component_imp_winding_number_zero:
+ assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
+ and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
+ shows "winding_number g z = 0"
+apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
+by (simp add: bounded_subset nb path_component_subset_connected_component)
+
+
+subsection\<open> Uniform convergence of path integral\<close>
+
+text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
+
+proposition contour_integral_uniform_limit:
+ assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
+ and ul_f: "uniform_limit (path_image \<gamma>) f l F"
+ and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
+ and \<gamma>: "valid_path \<gamma>"
+ and [simp]: "\<not> trivial_limit F"
+ shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
+proof -
+ have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
+ { fix e::real
+ assume "0 < e"
+ then have "0 < e / (\<bar>B\<bar> + 1)" by simp
+ then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
+ using ul_f [unfolded uniform_limit_iff dist_norm] by auto
+ with ev_fint
+ obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
+ and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
+ using eventually_happens [OF eventually_conj]
+ by (fastforce simp: contour_integrable_on path_image_def)
+ have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
+ using \<open>0 \<le> B\<close> \<open>0 < e\<close> by (simp add: field_split_simps)
+ have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
+ proof (intro exI conjI ballI)
+ show "cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - f a (\<gamma> x) * vector_derivative \<gamma> (at x)) \<le> e"
+ if "x \<in> {0..1}" for x
+ apply (rule order_trans [OF _ Ble])
+ using noleB [OF that] fga [OF that] \<open>0 \<le> B\<close> \<open>0 < e\<close>
+ apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
+ apply (fastforce simp: mult_ac dest: mult_mono [OF less_imp_le])
+ done
+ qed (rule inta)
+ }
+ then show lintg: "l contour_integrable_on \<gamma>"
+ unfolding contour_integrable_on by (metis (mono_tags, lifting)integrable_uniform_limit_real)
+ { fix e::real
+ define B' where "B' = B + 1"
+ have B': "B' > 0" "B' > B" using \<open>0 \<le> B\<close> by (auto simp: B'_def)
+ assume "0 < e"
+ then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
+ using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
+ by (simp add: field_simps)
+ have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
+ have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
+ if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
+ proof -
+ have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
+ using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
+ also have "\<dots> < e"
+ by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
+ finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
+ then show ?thesis
+ by (simp add: left_diff_distrib [symmetric] norm_mult)
+ qed
+ have le_e: "\<And>x. \<lbrakk>\<forall>xa\<in>{0..1}. 2 * cmod (f x (\<gamma> xa) - l (\<gamma> xa)) < e / B'; f x contour_integrable_on \<gamma>\<rbrakk>
+ \<Longrightarrow> cmod (integral {0..1}
+ (\<lambda>u. f x (\<gamma> u) * vector_derivative \<gamma> (at u) - l (\<gamma> u) * vector_derivative \<gamma> (at u))) < e"
+ apply (rule le_less_trans [OF integral_norm_bound_integral ie])
+ apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
+ apply (blast intro: *)+
+ done
+ have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
+ apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
+ apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
+ apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric] le_e)
+ done
+ }
+ then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
+ by (rule tendstoI)
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> contour_integral_uniform_limit_circlepath:
+ assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
+ and "uniform_limit (sphere z r) f l F"
+ and "\<not> trivial_limit F" "0 < r"
+ shows "l contour_integrable_on (circlepath z r)"
+ "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
+ using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>General stepping result for derivative formulas\<close>
+
+lemma Cauchy_next_derivative:
+ assumes "continuous_on (path_image \<gamma>) f'"
+ and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
+ and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
+ and k: "k \<noteq> 0"
+ and "open s"
+ and \<gamma>: "valid_path \<gamma>"
+ and w: "w \<in> s - path_image \<gamma>"
+ shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
+ and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
+ (at w)" (is "?thes2")
+proof -
+ have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
+ then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
+ using open_contains_ball by blast
+ have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
+ by (metis norm_of_nat of_nat_Suc)
+ have cint: "\<And>x. \<lbrakk>x \<noteq> w; cmod (x - w) < d\<rbrakk>
+ \<Longrightarrow> (\<lambda>z. (f' z / (z - x) ^ k - f' z / (z - w) ^ k) / (x * k - w * k)) contour_integrable_on \<gamma>"
+ apply (rule contour_integrable_div [OF contour_integrable_diff])
+ using int w d
+ by (force simp: dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
+ have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
+ contour_integrable_on \<gamma>"
+ unfolding eventually_at
+ apply (rule_tac x=d in exI)
+ apply (simp add: \<open>d > 0\<close> dist_norm field_simps cint)
+ done
+ have bim_g: "bounded (image f' (path_image \<gamma>))"
+ by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
+ then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
+ by (force simp: bounded_pos path_image_def)
+ have twom: "\<forall>\<^sub>F n in at w.
+ \<forall>x\<in>path_image \<gamma>.
+ cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
+ if "0 < e" for e
+ proof -
+ have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k) < e"
+ if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
+ and uw_less: "cmod (u - w) < e * (d/2) ^ (k+2) / (1 + real k)"
+ for u x
+ proof -
+ define ff where [abs_def]:
+ "ff n w =
+ (if n = 0 then inverse(x - w)^k
+ else if n = 1 then k / (x - w)^(Suc k)
+ else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
+ have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
+ by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
+ have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d/2))"
+ if "z \<in> ball w (d/2)" "i \<le> 1" for i z
+ proof -
+ have "z \<notin> path_image \<gamma>"
+ using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
+ then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
+ then have neq: "x * x + z * z \<noteq> x * (z * 2)"
+ by (blast intro: dest!: sum_sqs_eq)
+ with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
+ then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
+ by (simp add: algebra_simps)
+ show ?thesis using \<open>i \<le> 1\<close>
+ apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
+ apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
+ done
+ qed
+ { fix a::real and b::real assume ab: "a > 0" "b > 0"
+ then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
+ by (subst mult_le_cancel_left_pos)
+ (use \<open>k \<noteq> 0\<close> in \<open>auto simp: divide_simps\<close>)
+ with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
+ by (simp add: field_simps)
+ } note canc = this
+ have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d/2) ^ (k + 2)"
+ if "v \<in> ball w (d/2)" for v
+ proof -
+ have lessd: "\<And>z. cmod (\<gamma> z - v) < d/2 \<Longrightarrow> cmod (w - \<gamma> z) < d"
+ by (metis that norm_minus_commute norm_triangle_half_r dist_norm mem_ball)
+ have "d/2 \<le> cmod (x - v)" using d x that
+ using lessd d x
+ by (auto simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps)
+ then have "d \<le> cmod (x - v) * 2"
+ by (simp add: field_split_simps)
+ then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
+ using \<open>0 < d\<close> order_less_imp_le power_mono by blast
+ have "x \<noteq> v" using that
+ using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
+ then show ?thesis
+ using \<open>d > 0\<close> apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
+ using dpow_le apply (simp add: field_split_simps)
+ done
+ qed
+ have ub: "u \<in> ball w (d/2)"
+ using uwd by (simp add: dist_commute dist_norm)
+ have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d/2) ^ k))"
+ using complex_Taylor [OF _ ff1 ff2 _ ub, of w, simplified]
+ by (simp add: ff_def \<open>0 < d\<close>)
+ then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+ by (simp add: field_simps)
+ then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
+ / (cmod (u - w) * real k)
+ \<le> (1 + real k) * cmod (u - w) / (d/2) ^ (k+2)"
+ using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
+ also have "\<dots> < e"
+ using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
+ finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
+ / cmod ((u - w) * real k) < e"
+ by (simp add: norm_mult)
+ have "x \<noteq> u"
+ using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
+ show ?thesis
+ apply (rule le_less_trans [OF _ e])
+ using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close> \<open>u \<noteq> w\<close>
+ apply (simp add: field_simps norm_divide [symmetric])
+ done
+ qed
+ show ?thesis
+ unfolding eventually_at
+ apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
+ apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
+ done
+ qed
+ have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
+ if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
+ and x: "0 \<le> x" "x \<le> 1"
+ for u x
+ proof (cases "(f' (\<gamma> x)) = 0")
+ case True then show ?thesis by (simp add: \<open>0 < e\<close>)
+ next
+ case False
+ have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
+ cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
+ by (simp add: field_simps)
+ also have "\<dots> = cmod (f' (\<gamma> x)) *
+ cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
+ inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
+ by (simp add: norm_mult)
+ also have "\<dots> < cmod (f' (\<gamma> x)) * (e/C)"
+ using False mult_strict_left_mono [OF ec] by force
+ also have "\<dots> \<le> e" using C
+ by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
+ finally show ?thesis .
+ qed
+ show "\<forall>\<^sub>F n in at w.
+ \<forall>x\<in>path_image \<gamma>.
+ cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
+ using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]] unfolding path_image_def
+ by (force intro: * elim: eventually_mono)
+ qed
+ show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
+ by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+ have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
+ \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
+ by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
+ have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
+ (f u - f w) / (u - w) / k"
+ if "dist u w < d" for u
+ proof -
+ have u: "u \<in> s - path_image \<gamma>"
+ by (metis subsetD d dist_commute mem_ball that)
+ show ?thesis
+ apply (rule contour_integral_unique)
+ apply (simp add: diff_divide_distrib algebra_simps)
+ apply (intro has_contour_integral_diff has_contour_integral_div)
+ using u w apply (simp_all add: field_simps int)
+ done
+ qed
+ show ?thes2
+ apply (simp add: has_field_derivative_iff del: power_Suc)
+ apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
+ apply (simp add: \<open>k \<noteq> 0\<close> **)
+ done
+qed
+
+lemma Cauchy_next_derivative_circlepath:
+ assumes contf: "continuous_on (path_image (circlepath z r)) f"
+ and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
+ and k: "k \<noteq> 0"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
+ (is "?thes2")
+proof -
+ have "r > 0" using w
+ using ball_eq_empty by fastforce
+ have wim: "w \<in> ball z r - path_image (circlepath z r)"
+ using w by (auto simp: dist_norm)
+ show ?thes1 ?thes2
+ by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
+ auto simp: vector_derivative_circlepath norm_mult)+
+qed
+
+
+text\<open> In particular, the first derivative formula.\<close>
+
+lemma Cauchy_derivative_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
+ (is "?thes2")
+proof -
+ have [simp]: "r \<ge> 0" using w
+ using ball_eq_empty by fastforce
+ have f: "continuous_on (path_image (circlepath z r)) f"
+ by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def)
+ have int: "\<And>w. dist z w < r \<Longrightarrow>
+ ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
+ by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
+ show ?thes1
+ apply (simp add: power2_eq_square)
+ apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
+ apply (blast intro: int)
+ done
+ have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
+ apply (simp add: power2_eq_square)
+ apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
+ apply (blast intro: int)
+ done
+ then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
+ by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
+ show ?thes2
+ by simp (rule fder)
+qed
+
+subsection\<open>Existence of all higher derivatives\<close>
+
+proposition derivative_is_holomorphic:
+ assumes "open S"
+ and fder: "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z)"
+ shows "f' holomorphic_on S"
+proof -
+ have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> S" for z
+ proof -
+ obtain r where "r > 0" and r: "cball z r \<subseteq> S"
+ using open_contains_cball \<open>z \<in> S\<close> \<open>open S\<close> by blast
+ then have holf_cball: "f holomorphic_on cball z r"
+ apply (simp add: holomorphic_on_def)
+ using field_differentiable_at_within field_differentiable_def fder by blast
+ then have "continuous_on (path_image (circlepath z r)) f"
+ using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
+ then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
+ by (auto intro: continuous_intros)+
+ have contf_cball: "continuous_on (cball z r) f" using holf_cball
+ by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
+ have holf_ball: "f holomorphic_on ball z r" using holf_cball
+ using ball_subset_cball holomorphic_on_subset by blast
+ { fix w assume w: "w \<in> ball z r"
+ have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
+ by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+ have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
+ (at w)"
+ by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
+ have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
+ using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
+ have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
+ contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+ (circlepath z r)"
+ by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
+ then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
+ contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
+ (circlepath z r)"
+ by (simp add: algebra_simps)
+ then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
+ by (simp add: f'_eq)
+ } note * = this
+ show ?thesis
+ apply (rule exI)
+ apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
+ apply (simp_all add: \<open>0 < r\<close> * dist_norm)
+ done
+ qed
+ show ?thesis
+ by (simp add: holomorphic_on_open [OF \<open>open S\<close>] *)
+qed
+
+lemma holomorphic_deriv [holomorphic_intros]:
+ "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on S"
+by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
+
+lemma analytic_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv f) analytic_on S"
+ using analytic_on_holomorphic holomorphic_deriv by auto
+
+lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on S; open S\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on S"
+ by (induction n) (auto simp: holomorphic_deriv)
+
+lemma analytic_higher_deriv [analytic_intros]: "f analytic_on S \<Longrightarrow> (deriv ^^ n) f analytic_on S"
+ unfolding analytic_on_def using holomorphic_higher_deriv by blast
+
+lemma has_field_derivative_higher_deriv:
+ "\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
+ \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
+by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
+ funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
+
+lemma valid_path_compose_holomorphic:
+ assumes "valid_path g" and holo:"f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
+ shows "valid_path (f \<circ> g)"
+proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
+ fix x assume "x \<in> path_image g"
+ then show "f field_differentiable at x"
+ using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
+next
+ have "deriv f holomorphic_on S"
+ using holomorphic_deriv holo \<open>open S\<close> by auto
+ then show "continuous_on (path_image g) (deriv f)"
+ using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
+qed
+
+
+subsection\<open>Morera's theorem\<close>
+
+lemma Morera_local_triangle_ball:
+ assumes "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+ (\<forall>b c. closed_segment b c \<subseteq> ball a e
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ shows "f analytic_on S"
+proof -
+ { fix z assume "z \<in> S"
+ with assms obtain e a where
+ "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
+ and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
+ \<Longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ by blast
+ have az: "dist a z < e" using mem_ball z by blast
+ have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
+ by (simp add: dist_commute ball_subset_ball_iff)
+ have "\<exists>e>0. f holomorphic_on ball z e"
+ proof (intro exI conjI)
+ have sub_ball: "\<And>y. dist a y < e \<Longrightarrow> closed_segment a y \<subseteq> ball a e"
+ by (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
+ show "f holomorphic_on ball z (e - dist a z)"
+ apply (rule holomorphic_on_subset [OF _ sb_ball])
+ apply (rule derivative_is_holomorphic[OF open_ball])
+ apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
+ apply (simp_all add: 0 \<open>0 < e\<close> sub_ball)
+ done
+ qed (simp add: az)
+ }
+ then show ?thesis
+ by (simp add: analytic_on_def)
+qed
+
+lemma Morera_local_triangle:
+ assumes "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and>
+ (\<forall>a b c. convex hull {a,b,c} \<subseteq> t
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ shows "f analytic_on S"
+proof -
+ { fix z assume "z \<in> S"
+ with assms obtain t where
+ "open t" and z: "z \<in> t" and contf: "continuous_on t f"
+ and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t
+ \<Longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+ by force
+ then obtain e where "e>0" and e: "ball z e \<subseteq> t"
+ using open_contains_ball by blast
+ have [simp]: "continuous_on (ball z e) f" using contf
+ using continuous_on_subset e by blast
+ have eq0: "\<And>b c. closed_segment b c \<subseteq> ball z e \<Longrightarrow>
+ contour_integral (linepath z b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c z) f = 0"
+ by (meson 0 z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
+ have "\<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
+ (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
+ contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
+ using \<open>e > 0\<close> eq0 by force
+ }
+ then show ?thesis
+ by (simp add: Morera_local_triangle_ball)
+qed
+
+proposition Morera_triangle:
+ "\<lbrakk>continuous_on S f; open S;
+ \<And>a b c. convex hull {a,b,c} \<subseteq> S
+ \<longrightarrow> contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0\<rbrakk>
+ \<Longrightarrow> f analytic_on S"
+ using Morera_local_triangle by blast
+
+subsection\<open>Combining theorems for higher derivatives including Leibniz rule\<close>
+
+lemma higher_deriv_linear [simp]:
+ "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
+ by (induction n) auto
+
+lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
+ by (induction n) auto
+
+lemma higher_deriv_ident [simp]:
+ "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
+ apply (induction n, simp)
+ apply (metis higher_deriv_linear lambda_one)
+ done
+
+lemma higher_deriv_id [simp]:
+ "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
+ by (simp add: id_def)
+
+lemma has_complex_derivative_funpow_1:
+ "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
+ apply (induction n, auto)
+ apply (simp add: id_def)
+ by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)
+
+lemma higher_deriv_uminus:
+ assumes "f holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have "((deriv ^^ n) (\<lambda>w. - f w) has_field_derivative - deriv ((deriv ^^ n) f) z) (at z)"
+ apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
+ apply (rule derivative_eq_intros | rule * refl assms)+
+ apply (auto simp add: Suc)
+ done
+ then show ?case
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_add:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have "((deriv ^^ n) (\<lambda>w. f w + g w) has_field_derivative
+ deriv ((deriv ^^ n) f) z + deriv ((deriv ^^ n) g) z) (at z)"
+ apply (rule has_field_derivative_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
+ apply (rule derivative_eq_intros | rule * refl assms)+
+ apply (auto simp add: Suc)
+ done
+ then show ?case
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_diff:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+ apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
+ apply (subst higher_deriv_add)
+ using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
+ done
+
+lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
+ by (cases k) simp_all
+
+lemma higher_deriv_mult:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+ (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
+ "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ have sumeq: "(\<Sum>i = 0..n.
+ of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
+ g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
+ apply (simp add: bb algebra_simps sum.distrib)
+ apply (subst (4) sum_Suc_reindex)
+ apply (auto simp: algebra_simps Suc_diff_le intro: sum.cong)
+ done
+ have "((deriv ^^ n) (\<lambda>w. f w * g w) has_field_derivative
+ (\<Sum>i = 0..Suc n. (Suc n choose i) * (deriv ^^ i) f z * (deriv ^^ (Suc n - i)) g z))
+ (at z)"
+ apply (rule has_field_derivative_transform_within_open
+ [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
+ apply (simp add: algebra_simps)
+ apply (rule DERIV_cong [OF DERIV_sum])
+ apply (rule DERIV_cmult)
+ apply (auto intro: DERIV_mult * sumeq \<open>open S\<close> Suc.prems Suc.IH [symmetric])
+ done
+ then show ?case
+ unfolding funpow.simps o_apply
+ by (simp add: DERIV_imp_deriv)
+qed
+
+lemma higher_deriv_transform_within_open:
+ fixes z::complex
+ assumes "f holomorphic_on S" "g holomorphic_on S" "open S" and z: "z \<in> S"
+ and fg: "\<And>w. w \<in> S \<Longrightarrow> f w = g w"
+ shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
+using z
+by (induction i arbitrary: z)
+ (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)
+
+lemma higher_deriv_compose_linear:
+ fixes z::complex
+ assumes f: "f holomorphic_on T" and S: "open S" and T: "open T" and z: "z \<in> S"
+ and fg: "\<And>w. w \<in> S \<Longrightarrow> u * w \<in> T"
+ shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
+using z
+proof (induction n arbitrary: z)
+ case 0 then show ?case by simp
+next
+ case (Suc n z)
+ have holo0: "f holomorphic_on (*) u ` S"
+ by (meson fg f holomorphic_on_subset image_subset_iff)
+ have holo2: "(deriv ^^ n) f holomorphic_on (*) u ` S"
+ by (meson f fg holomorphic_higher_deriv holomorphic_on_subset image_subset_iff T)
+ have holo3: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
+ by (intro holo2 holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def] holomorphic_intros)
+ have holo1: "(\<lambda>w. f (u * w)) holomorphic_on S"
+ apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
+ apply (rule holo0 holomorphic_intros)+
+ done
+ have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
+ apply (rule complex_derivative_transform_within_open [OF _ holo3 S Suc.prems])
+ apply (rule holomorphic_higher_deriv [OF holo1 S])
+ apply (simp add: Suc.IH)
+ done
+ also have "\<dots> = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
+ apply (rule deriv_cmult)
+ apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
+ apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and T=T, unfolded o_def])
+ apply (simp)
+ apply (simp add: analytic_on_open f holomorphic_higher_deriv T)
+ apply (blast intro: fg)
+ done
+ also have "\<dots> = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
+ apply (subst deriv_chain [where g = "(deriv ^^ n) f" and f = "(*) u", unfolded o_def])
+ apply (rule derivative_intros)
+ using Suc.prems field_differentiable_def f fg has_field_derivative_higher_deriv T apply blast
+ apply (simp)
+ done
+ finally show ?case
+ by simp
+qed
+
+lemma higher_deriv_add_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_add show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_diff_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_diff show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+lemma higher_deriv_uminus_at:
+ "f analytic_on {z} \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
+ using higher_deriv_uminus
+ by (auto simp: analytic_at)
+
+lemma higher_deriv_mult_at:
+ assumes "f analytic_on {z}" "g analytic_on {z}"
+ shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
+ (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
+proof -
+ have "f analytic_on {z} \<and> g analytic_on {z}"
+ using assms by blast
+ with higher_deriv_mult show ?thesis
+ by (auto simp: analytic_at_two)
+qed
+
+
+text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>
+
+proposition no_isolated_singularity:
+ fixes z::complex
+ assumes f: "continuous_on S f" and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+ shows "f holomorphic_on S"
+proof -
+ { fix z
+ assume "z \<in> S" and cdf: "\<And>x. x \<in> S - K \<Longrightarrow> f field_differentiable at x"
+ have "f field_differentiable at z"
+ proof (cases "z \<in> K")
+ case False then show ?thesis by (blast intro: cdf \<open>z \<in> S\<close>)
+ next
+ case True
+ with finite_set_avoid [OF K, of z]
+ obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>K; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
+ by blast
+ obtain e where "e>0" and e: "ball z e \<subseteq> S"
+ using S \<open>z \<in> S\<close> by (force simp: open_contains_ball)
+ have fde: "continuous_on (ball z (min d e)) f"
+ by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
+ have cont: "{a,b,c} \<subseteq> ball z (min d e) \<Longrightarrow> continuous_on (convex hull {a, b, c}) f" for a b c
+ by (simp add: hull_minimal continuous_on_subset [OF fde])
+ have fd: "\<lbrakk>{a,b,c} \<subseteq> ball z (min d e); x \<in> interior (convex hull {a, b, c}) - K\<rbrakk>
+ \<Longrightarrow> f field_differentiable at x" for a b c x
+ by (metis cdf Diff_iff Int_iff ball_min_Int subsetD convex_ball e interior_mono interior_subset subset_hull)
+ obtain g where "\<And>w. w \<in> ball z (min d e) \<Longrightarrow> (g has_field_derivative f w) (at w within ball z (min d e))"
+ apply (rule contour_integral_convex_primitive
+ [OF convex_ball fde Cauchy_theorem_triangle_cofinite [OF _ K]])
+ using cont fd by auto
+ then have "f holomorphic_on ball z (min d e)"
+ by (metis open_ball at_within_open derivative_is_holomorphic)
+ then show ?thesis
+ unfolding holomorphic_on_def
+ by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
+ qed
+ }
+ with holf S K show ?thesis
+ by (simp add: holomorphic_on_open open_Diff finite_imp_closed field_differentiable_def [symmetric])
+qed
+
+lemma no_isolated_singularity':
+ fixes z::complex
+ assumes f: "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"
+ and holf: "f holomorphic_on (S - K)" and S: "open S" and K: "finite K"
+ shows "f holomorphic_on S"
+proof (rule no_isolated_singularity[OF _ assms(2-)])
+ show "continuous_on S f" unfolding continuous_on_def
+ proof
+ fix z assume z: "z \<in> S"
+ show "(f \<longlongrightarrow> f z) (at z within S)"
+ proof (cases "z \<in> K")
+ case False
+ from holf have "continuous_on (S - K) f"
+ by (rule holomorphic_on_imp_continuous_on)
+ with z False have "(f \<longlongrightarrow> f z) (at z within (S - K))"
+ by (simp add: continuous_on_def)
+ also from z K S False have "at z within (S - K) = at z within S"
+ by (subst (1 2) at_within_open) (auto intro: finite_imp_closed)
+ finally show "(f \<longlongrightarrow> f z) (at z within S)" .
+ qed (insert assms z, simp_all)
+ qed
+qed
+
+proposition Cauchy_integral_formula_convex:
+ assumes S: "convex S" and K: "finite K" and contf: "continuous_on S f"
+ and fcd: "(\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x)"
+ and z: "z \<in> interior S" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ have *: "\<And>x. x \<in> interior S \<Longrightarrow> f field_differentiable at x"
+ unfolding holomorphic_on_open [symmetric] field_differentiable_def
+ using no_isolated_singularity [where S = "interior S"]
+ by (meson K contf continuous_at_imp_continuous_on continuous_on_interior fcd
+ field_differentiable_at_within field_differentiable_def holomorphic_onI
+ holomorphic_on_imp_differentiable_at open_interior)
+ show ?thesis
+ by (rule Cauchy_integral_formula_weak [OF S finite.emptyI contf]) (use * assms in auto)
+qed
+
+text\<open> Formula for higher derivatives.\<close>
+
+lemma Cauchy_has_contour_integral_higher_derivative_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * \<i>) / (fact k) * (deriv ^^ k) f w))
+ (circlepath z r)"
+using w
+proof (induction k arbitrary: w)
+ case 0 then show ?case
+ using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
+next
+ case (Suc k)
+ have [simp]: "r > 0" using w
+ using ball_eq_empty by fastforce
+ have f: "continuous_on (path_image (circlepath z r)) f"
+ by (rule continuous_on_subset [OF contf]) (force simp: cball_def sphere_def less_imp_le)
+ obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
+ using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
+ by (auto simp: contour_integrable_on_def)
+ then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
+ by (rule contour_integral_unique)
+ have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
+ using Suc.prems assms has_field_derivative_higher_deriv by auto
+ then have dnf_diff: "\<And>n. (deriv ^^ n) f field_differentiable (at w)"
+ by (force simp: field_differentiable_def)
+ have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
+ of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
+ by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
+ also have "\<dots> = of_nat (Suc k) * X"
+ by (simp only: con)
+ finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
+ then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
+ by (metis deriv_cmult dnf_diff)
+ then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
+ by (simp add: field_simps)
+ then show ?case
+ using of_nat_eq_0_iff X by fastforce
+qed
+
+lemma Cauchy_higher_derivative_integral_circlepath:
+ assumes contf: "continuous_on (cball z r) f"
+ and holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
+ (is "?thes1")
+ and "(deriv ^^ k) f w = (fact k) / (2 * pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
+ (is "?thes2")
+proof -
+ have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
+ (circlepath z r)"
+ using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
+ by simp
+ show ?thes1 using *
+ using contour_integrable_on_def by blast
+ show ?thes2
+ unfolding contour_integral_unique [OF *] by (simp add: field_split_simps)
+qed
+
+corollary Cauchy_contour_integral_circlepath:
+ assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+ shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * \<i>) * (deriv ^^ k) f w / (fact k)"
+by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])
+
+lemma Cauchy_contour_integral_circlepath_2:
+ assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
+ shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * \<i>) * deriv f w"
+ using Cauchy_contour_integral_circlepath [OF assms, of 1]
+ by (simp add: power2_eq_square)
+
+
+subsection\<open>A holomorphic function is analytic, i.e. has local power series\<close>
+
+theorem holomorphic_power_series:
+ assumes holf: "f holomorphic_on ball z r"
+ and w: "w \<in> ball z r"
+ shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+proof -
+ \<comment> \<open>Replacing \<^term>\<open>r\<close> and the original (weak) premises with stronger ones\<close>
+ obtain r where "r > 0" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
+ proof
+ have "cball z ((r + dist w z) / 2) \<subseteq> ball z r"
+ using w by (simp add: dist_commute field_sum_of_halves subset_eq)
+ then show "f holomorphic_on cball z ((r + dist w z) / 2)"
+ by (rule holomorphic_on_subset [OF holf])
+ have "r > 0"
+ using w by clarsimp (metis dist_norm le_less_trans norm_ge_zero)
+ then show "0 < (r + dist w z) / 2"
+ by simp (use zero_le_dist [of w z] in linarith)
+ qed (use w in \<open>auto simp: dist_commute\<close>)
+ then have holf: "f holomorphic_on ball z r"
+ using ball_subset_cball holomorphic_on_subset by blast
+ have contf: "continuous_on (cball z r) f"
+ by (simp add: holfc holomorphic_on_imp_continuous_on)
+ have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
+ by (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf]) (simp add: \<open>0 < r\<close>)
+ obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
+ by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
+ obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
+ and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
+ proof
+ show "\<And>u. cmod (u - z) = r \<Longrightarrow> r - dist z w \<le> cmod (u - w)"
+ by (metis add_diff_eq diff_add_cancel dist_norm norm_diff_ineq)
+ qed (use w in \<open>auto simp: dist_norm norm_minus_commute\<close>)
+ have ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using k by auto
+ obtain n where n: "((r - k) / r) ^ n < e / B * k"
+ using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
+ have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
+ if "n \<le> N" and r: "r = dist z u" for N u
+ proof -
+ have N: "((r - k) / r) ^ N < e / B * k"
+ apply (rule le_less_trans [OF power_decreasing n])
+ using \<open>n \<le> N\<close> k by auto
+ have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
+ using \<open>0 < r\<close> r w by auto
+ have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
+ by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
+ have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
+ = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
+ unfolding sum_distrib_right sum_divide_distrib power_divide by (simp add: algebra_simps)
+ also have "\<dots> = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
+ using \<open>0 < B\<close>
+ apply (auto simp: geometric_sum [OF wzu_not1])
+ apply (simp add: field_simps norm_mult [symmetric])
+ done
+ also have "\<dots> = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
+ using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
+ also have "\<dots> = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
+ by (simp add: algebra_simps)
+ also have "\<dots> = norm (w - z) ^ N * norm (f u) / r ^ N"
+ by (simp add: norm_mult norm_power norm_minus_commute)
+ also have "\<dots> \<le> (((r - k)/r)^N) * B"
+ using \<open>0 < r\<close> w k
+ apply (simp add: divide_simps)
+ apply (rule mult_mono [OF power_mono])
+ apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
+ done
+ also have "\<dots> < e * k"
+ using \<open>0 < B\<close> N by (simp add: divide_simps)
+ also have "\<dots> \<le> e * norm (u - w)"
+ using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
+ finally show ?thesis
+ by (simp add: field_split_simps norm_divide del: power_Suc)
+ qed
+ with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
+ norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
+ by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
+ qed
+ have eq: "\<forall>\<^sub>F x in sequentially.
+ contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
+ (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
+ apply (rule eventuallyI)
+ apply (subst contour_integral_sum, simp)
+ using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
+ apply (simp only: contour_integral_lmul cint algebra_simps)
+ done
+ have cic: "\<And>u. (\<lambda>y. \<Sum>k<u. (w - z) ^ k * (f y / (y - z) ^ Suc k)) contour_integrable_on circlepath z r"
+ apply (intro contour_integrable_sum contour_integrable_lmul, simp)
+ using \<open>0 < r\<close> by (force intro!: Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+ have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+ sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
+ unfolding sums_def
+ apply (intro Lim_transform_eventually [OF _ eq] contour_integral_uniform_limit_circlepath [OF eventuallyI ul] cic)
+ using \<open>0 < r\<close> apply auto
+ done
+ then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
+ sums (2 * of_real pi * \<i> * f w)"
+ using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
+ then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
+ sums ((2 * of_real pi * \<i> * f w) / (\<i> * (complex_of_real pi * 2)))"
+ by (rule sums_divide)
+ then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
+ sums f w"
+ by (simp add: field_simps)
+ then show ?thesis
+ by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
+qed
+
+
+subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra\<close>
+
+text\<open> These weak Liouville versions don't even need the derivative formula.\<close>
+
+lemma Liouville_weak_0:
+ assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
+ shows "f z = 0"
+proof (rule ccontr)
+ assume fz: "f z \<noteq> 0"
+ with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
+ obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
+ by (auto simp: dist_norm)
+ define R where "R = 1 + \<bar>B\<bar> + norm z"
+ have "R > 0" unfolding R_def
+ proof -
+ have "0 \<le> cmod z + \<bar>B\<bar>"
+ by (metis (full_types) add_nonneg_nonneg norm_ge_zero real_norm_def)
+ then show "0 < 1 + \<bar>B\<bar> + cmod z"
+ by linarith
+ qed
+ have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
+ apply (rule Cauchy_integral_circlepath)
+ using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
+ done
+ have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 < cmod (f z)" for x
+ unfolding R_def
+ by (rule B) (use norm_triangle_ineq4 [of x z] in auto)
+ with \<open>R > 0\<close> fz show False
+ using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
+ by (auto simp: less_imp_le norm_mult norm_divide field_split_simps)
+qed
+
+proposition Liouville_weak:
+ assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
+ shows "f z = l"
+ using Liouville_weak_0 [of "\<lambda>z. f z - l"]
+ by (simp add: assms holomorphic_on_diff LIM_zero)
+
+proposition Liouville_weak_inverse:
+ assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
+ obtains z where "f z = 0"
+proof -
+ { assume f: "\<And>z. f z \<noteq> 0"
+ have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
+ by (simp add: holomorphic_on_divide assms f)
+ have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
+ apply (rule tendstoI [OF eventually_mono])
+ apply (rule_tac B="2/e" in unbounded)
+ apply (simp add: dist_norm norm_divide field_split_simps)
+ done
+ have False
+ using Liouville_weak_0 [OF 1 2] f by simp
+ }
+ then show ?thesis
+ using that by blast
+qed
+
+text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>
+
+theorem fundamental_theorem_of_algebra:
+ fixes a :: "nat \<Rightarrow> complex"
+ assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
+ obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
+using assms
+proof (elim disjE bexE)
+ assume "a 0 = 0" then show ?thesis
+ by (auto simp: that [of 0])
+next
+ fix i
+ assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
+ have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
+ by (rule holomorphic_intros)+
+ show thesis
+ proof (rule Liouville_weak_inverse [OF 1])
+ show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
+ using i nz by (intro polyfun_extremal exI[of _ i]) auto
+ qed (use that in auto)
+qed
+
+subsection\<open>Weierstrass convergence theorem\<close>
+
+lemma holomorphic_uniform_limit:
+ assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
+ and ulim: "uniform_limit (cball z r) f g F"
+ and F: "\<not> trivial_limit F"
+ obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+proof (cases r "0::real" rule: linorder_cases)
+ case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
+next
+ case equal then show ?thesis
+ by (force simp: holomorphic_on_def intro: that)
+next
+ case greater
+ have contg: "continuous_on (cball z r) g"
+ using cont uniform_limit_theorem [OF eventually_mono ulim F] by blast
+ have "path_image (circlepath z r) \<subseteq> cball z r"
+ using \<open>0 < r\<close> by auto
+ then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
+ by (intro continuous_intros continuous_on_subset [OF contg])
+ have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
+ if w: "w \<in> ball z r" for w
+ proof -
+ define d where "d = (r - norm(w - z))"
+ have "0 < d" "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
+ have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
+ unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
+ have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
+ apply (rule eventually_mono [OF cont])
+ using w
+ apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
+ done
+ have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
+ using greater \<open>0 < d\<close>
+ apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
+ apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
+ apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
+ done
+ have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
+ by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+ have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
+ by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_less F \<open>0 < r\<close>])
+ have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
+ proof (rule Lim_transform_eventually)
+ show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
+ = 2 * of_real pi * \<i> * f x w"
+ apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
+ using w\<open>0 < d\<close> d_def by auto
+ qed (auto simp: cif_tends_cig)
+ have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
+ by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
+ then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
+ by (rule tendsto_mult_left [OF tendstoI])
+ then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
+ using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
+ by fastforce
+ then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
+ using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
+ by (force simp: field_simps)
+ then show ?thesis
+ by (simp add: dist_norm)
+ qed
+ show ?thesis
+ using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
+ by (fastforce simp add: holomorphic_on_open contg intro: that)
+qed
+
+
+text\<open> Version showing that the limit is the limit of the derivatives.\<close>
+
+proposition has_complex_derivative_uniform_limit:
+ fixes z::complex
+ assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
+ (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
+ and ulim: "uniform_limit (cball z r) f g F"
+ and F: "\<not> trivial_limit F" and "0 < r"
+ obtains g' where
+ "continuous_on (cball z r) g"
+ "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+proof -
+ let ?conint = "contour_integral (circlepath z r)"
+ have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
+ by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
+ auto simp: holomorphic_on_open field_differentiable_def)+
+ then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
+ using DERIV_deriv_iff_has_field_derivative
+ by (fastforce simp add: holomorphic_on_open)
+ then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
+ by (simp add: DERIV_imp_deriv)
+ have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
+ proof -
+ have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
+ if cont_fn: "continuous_on (cball z r) (f n)"
+ and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
+ proof -
+ have hol_fn: "f n holomorphic_on ball z r"
+ using fnd by (force simp: holomorphic_on_open)
+ have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
+ by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
+ then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
+ using DERIV_unique [OF fnd] w by blast
+ show ?thesis
+ by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] field_split_simps)
+ qed
+ define d where "d = (r - norm(w - z))^2"
+ have "d > 0"
+ using w by (simp add: dist_commute dist_norm d_def)
+ have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
+ proof -
+ have "w \<in> ball z (cmod (z - y))"
+ using that w by fastforce
+ then have "cmod (w - z) \<le> cmod (z - y)"
+ by (simp add: dist_complex_def norm_minus_commute)
+ moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
+ by (metis diff_add_cancel diff_add_eq_diff_diff_swap norm_minus_commute norm_triangle_ineq2)
+ ultimately show ?thesis
+ using that by (simp add: d_def norm_power power_mono)
+ qed
+ have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
+ by (force simp: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
+ have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
+ unfolding uniform_limit_iff
+ proof clarify
+ fix e::real
+ assume "0 < e"
+ with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
+ apply (simp add: norm_divide field_split_simps sphere_def dist_norm)
+ apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
+ apply (simp add: \<open>0 < d\<close>)
+ apply (force simp: dist_norm dle intro: less_le_trans)
+ done
+ qed
+ have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
+ \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
+ by (rule contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
+ then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
+ using Lim_null by (force intro!: tendsto_mult_right_zero)
+ have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
+ apply (rule Lim_transform_eventually [OF tendsto_0])
+ apply (force simp: divide_simps intro: eq_f' eventually_mono [OF cont])
+ done
+ then show ?thesis using Lim_null by blast
+ qed
+ obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
+ by (blast intro: tends_f'n_g' g')
+ then show ?thesis using g
+ using that by blast
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some more simple/convenient versions for applications\<close>
+
+lemma holomorphic_uniform_sequence:
+ assumes S: "open S"
+ and hol_fn: "\<And>n. (f n) holomorphic_on S"
+ and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+ shows "g holomorphic_on S"
+proof -
+ have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> S" for z
+ proof -
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ and ul: "uniform_limit (cball z r) f g sequentially"
+ using ulim_g [OF \<open>z \<in> S\<close>] by blast
+ have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
+ proof (intro eventuallyI conjI)
+ show "continuous_on (cball z r) (f x)" for x
+ using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
+ show "f x holomorphic_on ball z r" for x
+ by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
+ qed
+ show ?thesis
+ apply (rule holomorphic_uniform_limit [OF *])
+ using \<open>0 < r\<close> centre_in_ball ul
+ apply (auto simp: holomorphic_on_open)
+ done
+ qed
+ with S show ?thesis
+ by (simp add: holomorphic_on_open)
+qed
+
+lemma has_complex_derivative_uniform_sequence:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
+ and ulim_g: "\<And>x. x \<in> S
+ \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+ shows "\<exists>g'. \<forall>x \<in> S. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
+proof -
+ have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> S" for z
+ proof -
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ and ul: "uniform_limit (cball z r) f g sequentially"
+ using ulim_g [OF \<open>z \<in> S\<close>] by blast
+ have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
+ (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
+ proof (intro eventuallyI conjI ballI)
+ show "continuous_on (cball z r) (f x)" for x
+ by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
+ show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
+ using ball_subset_cball hfd r by blast
+ qed
+ show ?thesis
+ by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
+ qed
+ show ?thesis
+ by (rule bchoice) (blast intro: y)
+qed
+
+subsection\<open>On analytic functions defined by a series\<close>
+
+lemma series_and_derivative_comparison:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and h: "summable h"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. norm (f n x) \<le> h n"
+ obtains g g' where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+ obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ using Weierstrass_m_test_ev [OF to_g h] by force
+ have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ if "x \<in> S" for x
+ proof -
+ obtain d where "d>0" and d: "cball x d \<subseteq> S"
+ using open_contains_cball [of "S"] \<open>x \<in> S\<close> S by blast
+ show ?thesis
+ proof (intro conjI exI)
+ show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
+ qed (use \<open>d > 0\<close> d in auto)
+ qed
+ have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
+ by (metis tendsto_uniform_limitI [OF g])
+ moreover have "\<exists>g'. \<forall>x\<in>S. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
+ by (rule has_complex_derivative_uniform_sequence [OF S]) (auto intro: * hfd DERIV_sum)+
+ ultimately show ?thesis
+ by (metis sums_def that)
+qed
+
+text\<open>A version where we only have local uniform/comparative convergence.\<close>
+
+lemma series_and_derivative_comparison_local:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. norm (f n y) \<le> h n)"
+ shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+proof -
+ have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
+ if "z \<in> S" for z
+ proof -
+ obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. norm (f n y) \<le> h n"
+ using to_g \<open>z \<in> S\<close> by meson
+ then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> S" using \<open>z \<in> S\<close> S
+ by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
+ have 1: "open (ball z d \<inter> S)"
+ by (simp add: open_Int S)
+ have 2: "\<And>n x. x \<in> ball z d \<inter> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ by (auto simp: hfd)
+ obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> S. ((\<lambda>n. f n x) sums g x) \<and>
+ ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+ by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
+ then have "(\<lambda>n. f' n z) sums g' z"
+ by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
+ moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
+ using summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
+ by (metis (full_types) Int_iff gg' summable_def that)
+ moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
+ proof (rule has_field_derivative_transform_within)
+ show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
+ by (metis subsetD dist_commute gg' mem_ball r sums_unique)
+ qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
+ ultimately show ?thesis by auto
+ qed
+ then show ?thesis
+ by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
+qed
+
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+
+lemma series_and_derivative_comparison_complex:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+ shows "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+apply (rule series_and_derivative_comparison_local [OF S hfd], assumption)
+apply (rule ex_forward [OF to_g], assumption)
+apply (erule exE)
+apply (rule_tac x="Re \<circ> h" in exI)
+apply (force simp: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
+done
+
+text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>
+lemma series_differentiable_comparison_complex:
+ fixes S :: "complex set"
+ assumes S: "open S"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> f n field_differentiable (at x)"
+ and to_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod(f n y) \<le> cmod (h n))"
+ obtains g where "\<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> g field_differentiable (at x)"
+proof -
+ have hfd': "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative deriv (f n) x) (at x)"
+ using hfd field_differentiable_derivI by blast
+ have "\<exists>g g'. \<forall>x \<in> S. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. deriv (f n) x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
+ by (metis series_and_derivative_comparison_complex [OF S hfd' to_g])
+ then show ?thesis
+ using field_differentiable_def that by blast
+qed
+
+text\<open>In particular, a power series is analytic inside circle of convergence.\<close>
+
+lemma power_series_and_derivative_0:
+ fixes a :: "nat \<Rightarrow> complex" and r::real
+ assumes "summable (\<lambda>n. a n * r^n)"
+ shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
+ ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
+proof (cases "0 < r")
+ case True
+ have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
+ by (rule derivative_eq_intros | simp)+
+ have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
+ using \<open>r > 0\<close>
+ apply (auto simp: algebra_simps norm_mult norm_divide norm_power simp flip: of_real_add)
+ using norm_triangle_ineq2 [of y z]
+ apply (simp only: diff_le_eq norm_minus_commute mult_2)
+ done
+ have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
+ using assms \<open>r > 0\<close> by simp
+ moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
+ using \<open>r > 0\<close>
+ by (simp flip: of_real_add)
+ ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
+ by (rule power_series_conv_imp_absconv_weak)
+ have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n. (a n) * z ^ n) sums g z \<and>
+ (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
+ apply (rule series_and_derivative_comparison_complex [OF open_ball der])
+ apply (rule_tac x="(r - norm z)/2" in exI)
+ apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
+ using \<open>r > 0\<close>
+ apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
+ done
+ then show ?thesis
+ by (simp add: ball_def)
+next
+ case False then show ?thesis
+ apply (simp add: not_less)
+ using less_le_trans norm_not_less_zero by blast
+qed
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_and_derivative:
+ fixes a :: "nat \<Rightarrow> complex" and r::real
+ assumes "summable (\<lambda>n. a n * r^n)"
+ obtains g g' where "\<forall>z \<in> ball w r.
+ ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
+ (g has_field_derivative g' z) (at z)"
+ using power_series_and_derivative_0 [OF assms]
+ apply clarify
+ apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
+ using DERIV_shift [where z="-w"]
+ apply (auto simp: norm_minus_commute Ball_def dist_norm)
+ done
+
+proposition\<^marker>\<open>tag unimportant\<close> power_series_holomorphic:
+ assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
+ shows "f holomorphic_on ball z r"
+proof -
+ have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
+ proof -
+ have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
+ proof -
+ have wz: "cmod (w - z) < r" using w
+ by (auto simp: field_split_simps dist_norm norm_minus_commute)
+ then have "0 \<le> r"
+ by (meson less_eq_real_def norm_ge_zero order_trans)
+ show ?thesis
+ using w by (simp add: dist_norm \<open>0\<le>r\<close> flip: of_real_add)
+ qed
+ have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
+ using assms [OF inb] by (force simp: summable_def dist_norm)
+ obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
+ (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
+ (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
+ by (rule power_series_and_derivative [OF sum, of z]) fastforce
+ have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
+ proof -
+ have less: "cmod (z - u) * 2 < cmod (z - w) + r"
+ using that dist_triangle2 [of z u w]
+ by (simp add: dist_norm [symmetric] algebra_simps)
+ show ?thesis
+ apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
+ using gg' [of u] less w
+ apply (auto simp: assms dist_norm)
+ done
+ qed
+ have "(f has_field_derivative g' w) (at w)"
+ by (rule has_field_derivative_transform_within [where d="(r - norm(z - w))/2"])
+ (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
+ then show ?thesis ..
+ qed
+ then show ?thesis by (simp add: holomorphic_on_open)
+qed
+
+corollary holomorphic_iff_power_series:
+ "f holomorphic_on ball z r \<longleftrightarrow>
+ (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+ apply (intro iffI ballI holomorphic_power_series, assumption+)
+ apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
+ done
+
+lemma power_series_analytic:
+ "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
+ by (force simp: analytic_on_open intro!: power_series_holomorphic)
+
+lemma analytic_iff_power_series:
+ "f analytic_on ball z r \<longleftrightarrow>
+ (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
+ by (simp add: analytic_on_open holomorphic_iff_power_series)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Equality between holomorphic functions, on open ball then connected set\<close>
+
+lemma holomorphic_fun_eq_on_ball:
+ "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
+ w \<in> ball z r;
+ \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
+ \<Longrightarrow> f w = g w"
+ apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+ apply (auto simp: holomorphic_iff_power_series)
+ done
+
+lemma holomorphic_fun_eq_0_on_ball:
+ "\<lbrakk>f holomorphic_on ball z r; w \<in> ball z r;
+ \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
+ \<Longrightarrow> f w = 0"
+ apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
+ apply (auto simp: holomorphic_iff_power_series)
+ done
+
+lemma holomorphic_fun_eq_0_on_connected:
+ assumes holf: "f holomorphic_on S" and "open S"
+ and cons: "connected S"
+ and der: "\<And>n. (deriv ^^ n) f z = 0"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = 0"
+proof -
+ have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
+ proof -
+ have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
+ apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
+ apply (rule holomorphic_on_subset [OF holf])
+ using that apply simp_all
+ by (metis funpow_add o_apply)
+ with that show ?thesis by auto
+ qed
+ have 1: "openin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ apply (rule open_subset, force)
+ using \<open>open S\<close>
+ apply (simp add: open_contains_ball Ball_def)
+ apply (erule all_forward)
+ using "*" by auto blast+
+ have 2: "closedin (top_of_set S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+ using assms
+ by (auto intro: continuous_closedin_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
+ obtain e where "e>0" and e: "ball w e \<subseteq> S" using openE [OF \<open>open S\<close> \<open>w \<in> S\<close>] .
+ then have holfb: "f holomorphic_on ball w e"
+ using holf holomorphic_on_subset by blast
+ have 3: "(\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}) = S \<Longrightarrow> f w = 0"
+ using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
+ show ?thesis
+ using cons der \<open>z \<in> S\<close>
+ apply (simp add: connected_clopen)
+ apply (drule_tac x="\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0}" in spec)
+ apply (auto simp: 1 2 3)
+ done
+qed
+
+lemma holomorphic_fun_eq_on_connected:
+ assumes "f holomorphic_on S" "g holomorphic_on S" and "open S" "connected S"
+ and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = g w"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
+ show "(\<lambda>x. f x - g x) holomorphic_on S"
+ by (intro assms holomorphic_intros)
+ show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
+ using assms higher_deriv_diff by auto
+qed (use assms in auto)
+
+lemma holomorphic_fun_eq_const_on_connected:
+ assumes holf: "f holomorphic_on S" and "open S"
+ and cons: "connected S"
+ and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
+ and "z \<in> S" "w \<in> S"
+ shows "f w = f z"
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
+ show "(\<lambda>w. f w - f z) holomorphic_on S"
+ by (intro assms holomorphic_intros)
+ show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
+ by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
+qed (use assms in auto)
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Some basic lemmas about poles/singularities\<close>
+
+lemma pole_lemma:
+ assumes holf: "f holomorphic_on S" and a: "a \<in> interior S"
+ shows "(\<lambda>z. if z = a then deriv f a
+ else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
+proof -
+ have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
+ proof -
+ have fcd: "f field_differentiable at u within S"
+ using holf holomorphic_on_def by (simp add: \<open>u \<in> S\<close>)
+ have cd: "(\<lambda>z. (f z - f a) / (z - a)) field_differentiable at u within S"
+ by (rule fcd derivative_intros | simp add: that)+
+ have "0 < dist a u" using that dist_nz by blast
+ then show ?thesis
+ by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
+ qed
+ have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> S" for e
+ proof -
+ have holfb: "f holomorphic_on ball a e"
+ by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> S\<close>])
+ have 2: "?F holomorphic_on ball a e - {a}"
+ apply (simp add: holomorphic_on_def flip: field_differentiable_def)
+ using mem_ball that
+ apply (auto intro: F1 field_differentiable_within_subset)
+ done
+ have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
+ if "dist a x < e" for x
+ proof (cases "x=a")
+ case True
+ then have "f field_differentiable at a"
+ using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
+ with True show ?thesis
+ by (auto simp: continuous_at has_field_derivative_iff simp flip: DERIV_deriv_iff_field_differentiable
+ elim: rev_iffD1 [OF _ LIM_equal])
+ next
+ case False with 2 that show ?thesis
+ by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
+ qed
+ then have 1: "continuous_on (ball a e) ?F"
+ by (clarsimp simp: continuous_on_eq_continuous_at)
+ have "?F holomorphic_on ball a e"
+ by (auto intro: no_isolated_singularity [OF 1 2])
+ with that show ?thesis
+ by (simp add: holomorphic_on_open field_differentiable_def [symmetric]
+ field_differentiable_at_within)
+ qed
+ show ?thesis
+ proof
+ fix x assume "x \<in> S" show "?F field_differentiable at x within S"
+ proof (cases "x=a")
+ case True then show ?thesis
+ using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
+ next
+ case False with F1 \<open>x \<in> S\<close>
+ show ?thesis by blast
+ qed
+ qed
+qed
+
+lemma pole_theorem:
+ assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) holomorphic_on S"
+ using pole_lemma [OF holg a]
+ by (rule holomorphic_transform) (simp add: eq field_split_simps)
+
+lemma pole_lemma_open:
+ assumes "f holomorphic_on S" "open S"
+ shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on S"
+proof (cases "a \<in> S")
+ case True with assms interior_eq pole_lemma
+ show ?thesis by fastforce
+next
+ case False with assms show ?thesis
+ apply (simp add: holomorphic_on_def field_differentiable_def [symmetric], clarify)
+ apply (rule field_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
+ apply (rule derivative_intros | force)+
+ done
+qed
+
+lemma pole_theorem_open:
+ assumes holg: "g holomorphic_on S" and S: "open S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) holomorphic_on S"
+ using pole_lemma_open [OF holg S]
+ by (rule holomorphic_transform) (auto simp: eq divide_simps)
+
+lemma pole_theorem_0:
+ assumes holg: "g holomorphic_on S" and a: "a \<in> interior S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f holomorphic_on S"
+ using pole_theorem [OF holg a eq]
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_open_0:
+ assumes holg: "g holomorphic_on S" and S: "open S"
+ and eq: "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f holomorphic_on S"
+ using pole_theorem_open [OF holg S eq]
+ by (rule holomorphic_transform) (auto simp: eq field_split_simps)
+
+lemma pole_theorem_analytic:
+ assumes g: "g analytic_on S"
+ and eq: "\<And>z. z \<in> S
+ \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+ shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
+ unfolding analytic_on_def
+proof
+ fix x
+ assume "x \<in> S"
+ with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
+ by (auto simp add: analytic_on_def)
+ obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
+ using \<open>x \<in> S\<close> eq by blast
+ have "?F holomorphic_on ball x (min d e)"
+ using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
+ then show "\<exists>e>0. ?F holomorphic_on ball x e"
+ using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
+qed
+
+lemma pole_theorem_analytic_0:
+ assumes g: "g analytic_on S"
+ and eq: "\<And>z. z \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f analytic_on S"
+proof -
+ have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+ by auto
+ show ?thesis
+ using pole_theorem_analytic [OF g eq] by simp
+qed
+
+lemma pole_theorem_analytic_open_superset:
+ assumes g: "g analytic_on S" and "S \<subseteq> T" "open T"
+ and eq: "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+ shows "(\<lambda>z. if z = a then deriv g a
+ else f z - g a/(z - a)) analytic_on S"
+proof (rule pole_theorem_analytic [OF g])
+ fix z
+ assume "z \<in> S"
+ then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
+ using assms openE by blast
+ then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
+ using eq by auto
+qed
+
+lemma pole_theorem_analytic_open_superset_0:
+ assumes g: "g analytic_on S" "S \<subseteq> T" "open T" "\<And>z. z \<in> T - {a} \<Longrightarrow> g z = (z - a) * f z"
+ and [simp]: "f a = deriv g a" "g a = 0"
+ shows "f analytic_on S"
+proof -
+ have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
+ by auto
+ have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S"
+ by (rule pole_theorem_analytic_open_superset [OF g])
+ then show ?thesis by simp
+qed
+
+
+subsection\<open>General, homology form of Cauchy's theorem\<close>
+
+text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>
+
+lemma contour_integral_continuous_on_linepath_2D:
+ assumes "open U" and cont_dw: "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on (linepath a b)"
+ and cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). F x y)"
+ and abu: "closed_segment a b \<subseteq> U"
+ shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
+proof -
+ have *: "\<exists>d>0. \<forall>x'\<in>U. dist x' w < d \<longrightarrow>
+ dist (contour_integral (linepath a b) (F x'))
+ (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
+ if "w \<in> U" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
+ proof -
+ obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> U" using open_contains_cball \<open>open U\<close> \<open>w \<in> U\<close> by force
+ let ?TZ = "cball w \<delta> \<times> closed_segment a b"
+ have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
+ proof (rule compact_uniformly_continuous)
+ show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
+ by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
+ show "compact ?TZ"
+ by (simp add: compact_Times)
+ qed
+ then obtain \<eta> where "\<eta>>0"
+ and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
+ dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
+ apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
+ using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
+ have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>; x2 \<in> closed_segment a b;
+ norm (w - x1') \<le> \<delta>; x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
+ \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
+ for x1 x2 x1' x2'
+ using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: dist_norm)
+ have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
+ if "x' \<in> U" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>" for x'
+ proof -
+ have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
+ by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
+ then have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
+ apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
+ using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
+ done
+ also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
+ finally show ?thesis .
+ qed
+ show ?thesis
+ apply (rule_tac x="min \<delta> \<eta>" in exI)
+ using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
+ apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> U\<close> intro: le_ee)
+ done
+ qed
+ show ?thesis
+ proof (cases "a=b")
+ case True
+ then show ?thesis by simp
+ next
+ case False
+ show ?thesis
+ by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
+ qed
+qed
+
+text\<open>This version has \<^term>\<open>polynomial_function \<gamma>\<close> as an additional assumption.\<close>
+lemma Cauchy_integral_formula_global_weak:
+ assumes "open U" and holf: "f holomorphic_on U"
+ and z: "z \<in> U" and \<gamma>: "polynomial_function \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> U - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and zero: "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
+ using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
+ then have "bounded(path_image \<gamma>')"
+ by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
+ then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
+ using bounded_pos by force
+ define d where [abs_def]: "d z w = (if w = z then deriv f z else (f w - f z)/(w - z))" for z w
+ define v where "v = {w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
+ have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
+ by (auto simp: path_polynomial_function valid_path_polynomial_function)
+ then have ov: "open v"
+ by (simp add: v_def open_winding_number_levelsets loop)
+ have uv_Un: "U \<union> v = UNIV"
+ using pasz zero by (auto simp: v_def)
+ have conf: "continuous_on U f"
+ by (metis holf holomorphic_on_imp_continuous_on)
+ have hol_d: "(d y) holomorphic_on U" if "y \<in> U" for y
+ proof -
+ have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on U"
+ by (simp add: holf pole_lemma_open \<open>open U\<close>)
+ then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
+ using at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
+ then have "continuous_on U (d y)"
+ apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
+ using * holomorphic_on_def
+ by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
+ moreover have "d y holomorphic_on U - {y}"
+ proof -
+ have "\<And>w. w \<in> U - {y} \<Longrightarrow>
+ (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
+ apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
+ apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
+ using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
+ then show ?thesis
+ unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
+ qed
+ ultimately show ?thesis
+ by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
+ qed
+ have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
+ proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
+ show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
+ by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+ show "path_image \<gamma> \<subseteq> U - {y}"
+ using pasz that by blast
+ qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
+ define h where
+ "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
+ have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
+ proof -
+ have "d z holomorphic_on U"
+ by (simp add: hol_d that)
+ with that show ?thesis
+ apply (simp add: h_def)
+ by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
+ qed
+ have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
+ proof -
+ have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
+ using v_def z by auto
+ then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
+ using z v_def has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
+ then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
+ using has_contour_integral_lmul by fastforce
+ then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
+ by (simp add: field_split_simps)
+ moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+ using z
+ apply (auto simp: v_def)
+ apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
+ done
+ ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
+ by (rule has_contour_integral_add)
+ have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
+ if "z \<in> U"
+ using * by (auto simp: divide_simps has_contour_integral_eq)
+ moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
+ if "z \<notin> U"
+ apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where S=U]])
+ using U pasz \<open>valid_path \<gamma>\<close> that
+ apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
+ apply (rule continuous_intros conf holomorphic_intros holf assms | force)+
+ done
+ ultimately show ?thesis
+ using z by (simp add: h_def)
+ qed
+ have znot: "z \<notin> path_image \<gamma>"
+ using pasz by blast
+ obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - U \<Longrightarrow> d0 \<le> dist x y"
+ using separate_compact_closed [of "path_image \<gamma>" "-U"] pasz \<open>open U\<close> \<open>path \<gamma>\<close> compact_path_image
+ by blast
+ obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
+ apply (rule that [of "d0/2"])
+ using \<open>0 < d0\<close>
+ apply (auto simp: dist_norm dest: d0)
+ done
+ have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
+ apply (rule_tac x=x in exI)
+ apply (rule_tac x="x'-x" in exI)
+ apply (force simp: dist_norm)
+ done
+ then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
+ apply (clarsimp simp add: mem_interior)
+ using \<open>0 < dd\<close>
+ apply (rule_tac x="dd/2" in exI, auto)
+ done
+ obtain T where "compact T" and subt: "path_image \<gamma> \<subseteq> interior T" and T: "T \<subseteq> U"
+ apply (rule that [OF _ 1])
+ apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
+ apply (rule order_trans [OF _ dd])
+ using \<open>0 < dd\<close> by fastforce
+ obtain L where "L>0"
+ and L: "\<And>f B. \<lbrakk>f holomorphic_on interior T; \<And>z. z\<in>interior T \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
+ cmod (contour_integral \<gamma> f) \<le> L * B"
+ using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
+ by blast
+ have "bounded(f ` T)"
+ by (meson \<open>compact T\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset T)
+ then obtain D where "D>0" and D: "\<And>x. x \<in> T \<Longrightarrow> norm (f x) \<le> D"
+ by (auto simp: bounded_pos)
+ obtain C where "C>0" and C: "\<And>x. x \<in> T \<Longrightarrow> norm x \<le> C"
+ using \<open>compact T\<close> bounded_pos compact_imp_bounded by force
+ have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
+ proof -
+ have "D * L / e > 0" using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
+ with le have ybig: "norm y > C" by force
+ with C have "y \<notin> T" by force
+ then have ynot: "y \<notin> path_image \<gamma>"
+ using subt interior_subset by blast
+ have [simp]: "winding_number \<gamma> y = 0"
+ apply (rule winding_number_zero_outside [of _ "cball 0 C"])
+ using ybig interior_subset subt
+ apply (force simp: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
+ done
+ have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
+ by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
+ have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior T"
+ apply (rule holomorphic_on_divide)
+ using holf holomorphic_on_subset interior_subset T apply blast
+ apply (rule holomorphic_intros)+
+ using \<open>y \<notin> T\<close> interior_subset by auto
+ have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior T" for z
+ proof -
+ have "D * L / e + cmod z \<le> cmod y"
+ using le C [of z] z using interior_subset by force
+ then have DL2: "D * L / e \<le> cmod (z - y)"
+ using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
+ have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
+ by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
+ also have "\<dots> \<le> D * (e / L / D)"
+ apply (rule mult_mono)
+ using that D interior_subset apply blast
+ using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
+ apply (auto simp: norm_divide field_split_simps)
+ done
+ finally show ?thesis .
+ qed
+ have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
+ by (simp add: dist_norm)
+ also have "\<dots> \<le> L * (D * (e / L / D))"
+ by (rule L [OF holint leD])
+ also have "\<dots> = e"
+ using \<open>L>0\<close> \<open>0 < D\<close> by auto
+ finally show ?thesis .
+ qed
+ then have "(h \<longlongrightarrow> 0) at_infinity"
+ by (meson Lim_at_infinityI)
+ moreover have "h holomorphic_on UNIV"
+ proof -
+ have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
+ if "x \<in> U" "z \<in> U" "x \<noteq> z" for x z
+ using that conf
+ apply (simp add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
+ apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
+ done
+ have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
+ by (rule continuous_intros)+
+ have open_uu_Id: "open (U \<times> U - Id)"
+ apply (rule open_Diff)
+ apply (simp add: open_Times \<open>open U\<close>)
+ using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
+ apply (auto simp: Id_fstsnd_eq algebra_simps)
+ done
+ have con_derf: "continuous (at z) (deriv f)" if "z \<in> U" for z
+ apply (rule continuous_on_interior [of U])
+ apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open U\<close>)
+ by (simp add: interior_open that \<open>open U\<close>)
+ have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
+ else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
+ (at (x, x) within U \<times> U)" if "x \<in> U" for x
+ proof (rule Lim_withinI)
+ fix e::real assume "0 < e"
+ obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
+ using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> U\<close>]]
+ by (metis UNIV_I dist_norm)
+ obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
+ by (blast intro: openE [OF \<open>open U\<close>] \<open>x \<in> U\<close>)
+ have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
+ if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
+ for x' z'
+ proof -
+ have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
+ apply (drule segment_furthest_le [where y=x])
+ by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
+ have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
+ by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
+ have f_has_der: "\<And>x. x \<in> U \<Longrightarrow> (f has_field_derivative deriv f x) (at x within U)"
+ by (metis DERIV_deriv_iff_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
+ have "closed_segment x' z' \<subseteq> U"
+ by (rule order_trans [OF _ k2]) (simp add: cs_less le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
+ then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
+ using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz by simp
+ then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
+ by (rule has_contour_integral_div)
+ have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
+ apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
+ using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
+ \<open>e > 0\<close> \<open>z' \<noteq> x'\<close>
+ apply (auto simp: norm_divide divide_simps derf_le)
+ done
+ also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
+ finally show ?thesis .
+ qed
+ show "\<exists>d>0. \<forall>xa\<in>U \<times> U.
+ 0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
+ dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
+ apply (rule_tac x="min k1 k2" in exI)
+ using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
+ apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
+ done
+ qed
+ have con_pa_f: "continuous_on (path_image \<gamma>) f"
+ by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt T)
+ have le_B: "\<And>T. T \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at T)) \<le> B"
+ apply (rule B)
+ using \<gamma>' using path_image_def vector_derivative_at by fastforce
+ have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
+ by (simp add: V)
+ have cond_uu: "continuous_on (U \<times> U) (\<lambda>(x,y). d x y)"
+ apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
+ apply (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, clarify)
+ apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
+ using con_ff
+ apply (auto simp: continuous_within)
+ done
+ have hol_dw: "(\<lambda>z. d z w) holomorphic_on U" if "w \<in> U" for w
+ proof -
+ have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<lambda>z. (w,z)))"
+ by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
+ then have *: "continuous_on U (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
+ by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
+ have **: "\<And>x. \<lbrakk>x \<in> U; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) field_differentiable at x"
+ apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
+ apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
+ done
+ show ?thesis
+ unfolding d_def
+ apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
+ apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
+ done
+ qed
+ { fix a b
+ assume abu: "closed_segment a b \<subseteq> U"
+ then have "\<And>w. w \<in> U \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
+ by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
+ then have cont_cint_d: "continuous_on U (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open U\<close> _ _ abu])
+ apply (auto intro: continuous_on_swap_args cond_uu)
+ done
+ have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) \<circ> \<gamma>)"
+ proof (rule continuous_on_compose)
+ show "continuous_on {0..1} \<gamma>"
+ using \<open>path \<gamma>\<close> path_def by blast
+ show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ using pasz unfolding path_image_def
+ by (auto intro!: continuous_on_subset [OF cont_cint_d])
+ qed
+ have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
+ apply (simp add: contour_integrable_on)
+ apply (rule integrable_continuous_real)
+ apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
+ using pf\<gamma>'
+ by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
+ have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
+ using abu by (force simp: h_def intro: contour_integral_eq)
+ also have "\<dots> = contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+ apply (rule contour_integral_swap)
+ apply (rule continuous_on_subset [OF cond_uu])
+ using abu pasz \<open>valid_path \<gamma>\<close>
+ apply (auto intro!: continuous_intros)
+ by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
+ finally have cint_h_eq:
+ "contour_integral (linepath a b) h =
+ contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
+ note cint_cint cint_h_eq
+ } note cint_h = this
+ have conthu: "continuous_on U h"
+ proof (simp add: continuous_on_sequentially, clarify)
+ fix a x
+ assume x: "x \<in> U" and au: "\<forall>n. a n \<in> U" and ax: "a \<longlonglongrightarrow> x"
+ then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
+ by (meson U contour_integrable_on_def eventuallyI)
+ obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
+ have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix ee::real
+ assume "0 < ee"
+ show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < ee"
+ proof -
+ let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
+ have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
+ apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
+ using dd pasz \<open>valid_path \<gamma>\<close>
+ apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
+ done
+ then obtain kk where "kk>0"
+ and kk: "\<And>x x'. \<lbrakk>x \<in> ?ddpa; x' \<in> ?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
+ dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
+ by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in auto)
+ have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
+ for w z
+ using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp: norm_minus_commute dist_norm)
+ show ?thesis
+ using ax unfolding lim_sequentially eventually_sequentially
+ apply (drule_tac x="min dd kk" in spec)
+ using \<open>dd > 0\<close> \<open>kk > 0\<close>
+ apply (fastforce simp: kk dist_norm)
+ done
+ qed
+ qed
+ have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
+ by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
+ then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
+ by (simp add: h_def x)
+ then show "(h \<circ> a) \<longlonglongrightarrow> h x"
+ by (simp add: h_def x au o_def)
+ qed
+ show ?thesis
+ proof (simp add: holomorphic_on_open field_differentiable_def [symmetric], clarify)
+ fix z0
+ consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
+ then show "h field_differentiable at z0"
+ proof cases
+ assume "z0 \<in> v" then show ?thesis
+ using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
+ by (auto simp: field_differentiable_def v_def)
+ next
+ assume "z0 \<in> U" then
+ obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
+ have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
+ if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e" for a b c
+ proof -
+ have *: "\<And>x1 x2 z. z \<in> U \<Longrightarrow> closed_segment x1 x2 \<subseteq> U \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
+ using hol_dw holomorphic_on_imp_continuous_on \<open>open U\<close>
+ by (auto intro!: contour_integrable_holomorphic_simple)
+ have abc: "closed_segment a b \<subseteq> U" "closed_segment b c \<subseteq> U" "closed_segment c a \<subseteq> U"
+ using that e segments_subset_convex_hull by fastforce+
+ have eq0: "\<And>w. w \<in> U \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
+ apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
+ apply (rule holomorphic_on_subset [OF hol_dw])
+ using e abc_subset by auto
+ have "contour_integral \<gamma>
+ (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
+ (contour_integral (linepath b c) (\<lambda>z. d z x) +
+ contour_integral (linepath c a) (\<lambda>z. d z x))) = 0"
+ apply (rule contour_integral_eq_0)
+ using abc pasz U
+ apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
+ done
+ then show ?thesis
+ by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
+ qed
+ show ?thesis
+ using e \<open>e > 0\<close>
+ by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
+ Morera_triangle continuous_on_subset [OF conthu] *)
+ qed
+ qed
+ qed
+ ultimately have [simp]: "h z = 0" for z
+ by (meson Liouville_weak)
+ have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
+ by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
+ then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+ by (metis mult.commute has_contour_integral_lmul)
+ then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
+ by (simp add: field_split_simps)
+ moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
+ using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
+ show ?thesis
+ using has_contour_integral_add [OF 1 2] by (simp add: diff_divide_distrib)
+qed
+
+theorem Cauchy_integral_formula_global:
+ assumes S: "open S" and holf: "f holomorphic_on S"
+ and z: "z \<in> S" and vpg: "valid_path \<gamma>"
+ and pasz: "path_image \<gamma> \<subseteq> S - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
+proof -
+ have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
+ have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on S - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on S - {z}"
+ by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
+ then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
+ by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete S vpg pasz)
+ obtain d where "d>0"
+ and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
+ pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
+ \<Longrightarrow> path_image h \<subseteq> S - {z} \<and> (\<forall>f. f holomorphic_on S - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
+ using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] S by (simp add: open_Diff) metis
+ obtain p where polyp: "polynomial_function p"
+ and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
+ using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
+ then have ploop: "pathfinish p = pathstart p" using loop by auto
+ have vpp: "valid_path p" using polyp valid_path_polynomial_function by blast
+ have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
+ have paps: "path_image p \<subseteq> S - {z}" and cint_eq: "(\<And>f. f holomorphic_on S - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
+ using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
+ have wn_eq: "winding_number p z = winding_number \<gamma> z"
+ using vpp paps
+ by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
+ have "winding_number p w = winding_number \<gamma> w" if "w \<notin> S" for w
+ proof -
+ have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on S - {z}"
+ using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+ have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
+ then show ?thesis
+ using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
+ qed
+ then have wn0: "\<And>w. w \<notin> S \<Longrightarrow> winding_number p w = 0"
+ by (simp add: zero)
+ show ?thesis
+ using Cauchy_integral_formula_global_weak [OF S holf z polyp paps ploop wn0] hols
+ by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
+qed
+
+theorem Cauchy_theorem_global:
+ assumes S: "open S" and holf: "f holomorphic_on S"
+ and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
+ and pas: "path_image \<gamma> \<subseteq> S"
+ and zero: "\<And>w. w \<notin> S \<Longrightarrow> winding_number \<gamma> w = 0"
+ shows "(f has_contour_integral 0) \<gamma>"
+proof -
+ obtain z where "z \<in> S" and znot: "z \<notin> path_image \<gamma>"
+ proof -
+ have "compact (path_image \<gamma>)"
+ using compact_valid_path_image vpg by blast
+ then have "path_image \<gamma> \<noteq> S"
+ by (metis (no_types) compact_open path_image_nonempty S)
+ with pas show ?thesis by (blast intro: that)
+ qed
+ then have pasz: "path_image \<gamma> \<subseteq> S - {z}" using pas by blast
+ have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on S"
+ by (rule holomorphic_intros holf)+
+ show ?thesis
+ using Cauchy_integral_formula_global [OF S hol \<open>z \<in> S\<close> vpg pasz loop zero]
+ by (auto simp: znot elim!: has_contour_integral_eq)
+qed
+
+corollary Cauchy_theorem_global_outside:
+ assumes "open S" "f holomorphic_on S" "valid_path \<gamma>" "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> S"
+ "\<And>w. w \<notin> S \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
+ shows "(f has_contour_integral 0) \<gamma>"
+by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)
+
+lemma simply_connected_imp_winding_number_zero:
+ assumes "simply_connected S" "path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
+ shows "winding_number g z = 0"
+proof -
+ have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
+ by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
+ then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
+ by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
+ then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
+ by (rule winding_number_homotopic_paths)
+ also have "\<dots> = 0"
+ using assms by (force intro: winding_number_trivial)
+ finally show ?thesis .
+qed
+
+lemma Cauchy_theorem_simply_connected:
+ assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g"
+ shows "(f has_contour_integral 0) g"
+using assms
+apply (simp add: simply_connected_eq_contractible_path)
+apply (auto intro!: Cauchy_theorem_null_homotopic [where a = "pathstart g"]
+ homotopic_paths_imp_homotopic_loops)
+using valid_path_imp_path by blast
+
+proposition\<^marker>\<open>tag unimportant\<close> holomorphic_logarithm_exists:
+ assumes A: "convex A" "open A"
+ and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
+ and z0: "z0 \<in> A"
+ obtains g where "g holomorphic_on A" and "\<And>x. x \<in> A \<Longrightarrow> exp (g x) = f x"
+proof -
+ note f' = holomorphic_derivI [OF f(1) A(2)]
+ obtain g where g: "\<And>x. x \<in> A \<Longrightarrow> (g has_field_derivative deriv f x / f x) (at x)"
+ proof (rule holomorphic_convex_primitive' [OF A])
+ show "(\<lambda>x. deriv f x / f x) holomorphic_on A"
+ by (intro holomorphic_intros f A)
+ qed (auto simp: A at_within_open[of _ A])
+ define h where "h = (\<lambda>x. -g z0 + ln (f z0) + g x)"
+ from g and A have g_holo: "g holomorphic_on A"
+ by (auto simp: holomorphic_on_def at_within_open[of _ A] field_differentiable_def)
+ hence h_holo: "h holomorphic_on A"
+ by (auto simp: h_def intro!: holomorphic_intros)
+ have "\<exists>c. \<forall>x\<in>A. f x / exp (h x) - 1 = c"
+ proof (rule has_field_derivative_zero_constant, goal_cases)
+ case (2 x)
+ note [simp] = at_within_open[OF _ \<open>open A\<close>]
+ from 2 and z0 and f show ?case
+ by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
+ qed fact+
+ then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
+ by blast
+ from c[OF z0] and z0 and f have "c = 0"
+ by (simp add: h_def)
+ with c have "\<And>x. x \<in> A \<Longrightarrow> exp (h x) = f x" by simp
+ from that[OF h_holo this] show ?thesis .
+qed
+
+subsection \<open>Complex functions and power series\<close>
+
+text \<open>
+ The following defines the power series expansion of a complex function at a given point
+ (assuming that it is analytic at that point).
+\<close>
+definition\<^marker>\<open>tag important\<close> fps_expansion :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex fps" where
+ "fps_expansion f z0 = Abs_fps (\<lambda>n. (deriv ^^ n) f z0 / fact n)"
+
+lemma
+ fixes r :: ereal
+ assumes "f holomorphic_on eball z0 r"
+ shows conv_radius_fps_expansion: "fps_conv_radius (fps_expansion f z0) \<ge> r"
+ and eval_fps_expansion: "\<And>z. z \<in> eball z0 r \<Longrightarrow> eval_fps (fps_expansion f z0) (z - z0) = f z"
+ and eval_fps_expansion': "\<And>z. norm z < r \<Longrightarrow> eval_fps (fps_expansion f z0) z = f (z0 + z)"
+proof -
+ have "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+ if "z \<in> ball z0 r'" "ereal r' < r" for z r'
+ proof -
+ from that(2) have "ereal r' \<le> r" by simp
+ from assms(1) and this have "f holomorphic_on ball z0 r'"
+ by (rule holomorphic_on_subset[OF _ ball_eball_mono])
+ from holomorphic_power_series [OF this that(1)]
+ show ?thesis by (simp add: fps_expansion_def)
+ qed
+ hence *: "(\<lambda>n. fps_nth (fps_expansion f z0) n * (z - z0) ^ n) sums f z"
+ if "z \<in> eball z0 r" for z
+ using that by (subst (asm) eball_conv_UNION_balls) blast
+ show "fps_conv_radius (fps_expansion f z0) \<ge> r" unfolding fps_conv_radius_def
+ proof (rule conv_radius_geI_ex)
+ fix r' :: real assume r': "r' > 0" "ereal r' < r"
+ thus "\<exists>z. norm z = r' \<and> summable (\<lambda>n. fps_nth (fps_expansion f z0) n * z ^ n)"
+ using *[of "z0 + of_real r'"]
+ by (intro exI[of _ "of_real r'"]) (auto simp: summable_def dist_norm)
+ qed
+ show "eval_fps (fps_expansion f z0) (z - z0) = f z" if "z \<in> eball z0 r" for z
+ using *[OF that] by (simp add: eval_fps_def sums_iff)
+ show "eval_fps (fps_expansion f z0) z = f (z0 + z)" if "ereal (norm z) < r" for z
+ using *[of "z0 + z"] and that by (simp add: eval_fps_def sums_iff dist_norm)
+qed
+
+
+text \<open>
+ We can now show several more facts about power series expansions (at least in the complex case)
+ with relative ease that would have been trickier without complex analysis.
+\<close>
+lemma
+ fixes f :: "complex fps" and r :: ereal
+ assumes "\<And>z. ereal (norm z) < r \<Longrightarrow> eval_fps f z \<noteq> 0"
+ shows fps_conv_radius_inverse: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)"
+ and eval_fps_inverse: "\<And>z. ereal (norm z) < fps_conv_radius f \<Longrightarrow> ereal (norm z) < r \<Longrightarrow>
+ eval_fps (inverse f) z = inverse (eval_fps f z)"
+proof -
+ define R where "R = min (fps_conv_radius f) r"
+ have *: "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f) \<and>
+ (\<forall>z\<in>eball 0 (min (fps_conv_radius f) r). eval_fps (inverse f) z = inverse (eval_fps f z))"
+ proof (cases "min r (fps_conv_radius f) > 0")
+ case True
+ define f' where "f' = fps_expansion (\<lambda>z. inverse (eval_fps f z)) 0"
+ have holo: "(\<lambda>z. inverse (eval_fps f z)) holomorphic_on eball 0 (min r (fps_conv_radius f))"
+ using assms by (intro holomorphic_intros) auto
+ from holo have radius: "fps_conv_radius f' \<ge> min r (fps_conv_radius f)"
+ unfolding f'_def by (rule conv_radius_fps_expansion)
+ have eval_f': "eval_fps f' z = inverse (eval_fps f z)"
+ if "norm z < fps_conv_radius f" "norm z < r" for z
+ using that unfolding f'_def by (subst eval_fps_expansion'[OF holo]) auto
+
+ have "f * f' = 1"
+ proof (rule eval_fps_eqD)
+ from radius and True have "0 < min (fps_conv_radius f) (fps_conv_radius f')"
+ by (auto simp: min_def split: if_splits)
+ also have "\<dots> \<le> fps_conv_radius (f * f')" by (rule fps_conv_radius_mult)
+ finally show "\<dots> > 0" .
+ next
+ from True have "R > 0" by (auto simp: R_def)
+ hence "eventually (\<lambda>z. z \<in> eball 0 R) (nhds 0)"
+ by (intro eventually_nhds_in_open) (auto simp: zero_ereal_def)
+ thus "eventually (\<lambda>z. eval_fps (f * f') z = eval_fps 1 z) (nhds 0)"
+ proof eventually_elim
+ case (elim z)
+ hence "eval_fps (f * f') z = eval_fps f z * eval_fps f' z"
+ using radius by (intro eval_fps_mult)
+ (auto simp: R_def min_def split: if_splits intro: less_trans)
+ also have "eval_fps f' z = inverse (eval_fps f z)"
+ using elim by (intro eval_f') (auto simp: R_def)
+ also from elim have "eval_fps f z \<noteq> 0"
+ by (intro assms) (auto simp: R_def)
+ hence "eval_fps f z * inverse (eval_fps f z) = eval_fps 1 z"
+ by simp
+ finally show "eval_fps (f * f') z = eval_fps 1 z" .
+ qed
+ qed simp_all
+ hence "f' = inverse f"
+ by (intro fps_inverse_unique [symmetric]) (simp_all add: mult_ac)
+ with eval_f' and radius show ?thesis by simp
+ next
+ case False
+ hence *: "eball 0 R = {}"
+ by (intro eball_empty) (auto simp: R_def min_def split: if_splits)
+ show ?thesis
+ proof safe
+ from False have "min r (fps_conv_radius f) \<le> 0"
+ by (simp add: min_def)
+ also have "0 \<le> fps_conv_radius (inverse f)"
+ by (simp add: fps_conv_radius_def conv_radius_nonneg)
+ finally show "min r (fps_conv_radius f) \<le> \<dots>" .
+ qed (unfold * [unfolded R_def], auto)
+ qed
+
+ from * show "fps_conv_radius (inverse f) \<ge> min r (fps_conv_radius f)" by blast
+ from * show "eval_fps (inverse f) z = inverse (eval_fps f z)"
+ if "ereal (norm z) < fps_conv_radius f" "ereal (norm z) < r" for z
+ using that by auto
+qed
+
+lemma
+ fixes f g :: "complex fps" and r :: ereal
+ defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+ assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+ assumes nz: "\<And>z. z \<in> eball 0 r \<Longrightarrow> eval_fps g z \<noteq> 0"
+ shows fps_conv_radius_divide': "fps_conv_radius (f / g) \<ge> R"
+ and eval_fps_divide':
+ "ereal (norm z) < R \<Longrightarrow> eval_fps (f / g) z = eval_fps f z / eval_fps g z"
+proof -
+ from nz[of 0] and \<open>r > 0\<close> have nz': "fps_nth g 0 \<noteq> 0"
+ by (auto simp: eval_fps_at_0 zero_ereal_def)
+ have "R \<le> min r (fps_conv_radius g)"
+ by (auto simp: R_def intro: min.coboundedI2)
+ also have "min r (fps_conv_radius g) \<le> fps_conv_radius (inverse g)"
+ by (intro fps_conv_radius_inverse assms) (auto simp: zero_ereal_def)
+ finally have radius: "fps_conv_radius (inverse g) \<ge> R" .
+ have "R \<le> min (fps_conv_radius f) (fps_conv_radius (inverse g))"
+ by (intro radius min.boundedI) (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "\<dots> \<le> fps_conv_radius (f * inverse g)"
+ by (rule fps_conv_radius_mult)
+ also have "f * inverse g = f / g"
+ by (intro fps_divide_unit [symmetric] nz')
+ finally show "fps_conv_radius (f / g) \<ge> R" .
+
+ assume z: "ereal (norm z) < R"
+ have "eval_fps (f * inverse g) z = eval_fps f z * eval_fps (inverse g) z"
+ using radius by (intro eval_fps_mult less_le_trans[OF z])
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "eval_fps (inverse g) z = inverse (eval_fps g z)" using \<open>r > 0\<close>
+ by (intro eval_fps_inverse[where r = r] less_le_trans[OF z] nz)
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ also have "f * inverse g = f / g" by fact
+ finally show "eval_fps (f / g) z = eval_fps f z / eval_fps g z" by (simp add: field_split_simps)
+qed
+
+lemma
+ fixes f g :: "complex fps" and r :: ereal
+ defines "R \<equiv> Min {r, fps_conv_radius f, fps_conv_radius g}"
+ assumes "subdegree g \<le> subdegree f"
+ assumes "fps_conv_radius f > 0" "fps_conv_radius g > 0" "r > 0"
+ assumes "\<And>z. z \<in> eball 0 r \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> eval_fps g z \<noteq> 0"
+ shows fps_conv_radius_divide: "fps_conv_radius (f / g) \<ge> R"
+ and eval_fps_divide:
+ "ereal (norm z) < R \<Longrightarrow> c = fps_nth f (subdegree g) / fps_nth g (subdegree g) \<Longrightarrow>
+ eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+proof -
+ define f' g' where "f' = fps_shift (subdegree g) f" and "g' = fps_shift (subdegree g) g"
+ have f_eq: "f = f' * fps_X ^ subdegree g" and g_eq: "g = g' * fps_X ^ subdegree g"
+ unfolding f'_def g'_def by (rule subdegree_decompose' le_refl | fact)+
+ have subdegree: "subdegree f' = subdegree f - subdegree g" "subdegree g' = 0"
+ using assms(2) by (simp_all add: f'_def g'_def)
+ have [simp]: "fps_conv_radius f' = fps_conv_radius f" "fps_conv_radius g' = fps_conv_radius g"
+ by (simp_all add: f'_def g'_def)
+ have [simp]: "fps_nth f' 0 = fps_nth f (subdegree g)"
+ "fps_nth g' 0 = fps_nth g (subdegree g)" by (simp_all add: f'_def g'_def)
+ have g_nz: "g \<noteq> 0"
+ proof -
+ define z :: complex where "z = (if r = \<infinity> then 1 else of_real (real_of_ereal r / 2))"
+ from \<open>r > 0\<close> have "z \<in> eball 0 r"
+ by (cases r) (auto simp: z_def eball_def)
+ moreover have "z \<noteq> 0" using \<open>r > 0\<close>
+ by (cases r) (auto simp: z_def)
+ ultimately have "eval_fps g z \<noteq> 0" by (rule assms(6))
+ thus "g \<noteq> 0" by auto
+ qed
+ have fg: "f / g = f' * inverse g'"
+ by (subst f_eq, subst (2) g_eq) (insert g_nz, simp add: fps_divide_unit)
+
+ have g'_nz: "eval_fps g' z \<noteq> 0" if z: "norm z < min r (fps_conv_radius g)" for z
+ proof (cases "z = 0")
+ case False
+ with assms and z have "eval_fps g z \<noteq> 0" by auto
+ also from z have "eval_fps g z = eval_fps g' z * z ^ subdegree g"
+ by (subst g_eq) (auto simp: eval_fps_mult)
+ finally show ?thesis by auto
+ qed (insert \<open>g \<noteq> 0\<close>, auto simp: g'_def eval_fps_at_0)
+
+ have "R \<le> min (min r (fps_conv_radius g)) (fps_conv_radius g')"
+ by (auto simp: R_def min.coboundedI1 min.coboundedI2)
+ also have "\<dots> \<le> fps_conv_radius (inverse g')"
+ using g'_nz by (rule fps_conv_radius_inverse)
+ finally have conv_radius_inv: "R \<le> fps_conv_radius (inverse g')" .
+ hence "R \<le> fps_conv_radius (f' * inverse g')"
+ by (intro order.trans[OF _ fps_conv_radius_mult])
+ (auto simp: R_def intro: min.coboundedI1 min.coboundedI2)
+ thus "fps_conv_radius (f / g) \<ge> R" by (simp add: fg)
+
+ fix z c :: complex assume z: "ereal (norm z) < R"
+ assume c: "c = fps_nth f (subdegree g) / fps_nth g (subdegree g)"
+ show "eval_fps (f / g) z = (if z = 0 then c else eval_fps f z / eval_fps g z)"
+ proof (cases "z = 0")
+ case False
+ from z and conv_radius_inv have "ereal (norm z) < fps_conv_radius (inverse g')"
+ by simp
+ with z have "eval_fps (f / g) z = eval_fps f' z * eval_fps (inverse g') z"
+ unfolding fg by (subst eval_fps_mult) (auto simp: R_def)
+ also have "eval_fps (inverse g') z = inverse (eval_fps g' z)"
+ using z by (intro eval_fps_inverse[of "min r (fps_conv_radius g')"] g'_nz) (auto simp: R_def)
+ also have "eval_fps f' z * \<dots> = eval_fps f z / eval_fps g z"
+ using z False assms(2) by (simp add: f'_def g'_def eval_fps_shift R_def)
+ finally show ?thesis using False by simp
+ qed (simp_all add: eval_fps_at_0 fg field_simps c)
+qed
+
+lemma has_fps_expansion_fps_expansion [intro]:
+ assumes "open A" "0 \<in> A" "f holomorphic_on A"
+ shows "f has_fps_expansion fps_expansion f 0"
+proof -
+ from assms(1,2) obtain r where r: "r > 0 " "ball 0 r \<subseteq> A"
+ by (auto simp: open_contains_ball)
+ have holo: "f holomorphic_on eball 0 (ereal r)"
+ using r(2) and assms(3) by auto
+ from r(1) have "0 < ereal r" by simp
+ also have "r \<le> fps_conv_radius (fps_expansion f 0)"
+ using holo by (intro conv_radius_fps_expansion) auto
+ finally have "\<dots> > 0" .
+ moreover have "eventually (\<lambda>z. z \<in> ball 0 r) (nhds 0)"
+ using r(1) by (intro eventually_nhds_in_open) auto
+ hence "eventually (\<lambda>z. eval_fps (fps_expansion f 0) z = f z) (nhds 0)"
+ by eventually_elim (subst eval_fps_expansion'[OF holo], auto)
+ ultimately show ?thesis using r(1) by (auto simp: has_fps_expansion_def)
+qed
+
+lemma fps_conv_radius_tan:
+ fixes c :: complex
+ assumes "c \<noteq> 0"
+ shows "fps_conv_radius (fps_tan c) \<ge> pi / (2 * norm c)"
+proof -
+ have "fps_conv_radius (fps_tan c) \<ge>
+ Min {pi / (2 * norm c), fps_conv_radius (fps_sin c), fps_conv_radius (fps_cos c)}"
+ unfolding fps_tan_def
+ proof (rule fps_conv_radius_divide)
+ fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+ with cos_eq_zero_imp_norm_ge[of "c*z"] assms
+ show "eval_fps (fps_cos c) z \<noteq> 0" by (auto simp: norm_mult field_simps)
+ qed (insert assms, auto)
+ thus ?thesis by (simp add: min_def)
+qed
+
+lemma eval_fps_tan:
+ fixes c :: complex
+ assumes "norm z < pi / (2 * norm c)"
+ shows "eval_fps (fps_tan c) z = tan (c * z)"
+proof (cases "c = 0")
+ case False
+ show ?thesis unfolding fps_tan_def
+ proof (subst eval_fps_divide'[where r = "pi / (2 * norm c)"])
+ fix z :: complex assume "z \<in> eball 0 (pi / (2 * norm c))"
+ with cos_eq_zero_imp_norm_ge[of "c*z"] assms
+ show "eval_fps (fps_cos c) z \<noteq> 0" using False by (auto simp: norm_mult field_simps)
+ qed (insert False assms, auto simp: field_simps tan_def)
+qed simp_all
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Complex_Analysis.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,6 @@
+theory Complex_Analysis
+ imports
+ Winding_Numbers
+begin
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Conformal_Mappings.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,5116 @@
+section \<open>Conformal Mappings and Consequences of Cauchy's Integral Theorem\<close>
+
+text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2016)\<close>
+
+text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
+
+theory Conformal_Mappings
+imports Cauchy_Integral_Theorem
+
+begin
+
+(* FIXME mv to Cauchy_Integral_Theorem.thy *)
+subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
+
+lemma Cauchy_higher_deriv_bound:
+ assumes holf: "f holomorphic_on (ball z r)"
+ and contf: "continuous_on (cball z r) f"
+ and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
+ and "0 < r" and "0 < n"
+ shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
+proof -
+ have "0 < B0" using \<open>0 < r\<close> fin [of z]
+ by (metis ball_eq_empty ex_in_conv fin not_less)
+ have le_B0: "\<And>w. cmod (w - z) \<le> r \<Longrightarrow> cmod (f w - y) \<le> B0"
+ apply (rule continuous_on_closure_norm_le [of "ball z r" "\<lambda>w. f w - y"])
+ apply (auto simp: \<open>0 < r\<close> dist_norm norm_minus_commute)
+ apply (rule continuous_intros contf)+
+ using fin apply (simp add: dist_commute dist_norm less_eq_real_def)
+ done
+ have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w) z - (deriv ^^ n) (\<lambda>w. y) z"
+ using \<open>0 < n\<close> by simp
+ also have "... = (deriv ^^ n) (\<lambda>w. f w - y) z"
+ by (rule higher_deriv_diff [OF holf, symmetric]) (auto simp: \<open>0 < r\<close>)
+ finally have "(deriv ^^ n) f z = (deriv ^^ n) (\<lambda>w. f w - y) z" .
+ have contf': "continuous_on (cball z r) (\<lambda>u. f u - y)"
+ by (rule contf continuous_intros)+
+ have holf': "(\<lambda>u. (f u - y)) holomorphic_on (ball z r)"
+ by (simp add: holf holomorphic_on_diff)
+ define a where "a = (2 * pi)/(fact n)"
+ have "0 < a" by (simp add: a_def)
+ have "B0/r^(Suc n)*2 * pi * r = a*((fact n)*B0/r^n)"
+ using \<open>0 < r\<close> by (simp add: a_def field_split_simps)
+ have der_dif: "(deriv ^^ n) (\<lambda>w. f w - y) z = (deriv ^^ n) f z"
+ using \<open>0 < r\<close> \<open>0 < n\<close>
+ by (auto simp: higher_deriv_diff [OF holf holomorphic_on_const])
+ have "norm ((2 * of_real pi * \<i>)/(fact n) * (deriv ^^ n) (\<lambda>w. f w - y) z)
+ \<le> (B0/r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath [of "(\<lambda>u. (f u - y)/(u - z)^(Suc n))" _ z])
+ using Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf' holf']
+ using \<open>0 < B0\<close> \<open>0 < r\<close>
+ apply (auto simp: norm_divide norm_mult norm_power divide_simps le_B0)
+ done
+ then show ?thesis
+ using \<open>0 < r\<close>
+ by (auto simp: norm_divide norm_mult norm_power field_simps der_dif le_B0)
+qed
+
+lemma Cauchy_inequality:
+ assumes holf: "f holomorphic_on (ball \<xi> r)"
+ and contf: "continuous_on (cball \<xi> r) f"
+ and "0 < r"
+ and nof: "\<And>x. norm(\<xi>-x) = r \<Longrightarrow> norm(f x) \<le> B"
+ shows "norm ((deriv ^^ n) f \<xi>) \<le> (fact n) * B / r^n"
+proof -
+ obtain x where "norm (\<xi>-x) = r"
+ by (metis abs_of_nonneg add_diff_cancel_left' \<open>0 < r\<close> diff_add_cancel
+ dual_order.strict_implies_order norm_of_real)
+ then have "0 \<le> B"
+ by (metis nof norm_not_less_zero not_le order_trans)
+ have "((\<lambda>u. f u / (u - \<xi>) ^ Suc n) has_contour_integral (2 * pi) * \<i> / fact n * (deriv ^^ n) f \<xi>)
+ (circlepath \<xi> r)"
+ apply (rule Cauchy_has_contour_integral_higher_derivative_circlepath [OF contf holf])
+ using \<open>0 < r\<close> by simp
+ then have "norm ((2 * pi * \<i>)/(fact n) * (deriv ^^ n) f \<xi>) \<le> (B / r^(Suc n)) * (2 * pi * r)"
+ apply (rule has_contour_integral_bound_circlepath)
+ using \<open>0 \<le> B\<close> \<open>0 < r\<close>
+ apply (simp_all add: norm_divide norm_power nof frac_le norm_minus_commute del: power_Suc)
+ done
+ then show ?thesis using \<open>0 < r\<close>
+ by (simp add: norm_divide norm_mult field_simps)
+qed
+
+lemma Liouville_polynomial:
+ assumes holf: "f holomorphic_on UNIV"
+ and nof: "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) \<le> B * norm z ^ n"
+ shows "f \<xi> = (\<Sum>k\<le>n. (deriv^^k) f 0 / fact k * \<xi> ^ k)"
+proof (cases rule: le_less_linear [THEN disjE])
+ assume "B \<le> 0"
+ then have "\<And>z. A \<le> norm z \<Longrightarrow> norm(f z) = 0"
+ by (metis nof less_le_trans zero_less_mult_iff neqE norm_not_less_zero norm_power not_le)
+ then have f0: "(f \<longlongrightarrow> 0) at_infinity"
+ using Lim_at_infinity by force
+ then have [simp]: "f = (\<lambda>w. 0)"
+ using Liouville_weak [OF holf, of 0]
+ by (simp add: eventually_at_infinity f0) meson
+ show ?thesis by simp
+next
+ assume "0 < B"
+ have "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * (\<xi> - 0)^k) sums f \<xi>)"
+ apply (rule holomorphic_power_series [where r = "norm \<xi> + 1"])
+ using holf holomorphic_on_subset apply auto
+ done
+ then have sumsf: "((\<lambda>k. (deriv ^^ k) f 0 / (fact k) * \<xi>^k) sums f \<xi>)" by simp
+ have "(deriv ^^ k) f 0 / fact k * \<xi> ^ k = 0" if "k>n" for k
+ proof (cases "(deriv ^^ k) f 0 = 0")
+ case True then show ?thesis by simp
+ next
+ case False
+ define w where "w = complex_of_real (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ have "1 \<le> abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge1: "1 \<le> norm w"
+ by (metis norm_of_real w_def)
+ then have "w \<noteq> 0" by auto
+ have kB: "0 < fact k * B"
+ using \<open>0 < B\<close> by simp
+ then have "0 \<le> fact k * B / cmod ((deriv ^^ k) f 0)"
+ by simp
+ then have wgeA: "A \<le> cmod w"
+ by (simp only: w_def norm_of_real)
+ have "fact k * B / cmod ((deriv ^^ k) f 0) < abs (fact k * B / cmod ((deriv ^^ k) f 0) + (\<bar>A\<bar> + 1))"
+ using \<open>0 < B\<close> by simp
+ then have wge: "fact k * B / cmod ((deriv ^^ k) f 0) < norm w"
+ by (metis norm_of_real w_def)
+ then have "fact k * B / norm w < cmod ((deriv ^^ k) f 0)"
+ using False by (simp add: field_split_simps mult.commute split: if_split_asm)
+ also have "... \<le> fact k * (B * norm w ^ n) / norm w ^ k"
+ apply (rule Cauchy_inequality)
+ using holf holomorphic_on_subset apply force
+ using holf holomorphic_on_imp_continuous_on holomorphic_on_subset apply blast
+ using \<open>w \<noteq> 0\<close> apply simp
+ by (metis nof wgeA dist_0_norm dist_norm)
+ also have "... = fact k * (B * 1 / cmod w ^ (k-n))"
+ apply (simp only: mult_cancel_left times_divide_eq_right [symmetric])
+ using \<open>k>n\<close> \<open>w \<noteq> 0\<close> \<open>0 < B\<close> apply (simp add: field_split_simps semiring_normalization_rules)
+ done
+ also have "... = fact k * B / cmod w ^ (k-n)"
+ by simp
+ finally have "fact k * B / cmod w < fact k * B / cmod w ^ (k - n)" .
+ then have "1 / cmod w < 1 / cmod w ^ (k - n)"
+ by (metis kB divide_inverse inverse_eq_divide mult_less_cancel_left_pos)
+ then have "cmod w ^ (k - n) < cmod w"
+ by (metis frac_le le_less_trans norm_ge_zero norm_one not_less order_refl wge1 zero_less_one)
+ with self_le_power [OF wge1] have False
+ by (meson diff_is_0_eq not_gr0 not_le that)
+ then show ?thesis by blast
+ qed
+ then have "(deriv ^^ (k + Suc n)) f 0 / fact (k + Suc n) * \<xi> ^ (k + Suc n) = 0" for k
+ using not_less_eq by blast
+ then have "(\<lambda>i. (deriv ^^ (i + Suc n)) f 0 / fact (i + Suc n) * \<xi> ^ (i + Suc n)) sums 0"
+ by (rule sums_0)
+ with sums_split_initial_segment [OF sumsf, where n = "Suc n"]
+ show ?thesis
+ using atLeast0AtMost lessThan_Suc_atMost sums_unique2 by fastforce
+qed
+
+text\<open>Every bounded entire function is a constant function.\<close>
+theorem Liouville_theorem:
+ assumes holf: "f holomorphic_on UNIV"
+ and bf: "bounded (range f)"
+ obtains c where "\<And>z. f z = c"
+proof -
+ obtain B where "\<And>z. cmod (f z) \<le> B"
+ by (meson bf bounded_pos rangeI)
+ then show ?thesis
+ using Liouville_polynomial [OF holf, of 0 B 0, simplified] that by blast
+qed
+
+text\<open>A holomorphic function f has only isolated zeros unless f is 0.\<close>
+
+lemma powser_0_nonzero:
+ fixes a :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
+ assumes r: "0 < r"
+ and sm: "\<And>x. norm (x - \<xi>) < r \<Longrightarrow> (\<lambda>n. a n * (x - \<xi>) ^ n) sums (f x)"
+ and [simp]: "f \<xi> = 0"
+ and m0: "a m \<noteq> 0" and "m>0"
+ obtains s where "0 < s" and "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+proof -
+ have "r \<le> conv_radius a"
+ using sm sums_summable by (auto simp: le_conv_radius_iff [where \<xi>=\<xi>])
+ obtain m where am: "a m \<noteq> 0" and az [simp]: "(\<And>n. n<m \<Longrightarrow> a n = 0)"
+ apply (rule_tac m = "LEAST n. a n \<noteq> 0" in that)
+ using m0
+ apply (rule LeastI2)
+ apply (fastforce intro: dest!: not_less_Least)+
+ done
+ define b where "b i = a (i+m) / a m" for i
+ define g where "g x = suminf (\<lambda>i. b i * (x - \<xi>) ^ i)" for x
+ have [simp]: "b 0 = 1"
+ by (simp add: am b_def)
+ { fix x::'a
+ assume "norm (x - \<xi>) < r"
+ then have "(\<lambda>n. (a m * (x - \<xi>)^m) * (b n * (x - \<xi>)^n)) sums (f x)"
+ using am az sm sums_zero_iff_shift [of m "(\<lambda>n. a n * (x - \<xi>) ^ n)" "f x"]
+ by (simp add: b_def monoid_mult_class.power_add algebra_simps)
+ then have "x \<noteq> \<xi> \<Longrightarrow> (\<lambda>n. b n * (x - \<xi>)^n) sums (f x / (a m * (x - \<xi>)^m))"
+ using am by (simp add: sums_mult_D)
+ } note bsums = this
+ then have "norm (x - \<xi>) < r \<Longrightarrow> summable (\<lambda>n. b n * (x - \<xi>)^n)" for x
+ using sums_summable by (cases "x=\<xi>") auto
+ then have "r \<le> conv_radius b"
+ by (simp add: le_conv_radius_iff [where \<xi>=\<xi>])
+ then have "r/2 < conv_radius b"
+ using not_le order_trans r by fastforce
+ then have "continuous_on (cball \<xi> (r/2)) g"
+ using powser_continuous_suminf [of "r/2" b \<xi>] by (simp add: g_def)
+ then obtain s where "s>0" "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> dist (g x) (g \<xi>) < 1/2"
+ apply (rule continuous_onE [where x=\<xi> and e = "1/2"])
+ using r apply (auto simp: norm_minus_commute dist_norm)
+ done
+ moreover have "g \<xi> = 1"
+ by (simp add: g_def)
+ ultimately have gnz: "\<And>x. \<lbrakk>norm (x - \<xi>) \<le> s; norm (x - \<xi>) \<le> r/2\<rbrakk> \<Longrightarrow> (g x) \<noteq> 0"
+ by fastforce
+ have "f x \<noteq> 0" if "x \<noteq> \<xi>" "norm (x - \<xi>) \<le> s" "norm (x - \<xi>) \<le> r/2" for x
+ using bsums [of x] that gnz [of x]
+ apply (auto simp: g_def)
+ using r sums_iff by fastforce
+ then show ?thesis
+ apply (rule_tac s="min s (r/2)" in that)
+ using \<open>0 < r\<close> \<open>0 < s\<close> by (auto simp: dist_commute dist_norm)
+qed
+
+subsection \<open>Analytic continuation\<close>
+
+proposition isolated_zeros:
+ assumes holf: "f holomorphic_on S"
+ and "open S" "connected S" "\<xi> \<in> S" "f \<xi> = 0" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
+ obtains r where "0 < r" and "ball \<xi> r \<subseteq> S" and
+ "\<And>z. z \<in> ball \<xi> r - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+proof -
+ obtain r where "0 < r" and r: "ball \<xi> r \<subseteq> S"
+ using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_ball_eq by blast
+ have powf: "((\<lambda>n. (deriv ^^ n) f \<xi> / (fact n) * (z - \<xi>)^n) sums f z)" if "z \<in> ball \<xi> r" for z
+ apply (rule holomorphic_power_series [OF _ that])
+ apply (rule holomorphic_on_subset [OF holf r])
+ done
+ obtain m where m: "(deriv ^^ m) f \<xi> / (fact m) \<noteq> 0"
+ using holomorphic_fun_eq_0_on_connected [OF holf \<open>open S\<close> \<open>connected S\<close> _ \<open>\<xi> \<in> S\<close> \<open>\<beta> \<in> S\<close>] \<open>f \<beta> \<noteq> 0\<close>
+ by auto
+ then have "m \<noteq> 0" using assms(5) funpow_0 by fastforce
+ obtain s where "0 < s" and s: "\<And>z. z \<in> cball \<xi> s - {\<xi>} \<Longrightarrow> f z \<noteq> 0"
+ apply (rule powser_0_nonzero [OF \<open>0 < r\<close> powf \<open>f \<xi> = 0\<close> m])
+ using \<open>m \<noteq> 0\<close> by (auto simp: dist_commute dist_norm)
+ have "0 < min r s" by (simp add: \<open>0 < r\<close> \<open>0 < s\<close>)
+ then show ?thesis
+ apply (rule that)
+ using r s by auto
+qed
+
+proposition analytic_continuation:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and "connected S"
+ and "U \<subseteq> S" and "\<xi> \<in> S"
+ and "\<xi> islimpt U"
+ and fU0 [simp]: "\<And>z. z \<in> U \<Longrightarrow> f z = 0"
+ and "w \<in> S"
+ shows "f w = 0"
+proof -
+ obtain e where "0 < e" and e: "cball \<xi> e \<subseteq> S"
+ using \<open>open S\<close> \<open>\<xi> \<in> S\<close> open_contains_cball_eq by blast
+ define T where "T = cball \<xi> e \<inter> U"
+ have contf: "continuous_on (closure T) f"
+ by (metis T_def closed_cball closure_minimal e holf holomorphic_on_imp_continuous_on
+ holomorphic_on_subset inf.cobounded1)
+ have fT0 [simp]: "\<And>x. x \<in> T \<Longrightarrow> f x = 0"
+ by (simp add: T_def)
+ have "\<And>r. \<lbrakk>\<forall>e>0. \<exists>x'\<in>U. x' \<noteq> \<xi> \<and> dist x' \<xi> < e; 0 < r\<rbrakk> \<Longrightarrow> \<exists>x'\<in>cball \<xi> e \<inter> U. x' \<noteq> \<xi> \<and> dist x' \<xi> < r"
+ by (metis \<open>0 < e\<close> IntI dist_commute less_eq_real_def mem_cball min_less_iff_conj)
+ then have "\<xi> islimpt T" using \<open>\<xi> islimpt U\<close>
+ by (auto simp: T_def islimpt_approachable)
+ then have "\<xi> \<in> closure T"
+ by (simp add: closure_def)
+ then have "f \<xi> = 0"
+ by (auto simp: continuous_constant_on_closure [OF contf])
+ show ?thesis
+ apply (rule ccontr)
+ apply (rule isolated_zeros [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>\<xi> \<in> S\<close> \<open>f \<xi> = 0\<close> \<open>w \<in> S\<close>], assumption)
+ by (metis open_ball \<open>\<xi> islimpt T\<close> centre_in_ball fT0 insertE insert_Diff islimptE)
+qed
+
+corollary analytic_continuation_open:
+ assumes "open s" and "open s'" and "s \<noteq> {}" and "connected s'"
+ and "s \<subseteq> s'"
+ assumes "f holomorphic_on s'" and "g holomorphic_on s'"
+ and "\<And>z. z \<in> s \<Longrightarrow> f z = g z"
+ assumes "z \<in> s'"
+ shows "f z = g z"
+proof -
+ from \<open>s \<noteq> {}\<close> obtain \<xi> where "\<xi> \<in> s" by auto
+ with \<open>open s\<close> have \<xi>: "\<xi> islimpt s"
+ by (intro interior_limit_point) (auto simp: interior_open)
+ have "f z - g z = 0"
+ by (rule analytic_continuation[of "\<lambda>z. f z - g z" s' s \<xi>])
+ (insert assms \<open>\<xi> \<in> s\<close> \<xi>, auto intro: holomorphic_intros)
+ thus ?thesis by simp
+qed
+
+subsection\<open>Open mapping theorem\<close>
+
+lemma holomorphic_contract_to_zero:
+ assumes contf: "continuous_on (cball \<xi> r) f"
+ and holf: "f holomorphic_on ball \<xi> r"
+ and "0 < r"
+ and norm_less: "\<And>z. norm(\<xi> - z) = r \<Longrightarrow> norm(f \<xi>) < norm(f z)"
+ obtains z where "z \<in> ball \<xi> r" "f z = 0"
+proof -
+ { assume fnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w \<noteq> 0"
+ then have "0 < norm (f \<xi>)"
+ by (simp add: \<open>0 < r\<close>)
+ have fnz': "\<And>w. w \<in> cball \<xi> r \<Longrightarrow> f w \<noteq> 0"
+ by (metis norm_less dist_norm fnz less_eq_real_def mem_ball mem_cball norm_not_less_zero norm_zero)
+ have "frontier(cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by simp
+ define g where [abs_def]: "g z = inverse (f z)" for z
+ have contg: "continuous_on (cball \<xi> r) g"
+ unfolding g_def using contf continuous_on_inverse fnz' by blast
+ have holg: "g holomorphic_on ball \<xi> r"
+ unfolding g_def using fnz holf holomorphic_on_inverse by blast
+ have "frontier (cball \<xi> r) \<subseteq> cball \<xi> r"
+ by (simp add: subset_iff)
+ then have contf': "continuous_on (frontier (cball \<xi> r)) f"
+ and contg': "continuous_on (frontier (cball \<xi> r)) g"
+ by (blast intro: contf contg continuous_on_subset)+
+ have froc: "frontier(cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by simp
+ moreover have "continuous_on (frontier (cball \<xi> r)) (norm o f)"
+ using contf' continuous_on_compose continuous_on_norm_id by blast
+ ultimately obtain w where w: "w \<in> frontier(cball \<xi> r)"
+ and now: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (f w) \<le> norm (f x)"
+ apply (rule bexE [OF continuous_attains_inf [OF compact_frontier [OF compact_cball]]])
+ apply simp
+ done
+ then have fw: "0 < norm (f w)"
+ by (simp add: fnz')
+ have "continuous_on (frontier (cball \<xi> r)) (norm o g)"
+ using contg' continuous_on_compose continuous_on_norm_id by blast
+ then obtain v where v: "v \<in> frontier(cball \<xi> r)"
+ and nov: "\<And>x. x \<in> frontier(cball \<xi> r) \<Longrightarrow> norm (g v) \<ge> norm (g x)"
+ apply (rule bexE [OF continuous_attains_sup [OF compact_frontier [OF compact_cball] froc]])
+ apply simp
+ done
+ then have fv: "0 < norm (f v)"
+ by (simp add: fnz')
+ have "norm ((deriv ^^ 0) g \<xi>) \<le> fact 0 * norm (g v) / r ^ 0"
+ by (rule Cauchy_inequality [OF holg contg \<open>0 < r\<close>]) (simp add: dist_norm nov)
+ then have "cmod (g \<xi>) \<le> norm (g v)"
+ by simp
+ with w have wr: "norm (\<xi> - w) = r" and nfw: "norm (f w) \<le> norm (f \<xi>)"
+ apply (simp_all add: dist_norm)
+ by (metis \<open>0 < cmod (f \<xi>)\<close> g_def less_imp_inverse_less norm_inverse not_le now order_trans v)
+ with fw have False
+ using norm_less by force
+ }
+ with that show ?thesis by blast
+qed
+
+theorem open_mapping_thm:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S" and "connected S"
+ and "open U" and "U \<subseteq> S"
+ and fne: "\<not> f constant_on S"
+ shows "open (f ` U)"
+proof -
+ have *: "open (f ` U)"
+ if "U \<noteq> {}" and U: "open U" "connected U" and "f holomorphic_on U" and fneU: "\<And>x. \<exists>y \<in> U. f y \<noteq> x"
+ for U
+ proof (clarsimp simp: open_contains_ball)
+ fix \<xi> assume \<xi>: "\<xi> \<in> U"
+ show "\<exists>e>0. ball (f \<xi>) e \<subseteq> f ` U"
+ proof -
+ have hol: "(\<lambda>z. f z - f \<xi>) holomorphic_on U"
+ by (rule holomorphic_intros that)+
+ obtain s where "0 < s" and sbU: "ball \<xi> s \<subseteq> U"
+ and sne: "\<And>z. z \<in> ball \<xi> s - {\<xi>} \<Longrightarrow> (\<lambda>z. f z - f \<xi>) z \<noteq> 0"
+ using isolated_zeros [OF hol U \<xi>] by (metis fneU right_minus_eq)
+ obtain r where "0 < r" and r: "cball \<xi> r \<subseteq> ball \<xi> s"
+ apply (rule_tac r="s/2" in that)
+ using \<open>0 < s\<close> by auto
+ have "cball \<xi> r \<subseteq> U"
+ using sbU r by blast
+ then have frsbU: "frontier (cball \<xi> r) \<subseteq> U"
+ using Diff_subset frontier_def order_trans by fastforce
+ then have cof: "compact (frontier(cball \<xi> r))"
+ by blast
+ have frne: "frontier (cball \<xi> r) \<noteq> {}"
+ using \<open>0 < r\<close> by auto
+ have contfr: "continuous_on (frontier (cball \<xi> r)) (\<lambda>z. norm (f z - f \<xi>))"
+ by (metis continuous_on_norm continuous_on_subset frsbU hol holomorphic_on_imp_continuous_on)
+ obtain w where "norm (\<xi> - w) = r"
+ and w: "(\<And>z. norm (\<xi> - z) = r \<Longrightarrow> norm (f w - f \<xi>) \<le> norm(f z - f \<xi>))"
+ apply (rule bexE [OF continuous_attains_inf [OF cof frne contfr]])
+ apply (simp add: dist_norm)
+ done
+ moreover define \<epsilon> where "\<epsilon> \<equiv> norm (f w - f \<xi>) / 3"
+ ultimately have "0 < \<epsilon>"
+ using \<open>0 < r\<close> dist_complex_def r sne by auto
+ have "ball (f \<xi>) \<epsilon> \<subseteq> f ` U"
+ proof
+ fix \<gamma>
+ assume \<gamma>: "\<gamma> \<in> ball (f \<xi>) \<epsilon>"
+ have *: "cmod (\<gamma> - f \<xi>) < cmod (\<gamma> - f z)" if "cmod (\<xi> - z) = r" for z
+ proof -
+ have lt: "cmod (f w - f \<xi>) / 3 < cmod (\<gamma> - f z)"
+ using w [OF that] \<gamma>
+ using dist_triangle2 [of "f \<xi>" "\<gamma>" "f z"] dist_triangle2 [of "f \<xi>" "f z" \<gamma>]
+ by (simp add: \<epsilon>_def dist_norm norm_minus_commute)
+ show ?thesis
+ by (metis \<epsilon>_def dist_commute dist_norm less_trans lt mem_ball \<gamma>)
+ qed
+ have "continuous_on (cball \<xi> r) (\<lambda>z. \<gamma> - f z)"
+ apply (rule continuous_intros)+
+ using \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close>
+ apply (blast intro: continuous_on_subset holomorphic_on_imp_continuous_on)
+ done
+ moreover have "(\<lambda>z. \<gamma> - f z) holomorphic_on ball \<xi> r"
+ apply (rule holomorphic_intros)+
+ apply (metis \<open>cball \<xi> r \<subseteq> U\<close> \<open>f holomorphic_on U\<close> holomorphic_on_subset interior_cball interior_subset)
+ done
+ ultimately obtain z where "z \<in> ball \<xi> r" "\<gamma> - f z = 0"
+ apply (rule holomorphic_contract_to_zero)
+ apply (blast intro!: \<open>0 < r\<close> *)+
+ done
+ then show "\<gamma> \<in> f ` U"
+ using \<open>cball \<xi> r \<subseteq> U\<close> by fastforce
+ qed
+ then show ?thesis using \<open>0 < \<epsilon>\<close> by blast
+ qed
+ qed
+ have "open (f ` X)" if "X \<in> components U" for X
+ proof -
+ have holfU: "f holomorphic_on U"
+ using \<open>U \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ have "X \<noteq> {}"
+ using that by (simp add: in_components_nonempty)
+ moreover have "open X"
+ using that \<open>open U\<close> open_components by auto
+ moreover have "connected X"
+ using that in_components_maximal by blast
+ moreover have "f holomorphic_on X"
+ by (meson that holfU holomorphic_on_subset in_components_maximal)
+ moreover have "\<exists>y\<in>X. f y \<noteq> x" for x
+ proof (rule ccontr)
+ assume not: "\<not> (\<exists>y\<in>X. f y \<noteq> x)"
+ have "X \<subseteq> S"
+ using \<open>U \<subseteq> S\<close> in_components_subset that by blast
+ obtain w where w: "w \<in> X" using \<open>X \<noteq> {}\<close> by blast
+ have wis: "w islimpt X"
+ using w \<open>open X\<close> interior_eq by auto
+ have hol: "(\<lambda>z. f z - x) holomorphic_on S"
+ by (simp add: holf holomorphic_on_diff)
+ with fne [unfolded constant_on_def]
+ analytic_continuation[OF hol S \<open>connected S\<close> \<open>X \<subseteq> S\<close> _ wis] not \<open>X \<subseteq> S\<close> w
+ show False by auto
+ qed
+ ultimately show ?thesis
+ by (rule *)
+ qed
+ then have "open (f ` \<Union>(components U))"
+ by (metis (no_types, lifting) imageE image_Union open_Union)
+ then show ?thesis
+ by force
+qed
+
+text\<open>No need for \<^term>\<open>S\<close> to be connected. But the nonconstant condition is stronger.\<close>
+corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm2:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S"
+ and "open U" "U \<subseteq> S"
+ and fnc: "\<And>X. \<lbrakk>open X; X \<subseteq> S; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<not> f constant_on X"
+ shows "open (f ` U)"
+proof -
+ have "S = \<Union>(components S)" by simp
+ with \<open>U \<subseteq> S\<close> have "U = (\<Union>C \<in> components S. C \<inter> U)" by auto
+ then have "f ` U = (\<Union>C \<in> components S. f ` (C \<inter> U))"
+ using image_UN by fastforce
+ moreover
+ { fix C assume "C \<in> components S"
+ with S \<open>C \<in> components S\<close> open_components in_components_connected
+ have C: "open C" "connected C" by auto
+ have "C \<subseteq> S"
+ by (metis \<open>C \<in> components S\<close> in_components_maximal)
+ have nf: "\<not> f constant_on C"
+ apply (rule fnc)
+ using C \<open>C \<subseteq> S\<close> \<open>C \<in> components S\<close> in_components_nonempty by auto
+ have "f holomorphic_on C"
+ by (metis holf holomorphic_on_subset \<open>C \<subseteq> S\<close>)
+ then have "open (f ` (C \<inter> U))"
+ apply (rule open_mapping_thm [OF _ C _ _ nf])
+ apply (simp add: C \<open>open U\<close> open_Int, blast)
+ done
+ } ultimately show ?thesis
+ by force
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> open_mapping_thm3:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ shows "open (f ` S)"
+apply (rule open_mapping_thm2 [OF holf])
+using assms
+apply (simp_all add:)
+using injective_not_constant subset_inj_on by blast
+
+subsection\<open>Maximum modulus principle\<close>
+
+text\<open>If \<^term>\<open>f\<close> is holomorphic, then its norm (modulus) cannot exhibit a true local maximum that is
+ properly within the domain of \<^term>\<open>f\<close>.\<close>
+
+proposition maximum_modulus_principle:
+ assumes holf: "f holomorphic_on S"
+ and S: "open S" and "connected S"
+ and "open U" and "U \<subseteq> S" and "\<xi> \<in> U"
+ and no: "\<And>z. z \<in> U \<Longrightarrow> norm(f z) \<le> norm(f \<xi>)"
+ shows "f constant_on S"
+proof (rule ccontr)
+ assume "\<not> f constant_on S"
+ then have "open (f ` U)"
+ using open_mapping_thm assms by blast
+ moreover have "\<not> open (f ` U)"
+ proof -
+ have "\<exists>t. cmod (f \<xi> - t) < e \<and> t \<notin> f ` U" if "0 < e" for e
+ apply (rule_tac x="if 0 < Re(f \<xi>) then f \<xi> + (e/2) else f \<xi> - (e/2)" in exI)
+ using that
+ apply (simp add: dist_norm)
+ apply (fastforce simp: cmod_Re_le_iff dest!: no dest: sym)
+ done
+ then show ?thesis
+ unfolding open_contains_ball by (metis \<open>\<xi> \<in> U\<close> contra_subsetD dist_norm imageI mem_ball)
+ qed
+ ultimately show False
+ by blast
+qed
+
+proposition maximum_modulus_frontier:
+ assumes holf: "f holomorphic_on (interior S)"
+ and contf: "continuous_on (closure S) f"
+ and bos: "bounded S"
+ and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> norm(f z) \<le> B"
+ and "\<xi> \<in> S"
+ shows "norm(f \<xi>) \<le> B"
+proof -
+ have "compact (closure S)" using bos
+ by (simp add: bounded_closure compact_eq_bounded_closed)
+ moreover have "continuous_on (closure S) (cmod \<circ> f)"
+ using contf continuous_on_compose continuous_on_norm_id by blast
+ ultimately obtain z where zin: "z \<in> closure S" and z: "\<And>y. y \<in> closure S \<Longrightarrow> (cmod \<circ> f) y \<le> (cmod \<circ> f) z"
+ using continuous_attains_sup [of "closure S" "norm o f"] \<open>\<xi> \<in> S\<close> by auto
+ then consider "z \<in> frontier S" | "z \<in> interior S" using frontier_def by auto
+ then have "norm(f z) \<le> B"
+ proof cases
+ case 1 then show ?thesis using leB by blast
+ next
+ case 2
+ have zin: "z \<in> connected_component_set (interior S) z"
+ by (simp add: 2)
+ have "f constant_on (connected_component_set (interior S) z)"
+ apply (rule maximum_modulus_principle [OF _ _ _ _ _ zin])
+ apply (metis connected_component_subset holf holomorphic_on_subset)
+ apply (simp_all add: open_connected_component)
+ by (metis closure_subset comp_eq_dest_lhs interior_subset subsetCE z connected_component_in)
+ then obtain c where c: "\<And>w. w \<in> connected_component_set (interior S) z \<Longrightarrow> f w = c"
+ by (auto simp: constant_on_def)
+ have "f ` closure(connected_component_set (interior S) z) \<subseteq> {c}"
+ apply (rule image_closure_subset)
+ apply (meson closure_mono connected_component_subset contf continuous_on_subset interior_subset)
+ using c
+ apply auto
+ done
+ then have cc: "\<And>w. w \<in> closure(connected_component_set (interior S) z) \<Longrightarrow> f w = c" by blast
+ have "frontier(connected_component_set (interior S) z) \<noteq> {}"
+ apply (simp add: frontier_eq_empty)
+ by (metis "2" bos bounded_interior connected_component_eq_UNIV connected_component_refl not_bounded_UNIV)
+ then obtain w where w: "w \<in> frontier(connected_component_set (interior S) z)"
+ by auto
+ then have "norm (f z) = norm (f w)" by (simp add: "2" c cc frontier_def)
+ also have "... \<le> B"
+ apply (rule leB)
+ using w
+using frontier_interior_subset frontier_of_connected_component_subset by blast
+ finally show ?thesis .
+ qed
+ then show ?thesis
+ using z \<open>\<xi> \<in> S\<close> closure_subset by fastforce
+qed
+
+corollary\<^marker>\<open>tag unimportant\<close> maximum_real_frontier:
+ assumes holf: "f holomorphic_on (interior S)"
+ and contf: "continuous_on (closure S) f"
+ and bos: "bounded S"
+ and leB: "\<And>z. z \<in> frontier S \<Longrightarrow> Re(f z) \<le> B"
+ and "\<xi> \<in> S"
+ shows "Re(f \<xi>) \<le> B"
+using maximum_modulus_frontier [of "exp o f" S "exp B"]
+ Transcendental.continuous_on_exp holomorphic_on_compose holomorphic_on_exp assms
+by auto
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Factoring out a zero according to its order\<close>
+
+lemma holomorphic_factor_order_of_zero:
+ assumes holf: "f holomorphic_on S"
+ and os: "open S"
+ and "\<xi> \<in> S" "0 < n"
+ and dnz: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ and dfz: "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
+ obtains g r where "0 < r"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof -
+ obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
+ then have holfb: "f holomorphic_on ball \<xi> r"
+ using holf holomorphic_on_subset by blast
+ define g where "g w = suminf (\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i)" for w
+ have sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
+ and feq: "f w - f \<xi> = (w - \<xi>)^n * g w"
+ if w: "w \<in> ball \<xi> r" for w
+ proof -
+ define powf where "powf = (\<lambda>i. (deriv ^^ i) f \<xi>/(fact i) * (w - \<xi>)^i)"
+ have sing: "{..<n} - {i. powf i = 0} = (if f \<xi> = 0 then {} else {0})"
+ unfolding powf_def using \<open>0 < n\<close> dfz by (auto simp: dfz; metis funpow_0 not_gr0)
+ have "powf sums f w"
+ unfolding powf_def by (rule holomorphic_power_series [OF holfb w])
+ moreover have "(\<Sum>i<n. powf i) = f \<xi>"
+ apply (subst Groups_Big.comm_monoid_add_class.sum.setdiff_irrelevant [symmetric])
+ apply simp
+ apply (simp only: dfz sing)
+ apply (simp add: powf_def)
+ done
+ ultimately have fsums: "(\<lambda>i. powf (i+n)) sums (f w - f \<xi>)"
+ using w sums_iff_shift' by metis
+ then have *: "summable (\<lambda>i. (w - \<xi>) ^ n * ((deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n)))"
+ unfolding powf_def using sums_summable
+ by (auto simp: power_add mult_ac)
+ have "summable (\<lambda>i. (deriv ^^ (i + n)) f \<xi> * (w - \<xi>) ^ i / fact (i + n))"
+ proof (cases "w=\<xi>")
+ case False then show ?thesis
+ using summable_mult [OF *, of "1 / (w - \<xi>) ^ n"] by simp
+ next
+ case True then show ?thesis
+ by (auto simp: Power.semiring_1_class.power_0_left intro!: summable_finite [of "{0}"]
+ split: if_split_asm)
+ qed
+ then show sumsg: "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / (fact(i + n)) * (w - \<xi>)^i) sums g w"
+ by (simp add: summable_sums_iff g_def)
+ show "f w - f \<xi> = (w - \<xi>)^n * g w"
+ apply (rule sums_unique2)
+ apply (rule fsums [unfolded powf_def])
+ using sums_mult [OF sumsg, of "(w - \<xi>) ^ n"]
+ by (auto simp: power_add mult_ac)
+ qed
+ then have holg: "g holomorphic_on ball \<xi> r"
+ by (meson sumsg power_series_holomorphic)
+ then have contg: "continuous_on (ball \<xi> r) g"
+ by (blast intro: holomorphic_on_imp_continuous_on)
+ have "g \<xi> \<noteq> 0"
+ using dnz unfolding g_def
+ by (subst suminf_finite [of "{0}"]) auto
+ obtain d where "0 < d" and d: "\<And>w. w \<in> ball \<xi> d \<Longrightarrow> g w \<noteq> 0"
+ apply (rule exE [OF continuous_on_avoid [OF contg _ \<open>g \<xi> \<noteq> 0\<close>]])
+ using \<open>0 < r\<close>
+ apply force
+ by (metis \<open>0 < r\<close> less_trans mem_ball not_less_iff_gr_or_eq)
+ show ?thesis
+ apply (rule that [where g=g and r ="min r d"])
+ using \<open>0 < r\<close> \<open>0 < d\<close> holg
+ apply (auto simp: feq holomorphic_on_subset subset_ball d)
+ done
+qed
+
+
+lemma holomorphic_factor_order_of_zero_strong:
+ assumes holf: "f holomorphic_on S" "open S" "\<xi> \<in> S" "0 < n"
+ and "(deriv ^^ n) f \<xi> \<noteq> 0"
+ and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0"
+ obtains g r where "0 < r"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof -
+ obtain g r where "0 < r"
+ and holg: "g holomorphic_on ball \<xi> r"
+ and feq: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>)^n * g w"
+ and gne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF assms])
+ have con: "continuous_on (ball \<xi> r) (\<lambda>z. deriv g z / g z)"
+ by (rule continuous_intros) (auto simp: gne holg holomorphic_deriv holomorphic_on_imp_continuous_on)
+ have cd: "\<And>x. dist \<xi> x < r \<Longrightarrow> (\<lambda>z. deriv g z / g z) field_differentiable at x"
+ apply (rule derivative_intros)+
+ using holg mem_ball apply (blast intro: holomorphic_deriv holomorphic_on_imp_differentiable_at)
+ apply (metis open_ball at_within_open holg holomorphic_on_def mem_ball)
+ using gne mem_ball by blast
+ obtain h where h: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> (h has_field_derivative deriv g x / g x) (at x)"
+ apply (rule exE [OF holomorphic_convex_primitive [of "ball \<xi> r" "{}" "\<lambda>z. deriv g z / g z"]])
+ apply (auto simp: con cd)
+ apply (metis open_ball at_within_open mem_ball)
+ done
+ then have "continuous_on (ball \<xi> r) h"
+ by (metis open_ball holomorphic_on_imp_continuous_on holomorphic_on_open)
+ then have con: "continuous_on (ball \<xi> r) (\<lambda>x. exp (h x) / g x)"
+ by (auto intro!: continuous_intros simp add: holg holomorphic_on_imp_continuous_on gne)
+ have 0: "dist \<xi> x < r \<Longrightarrow> ((\<lambda>x. exp (h x) / g x) has_field_derivative 0) (at x)" for x
+ apply (rule h derivative_eq_intros | simp)+
+ apply (rule DERIV_deriv_iff_field_differentiable [THEN iffD2])
+ using holg apply (auto simp: holomorphic_on_imp_differentiable_at gne h)
+ done
+ obtain c where c: "\<And>x. x \<in> ball \<xi> r \<Longrightarrow> exp (h x) / g x = c"
+ by (rule DERIV_zero_connected_constant [of "ball \<xi> r" "{}" "\<lambda>x. exp(h x) / g x"]) (auto simp: con 0)
+ have hol: "(\<lambda>z. exp ((Ln (inverse c) + h z) / of_nat n)) holomorphic_on ball \<xi> r"
+ apply (rule holomorphic_on_compose [unfolded o_def, where g = exp])
+ apply (rule holomorphic_intros)+
+ using h holomorphic_on_open apply blast
+ apply (rule holomorphic_intros)+
+ using \<open>0 < n\<close> apply simp
+ apply (rule holomorphic_intros)+
+ done
+ show ?thesis
+ apply (rule that [where g="\<lambda>z. exp((Ln(inverse c) + h z)/n)" and r =r])
+ using \<open>0 < r\<close> \<open>0 < n\<close>
+ apply (auto simp: feq power_mult_distrib exp_divide_power_eq c [symmetric])
+ apply (rule hol)
+ apply (simp add: Transcendental.exp_add gne)
+ done
+qed
+
+
+lemma
+ fixes k :: "'a::wellorder"
+ assumes a_def: "a == LEAST x. P x" and P: "P k"
+ shows def_LeastI: "P a" and def_Least_le: "a \<le> k"
+unfolding a_def
+by (rule LeastI Least_le; rule P)+
+
+lemma holomorphic_factor_zero_nonconstant:
+ assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
+ and "\<xi> \<in> S" "f \<xi> = 0"
+ and nonconst: "\<not> f constant_on S"
+ obtains g r n
+ where "0 < n" "0 < r" "ball \<xi> r \<subseteq> S"
+ "g holomorphic_on ball \<xi> r"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w = (w - \<xi>)^n * g w"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
+ case True then show ?thesis
+ using holomorphic_fun_eq_const_on_connected [OF holf S _ \<open>\<xi> \<in> S\<close>] nonconst by (simp add: constant_on_def)
+next
+ case False
+ then obtain n0 where "n0 > 0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0" by blast
+ obtain r0 where "r0 > 0" "ball \<xi> r0 \<subseteq> S" using S openE \<open>\<xi> \<in> S\<close> by auto
+ define n where "n \<equiv> LEAST n. (deriv ^^ n) f \<xi> \<noteq> 0"
+ have n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ by (rule def_LeastI [OF n_def]) (rule n0)
+ then have "0 < n" using \<open>f \<xi> = 0\<close>
+ using funpow_0 by fastforce
+ have n_min: "\<And>k. k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ using def_Least_le [OF n_def] not_le by blast
+ then obtain g r1
+ where "0 < r1" "g holomorphic_on ball \<xi> r1"
+ "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> f w = (w - \<xi>) ^ n * g w"
+ "\<And>w. w \<in> ball \<xi> r1 \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne] simp: \<open>f \<xi> = 0\<close>)
+ then show ?thesis
+ apply (rule_tac g=g and r="min r0 r1" and n=n in that)
+ using \<open>0 < n\<close> \<open>0 < r0\<close> \<open>0 < r1\<close> \<open>ball \<xi> r0 \<subseteq> S\<close>
+ apply (auto simp: subset_ball intro: holomorphic_on_subset)
+ done
+qed
+
+
+lemma holomorphic_lower_bound_difference:
+ assumes holf: "f holomorphic_on S" and S: "open S" "connected S"
+ and "\<xi> \<in> S" and "\<phi> \<in> S"
+ and fne: "f \<phi> \<noteq> f \<xi>"
+ obtains k n r
+ where "0 < k" "0 < r"
+ "ball \<xi> r \<subseteq> S"
+ "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> k * norm(w - \<xi>)^n \<le> norm(f w - f \<xi>)"
+proof -
+ define n where "n = (LEAST n. 0 < n \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
+ obtain n0 where "0 < n0" and n0: "(deriv ^^ n0) f \<xi> \<noteq> 0"
+ using fne holomorphic_fun_eq_const_on_connected [OF holf S] \<open>\<xi> \<in> S\<close> \<open>\<phi> \<in> S\<close> by blast
+ then have "0 < n" and n_ne: "(deriv ^^ n) f \<xi> \<noteq> 0"
+ unfolding n_def by (metis (mono_tags, lifting) LeastI)+
+ have n_min: "\<And>k. \<lbrakk>0 < k; k < n\<rbrakk> \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ unfolding n_def by (blast dest: not_less_Least)
+ then obtain g r
+ where "0 < r" and holg: "g holomorphic_on ball \<xi> r"
+ and fne: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w"
+ and gnz: "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
+ by (auto intro: holomorphic_factor_order_of_zero [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> \<open>n > 0\<close> n_ne])
+ obtain e where "e>0" and e: "ball \<xi> e \<subseteq> S" using assms by (blast elim!: openE)
+ then have holfb: "f holomorphic_on ball \<xi> e"
+ using holf holomorphic_on_subset by blast
+ define d where "d = (min e r) / 2"
+ have "0 < d" using \<open>0 < r\<close> \<open>0 < e\<close> by (simp add: d_def)
+ have "d < r"
+ using \<open>0 < r\<close> by (auto simp: d_def)
+ then have cbb: "cball \<xi> d \<subseteq> ball \<xi> r"
+ by (auto simp: cball_subset_ball_iff)
+ then have "g holomorphic_on cball \<xi> d"
+ by (rule holomorphic_on_subset [OF holg])
+ then have "closed (g ` cball \<xi> d)"
+ by (simp add: compact_imp_closed compact_continuous_image holomorphic_on_imp_continuous_on)
+ moreover have "g ` cball \<xi> d \<noteq> {}"
+ using \<open>0 < d\<close> by auto
+ ultimately obtain x where x: "x \<in> g ` cball \<xi> d" and "\<And>y. y \<in> g ` cball \<xi> d \<Longrightarrow> dist 0 x \<le> dist 0 y"
+ by (rule distance_attains_inf) blast
+ then have leg: "\<And>w. w \<in> cball \<xi> d \<Longrightarrow> norm x \<le> norm (g w)"
+ by auto
+ have "ball \<xi> d \<subseteq> cball \<xi> d" by auto
+ also have "... \<subseteq> ball \<xi> e" using \<open>0 < d\<close> d_def by auto
+ also have "... \<subseteq> S" by (rule e)
+ finally have dS: "ball \<xi> d \<subseteq> S" .
+ moreover have "x \<noteq> 0" using gnz x \<open>d < r\<close> by auto
+ ultimately show ?thesis
+ apply (rule_tac k="norm x" and n=n and r=d in that)
+ using \<open>d < r\<close> leg
+ apply (auto simp: \<open>0 < d\<close> fne norm_mult norm_power algebra_simps mult_right_mono)
+ done
+qed
+
+lemma
+ assumes holf: "f holomorphic_on (S - {\<xi>})" and \<xi>: "\<xi> \<in> interior S"
+ shows holomorphic_on_extend_lim:
+ "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
+ ((\<lambda>z. (z - \<xi>) * f z) \<longlongrightarrow> 0) (at \<xi>)"
+ (is "?P = ?Q")
+ and holomorphic_on_extend_bounded:
+ "(\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S - {\<xi>}. g z = f z)) \<longleftrightarrow>
+ (\<exists>B. eventually (\<lambda>z. norm(f z) \<le> B) (at \<xi>))"
+ (is "?P = ?R")
+proof -
+ obtain \<delta> where "0 < \<delta>" and \<delta>: "ball \<xi> \<delta> \<subseteq> S"
+ using \<xi> mem_interior by blast
+ have "?R" if holg: "g holomorphic_on S" and gf: "\<And>z. z \<in> S - {\<xi>} \<Longrightarrow> g z = f z" for g
+ proof -
+ have *: "\<forall>\<^sub>F z in at \<xi>. dist (g z) (g \<xi>) < 1 \<longrightarrow> cmod (f z) \<le> cmod (g \<xi>) + 1"
+ apply (simp add: eventually_at)
+ apply (rule_tac x="\<delta>" in exI)
+ using \<delta> \<open>0 < \<delta>\<close>
+ apply (clarsimp simp:)
+ apply (drule_tac c=x in subsetD)
+ apply (simp add: dist_commute)
+ by (metis DiffI add.commute diff_le_eq dist_norm gf le_less_trans less_eq_real_def norm_triangle_ineq2 singletonD)
+ have "continuous_on (interior S) g"
+ by (meson continuous_on_subset holg holomorphic_on_imp_continuous_on interior_subset)
+ then have "\<And>x. x \<in> interior S \<Longrightarrow> (g \<longlongrightarrow> g x) (at x)"
+ using continuous_on_interior continuous_within holg holomorphic_on_imp_continuous_on by blast
+ then have "(g \<longlongrightarrow> g \<xi>) (at \<xi>)"
+ by (simp add: \<xi>)
+ then show ?thesis
+ apply (rule_tac x="norm(g \<xi>) + 1" in exI)
+ apply (rule eventually_mp [OF * tendstoD [where e=1]], auto)
+ done
+ qed
+ moreover have "?Q" if "\<forall>\<^sub>F z in at \<xi>. cmod (f z) \<le> B" for B
+ by (rule lim_null_mult_right_bounded [OF _ that]) (simp add: LIM_zero)
+ moreover have "?P" if "(\<lambda>z. (z - \<xi>) * f z) \<midarrow>\<xi>\<rightarrow> 0"
+ proof -
+ define h where [abs_def]: "h z = (z - \<xi>)^2 * f z" for z
+ have h0: "(h has_field_derivative 0) (at \<xi>)"
+ apply (simp add: h_def has_field_derivative_iff)
+ apply (rule Lim_transform_within [OF that, of 1])
+ apply (auto simp: field_split_simps power2_eq_square)
+ done
+ have holh: "h holomorphic_on S"
+ proof (simp add: holomorphic_on_def, clarify)
+ fix z assume "z \<in> S"
+ show "h field_differentiable at z within S"
+ proof (cases "z = \<xi>")
+ case True then show ?thesis
+ using field_differentiable_at_within field_differentiable_def h0 by blast
+ next
+ case False
+ then have "f field_differentiable at z within S"
+ using holomorphic_onD [OF holf, of z] \<open>z \<in> S\<close>
+ unfolding field_differentiable_def has_field_derivative_iff
+ by (force intro: exI [where x="dist \<xi> z"] elim: Lim_transform_within_set [unfolded eventually_at])
+ then show ?thesis
+ by (simp add: h_def power2_eq_square derivative_intros)
+ qed
+ qed
+ define g where [abs_def]: "g z = (if z = \<xi> then deriv h \<xi> else (h z - h \<xi>) / (z - \<xi>))" for z
+ have holg: "g holomorphic_on S"
+ unfolding g_def by (rule pole_lemma [OF holh \<xi>])
+ show ?thesis
+ apply (rule_tac x="\<lambda>z. if z = \<xi> then deriv g \<xi> else (g z - g \<xi>)/(z - \<xi>)" in exI)
+ apply (rule conjI)
+ apply (rule pole_lemma [OF holg \<xi>])
+ apply (auto simp: g_def power2_eq_square divide_simps)
+ using h0 apply (simp add: h0 DERIV_imp_deriv h_def power2_eq_square)
+ done
+ qed
+ ultimately show "?P = ?Q" and "?P = ?R"
+ by meson+
+qed
+
+lemma pole_at_infinity:
+ assumes holf: "f holomorphic_on UNIV" and lim: "((inverse o f) \<longlongrightarrow> l) at_infinity"
+ obtains a n where "\<And>z. f z = (\<Sum>i\<le>n. a i * z^i)"
+proof (cases "l = 0")
+ case False
+ with tendsto_inverse [OF lim] show ?thesis
+ apply (rule_tac a="(\<lambda>n. inverse l)" and n=0 in that)
+ apply (simp add: Liouville_weak [OF holf, of "inverse l"])
+ done
+next
+ case True
+ then have [simp]: "l = 0" .
+ show ?thesis
+ proof (cases "\<exists>r. 0 < r \<and> (\<forall>z \<in> ball 0 r - {0}. f(inverse z) \<noteq> 0)")
+ case True
+ then obtain r where "0 < r" and r: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> f(inverse z) \<noteq> 0"
+ by auto
+ have 1: "inverse \<circ> f \<circ> inverse holomorphic_on ball 0 r - {0}"
+ by (rule holomorphic_on_compose holomorphic_intros holomorphic_on_subset [OF holf] | force simp: r)+
+ have 2: "0 \<in> interior (ball 0 r)"
+ using \<open>0 < r\<close> by simp
+ have "\<exists>B. 0<B \<and> eventually (\<lambda>z. cmod ((inverse \<circ> f \<circ> inverse) z) \<le> B) (at 0)"
+ apply (rule exI [where x=1])
+ apply simp
+ using tendstoD [OF lim [unfolded lim_at_infinity_0] zero_less_one]
+ apply (rule eventually_mono)
+ apply (simp add: dist_norm)
+ done
+ with holomorphic_on_extend_bounded [OF 1 2]
+ obtain g where holg: "g holomorphic_on ball 0 r"
+ and geq: "\<And>z. z \<in> ball 0 r - {0} \<Longrightarrow> g z = (inverse \<circ> f \<circ> inverse) z"
+ by meson
+ have ifi0: "(inverse \<circ> f \<circ> inverse) \<midarrow>0\<rightarrow> 0"
+ using \<open>l = 0\<close> lim lim_at_infinity_0 by blast
+ have g2g0: "g \<midarrow>0\<rightarrow> g 0"
+ using \<open>0 < r\<close> centre_in_ball continuous_at continuous_on_eq_continuous_at holg
+ by (blast intro: holomorphic_on_imp_continuous_on)
+ have g2g1: "g \<midarrow>0\<rightarrow> 0"
+ apply (rule Lim_transform_within_open [OF ifi0 open_ball [of 0 r]])
+ using \<open>0 < r\<close> by (auto simp: geq)
+ have [simp]: "g 0 = 0"
+ by (rule tendsto_unique [OF _ g2g0 g2g1]) simp
+ have "ball 0 r - {0::complex} \<noteq> {}"
+ using \<open>0 < r\<close>
+ apply (clarsimp simp: ball_def dist_norm)
+ apply (drule_tac c="of_real r/2" in subsetD, auto)
+ done
+ then obtain w::complex where "w \<noteq> 0" and w: "norm w < r" by force
+ then have "g w \<noteq> 0" by (simp add: geq r)
+ obtain B n e where "0 < B" "0 < e" "e \<le> r"
+ and leg: "\<And>w. norm w < e \<Longrightarrow> B * cmod w ^ n \<le> cmod (g w)"
+ apply (rule holomorphic_lower_bound_difference [OF holg open_ball connected_ball, of 0 w])
+ using \<open>0 < r\<close> w \<open>g w \<noteq> 0\<close> by (auto simp: ball_subset_ball_iff)
+ have "cmod (f z) \<le> cmod z ^ n / B" if "2/e \<le> cmod z" for z
+ proof -
+ have ize: "inverse z \<in> ball 0 e - {0}" using that \<open>0 < e\<close>
+ by (auto simp: norm_divide field_split_simps algebra_simps)
+ then have [simp]: "z \<noteq> 0" and izr: "inverse z \<in> ball 0 r - {0}" using \<open>e \<le> r\<close>
+ by auto
+ then have [simp]: "f z \<noteq> 0"
+ using r [of "inverse z"] by simp
+ have [simp]: "f z = inverse (g (inverse z))"
+ using izr geq [of "inverse z"] by simp
+ show ?thesis using ize leg [of "inverse z"] \<open>0 < B\<close> \<open>0 < e\<close>
+ by (simp add: field_split_simps norm_divide algebra_simps)
+ qed
+ then show ?thesis
+ apply (rule_tac a = "\<lambda>k. (deriv ^^ k) f 0 / (fact k)" and n=n in that)
+ apply (rule_tac A = "2/e" and B = "1/B" in Liouville_polynomial [OF holf], simp)
+ done
+ next
+ case False
+ then have fi0: "\<And>r. r > 0 \<Longrightarrow> \<exists>z\<in>ball 0 r - {0}. f (inverse z) = 0"
+ by simp
+ have fz0: "f z = 0" if "0 < r" and lt1: "\<And>x. x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> inverse (cmod (f (inverse x))) < 1"
+ for z r
+ proof -
+ have f0: "(f \<longlongrightarrow> 0) at_infinity"
+ proof -
+ have DIM_complex[intro]: "2 \<le> DIM(complex)" \<comment> \<open>should not be necessary!\<close>
+ by simp
+ have "f (inverse x) \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> 1 < cmod (f (inverse x))" for x
+ using lt1[of x] by (auto simp: field_simps)
+ then have **: "cmod (f (inverse x)) \<le> 1 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> cmod x < r \<Longrightarrow> f (inverse x) = 0" for x
+ by force
+ then have *: "(f \<circ> inverse) ` (ball 0 r - {0}) \<subseteq> {0} \<union> - ball 0 1"
+ by force
+ have "continuous_on (inverse ` (ball 0 r - {0})) f"
+ using continuous_on_subset holf holomorphic_on_imp_continuous_on by blast
+ then have "connected ((f \<circ> inverse) ` (ball 0 r - {0}))"
+ apply (intro connected_continuous_image continuous_intros)
+ apply (force intro: connected_punctured_ball)+
+ done
+ then have "{0} \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {} \<or> - ball 0 1 \<inter> (f \<circ> inverse) ` (ball 0 r - {0}) = {}"
+ by (rule connected_closedD) (use * in auto)
+ then have "w \<noteq> 0 \<Longrightarrow> cmod w < r \<Longrightarrow> f (inverse w) = 0" for w
+ using fi0 **[of w] \<open>0 < r\<close>
+ apply (auto simp add: inf.commute [of "- ball 0 1"] Diff_eq [symmetric] image_subset_iff dest: less_imp_le)
+ apply fastforce
+ apply (drule bspec [of _ _ w])
+ apply (auto dest: less_imp_le)
+ done
+ then show ?thesis
+ apply (simp add: lim_at_infinity_0)
+ apply (rule tendsto_eventually)
+ apply (simp add: eventually_at)
+ apply (rule_tac x=r in exI)
+ apply (simp add: \<open>0 < r\<close> dist_norm)
+ done
+ qed
+ obtain w where "w \<in> ball 0 r - {0}" and "f (inverse w) = 0"
+ using False \<open>0 < r\<close> by blast
+ then show ?thesis
+ by (auto simp: f0 Liouville_weak [OF holf, of 0])
+ qed
+ show ?thesis
+ apply (rule that [of "\<lambda>n. 0" 0])
+ using lim [unfolded lim_at_infinity_0]
+ apply (simp add: Lim_at dist_norm norm_inverse)
+ apply (drule_tac x=1 in spec)
+ using fz0 apply auto
+ done
+ qed
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>Entire proper functions are precisely the non-trivial polynomials\<close>
+
+lemma proper_map_polyfun:
+ fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
+ assumes "closed S" and "compact K" and c: "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
+ shows "compact (S \<inter> {z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
+proof -
+ obtain B where "B > 0" and B: "\<And>x. x \<in> K \<Longrightarrow> norm x \<le> B"
+ by (metis compact_imp_bounded \<open>compact K\<close> bounded_pos)
+ have *: "norm x \<le> b"
+ if "\<And>x. b \<le> norm x \<Longrightarrow> B + 1 \<le> norm (\<Sum>i\<le>n. c i * x ^ i)"
+ "(\<Sum>i\<le>n. c i * x ^ i) \<in> K" for b x
+ proof -
+ have "norm (\<Sum>i\<le>n. c i * x ^ i) \<le> B"
+ using B that by blast
+ moreover have "\<not> B + 1 \<le> B"
+ by simp
+ ultimately show "norm x \<le> b"
+ using that by (metis (no_types) less_eq_real_def not_less order_trans)
+ qed
+ have "bounded {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
+ using Limits.polyfun_extremal [where c=c and B="B+1", OF c]
+ by (auto simp: bounded_pos eventually_at_infinity_pos *)
+ moreover have "closed ((\<lambda>z. (\<Sum>i\<le>n. c i * z ^ i)) -` K)"
+ apply (intro allI continuous_closed_vimage continuous_intros)
+ using \<open>compact K\<close> compact_eq_bounded_closed by blast
+ ultimately show ?thesis
+ using closed_Int_compact [OF \<open>closed S\<close>] compact_eq_bounded_closed
+ by (auto simp add: vimage_def)
+qed
+
+lemma proper_map_polyfun_univ:
+ fixes c :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,heine_borel}"
+ assumes "compact K" "c i \<noteq> 0" "1 \<le> i" "i \<le> n"
+ shows "compact ({z. (\<Sum>i\<le>n. c i * z^i) \<in> K})"
+ using proper_map_polyfun [of UNIV K c i n] assms by simp
+
+lemma proper_map_polyfun_eq:
+ assumes "f holomorphic_on UNIV"
+ shows "(\<forall>k. compact k \<longrightarrow> compact {z. f z \<in> k}) \<longleftrightarrow>
+ (\<exists>c n. 0 < n \<and> (c n \<noteq> 0) \<and> f = (\<lambda>z. \<Sum>i\<le>n. c i * z^i))"
+ (is "?lhs = ?rhs")
+proof
+ assume compf [rule_format]: ?lhs
+ have 2: "\<exists>k. 0 < k \<and> a k \<noteq> 0 \<and> f = (\<lambda>z. \<Sum>i \<le> k. a i * z ^ i)"
+ if "\<And>z. f z = (\<Sum>i\<le>n. a i * z ^ i)" for a n
+ proof (cases "\<forall>i\<le>n. 0<i \<longrightarrow> a i = 0")
+ case True
+ then have [simp]: "\<And>z. f z = a 0"
+ by (simp add: that sum.atMost_shift)
+ have False using compf [of "{a 0}"] by simp
+ then show ?thesis ..
+ next
+ case False
+ then obtain k where k: "0 < k" "k\<le>n" "a k \<noteq> 0" by force
+ define m where "m = (GREATEST k. k\<le>n \<and> a k \<noteq> 0)"
+ have m: "m\<le>n \<and> a m \<noteq> 0"
+ unfolding m_def
+ apply (rule GreatestI_nat [where b = n])
+ using k apply auto
+ done
+ have [simp]: "a i = 0" if "m < i" "i \<le> n" for i
+ using Greatest_le_nat [where b = "n" and P = "\<lambda>k. k\<le>n \<and> a k \<noteq> 0"]
+ using m_def not_le that by auto
+ have "k \<le> m"
+ unfolding m_def
+ apply (rule Greatest_le_nat [where b = "n"])
+ using k apply auto
+ done
+ with k m show ?thesis
+ by (rule_tac x=m in exI) (auto simp: that comm_monoid_add_class.sum.mono_neutral_right)
+ qed
+ have "((inverse \<circ> f) \<longlongrightarrow> 0) at_infinity"
+ proof (rule Lim_at_infinityI)
+ fix e::real assume "0 < e"
+ with compf [of "cball 0 (inverse e)"]
+ show "\<exists>B. \<forall>x. B \<le> cmod x \<longrightarrow> dist ((inverse \<circ> f) x) 0 \<le> e"
+ apply simp
+ apply (clarsimp simp add: compact_eq_bounded_closed bounded_pos norm_inverse)
+ apply (rule_tac x="b+1" in exI)
+ apply (metis inverse_inverse_eq less_add_same_cancel2 less_imp_inverse_less add.commute not_le not_less_iff_gr_or_eq order_trans zero_less_one)
+ done
+ qed
+ then show ?rhs
+ apply (rule pole_at_infinity [OF assms])
+ using 2 apply blast
+ done
+next
+ assume ?rhs
+ then obtain c n where "0 < n" "c n \<noteq> 0" "f = (\<lambda>z. \<Sum>i\<le>n. c i * z ^ i)" by blast
+ then have "compact {z. f z \<in> k}" if "compact k" for k
+ by (auto intro: proper_map_polyfun_univ [OF that])
+ then show ?lhs by blast
+qed
+
+subsection \<open>Relating invertibility and nonvanishing of derivative\<close>
+
+lemma has_complex_derivative_locally_injective:
+ assumes holf: "f holomorphic_on S"
+ and S: "\<xi> \<in> S" "open S"
+ and dnz: "deriv f \<xi> \<noteq> 0"
+ obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
+proof -
+ have *: "\<exists>d>0. \<forall>x. dist \<xi> x < d \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) < e" if "e > 0" for e
+ proof -
+ have contdf: "continuous_on S (deriv f)"
+ by (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on \<open>open S\<close>)
+ obtain \<delta> where "\<delta>>0" and \<delta>: "\<And>x. \<lbrakk>x \<in> S; dist x \<xi> \<le> \<delta>\<rbrakk> \<Longrightarrow> cmod (deriv f x - deriv f \<xi>) \<le> e/2"
+ using continuous_onE [OF contdf \<open>\<xi> \<in> S\<close>, of "e/2"] \<open>0 < e\<close>
+ by (metis dist_complex_def half_gt_zero less_imp_le)
+ obtain \<epsilon> where "\<epsilon>>0" "ball \<xi> \<epsilon> \<subseteq> S"
+ by (metis openE [OF \<open>open S\<close> \<open>\<xi> \<in> S\<close>])
+ with \<open>\<delta>>0\<close> have "\<exists>\<delta>>0. \<forall>x. dist \<xi> x < \<delta> \<longrightarrow> onorm (\<lambda>v. deriv f x * v - deriv f \<xi> * v) \<le> e/2"
+ apply (rule_tac x="min \<delta> \<epsilon>" in exI)
+ apply (intro conjI allI impI Operator_Norm.onorm_le)
+ apply simp
+ apply (simp only: Rings.ring_class.left_diff_distrib [symmetric] norm_mult)
+ apply (rule mult_right_mono [OF \<delta>])
+ apply (auto simp: dist_commute Rings.ordered_semiring_class.mult_right_mono \<delta>)
+ done
+ with \<open>e>0\<close> show ?thesis by force
+ qed
+ have "inj ((*) (deriv f \<xi>))"
+ using dnz by simp
+ then obtain g' where g': "linear g'" "g' \<circ> (*) (deriv f \<xi>) = id"
+ using linear_injective_left_inverse [of "(*) (deriv f \<xi>)"]
+ by (auto simp: linear_times)
+ show ?thesis
+ apply (rule has_derivative_locally_injective [OF S, where f=f and f' = "\<lambda>z h. deriv f z * h" and g' = g'])
+ using g' *
+ apply (simp_all add: linear_conv_bounded_linear that)
+ using DERIV_deriv_iff_field_differentiable has_field_derivative_imp_has_derivative holf
+ holomorphic_on_imp_differentiable_at \<open>open S\<close> apply blast
+ done
+qed
+
+lemma has_complex_derivative_locally_invertible:
+ assumes holf: "f holomorphic_on S"
+ and S: "\<xi> \<in> S" "open S"
+ and dnz: "deriv f \<xi> \<noteq> 0"
+ obtains r where "r > 0" "ball \<xi> r \<subseteq> S" "open (f ` (ball \<xi> r))" "inj_on f (ball \<xi> r)"
+proof -
+ obtain r where "r > 0" "ball \<xi> r \<subseteq> S" "inj_on f (ball \<xi> r)"
+ by (blast intro: that has_complex_derivative_locally_injective [OF assms])
+ then have \<xi>: "\<xi> \<in> ball \<xi> r" by simp
+ then have nc: "\<not> f constant_on ball \<xi> r"
+ using \<open>inj_on f (ball \<xi> r)\<close> injective_not_constant by fastforce
+ have holf': "f holomorphic_on ball \<xi> r"
+ using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ have "open (f ` ball \<xi> r)"
+ apply (rule open_mapping_thm [OF holf'])
+ using nc apply auto
+ done
+ then show ?thesis
+ using \<open>0 < r\<close> \<open>ball \<xi> r \<subseteq> S\<close> \<open>inj_on f (ball \<xi> r)\<close> that by blast
+qed
+
+lemma holomorphic_injective_imp_regular:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ and "\<xi> \<in> S"
+ shows "deriv f \<xi> \<noteq> 0"
+proof -
+ obtain r where "r>0" and r: "ball \<xi> r \<subseteq> S" using assms by (blast elim!: openE)
+ have holf': "f holomorphic_on ball \<xi> r"
+ using \<open>ball \<xi> r \<subseteq> S\<close> holf holomorphic_on_subset by blast
+ show ?thesis
+ proof (cases "\<forall>n>0. (deriv ^^ n) f \<xi> = 0")
+ case True
+ have fcon: "f w = f \<xi>" if "w \<in> ball \<xi> r" for w
+ apply (rule holomorphic_fun_eq_const_on_connected [OF holf'])
+ using True \<open>0 < r\<close> that by auto
+ have False
+ using fcon [of "\<xi> + r/2"] \<open>0 < r\<close> r injf unfolding inj_on_def
+ by (metis \<open>\<xi> \<in> S\<close> contra_subsetD dist_commute fcon mem_ball perfect_choose_dist)
+ then show ?thesis ..
+ next
+ case False
+ then obtain n0 where n0: "n0 > 0 \<and> (deriv ^^ n0) f \<xi> \<noteq> 0" by blast
+ define n where [abs_def]: "n = (LEAST n. n > 0 \<and> (deriv ^^ n) f \<xi> \<noteq> 0)"
+ have n_ne: "n > 0" "(deriv ^^ n) f \<xi> \<noteq> 0"
+ using def_LeastI [OF n_def n0] by auto
+ have n_min: "\<And>k. 0 < k \<Longrightarrow> k < n \<Longrightarrow> (deriv ^^ k) f \<xi> = 0"
+ using def_Least_le [OF n_def] not_le by auto
+ obtain g \<delta> where "0 < \<delta>"
+ and holg: "g holomorphic_on ball \<xi> \<delta>"
+ and fd: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> f w - f \<xi> = ((w - \<xi>) * g w) ^ n"
+ and gnz: "\<And>w. w \<in> ball \<xi> \<delta> \<Longrightarrow> g w \<noteq> 0"
+ apply (rule holomorphic_factor_order_of_zero_strong [OF holf \<open>open S\<close> \<open>\<xi> \<in> S\<close> n_ne])
+ apply (blast intro: n_min)+
+ done
+ show ?thesis
+ proof (cases "n=1")
+ case True
+ with n_ne show ?thesis by auto
+ next
+ case False
+ have holgw: "(\<lambda>w. (w - \<xi>) * g w) holomorphic_on ball \<xi> (min r \<delta>)"
+ apply (rule holomorphic_intros)+
+ using holg by (simp add: holomorphic_on_subset subset_ball)
+ have gd: "\<And>w. dist \<xi> w < \<delta> \<Longrightarrow> (g has_field_derivative deriv g w) (at w)"
+ using holg
+ by (simp add: DERIV_deriv_iff_field_differentiable holomorphic_on_def at_within_open_NO_MATCH)
+ have *: "\<And>w. w \<in> ball \<xi> (min r \<delta>)
+ \<Longrightarrow> ((\<lambda>w. (w - \<xi>) * g w) has_field_derivative ((w - \<xi>) * deriv g w + g w))
+ (at w)"
+ by (rule gd derivative_eq_intros | simp)+
+ have [simp]: "deriv (\<lambda>w. (w - \<xi>) * g w) \<xi> \<noteq> 0"
+ using * [of \<xi>] \<open>0 < \<delta>\<close> \<open>0 < r\<close> by (simp add: DERIV_imp_deriv gnz)
+ obtain T where "\<xi> \<in> T" "open T" and Tsb: "T \<subseteq> ball \<xi> (min r \<delta>)" and oimT: "open ((\<lambda>w. (w - \<xi>) * g w) ` T)"
+ apply (rule has_complex_derivative_locally_invertible [OF holgw, of \<xi>])
+ using \<open>0 < r\<close> \<open>0 < \<delta>\<close>
+ apply (simp_all add:)
+ by (meson open_ball centre_in_ball)
+ define U where "U = (\<lambda>w. (w - \<xi>) * g w) ` T"
+ have "open U" by (metis oimT U_def)
+ have "0 \<in> U"
+ apply (auto simp: U_def)
+ apply (rule image_eqI [where x = \<xi>])
+ apply (auto simp: \<open>\<xi> \<in> T\<close>)
+ done
+ then obtain \<epsilon> where "\<epsilon>>0" and \<epsilon>: "cball 0 \<epsilon> \<subseteq> U"
+ using \<open>open U\<close> open_contains_cball by blast
+ then have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> cball 0 \<epsilon>"
+ "\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> cball 0 \<epsilon>"
+ by (auto simp: norm_mult)
+ with \<epsilon> have "\<epsilon> * exp(2 * of_real pi * \<i> * (0/n)) \<in> U"
+ "\<epsilon> * exp(2 * of_real pi * \<i> * (1/n)) \<in> U" by blast+
+ then obtain y0 y1 where "y0 \<in> T" and y0: "(y0 - \<xi>) * g y0 = \<epsilon> * exp(2 * of_real pi * \<i> * (0/n))"
+ and "y1 \<in> T" and y1: "(y1 - \<xi>) * g y1 = \<epsilon> * exp(2 * of_real pi * \<i> * (1/n))"
+ by (auto simp: U_def)
+ then have "y0 \<in> ball \<xi> \<delta>" "y1 \<in> ball \<xi> \<delta>" using Tsb by auto
+ moreover have "y0 \<noteq> y1"
+ using y0 y1 \<open>\<epsilon> > 0\<close> complex_root_unity_eq_1 [of n 1] \<open>n > 0\<close> False by auto
+ moreover have "T \<subseteq> S"
+ by (meson Tsb min.cobounded1 order_trans r subset_ball)
+ ultimately have False
+ using inj_onD [OF injf, of y0 y1] \<open>y0 \<in> T\<close> \<open>y1 \<in> T\<close>
+ using fd [of y0] fd [of y1] complex_root_unity [of n 1] n_ne
+ apply (simp add: y0 y1 power_mult_distrib)
+ apply (force simp: algebra_simps)
+ done
+ then show ?thesis ..
+ qed
+ qed
+qed
+
+text\<open>Hence a nice clean inverse function theorem\<close>
+
+lemma has_field_derivative_inverse_strong:
+ fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
+ shows "\<lbrakk>DERIV f x :> f'; f' \<noteq> 0; open S; x \<in> S; continuous_on S f;
+ \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
+ \<Longrightarrow> DERIV g (f x) :> inverse (f')"
+ unfolding has_field_derivative_def
+ by (rule has_derivative_inverse_strong [of S x f g]) auto
+
+lemma has_field_derivative_inverse_strong_x:
+ fixes f :: "'a::{euclidean_space,real_normed_field} \<Rightarrow> 'a"
+ shows "\<lbrakk>DERIV f (g y) :> f'; f' \<noteq> 0; open S; continuous_on S f; g y \<in> S; f(g y) = y;
+ \<And>z. z \<in> S \<Longrightarrow> g (f z) = z\<rbrakk>
+ \<Longrightarrow> DERIV g y :> inverse (f')"
+ unfolding has_field_derivative_def
+ by (rule has_derivative_inverse_strong_x [of S g y f]) auto
+
+proposition holomorphic_has_inverse:
+ assumes holf: "f holomorphic_on S"
+ and "open S" and injf: "inj_on f S"
+ obtains g where "g holomorphic_on (f ` S)"
+ "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
+ "\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
+proof -
+ have ofs: "open (f ` S)"
+ by (rule open_mapping_thm3 [OF assms])
+ have contf: "continuous_on S f"
+ by (simp add: holf holomorphic_on_imp_continuous_on)
+ have *: "(the_inv_into S f has_field_derivative inverse (deriv f z)) (at (f z))" if "z \<in> S" for z
+ proof -
+ have 1: "(f has_field_derivative deriv f z) (at z)"
+ using DERIV_deriv_iff_field_differentiable \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_on_imp_differentiable_at
+ by blast
+ have 2: "deriv f z \<noteq> 0"
+ using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
+ show ?thesis
+ apply (rule has_field_derivative_inverse_strong [OF 1 2 \<open>open S\<close> \<open>z \<in> S\<close>])
+ apply (simp add: holf holomorphic_on_imp_continuous_on)
+ by (simp add: injf the_inv_into_f_f)
+ qed
+ show ?thesis
+ proof
+ show "the_inv_into S f holomorphic_on f ` S"
+ by (simp add: holomorphic_on_open ofs) (blast intro: *)
+ next
+ fix z assume "z \<in> S"
+ have "deriv f z \<noteq> 0"
+ using \<open>z \<in> S\<close> \<open>open S\<close> holf holomorphic_injective_imp_regular injf by blast
+ then show "deriv f z * deriv (the_inv_into S f) (f z) = 1"
+ using * [OF \<open>z \<in> S\<close>] by (simp add: DERIV_imp_deriv)
+ next
+ fix z assume "z \<in> S"
+ show "the_inv_into S f (f z) = z"
+ by (simp add: \<open>z \<in> S\<close> injf the_inv_into_f_f)
+ qed
+qed
+
+subsection\<open>The Schwarz Lemma\<close>
+
+lemma Schwarz1:
+ assumes holf: "f holomorphic_on S"
+ and contf: "continuous_on (closure S) f"
+ and S: "open S" "connected S"
+ and boS: "bounded S"
+ and "S \<noteq> {}"
+ obtains w where "w \<in> frontier S"
+ "\<And>z. z \<in> closure S \<Longrightarrow> norm (f z) \<le> norm (f w)"
+proof -
+ have connf: "continuous_on (closure S) (norm o f)"
+ using contf continuous_on_compose continuous_on_norm_id by blast
+ have coc: "compact (closure S)"
+ by (simp add: \<open>bounded S\<close> bounded_closure compact_eq_bounded_closed)
+ then obtain x where x: "x \<in> closure S" and xmax: "\<And>z. z \<in> closure S \<Longrightarrow> norm(f z) \<le> norm(f x)"
+ apply (rule bexE [OF continuous_attains_sup [OF _ _ connf]])
+ using \<open>S \<noteq> {}\<close> apply auto
+ done
+ then show ?thesis
+ proof (cases "x \<in> frontier S")
+ case True
+ then show ?thesis using that xmax by blast
+ next
+ case False
+ then have "x \<in> S"
+ using \<open>open S\<close> frontier_def interior_eq x by auto
+ then have "f constant_on S"
+ apply (rule maximum_modulus_principle [OF holf S \<open>open S\<close> order_refl])
+ using closure_subset apply (blast intro: xmax)
+ done
+ then have "f constant_on (closure S)"
+ by (rule constant_on_closureI [OF _ contf])
+ then obtain c where c: "\<And>x. x \<in> closure S \<Longrightarrow> f x = c"
+ by (meson constant_on_def)
+ obtain w where "w \<in> frontier S"
+ by (metis coc all_not_in_conv assms(6) closure_UNIV frontier_eq_empty not_compact_UNIV)
+ then show ?thesis
+ by (simp add: c frontier_def that)
+ qed
+qed
+
+lemma Schwarz2:
+ "\<lbrakk>f holomorphic_on ball 0 r;
+ 0 < s; ball w s \<subseteq> ball 0 r;
+ \<And>z. norm (w-z) < s \<Longrightarrow> norm(f z) \<le> norm(f w)\<rbrakk>
+ \<Longrightarrow> f constant_on ball 0 r"
+by (rule maximum_modulus_principle [where U = "ball w s" and \<xi> = w]) (simp_all add: dist_norm)
+
+lemma Schwarz3:
+ assumes holf: "f holomorphic_on (ball 0 r)" and [simp]: "f 0 = 0"
+ obtains h where "h holomorphic_on (ball 0 r)" and "\<And>z. norm z < r \<Longrightarrow> f z = z * (h z)" and "deriv f 0 = h 0"
+proof -
+ define h where "h z = (if z = 0 then deriv f 0 else f z / z)" for z
+ have d0: "deriv f 0 = h 0"
+ by (simp add: h_def)
+ moreover have "h holomorphic_on (ball 0 r)"
+ by (rule pole_theorem_open_0 [OF holf, of 0]) (auto simp: h_def)
+ moreover have "norm z < r \<Longrightarrow> f z = z * h z" for z
+ by (simp add: h_def)
+ ultimately show ?thesis
+ using that by blast
+qed
+
+proposition Schwarz_Lemma:
+ assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
+ and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
+ and \<xi>: "norm \<xi> < 1"
+ shows "norm (f \<xi>) \<le> norm \<xi>" and "norm(deriv f 0) \<le> 1"
+ and "((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
+ \<or> norm(deriv f 0) = 1)
+ \<Longrightarrow> \<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1"
+ (is "?P \<Longrightarrow> ?Q")
+proof -
+ obtain h where holh: "h holomorphic_on (ball 0 1)"
+ and fz_eq: "\<And>z. norm z < 1 \<Longrightarrow> f z = z * (h z)" and df0: "deriv f 0 = h 0"
+ by (rule Schwarz3 [OF holf]) auto
+ have noh_le: "norm (h z) \<le> 1" if z: "norm z < 1" for z
+ proof -
+ have "norm (h z) < a" if a: "1 < a" for a
+ proof -
+ have "max (inverse a) (norm z) < 1"
+ using z a by (simp_all add: inverse_less_1_iff)
+ then obtain r where r: "max (inverse a) (norm z) < r" and "r < 1"
+ using Rats_dense_in_real by blast
+ then have nzr: "norm z < r" and ira: "inverse r < a"
+ using z a less_imp_inverse_less by force+
+ then have "0 < r"
+ by (meson norm_not_less_zero not_le order.strict_trans2)
+ have holh': "h holomorphic_on ball 0 r"
+ by (meson holh \<open>r < 1\<close> holomorphic_on_subset less_eq_real_def subset_ball)
+ have conth': "continuous_on (cball 0 r) h"
+ by (meson \<open>r < 1\<close> dual_order.trans holh holomorphic_on_imp_continuous_on holomorphic_on_subset mem_ball_0 mem_cball_0 not_less subsetI)
+ obtain w where w: "norm w = r" and lenw: "\<And>z. norm z < r \<Longrightarrow> norm(h z) \<le> norm(h w)"
+ apply (rule Schwarz1 [OF holh']) using conth' \<open>0 < r\<close> by auto
+ have "h w = f w / w" using fz_eq \<open>r < 1\<close> nzr w by auto
+ then have "cmod (h z) < inverse r"
+ by (metis \<open>0 < r\<close> \<open>r < 1\<close> divide_strict_right_mono inverse_eq_divide
+ le_less_trans lenw no norm_divide nzr w)
+ then show ?thesis using ira by linarith
+ qed
+ then show "norm (h z) \<le> 1"
+ using not_le by blast
+ qed
+ show "cmod (f \<xi>) \<le> cmod \<xi>"
+ proof (cases "\<xi> = 0")
+ case True then show ?thesis by auto
+ next
+ case False
+ then show ?thesis
+ by (simp add: noh_le fz_eq \<xi> mult_left_le norm_mult)
+ qed
+ show no_df0: "norm(deriv f 0) \<le> 1"
+ by (simp add: \<open>\<And>z. cmod z < 1 \<Longrightarrow> cmod (h z) \<le> 1\<close> df0)
+ show "?Q" if "?P"
+ using that
+ proof
+ assume "\<exists>z. cmod z < 1 \<and> z \<noteq> 0 \<and> cmod (f z) = cmod z"
+ then obtain \<gamma> where \<gamma>: "cmod \<gamma> < 1" "\<gamma> \<noteq> 0" "cmod (f \<gamma>) = cmod \<gamma>" by blast
+ then have [simp]: "norm (h \<gamma>) = 1"
+ by (simp add: fz_eq norm_mult)
+ have "ball \<gamma> (1 - cmod \<gamma>) \<subseteq> ball 0 1"
+ by (simp add: ball_subset_ball_iff)
+ moreover have "\<And>z. cmod (\<gamma> - z) < 1 - cmod \<gamma> \<Longrightarrow> cmod (h z) \<le> cmod (h \<gamma>)"
+ apply (simp add: algebra_simps)
+ by (metis add_diff_cancel_left' diff_diff_eq2 le_less_trans noh_le norm_triangle_ineq4)
+ ultimately obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
+ using Schwarz2 [OF holh, of "1 - norm \<gamma>" \<gamma>, unfolded constant_on_def] \<gamma> by auto
+ then have "norm c = 1"
+ using \<gamma> by force
+ with c show ?thesis
+ using fz_eq by auto
+ next
+ assume [simp]: "cmod (deriv f 0) = 1"
+ then obtain c where c: "\<And>z. norm z < 1 \<Longrightarrow> h z = c"
+ using Schwarz2 [OF holh zero_less_one, of 0, unfolded constant_on_def] df0 noh_le
+ by auto
+ moreover have "norm c = 1" using df0 c by auto
+ ultimately show ?thesis
+ using fz_eq by auto
+ qed
+qed
+
+corollary Schwarz_Lemma':
+ assumes holf: "f holomorphic_on (ball 0 1)" and [simp]: "f 0 = 0"
+ and no: "\<And>z. norm z < 1 \<Longrightarrow> norm (f z) < 1"
+ shows "((\<forall>\<xi>. norm \<xi> < 1 \<longrightarrow> norm (f \<xi>) \<le> norm \<xi>)
+ \<and> norm(deriv f 0) \<le> 1)
+ \<and> (((\<exists>z. norm z < 1 \<and> z \<noteq> 0 \<and> norm(f z) = norm z)
+ \<or> norm(deriv f 0) = 1)
+ \<longrightarrow> (\<exists>\<alpha>. (\<forall>z. norm z < 1 \<longrightarrow> f z = \<alpha> * z) \<and> norm \<alpha> = 1))"
+ using Schwarz_Lemma [OF assms]
+ by (metis (no_types) norm_eq_zero zero_less_one)
+
+subsection\<open>The Schwarz reflection principle\<close>
+
+lemma hol_pal_lem0:
+ assumes "d \<bullet> a \<le> k" "k \<le> d \<bullet> b"
+ obtains c where
+ "c \<in> closed_segment a b" "d \<bullet> c = k"
+ "\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k"
+ "\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z"
+proof -
+ obtain c where cin: "c \<in> closed_segment a b" and keq: "k = d \<bullet> c"
+ using connected_ivt_hyperplane [of "closed_segment a b" a b d k]
+ by (auto simp: assms)
+ have "closed_segment a c \<subseteq> {z. d \<bullet> z \<le> k}" "closed_segment c b \<subseteq> {z. k \<le> d \<bullet> z}"
+ unfolding segment_convex_hull using assms keq
+ by (auto simp: convex_halfspace_le convex_halfspace_ge hull_minimal)
+ then show ?thesis using cin that by fastforce
+qed
+
+lemma hol_pal_lem1:
+ assumes "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k" "d \<bullet> c \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof -
+ have "interior (convex hull {a, b, c}) \<subseteq> interior(S \<inter> {x. d \<bullet> x \<le> k})"
+ apply (rule interior_mono)
+ apply (rule hull_minimal)
+ apply (simp add: abc lek)
+ apply (rule convex_Int [OF \<open>convex S\<close> convex_halfspace_le])
+ done
+ also have "... \<subseteq> {z \<in> S. d \<bullet> z < k}"
+ by (force simp: interior_open [OF \<open>open S\<close>] \<open>d \<noteq> 0\<close>)
+ finally have *: "interior (convex hull {a, b, c}) \<subseteq> {z \<in> S. d \<bullet> z < k}" .
+ have "continuous_on (convex hull {a,b,c}) f"
+ using \<open>convex S\<close> contf abc continuous_on_subset subset_hull
+ by fastforce
+ moreover have "f holomorphic_on interior (convex hull {a,b,c})"
+ by (rule holomorphic_on_subset [OF holf1 *])
+ ultimately show ?thesis
+ using Cauchy_theorem_triangle_interior has_chain_integral_chain_integral3
+ by blast
+qed
+
+lemma hol_pal_lem2:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k" "d \<bullet> b \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> c \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem1 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 contf])
+next
+ case False
+ then have "d \<bullet> c > k" by force
+ obtain a' where a': "a' \<in> closed_segment b c" and "d \<bullet> a' = k"
+ and ba': "\<And>z. z \<in> closed_segment b a' \<Longrightarrow> d \<bullet> z \<le> k"
+ and a'c: "\<And>z. z \<in> closed_segment a' c \<Longrightarrow> k \<le> d \<bullet> z"
+ apply (rule hol_pal_lem0 [of d b k c, OF \<open>d \<bullet> b \<le> k\<close>])
+ using False by auto
+ obtain b' where b': "b' \<in> closed_segment a c" and "d \<bullet> b' = k"
+ and ab': "\<And>z. z \<in> closed_segment a b' \<Longrightarrow> d \<bullet> z \<le> k"
+ and b'c: "\<And>z. z \<in> closed_segment b' c \<Longrightarrow> k \<le> d \<bullet> z"
+ apply (rule hol_pal_lem0 [of d a k c, OF \<open>d \<bullet> a \<le> k\<close>])
+ using False by auto
+ have a'b': "a' \<in> S \<and> b' \<in> S"
+ using a' abc b' convex_contains_segment \<open>convex S\<close> by auto
+ have "continuous_on (closed_segment c a) f"
+ by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
+ then have 1: "contour_integral (linepath c a) f =
+ contour_integral (linepath c b') f + contour_integral (linepath b' a) f"
+ apply (rule contour_integral_split_linepath)
+ using b' by (simp add: closed_segment_commute)
+ have "continuous_on (closed_segment b c) f"
+ by (meson abc contf continuous_on_subset convex_contains_segment \<open>convex S\<close>)
+ then have 2: "contour_integral (linepath b c) f =
+ contour_integral (linepath b a') f + contour_integral (linepath a' c) f"
+ by (rule contour_integral_split_linepath [OF _ a'])
+ have 3: "contour_integral (reversepath (linepath b' a')) f =
+ - contour_integral (linepath b' a') f"
+ by (rule contour_integral_reversepath [OF valid_path_linepath])
+ have fcd_le: "f field_differentiable at x"
+ if "x \<in> interior S \<and> x \<in> interior {x. d \<bullet> x \<le> k}" for x
+ proof -
+ have "f holomorphic_on S \<inter> {c. d \<bullet> c < k}"
+ by (metis (no_types) Collect_conj_eq Collect_mem_eq holf1)
+ then have "\<exists>C D. x \<in> interior C \<inter> interior D \<and> f holomorphic_on interior C \<inter> interior D"
+ using that
+ by (metis Collect_mem_eq Int_Collect \<open>d \<noteq> 0\<close> interior_halfspace_le interior_open \<open>open S\<close>)
+ then show "f field_differentiable at x"
+ by (metis at_within_interior holomorphic_on_def interior_Int interior_interior)
+ qed
+ have ab_le: "\<And>x. x \<in> closed_segment a b \<Longrightarrow> d \<bullet> x \<le> k"
+ proof -
+ fix x :: complex
+ assume "x \<in> closed_segment a b"
+ then have "\<And>C. x \<in> C \<or> b \<notin> C \<or> a \<notin> C \<or> \<not> convex C"
+ by (meson contra_subsetD convex_contains_segment)
+ then show "d \<bullet> x \<le> k"
+ by (metis lek convex_halfspace_le mem_Collect_eq)
+ qed
+ have "continuous_on (S \<inter> {x. d \<bullet> x \<le> k}) f" using contf
+ by (simp add: continuous_on_subset)
+ then have "(f has_contour_integral 0)
+ (linepath a b +++ linepath b a' +++ linepath a' b' +++ linepath b' a)"
+ apply (rule Cauchy_theorem_convex [where K = "{}"])
+ apply (simp_all add: path_image_join convex_Int convex_halfspace_le \<open>convex S\<close> fcd_le ab_le
+ closed_segment_subset abc a'b' ba')
+ by (metis \<open>d \<bullet> a' = k\<close> \<open>d \<bullet> b' = k\<close> convex_contains_segment convex_halfspace_le lek(1) mem_Collect_eq order_refl)
+ then have 4: "contour_integral (linepath a b) f +
+ contour_integral (linepath b a') f +
+ contour_integral (linepath a' b') f +
+ contour_integral (linepath b' a) f = 0"
+ by (rule has_chain_integral_chain_integral4)
+ have fcd_ge: "f field_differentiable at x"
+ if "x \<in> interior S \<and> x \<in> interior {x. k \<le> d \<bullet> x}" for x
+ proof -
+ have f2: "f holomorphic_on S \<inter> {c. k < d \<bullet> c}"
+ by (metis (full_types) Collect_conj_eq Collect_mem_eq holf2)
+ have f3: "interior S = S"
+ by (simp add: interior_open \<open>open S\<close>)
+ then have "x \<in> S \<inter> interior {c. k \<le> d \<bullet> c}"
+ using that by simp
+ then show "f field_differentiable at x"
+ using f3 f2 unfolding holomorphic_on_def
+ by (metis (no_types) \<open>d \<noteq> 0\<close> at_within_interior interior_Int interior_halfspace_ge interior_interior)
+ qed
+ have "continuous_on (S \<inter> {x. k \<le> d \<bullet> x}) f" using contf
+ by (simp add: continuous_on_subset)
+ then have "(f has_contour_integral 0) (linepath a' c +++ linepath c b' +++ linepath b' a')"
+ apply (rule Cauchy_theorem_convex [where K = "{}"])
+ apply (simp_all add: path_image_join convex_Int convex_halfspace_ge \<open>convex S\<close>
+ fcd_ge closed_segment_subset abc a'b' a'c)
+ by (metis \<open>d \<bullet> a' = k\<close> b'c closed_segment_commute convex_contains_segment
+ convex_halfspace_ge ends_in_segment(2) mem_Collect_eq order_refl)
+ then have 5: "contour_integral (linepath a' c) f + contour_integral (linepath c b') f + contour_integral (linepath b' a') f = 0"
+ by (rule has_chain_integral_chain_integral3)
+ show ?thesis
+ using 1 2 3 4 5 by (metis add.assoc eq_neg_iff_add_eq_0 reversepath_linepath)
+qed
+
+lemma hol_pal_lem3:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S"
+ and "d \<noteq> 0" and lek: "d \<bullet> a \<le> k"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> b \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem2 [OF S abc \<open>d \<noteq> 0\<close> lek True holf1 holf2 contf])
+next
+ case False
+ show ?thesis
+ proof (cases "d \<bullet> c \<le> k")
+ case True
+ have "contour_integral (linepath c a) f +
+ contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f = 0"
+ by (rule hol_pal_lem2 [OF S \<open>c \<in> S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close> \<open>d \<noteq> 0\<close> \<open>d \<bullet> c \<le> k\<close> lek holf1 holf2 contf])
+ then show ?thesis
+ by (simp add: algebra_simps)
+ next
+ case False
+ have "contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f +
+ contour_integral (linepath a b) f = 0"
+ apply (rule hol_pal_lem2 [OF S \<open>b \<in> S\<close> \<open>c \<in> S\<close> \<open>a \<in> S\<close>, of "-d" "-k"])
+ using \<open>d \<noteq> 0\<close> \<open>\<not> d \<bullet> b \<le> k\<close> False by (simp_all add: holf1 holf2 contf)
+ then show ?thesis
+ by (simp add: algebra_simps)
+ qed
+qed
+
+lemma hol_pal_lem4:
+ assumes S: "convex S" "open S"
+ and abc: "a \<in> S" "b \<in> S" "c \<in> S" and "d \<noteq> 0"
+ and holf1: "f holomorphic_on {z. z \<in> S \<and> d \<bullet> z < k}"
+ and holf2: "f holomorphic_on {z. z \<in> S \<and> k < d \<bullet> z}"
+ and contf: "continuous_on S f"
+ shows "contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0"
+proof (cases "d \<bullet> a \<le> k")
+ case True show ?thesis
+ by (rule hol_pal_lem3 [OF S abc \<open>d \<noteq> 0\<close> True holf1 holf2 contf])
+next
+ case False
+ show ?thesis
+ apply (rule hol_pal_lem3 [OF S abc, of "-d" "-k"])
+ using \<open>d \<noteq> 0\<close> False by (simp_all add: holf1 holf2 contf)
+qed
+
+lemma holomorphic_on_paste_across_line:
+ assumes S: "open S" and "d \<noteq> 0"
+ and holf1: "f holomorphic_on (S \<inter> {z. d \<bullet> z < k})"
+ and holf2: "f holomorphic_on (S \<inter> {z. k < d \<bullet> z})"
+ and contf: "continuous_on S f"
+ shows "f holomorphic_on S"
+proof -
+ have *: "\<exists>t. open t \<and> p \<in> t \<and> continuous_on t f \<and>
+ (\<forall>a b c. convex hull {a, b, c} \<subseteq> t \<longrightarrow>
+ contour_integral (linepath a b) f +
+ contour_integral (linepath b c) f +
+ contour_integral (linepath c a) f = 0)"
+ if "p \<in> S" for p
+ proof -
+ obtain e where "e>0" and e: "ball p e \<subseteq> S"
+ using \<open>p \<in> S\<close> openE S by blast
+ then have "continuous_on (ball p e) f"
+ using contf continuous_on_subset by blast
+ moreover have "f holomorphic_on {z. dist p z < e \<and> d \<bullet> z < k}"
+ apply (rule holomorphic_on_subset [OF holf1])
+ using e by auto
+ moreover have "f holomorphic_on {z. dist p z < e \<and> k < d \<bullet> z}"
+ apply (rule holomorphic_on_subset [OF holf2])
+ using e by auto
+ ultimately show ?thesis
+ apply (rule_tac x="ball p e" in exI)
+ using \<open>e > 0\<close> e \<open>d \<noteq> 0\<close>
+ apply (simp add:, clarify)
+ apply (rule hol_pal_lem4 [of "ball p e" _ _ _ d _ k])
+ apply (auto simp: subset_hull)
+ done
+ qed
+ show ?thesis
+ by (blast intro: * Morera_local_triangle analytic_imp_holomorphic)
+qed
+
+proposition Schwarz_reflection:
+ assumes "open S" and cnjs: "cnj ` S \<subseteq> S"
+ and holf: "f holomorphic_on (S \<inter> {z. 0 < Im z})"
+ and contf: "continuous_on (S \<inter> {z. 0 \<le> Im z}) f"
+ and f: "\<And>z. \<lbrakk>z \<in> S; z \<in> \<real>\<rbrakk> \<Longrightarrow> (f z) \<in> \<real>"
+ shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj(f(cnj z))) holomorphic_on S"
+proof -
+ have 1: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. 0 < Im z})"
+ by (force intro: iffD1 [OF holomorphic_cong [OF refl] holf])
+ have cont_cfc: "continuous_on (S \<inter> {z. Im z \<le> 0}) (cnj o f o cnj)"
+ apply (intro continuous_intros continuous_on_compose continuous_on_subset [OF contf])
+ using cnjs apply auto
+ done
+ have "cnj \<circ> f \<circ> cnj field_differentiable at x within S \<inter> {z. Im z < 0}"
+ if "x \<in> S" "Im x < 0" "f field_differentiable at (cnj x) within S \<inter> {z. 0 < Im z}" for x
+ using that
+ apply (simp add: field_differentiable_def has_field_derivative_iff Lim_within dist_norm, clarify)
+ apply (rule_tac x="cnj f'" in exI)
+ apply (elim all_forward ex_forward conj_forward imp_forward asm_rl, clarify)
+ apply (drule_tac x="cnj xa" in bspec)
+ using cnjs apply force
+ apply (metis complex_cnj_cnj complex_cnj_diff complex_cnj_divide complex_mod_cnj)
+ done
+ then have hol_cfc: "(cnj o f o cnj) holomorphic_on (S \<inter> {z. Im z < 0})"
+ using holf cnjs
+ by (force simp: holomorphic_on_def)
+ have 2: "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on (S \<inter> {z. Im z < 0})"
+ apply (rule iffD1 [OF holomorphic_cong [OF refl]])
+ using hol_cfc by auto
+ have [simp]: "(S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}) = S"
+ by force
+ have "continuous_on ((S \<inter> {z. 0 \<le> Im z}) \<union> (S \<inter> {z. Im z \<le> 0}))
+ (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
+ apply (rule continuous_on_cases_local)
+ using cont_cfc contf
+ apply (simp_all add: closedin_closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
+ using f Reals_cnj_iff complex_is_Real_iff apply auto
+ done
+ then have 3: "continuous_on S (\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z)))"
+ by force
+ show ?thesis
+ apply (rule holomorphic_on_paste_across_line [OF \<open>open S\<close>, of "- \<i>" _ 0])
+ using 1 2 3
+ apply auto
+ done
+qed
+
+subsection\<open>Bloch's theorem\<close>
+
+lemma Bloch_lemma_0:
+ assumes holf: "f holomorphic_on cball 0 r" and "0 < r"
+ and [simp]: "f 0 = 0"
+ and le: "\<And>z. norm z < r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f 0)"
+ shows "ball 0 ((3 - 2 * sqrt 2) * r * norm(deriv f 0)) \<subseteq> f ` ball 0 r"
+proof -
+ have "sqrt 2 < 3/2"
+ by (rule real_less_lsqrt) (auto simp: power2_eq_square)
+ then have sq3: "0 < 3 - 2 * sqrt 2" by simp
+ show ?thesis
+ proof (cases "deriv f 0 = 0")
+ case True then show ?thesis by simp
+ next
+ case False
+ define C where "C = 2 * norm(deriv f 0)"
+ have "0 < C" using False by (simp add: C_def)
+ have holf': "f holomorphic_on ball 0 r" using holf
+ using ball_subset_cball holomorphic_on_subset by blast
+ then have holdf': "deriv f holomorphic_on ball 0 r"
+ by (rule holomorphic_deriv [OF _ open_ball])
+ have "Le1": "norm(deriv f z - deriv f 0) \<le> norm z / (r - norm z) * C"
+ if "norm z < r" for z
+ proof -
+ have T1: "norm(deriv f z - deriv f 0) \<le> norm z / (R - norm z) * C"
+ if R: "norm z < R" "R < r" for R
+ proof -
+ have "0 < R" using R
+ by (metis less_trans norm_zero zero_less_norm_iff)
+ have df_le: "\<And>x. norm x < r \<Longrightarrow> norm (deriv f x) \<le> C"
+ using le by (simp add: C_def)
+ have hol_df: "deriv f holomorphic_on cball 0 R"
+ apply (rule holomorphic_on_subset) using R holdf' by auto
+ have *: "((\<lambda>w. deriv f w / (w - z)) has_contour_integral 2 * pi * \<i> * deriv f z) (circlepath 0 R)"
+ if "norm z < R" for z
+ using \<open>0 < R\<close> that Cauchy_integral_formula_convex_simple [OF convex_cball hol_df, of _ "circlepath 0 R"]
+ by (force simp: winding_number_circlepath)
+ have **: "((\<lambda>x. deriv f x / (x - z) - deriv f x / x) has_contour_integral
+ of_real (2 * pi) * \<i> * (deriv f z - deriv f 0))
+ (circlepath 0 R)"
+ using has_contour_integral_diff [OF * [of z] * [of 0]] \<open>0 < R\<close> that
+ by (simp add: algebra_simps)
+ have [simp]: "\<And>x. norm x = R \<Longrightarrow> x \<noteq> z" using that(1) by blast
+ have "norm (deriv f x / (x - z) - deriv f x / x)
+ \<le> C * norm z / (R * (R - norm z))"
+ if "norm x = R" for x
+ proof -
+ have [simp]: "norm (deriv f x * x - deriv f x * (x - z)) =
+ norm (deriv f x) * norm z"
+ by (simp add: norm_mult right_diff_distrib')
+ show ?thesis
+ using \<open>0 < R\<close> \<open>0 < C\<close> R that
+ apply (simp add: norm_mult norm_divide divide_simps)
+ using df_le norm_triangle_ineq2 \<open>0 < C\<close> apply (auto intro!: mult_mono)
+ done
+ qed
+ then show ?thesis
+ using has_contour_integral_bound_circlepath
+ [OF **, of "C * norm z/(R*(R - norm z))"]
+ \<open>0 < R\<close> \<open>0 < C\<close> R
+ apply (simp add: norm_mult norm_divide)
+ apply (simp add: divide_simps mult.commute)
+ done
+ qed
+ obtain r' where r': "norm z < r'" "r' < r"
+ using Rats_dense_in_real [of "norm z" r] \<open>norm z < r\<close> by blast
+ then have [simp]: "closure {r'<..<r} = {r'..r}" by simp
+ show ?thesis
+ apply (rule continuous_ge_on_closure
+ [where f = "\<lambda>r. norm z / (r - norm z) * C" and s = "{r'<..<r}",
+ OF _ _ T1])
+ apply (intro continuous_intros)
+ using that r'
+ apply (auto simp: not_le)
+ done
+ qed
+ have "*": "(norm z - norm z^2/(r - norm z)) * norm(deriv f 0) \<le> norm(f z)"
+ if r: "norm z < r" for z
+ proof -
+ have 1: "\<And>x. x \<in> ball 0 r \<Longrightarrow>
+ ((\<lambda>z. f z - deriv f 0 * z) has_field_derivative deriv f x - deriv f 0)
+ (at x within ball 0 r)"
+ by (rule derivative_eq_intros holomorphic_derivI holf' | simp)+
+ have 2: "closed_segment 0 z \<subseteq> ball 0 r"
+ by (metis \<open>0 < r\<close> convex_ball convex_contains_segment dist_self mem_ball mem_ball_0 that)
+ have 3: "(\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C) integrable_on {0..1}"
+ apply (rule integrable_on_cmult_right [where 'b=real, simplified])
+ apply (rule integrable_on_cdivide [where 'b=real, simplified])
+ apply (rule integrable_on_cmult_left [where 'b=real, simplified])
+ apply (rule ident_integrable_on)
+ done
+ have 4: "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm z * norm z * x * C / (r - norm z)"
+ if x: "0 \<le> x" "x \<le> 1" for x
+ proof -
+ have [simp]: "x * norm z < r"
+ using r x by (meson le_less_trans mult_le_cancel_right2 norm_not_less_zero)
+ have "norm (deriv f (x *\<^sub>R z) - deriv f 0) \<le> norm (x *\<^sub>R z) / (r - norm (x *\<^sub>R z)) * C"
+ apply (rule Le1) using r x \<open>0 < r\<close> by simp
+ also have "... \<le> norm (x *\<^sub>R z) / (r - norm z) * C"
+ using r x \<open>0 < r\<close>
+ apply (simp add: field_split_simps)
+ by (simp add: \<open>0 < C\<close> mult.assoc mult_left_le_one_le ordered_comm_semiring_class.comm_mult_left_mono)
+ finally have "norm (deriv f (x *\<^sub>R z) - deriv f 0) * norm z \<le> norm (x *\<^sub>R z) / (r - norm z) * C * norm z"
+ by (rule mult_right_mono) simp
+ with x show ?thesis by (simp add: algebra_simps)
+ qed
+ have le_norm: "abc \<le> norm d - e \<Longrightarrow> norm(f - d) \<le> e \<Longrightarrow> abc \<le> norm f" for abc d e and f::complex
+ by (metis add_diff_cancel_left' add_diff_eq diff_left_mono norm_diff_ineq order_trans)
+ have "norm (integral {0..1} (\<lambda>x. (deriv f (x *\<^sub>R z) - deriv f 0) * z))
+ \<le> integral {0..1} (\<lambda>t. (norm z)\<^sup>2 * t / (r - norm z) * C)"
+ apply (rule integral_norm_bound_integral)
+ using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
+ apply (simp add: has_contour_integral_linepath has_integral_integrable_integral)
+ apply (rule 3)
+ apply (simp add: norm_mult power2_eq_square 4)
+ done
+ then have int_le: "norm (f z - deriv f 0 * z) \<le> (norm z)\<^sup>2 * norm(deriv f 0) / ((r - norm z))"
+ using contour_integral_primitive [OF 1, of "linepath 0 z"] 2
+ apply (simp add: has_contour_integral_linepath has_integral_integrable_integral C_def)
+ done
+ show ?thesis
+ apply (rule le_norm [OF _ int_le])
+ using \<open>norm z < r\<close>
+ apply (simp add: power2_eq_square divide_simps C_def norm_mult)
+ proof -
+ have "norm z * (norm (deriv f 0) * (r - norm z - norm z)) \<le> norm z * (norm (deriv f 0) * (r - norm z) - norm (deriv f 0) * norm z)"
+ by (simp add: algebra_simps)
+ then show "(norm z * (r - norm z) - norm z * norm z) * norm (deriv f 0) \<le> norm (deriv f 0) * norm z * (r - norm z) - norm z * norm z * norm (deriv f 0)"
+ by (simp add: algebra_simps)
+ qed
+ qed
+ have sq201 [simp]: "0 < (1 - sqrt 2 / 2)" "(1 - sqrt 2 / 2) < 1"
+ by (auto simp: sqrt2_less_2)
+ have 1: "continuous_on (closure (ball 0 ((1 - sqrt 2 / 2) * r))) f"
+ apply (rule continuous_on_subset [OF holomorphic_on_imp_continuous_on [OF holf]])
+ apply (subst closure_ball)
+ using \<open>0 < r\<close> mult_pos_pos sq201
+ apply (auto simp: cball_subset_cball_iff)
+ done
+ have 2: "open (f ` interior (ball 0 ((1 - sqrt 2 / 2) * r)))"
+ apply (rule open_mapping_thm [OF holf' open_ball connected_ball], force)
+ using \<open>0 < r\<close> mult_pos_pos sq201 apply (simp add: ball_subset_ball_iff)
+ using False \<open>0 < r\<close> centre_in_ball holf' holomorphic_nonconstant by blast
+ have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv f 0)) =
+ ball (f 0) ((3 - 2 * sqrt 2) * r * norm (deriv f 0))"
+ by simp
+ also have "... \<subseteq> f ` ball 0 ((1 - sqrt 2 / 2) * r)"
+ proof -
+ have 3: "(3 - 2 * sqrt 2) * r * norm (deriv f 0) \<le> norm (f z)"
+ if "norm z = (1 - sqrt 2 / 2) * r" for z
+ apply (rule order_trans [OF _ *])
+ using \<open>0 < r\<close>
+ apply (simp_all add: field_simps power2_eq_square that)
+ apply (simp add: mult.assoc [symmetric])
+ done
+ show ?thesis
+ apply (rule ball_subset_open_map_image [OF 1 2 _ bounded_ball])
+ using \<open>0 < r\<close> sq201 3 apply simp_all
+ using C_def \<open>0 < C\<close> sq3 apply force
+ done
+ qed
+ also have "... \<subseteq> f ` ball 0 r"
+ apply (rule image_subsetI [OF imageI], simp)
+ apply (erule less_le_trans)
+ using \<open>0 < r\<close> apply (auto simp: field_simps)
+ done
+ finally show ?thesis .
+ qed
+qed
+
+lemma Bloch_lemma:
+ assumes holf: "f holomorphic_on cball a r" and "0 < r"
+ and le: "\<And>z. z \<in> ball a r \<Longrightarrow> norm(deriv f z) \<le> 2 * norm(deriv f a)"
+ shows "ball (f a) ((3 - 2 * sqrt 2) * r * norm(deriv f a)) \<subseteq> f ` ball a r"
+proof -
+ have fz: "(\<lambda>z. f (a + z)) = f o (\<lambda>z. (a + z))"
+ by (simp add: o_def)
+ have hol0: "(\<lambda>z. f (a + z)) holomorphic_on cball 0 r"
+ unfolding fz by (intro holomorphic_intros holf holomorphic_on_compose | simp)+
+ then have [simp]: "\<And>x. norm x < r \<Longrightarrow> (\<lambda>z. f (a + z)) field_differentiable at x"
+ by (metis open_ball at_within_open ball_subset_cball diff_0 dist_norm holomorphic_on_def holomorphic_on_subset mem_ball norm_minus_cancel)
+ have [simp]: "\<And>z. norm z < r \<Longrightarrow> f field_differentiable at (a + z)"
+ by (metis holf open_ball add_diff_cancel_left' dist_complex_def holomorphic_on_imp_differentiable_at holomorphic_on_subset interior_cball interior_subset mem_ball norm_minus_commute)
+ then have [simp]: "f field_differentiable at a"
+ by (metis add.comm_neutral \<open>0 < r\<close> norm_eq_zero)
+ have hol1: "(\<lambda>z. f (a + z) - f a) holomorphic_on cball 0 r"
+ by (intro holomorphic_intros hol0)
+ then have "ball 0 ((3 - 2 * sqrt 2) * r * norm (deriv (\<lambda>z. f (a + z) - f a) 0))
+ \<subseteq> (\<lambda>z. f (a + z) - f a) ` ball 0 r"
+ apply (rule Bloch_lemma_0)
+ apply (simp_all add: \<open>0 < r\<close>)
+ apply (simp add: fz deriv_chain)
+ apply (simp add: dist_norm le)
+ done
+ then show ?thesis
+ apply clarify
+ apply (drule_tac c="x - f a" in subsetD)
+ apply (force simp: fz \<open>0 < r\<close> dist_norm deriv_chain field_differentiable_compose)+
+ done
+qed
+
+proposition Bloch_unit:
+ assumes holf: "f holomorphic_on ball a 1" and [simp]: "deriv f a = 1"
+ obtains b r where "1/12 < r" and "ball b r \<subseteq> f ` (ball a 1)"
+proof -
+ define r :: real where "r = 249/256"
+ have "0 < r" "r < 1" by (auto simp: r_def)
+ define g where "g z = deriv f z * of_real(r - norm(z - a))" for z
+ have "deriv f holomorphic_on ball a 1"
+ by (rule holomorphic_deriv [OF holf open_ball])
+ then have "continuous_on (ball a 1) (deriv f)"
+ using holomorphic_on_imp_continuous_on by blast
+ then have "continuous_on (cball a r) (deriv f)"
+ by (rule continuous_on_subset) (simp add: cball_subset_ball_iff \<open>r < 1\<close>)
+ then have "continuous_on (cball a r) g"
+ by (simp add: g_def continuous_intros)
+ then have 1: "compact (g ` cball a r)"
+ by (rule compact_continuous_image [OF _ compact_cball])
+ have 2: "g ` cball a r \<noteq> {}"
+ using \<open>r > 0\<close> by auto
+ obtain p where pr: "p \<in> cball a r"
+ and pge: "\<And>y. y \<in> cball a r \<Longrightarrow> norm (g y) \<le> norm (g p)"
+ using distance_attains_sup [OF 1 2, of 0] by force
+ define t where "t = (r - norm(p - a)) / 2"
+ have "norm (p - a) \<noteq> r"
+ using pge [of a] \<open>r > 0\<close> by (auto simp: g_def norm_mult)
+ then have "norm (p - a) < r" using pr
+ by (simp add: norm_minus_commute dist_norm)
+ then have "0 < t"
+ by (simp add: t_def)
+ have cpt: "cball p t \<subseteq> ball a r"
+ using \<open>0 < t\<close> by (simp add: cball_subset_ball_iff dist_norm t_def field_simps)
+ have gen_le_dfp: "norm (deriv f y) * (r - norm (y - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
+ if "y \<in> cball a r" for y
+ proof -
+ have [simp]: "norm (y - a) \<le> r"
+ using that by (simp add: dist_norm norm_minus_commute)
+ have "norm (g y) \<le> norm (g p)"
+ using pge [OF that] by simp
+ then have "norm (deriv f y) * abs (r - norm (y - a)) \<le> norm (deriv f p) * abs (r - norm (p - a))"
+ by (simp only: dist_norm g_def norm_mult norm_of_real)
+ with that \<open>norm (p - a) < r\<close> show ?thesis
+ by (simp add: dist_norm field_split_simps)
+ qed
+ have le_norm_dfp: "r / (r - norm (p - a)) \<le> norm (deriv f p)"
+ using gen_le_dfp [of a] \<open>r > 0\<close> by auto
+ have 1: "f holomorphic_on cball p t"
+ apply (rule holomorphic_on_subset [OF holf])
+ using cpt \<open>r < 1\<close> order_subst1 subset_ball by auto
+ have 2: "norm (deriv f z) \<le> 2 * norm (deriv f p)" if "z \<in> ball p t" for z
+ proof -
+ have z: "z \<in> cball a r"
+ by (meson ball_subset_cball subsetD cpt that)
+ then have "norm(z - a) < r"
+ by (metis ball_subset_cball contra_subsetD cpt dist_norm mem_ball norm_minus_commute that)
+ have "norm (deriv f z) * (r - norm (z - a)) / (r - norm (p - a)) \<le> norm (deriv f p)"
+ using gen_le_dfp [OF z] by simp
+ with \<open>norm (z - a) < r\<close> \<open>norm (p - a) < r\<close>
+ have "norm (deriv f z) \<le> (r - norm (p - a)) / (r - norm (z - a)) * norm (deriv f p)"
+ by (simp add: field_simps)
+ also have "... \<le> 2 * norm (deriv f p)"
+ apply (rule mult_right_mono)
+ using that \<open>norm (p - a) < r\<close> \<open>norm(z - a) < r\<close>
+ apply (simp_all add: field_simps t_def dist_norm [symmetric])
+ using dist_triangle3 [of z a p] by linarith
+ finally show ?thesis .
+ qed
+ have sqrt2: "sqrt 2 < 2113/1494"
+ by (rule real_less_lsqrt) (auto simp: power2_eq_square)
+ then have sq3: "0 < 3 - 2 * sqrt 2" by simp
+ have "1 / 12 / ((3 - 2 * sqrt 2) / 2) < r"
+ using sq3 sqrt2 by (auto simp: field_simps r_def)
+ also have "... \<le> cmod (deriv f p) * (r - cmod (p - a))"
+ using \<open>norm (p - a) < r\<close> le_norm_dfp by (simp add: pos_divide_le_eq)
+ finally have "1 / 12 < cmod (deriv f p) * (r - cmod (p - a)) * ((3 - 2 * sqrt 2) / 2)"
+ using pos_divide_less_eq half_gt_zero_iff sq3 by blast
+ then have **: "1 / 12 < (3 - 2 * sqrt 2) * t * norm (deriv f p)"
+ using sq3 by (simp add: mult.commute t_def)
+ have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball p t"
+ by (rule Bloch_lemma [OF 1 \<open>0 < t\<close> 2])
+ also have "... \<subseteq> f ` ball a 1"
+ apply (rule image_mono)
+ apply (rule order_trans [OF ball_subset_cball])
+ apply (rule order_trans [OF cpt])
+ using \<open>0 < t\<close> \<open>r < 1\<close> apply (simp add: ball_subset_ball_iff dist_norm)
+ done
+ finally have "ball (f p) ((3 - 2 * sqrt 2) * t * norm (deriv f p)) \<subseteq> f ` ball a 1" .
+ with ** show ?thesis
+ by (rule that)
+qed
+
+theorem Bloch:
+ assumes holf: "f holomorphic_on ball a r" and "0 < r"
+ and r': "r' \<le> r * norm (deriv f a) / 12"
+ obtains b where "ball b r' \<subseteq> f ` (ball a r)"
+proof (cases "deriv f a = 0")
+ case True with r' show ?thesis
+ using ball_eq_empty that by fastforce
+next
+ case False
+ define C where "C = deriv f a"
+ have "0 < norm C" using False by (simp add: C_def)
+ have dfa: "f field_differentiable at a"
+ apply (rule holomorphic_on_imp_differentiable_at [OF holf])
+ using \<open>0 < r\<close> by auto
+ have fo: "(\<lambda>z. f (a + of_real r * z)) = f o (\<lambda>z. (a + of_real r * z))"
+ by (simp add: o_def)
+ have holf': "f holomorphic_on (\<lambda>z. a + complex_of_real r * z) ` ball 0 1"
+ apply (rule holomorphic_on_subset [OF holf])
+ using \<open>0 < r\<close> apply (force simp: dist_norm norm_mult)
+ done
+ have 1: "(\<lambda>z. f (a + r * z) / (C * r)) holomorphic_on ball 0 1"
+ apply (rule holomorphic_intros holomorphic_on_compose holf' | simp add: fo)+
+ using \<open>0 < r\<close> by (simp add: C_def False)
+ have "((\<lambda>z. f (a + of_real r * z) / (C * of_real r)) has_field_derivative
+ (deriv f (a + of_real r * z) / C)) (at z)"
+ if "norm z < 1" for z
+ proof -
+ have *: "((\<lambda>x. f (a + of_real r * x)) has_field_derivative
+ (deriv f (a + of_real r * z) * of_real r)) (at z)"
+ apply (simp add: fo)
+ apply (rule DERIV_chain [OF field_differentiable_derivI])
+ apply (rule holomorphic_on_imp_differentiable_at [OF holf], simp)
+ using \<open>0 < r\<close> apply (simp add: dist_norm norm_mult that)
+ apply (rule derivative_eq_intros | simp)+
+ done
+ show ?thesis
+ apply (rule derivative_eq_intros * | simp)+
+ using \<open>0 < r\<close> by (auto simp: C_def False)
+ qed
+ have 2: "deriv (\<lambda>z. f (a + of_real r * z) / (C * of_real r)) 0 = 1"
+ apply (subst deriv_cdivide_right)
+ apply (simp add: field_differentiable_def fo)
+ apply (rule exI)
+ apply (rule DERIV_chain [OF field_differentiable_derivI])
+ apply (simp add: dfa)
+ apply (rule derivative_eq_intros | simp add: C_def False fo)+
+ using \<open>0 < r\<close>
+ apply (simp add: C_def False fo)
+ apply (simp add: derivative_intros dfa deriv_chain)
+ done
+ have sb1: "(*) (C * r) ` (\<lambda>z. f (a + of_real r * z) / (C * r)) ` ball 0 1
+ \<subseteq> f ` ball a r"
+ using \<open>0 < r\<close> by (auto simp: dist_norm norm_mult C_def False)
+ have sb2: "ball (C * r * b) r' \<subseteq> (*) (C * r) ` ball b t"
+ if "1 / 12 < t" for b t
+ proof -
+ have *: "r * cmod (deriv f a) / 12 \<le> r * (t * cmod (deriv f a))"
+ using that \<open>0 < r\<close> less_eq_real_def mult.commute mult.right_neutral mult_left_mono norm_ge_zero times_divide_eq_right
+ by auto
+ show ?thesis
+ apply clarify
+ apply (rule_tac x="x / (C * r)" in image_eqI)
+ using \<open>0 < r\<close>
+ apply (simp_all add: dist_norm norm_mult norm_divide C_def False field_simps)
+ apply (erule less_le_trans)
+ apply (rule order_trans [OF r' *])
+ done
+ qed
+ show ?thesis
+ apply (rule Bloch_unit [OF 1 2])
+ apply (rename_tac t)
+ apply (rule_tac b="(C * of_real r) * b" in that)
+ apply (drule image_mono [where f = "\<lambda>z. (C * of_real r) * z"])
+ using sb1 sb2
+ apply force
+ done
+qed
+
+corollary Bloch_general:
+ assumes holf: "f holomorphic_on s" and "a \<in> s"
+ and tle: "\<And>z. z \<in> frontier s \<Longrightarrow> t \<le> dist a z"
+ and rle: "r \<le> t * norm(deriv f a) / 12"
+ obtains b where "ball b r \<subseteq> f ` s"
+proof -
+ consider "r \<le> 0" | "0 < t * norm(deriv f a) / 12" using rle by force
+ then show ?thesis
+ proof cases
+ case 1 then show ?thesis
+ by (simp add: ball_empty that)
+ next
+ case 2
+ show ?thesis
+ proof (cases "deriv f a = 0")
+ case True then show ?thesis
+ using rle by (simp add: ball_empty that)
+ next
+ case False
+ then have "t > 0"
+ using 2 by (force simp: zero_less_mult_iff)
+ have "\<not> ball a t \<subseteq> s \<Longrightarrow> ball a t \<inter> frontier s \<noteq> {}"
+ apply (rule connected_Int_frontier [of "ball a t" s], simp_all)
+ using \<open>0 < t\<close> \<open>a \<in> s\<close> centre_in_ball apply blast
+ done
+ with tle have *: "ball a t \<subseteq> s" by fastforce
+ then have 1: "f holomorphic_on ball a t"
+ using holf using holomorphic_on_subset by blast
+ show ?thesis
+ apply (rule Bloch [OF 1 \<open>t > 0\<close> rle])
+ apply (rule_tac b=b in that)
+ using * apply force
+ done
+ qed
+ qed
+qed
+
+subsection \<open>Cauchy's residue theorem\<close>
+
+text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
+ Interactive Theorem Proving\<close>
+
+definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
+ "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e
+ \<longrightarrow> (f has_contour_integral 2*pi* \<i> *int) (circlepath z \<epsilon>))"
+
+lemma Eps_cong:
+ assumes "\<And>x. P x = Q x"
+ shows "Eps P = Eps Q"
+ using ext[of P Q, OF assms] by simp
+
+lemma residue_cong:
+ assumes eq: "eventually (\<lambda>z. f z = g z) (at z)" and "z = z'"
+ shows "residue f z = residue g z'"
+proof -
+ from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
+ by (simp add: eq_commute)
+ let ?P = "\<lambda>f c e. (\<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow>
+ (f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>))"
+ have "residue f z = residue g z" unfolding residue_def
+ proof (rule Eps_cong)
+ fix c :: complex
+ have "\<exists>e>0. ?P g c e"
+ if "\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
+ proof -
+ from that(1) obtain e where e: "e > 0" "?P f c e"
+ by blast
+ from that(2) obtain e' where e': "e' > 0" "\<And>z'. z' \<noteq> z \<Longrightarrow> dist z' z < e' \<Longrightarrow> f z' = g z'"
+ unfolding eventually_at by blast
+ have "?P g c (min e e')"
+ proof (intro allI exI impI, goal_cases)
+ case (1 \<epsilon>)
+ hence "(f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>)"
+ using e(2) by auto
+ thus ?case
+ proof (rule has_contour_integral_eq)
+ fix z' assume "z' \<in> path_image (circlepath z \<epsilon>)"
+ hence "dist z' z < e'" and "z' \<noteq> z"
+ using 1 by (auto simp: dist_commute)
+ with e'(2)[of z'] show "f z' = g z'" by simp
+ qed
+ qed
+ moreover from e and e' have "min e e' > 0" by auto
+ ultimately show ?thesis by blast
+ qed
+ from this[OF _ eq] and this[OF _ eq']
+ show "(\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
+ by blast
+ qed
+ with assms show ?thesis by simp
+qed
+
+lemma contour_integral_circlepath_eq:
+ assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1\<le>e2"
+ and e2_cball:"cball z e2 \<subseteq> s"
+ shows
+ "f contour_integrable_on circlepath z e1"
+ "f contour_integrable_on circlepath z e2"
+ "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
+proof -
+ define l where "l \<equiv> linepath (z+e2) (z+e1)"
+ have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
+ have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
+ have zl_img:"z\<notin>path_image l"
+ proof
+ assume "z \<in> path_image l"
+ then have "e2 \<le> cmod (e2 - e1)"
+ using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
+ by (auto simp add:closed_segment_commute)
+ thus False using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ apply (subst (asm) norm_of_real)
+ by auto
+ qed
+ define g where "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
+ show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
+ proof -
+ show "f contour_integrable_on circlepath z e2"
+ apply (intro contour_integrable_continuous_circlepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ using \<open>e2>0\<close> e2_cball by auto
+ show "f contour_integrable_on (circlepath z e1)"
+ apply (intro contour_integrable_continuous_circlepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
+ qed
+ have [simp]:"f contour_integrable_on l"
+ proof -
+ have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ by (intro closed_segment_subset,auto simp add:dist_norm)
+ hence "closed_segment (z + e2) (z + e1) \<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
+ by auto
+ then show "f contour_integrable_on l" unfolding l_def
+ apply (intro contour_integrable_continuous_linepath[OF
+ continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
+ by auto
+ qed
+ let ?ig="\<lambda>g. contour_integral g f"
+ have "(f has_contour_integral 0) g"
+ proof (rule Cauchy_theorem_global[OF _ f_holo])
+ show "open (s - {z})" using \<open>open s\<close> by auto
+ show "valid_path g" unfolding g_def l_def by auto
+ show "pathfinish g = pathstart g" unfolding g_def l_def by auto
+ next
+ have path_img:"path_image g \<subseteq> cball z e2"
+ proof -
+ have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
+ by (intro closed_segment_subset,auto simp add:dist_norm)
+ moreover have "sphere z \<bar>e1\<bar> \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1\<le>e2\<close> \<open>e1>0\<close> by auto
+ ultimately show ?thesis unfolding g_def l_def using \<open>e2>0\<close>
+ by (simp add: path_image_join closed_segment_commute)
+ qed
+ show "path_image g \<subseteq> s - {z}"
+ proof -
+ have "z\<notin>path_image g" using zl_img
+ unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
+ moreover note \<open>cball z e2 \<subseteq> s\<close> and path_img
+ ultimately show ?thesis by auto
+ qed
+ show "winding_number g w = 0" when"w \<notin> s - {z}" for w
+ proof -
+ have "winding_number g w = 0" when "w\<notin>s" using that e2_cball
+ apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
+ by (auto simp add:g_def l_def)
+ moreover have "winding_number g z=0"
+ proof -
+ let ?Wz="\<lambda>g. winding_number g z"
+ have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
+ + ?Wz (reversepath l)"
+ using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_def
+ by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
+ also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
+ using zl_img
+ apply (subst (2) winding_number_reversepath)
+ by (auto simp add:l_def closed_segment_commute)
+ also have "... = 0"
+ proof -
+ have "?Wz (circlepath z e2) = 1" using \<open>e2>0\<close>
+ by (auto intro: winding_number_circlepath_centre)
+ moreover have "?Wz (reversepath (circlepath z e1)) = -1" using \<open>e1>0\<close>
+ apply (subst winding_number_reversepath)
+ by (auto intro: winding_number_circlepath_centre)
+ ultimately show ?thesis by auto
+ qed
+ finally show ?thesis .
+ qed
+ ultimately show ?thesis using that by auto
+ qed
+ qed
+ then have "0 = ?ig g" using contour_integral_unique by simp
+ also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
+ + ?ig (reversepath l)"
+ unfolding g_def
+ by (auto simp add:contour_integrable_reversepath_eq)
+ also have "... = ?ig (circlepath z e2) - ?ig (circlepath z e1)"
+ by (auto simp add:contour_integral_reversepath)
+ finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
+ by simp
+qed
+
+lemma base_residue:
+ assumes "open s" "z\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
+ and r_cball:"cball z r \<subseteq> s"
+ shows "(f has_contour_integral 2 * pi * \<i> * (residue f z)) (circlepath z r)"
+proof -
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s"
+ using open_contains_cball[of s] \<open>open s\<close> \<open>z\<in>s\<close> by auto
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define i where "i \<equiv> contour_integral (circlepath z e) f / c"
+ have "(f has_contour_integral c*i) (circlepath z \<epsilon>)" when "\<epsilon>>0" "\<epsilon><e" for \<epsilon>
+ proof -
+ have "contour_integral (circlepath z e) f = contour_integral (circlepath z \<epsilon>) f"
+ "f contour_integrable_on circlepath z \<epsilon>"
+ "f contour_integrable_on circlepath z e"
+ using \<open>\<epsilon><e\<close>
+ by (intro contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> _ e_cball],auto)+
+ then show ?thesis unfolding i_def c_def
+ by (auto intro:has_contour_integral_integral)
+ qed
+ then have "\<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ unfolding residue_def c_def
+ apply (rule_tac someI[of _ i],intro exI[where x=e])
+ by (auto simp add:\<open>e>0\<close> c_def)
+ then obtain e' where "e'>0"
+ and e'_def:"\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ by auto
+ let ?int="\<lambda>e. contour_integral (circlepath z e) f"
+ define \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2"
+ have "\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
+ have "(f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
+ using e'_def[rule_format,OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><e'\<close>] .
+ then show ?thesis unfolding c_def
+ using contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball]
+ by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z \<epsilon>" "circlepath z r"])
+qed
+
+lemma residue_holo:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s"
+ shows "residue f z = 0"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f has_contour_integral c*residue f z) (circlepath z e)"
+ using f_holo
+ by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ moreover have "(f has_contour_integral 0) (circlepath z e)"
+ using f_holo e_cball \<open>e>0\<close>
+ by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
+ ultimately have "c*residue f z =0"
+ using has_contour_integral_unique by blast
+ thus ?thesis unfolding c_def by auto
+qed
+
+lemma residue_const:"residue (\<lambda>_. c) z = 0"
+ by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)
+
+lemma residue_add:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ and g_holo:"g holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. f z + g z) z= residue f z + residue g z"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define fg where "fg \<equiv> (\<lambda>z. f z+g z)"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
+ unfolding fg_def using f_holo g_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ by (auto intro:holomorphic_intros)
+ moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
+ unfolding fg_def using f_holo g_holo
+ by (auto intro: has_contour_integral_add base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ ultimately have "c*(residue f z + residue g z) = c * residue fg z"
+ using has_contour_integral_unique by (auto simp add:distrib_left)
+ thus ?thesis unfolding fg_def
+ by (auto simp add:c_def)
+qed
+
+lemma residue_lmul:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. c * (f z)) z= c * residue f z"
+proof (cases "c=0")
+ case True
+ thus ?thesis using residue_const by auto
+next
+ case False
+ define c' where "c' \<equiv> 2 * pi * \<i>"
+ define f' where "f' \<equiv> (\<lambda>z. c * (f z))"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
+ unfolding f'_def using f_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
+ by (auto intro:holomorphic_intros)
+ moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
+ unfolding f'_def using f_holo
+ by (auto intro: has_contour_integral_lmul
+ base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
+ ultimately have "c' * residue f' z = c * (c' * residue f z)"
+ using has_contour_integral_unique by auto
+ thus ?thesis unfolding f'_def c'_def using False
+ by (auto simp add:field_simps)
+qed
+
+lemma residue_rmul:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. (f z) * c) z= residue f z * c"
+using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)
+
+lemma residue_div:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. (f z) / c) z= residue f z / c "
+using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)
+
+lemma residue_neg:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. - (f z)) z= - residue f z"
+using residue_lmul[OF assms,of "-1"] by auto
+
+lemma residue_diff:
+ assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
+ and g_holo:"g holomorphic_on s - {z}"
+ shows "residue (\<lambda>z. f z - g z) z= residue f z - residue g z"
+using residue_add[OF assms(1,2,3),of "\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
+by (auto intro:holomorphic_intros g_holo)
+
+lemma residue_simple:
+ assumes "open s" "z\<in>s" and f_holo:"f holomorphic_on s"
+ shows "residue (\<lambda>w. f w / (w - z)) z = f z"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
+ obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
+ using open_contains_cball_eq by blast
+ have "(f' has_contour_integral c * f z) (circlepath z e)"
+ unfolding f'_def c_def using \<open>e>0\<close> f_holo e_cball
+ by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
+ moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
+ unfolding f'_def using f_holo
+ apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
+ by (auto intro!:holomorphic_intros)
+ ultimately have "c * f z = c * residue f' z"
+ using has_contour_integral_unique by blast
+ thus ?thesis unfolding c_def f'_def by auto
+qed
+
+lemma residue_simple':
+ assumes s: "open s" "z \<in> s" and holo: "f holomorphic_on (s - {z})"
+ and lim: "((\<lambda>w. f w * (w - z)) \<longlongrightarrow> c) (at z)"
+ shows "residue f z = c"
+proof -
+ define g where "g = (\<lambda>w. if w = z then c else f w * (w - z))"
+ from holo have "(\<lambda>w. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
+ by (force intro: holomorphic_intros)
+ also have "?P \<longleftrightarrow> g holomorphic_on (s - {z})"
+ by (intro holomorphic_cong refl) (simp_all add: g_def)
+ finally have *: "g holomorphic_on (s - {z})" .
+
+ note lim
+ also have "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z\<rightarrow> g z"
+ by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
+ finally have **: "g \<midarrow>z\<rightarrow> g z" .
+
+ have g_holo: "g holomorphic_on s"
+ by (rule no_isolated_singularity'[where K = "{z}"])
+ (insert assms * **, simp_all add: at_within_open_NO_MATCH)
+ from s and this have "residue (\<lambda>w. g w / (w - z)) z = g z"
+ by (rule residue_simple)
+ also have "\<forall>\<^sub>F za in at z. g za / (za - z) = f za"
+ unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
+ hence "residue (\<lambda>w. g w / (w - z)) z = residue f z"
+ by (intro residue_cong refl)
+ finally show ?thesis
+ by (simp add: g_def)
+qed
+
+lemma residue_holomorphic_over_power:
+ assumes "open A" "z0 \<in> A" "f holomorphic_on A"
+ shows "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
+proof -
+ let ?f = "\<lambda>z. f z / (z - z0) ^ Suc n"
+ from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A"
+ by (auto simp: open_contains_cball)
+ have "(?f has_contour_integral 2 * pi * \<i> * residue ?f z0) (circlepath z0 r)"
+ using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
+ moreover have "(?f has_contour_integral 2 * pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
+ using assms r
+ by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
+ (auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
+ ultimately have "2 * pi * \<i> * residue ?f z0 = 2 * pi * \<i> / fact n * (deriv ^^ n) f z0"
+ by (rule has_contour_integral_unique)
+ thus ?thesis by (simp add: field_simps)
+qed
+
+lemma residue_holomorphic_over_power':
+ assumes "open A" "0 \<in> A" "f holomorphic_on A"
+ shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
+ using residue_holomorphic_over_power[OF assms] by simp
+
+theorem residue_fps_expansion_over_power_at_0:
+ assumes "f has_fps_expansion F"
+ shows "residue (\<lambda>z. f z / z ^ Suc n) 0 = fps_nth F n"
+proof -
+ from has_fps_expansion_imp_holomorphic[OF assms] guess s . note s = this
+ have "residue (\<lambda>z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
+ using assms s unfolding has_fps_expansion_def
+ by (intro residue_holomorphic_over_power[of s]) (auto simp: zero_ereal_def)
+ also from assms have "\<dots> = fps_nth F n"
+ by (subst fps_nth_fps_expansion) auto
+ finally show ?thesis by simp
+qed
+
+lemma get_integrable_path:
+ assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
+ obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
+ "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
+proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
+ case 1
+ obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
+ using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
+ valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
+ moreover have "f contour_integrable_on g"
+ using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
+ \<open>f holomorphic_on s - {}\<close>
+ by auto
+ ultimately show ?case using "1"(1)[of g] by auto
+next
+ case idt:(2 p pts)
+ obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
+ using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
+ \<open>a \<in> s - insert p pts\<close>
+ by auto
+ define a' where "a' \<equiv> a+e/2"
+ have "a'\<in>s-{p} -pts" using e[rule_format,of "a+e/2"] \<open>e>0\<close>
+ by (auto simp add:dist_complex_def a'_def)
+ then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
+ "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
+ using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
+ by (metis Diff_insert2 open_delete)
+ define g where "g \<equiv> linepath a a' +++ g'"
+ have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
+ moreover have "pathstart g = a" and "pathfinish g = b" unfolding g_def by auto
+ moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
+ proof (rule subset_path_image_join)
+ have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
+ by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
+ then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
+ by auto
+ next
+ show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
+ qed
+ moreover have "f contour_integrable_on g"
+ proof -
+ have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
+ by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
+ then have "continuous_on (closed_segment a a') f"
+ using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
+ apply (elim continuous_on_subset)
+ by auto
+ then have "f contour_integrable_on linepath a a'"
+ using contour_integrable_continuous_linepath by auto
+ then show ?thesis unfolding g_def
+ apply (rule contour_integrable_joinI)
+ by (auto simp add: \<open>e>0\<close>)
+ qed
+ ultimately show ?case using idt.prems(1)[of g] by auto
+qed
+
+lemma Cauchy_theorem_aux:
+ assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
+ "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
+ "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
+ "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ using assms
+proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
+ case 1
+ then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
+next
+ case (2 p pts)
+ note fin[simp] = \<open>finite (insert p pts)\<close>
+ and connected = \<open>connected (s - insert p pts)\<close>
+ and valid[simp] = \<open>valid_path g\<close>
+ and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
+ and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
+ and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
+ and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
+ and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
+ have "h p>0" and "p\<in>s"
+ and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
+ using h \<open>insert p pts \<subseteq> s\<close> by auto
+ obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
+ "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
+ proof -
+ have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
+ by (simp add: \<open>p \<in> s\<close> dist_norm)
+ then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
+ by fastforce
+ moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
+ ultimately show ?thesis
+ using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
+ by blast
+ qed
+ obtain n::int where "n=winding_number g p"
+ using integer_winding_number[OF _ g_loop,of p] valid path_img
+ by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
+ define p_circ where "p_circ \<equiv> circlepath p (h p)"
+ define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
+ define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
+ define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
+ have n_circ:"valid_path (n_circ k)"
+ "winding_number (n_circ k) p = k"
+ "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
+ "path_image (n_circ k) = (if k=0 then {p + h p} else sphere p (h p))"
+ "p \<notin> path_image (n_circ k)"
+ "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
+ "f contour_integrable_on (n_circ k)"
+ "contour_integral (n_circ k) f = k * contour_integral p_circ f"
+ for k
+ proof (induct k)
+ case 0
+ show "valid_path (n_circ 0)"
+ and "path_image (n_circ 0) = (if 0=0 then {p + h p} else sphere p (h p))"
+ and "winding_number (n_circ 0) p = of_nat 0"
+ and "pathstart (n_circ 0) = p + h p"
+ and "pathfinish (n_circ 0) = p + h p"
+ and "p \<notin> path_image (n_circ 0)"
+ unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
+ by (auto simp add: dist_norm)
+ show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
+ unfolding n_circ_def p_circ_pt_def
+ apply (auto intro!:winding_number_trivial)
+ by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
+ show "f contour_integrable_on (n_circ 0)"
+ unfolding n_circ_def p_circ_pt_def
+ by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
+ show "contour_integral (n_circ 0) f = of_nat 0 * contour_integral p_circ f"
+ unfolding n_circ_def p_circ_pt_def by auto
+ next
+ case (Suc k)
+ have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
+ have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
+ using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
+ have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
+ proof -
+ have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
+ then show ?thesis using h_p pcirc(1) by auto
+ qed
+ have pcirc_integrable:"f contour_integrable_on p_circ"
+ by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
+ contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
+ holomorphic_on_subset[OF holo])
+ show "valid_path (n_circ (Suc k))"
+ using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
+ show "path_image (n_circ (Suc k))
+ = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
+ proof -
+ have "path_image p_circ = sphere p (h p)"
+ unfolding p_circ_def using \<open>0 < h p\<close> by auto
+ then show ?thesis unfolding n_Suc using Suc.hyps(5) \<open>h p>0\<close>
+ by (auto simp add: path_image_join[OF pcirc(3)] dist_norm)
+ qed
+ then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
+ show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
+ proof -
+ have "winding_number p_circ p = 1"
+ by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
+ moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
+ then have "winding_number (p_circ +++ n_circ k) p
+ = winding_number p_circ p + winding_number (n_circ k) p"
+ using valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
+ apply (intro winding_number_join)
+ by auto
+ ultimately show ?thesis using Suc(2) unfolding n_circ_def
+ by auto
+ qed
+ show "pathstart (n_circ (Suc k)) = p + h p"
+ by (simp add: n_circ_def p_circ_def)
+ show "pathfinish (n_circ (Suc k)) = p + h p"
+ using Suc(4) unfolding n_circ_def by auto
+ show "winding_number (n_circ (Suc k)) p'=0 \<and> p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
+ proof -
+ have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
+ moreover have "p' \<notin> path_image (n_circ k)"
+ using Suc.hyps(7) that by blast
+ moreover have "winding_number p_circ p' = 0"
+ proof -
+ have "path_image p_circ \<subseteq> cball p (h p)"
+ using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
+ moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
+ ultimately show ?thesis unfolding p_circ_def
+ apply (intro winding_number_zero_outside)
+ by auto
+ qed
+ ultimately show ?thesis
+ unfolding n_Suc
+ apply (subst winding_number_join)
+ by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
+ qed
+ show "f contour_integrable_on (n_circ (Suc k))"
+ unfolding n_Suc
+ by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
+ show "contour_integral (n_circ (Suc k)) f = (Suc k) * contour_integral p_circ f"
+ unfolding n_Suc
+ by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
+ Suc(9) algebra_simps)
+ qed
+ have cp[simp]:"pathstart cp = p + h p" "pathfinish cp = p + h p"
+ "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
+ "winding_number cp p = - n"
+ "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
+ "f contour_integrable_on cp"
+ "contour_integral cp f = - n * contour_integral p_circ f"
+ proof -
+ show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
+ using n_circ unfolding cp_def by auto
+ next
+ have "sphere p (h p) \<subseteq> s - insert p pts"
+ using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
+ moreover have "p + complex_of_real (h p) \<in> s - insert p pts"
+ using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
+ ultimately show "path_image cp \<subseteq> s - insert p pts" unfolding cp_def
+ using n_circ(5) by auto
+ next
+ show "winding_number cp p = - n"
+ unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
+ by (auto simp: valid_path_imp_path)
+ next
+ show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
+ unfolding cp_def
+ apply (auto)
+ apply (subst winding_number_reversepath)
+ by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
+ next
+ show "f contour_integrable_on cp" unfolding cp_def
+ using contour_integrable_reversepath_eq n_circ(1,8) by auto
+ next
+ show "contour_integral cp f = - n * contour_integral p_circ f"
+ unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
+ by auto
+ qed
+ define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
+ have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
+ proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
+ show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
+ show "open (s - {p})" using \<open>open s\<close> by auto
+ show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close> by blast
+ show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
+ show "valid_path g'"
+ unfolding g'_def cp_def using n_circ valid pg g_loop
+ by (auto intro!:valid_path_join )
+ show "pathfinish g' = pathstart g'"
+ unfolding g'_def cp_def using pg(2) by simp
+ show "path_image g' \<subseteq> s - {p} - pts"
+ proof -
+ define s' where "s' \<equiv> s - {p} - pts"
+ have s':"s' = s-insert p pts " unfolding s'_def by auto
+ then show ?thesis using path_img pg(4) cp(4)
+ unfolding g'_def
+ apply (fold s'_def s')
+ apply (intro subset_path_image_join)
+ by auto
+ qed
+ note path_join_imp[simp]
+ show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
+ proof clarify
+ fix z assume z:"z\<notin>s - {p}"
+ have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
+ + winding_number (pg +++ cp +++ (reversepath pg)) z"
+ proof (rule winding_number_join)
+ show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
+ show "z \<notin> path_image g" using z path_img by auto
+ show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
+ by (simp add: valid_path_imp_path)
+ next
+ have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
+ using pg(4) cp(4) by (auto simp:subset_path_image_join)
+ then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
+ next
+ show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
+ qed
+ also have "... = winding_number g z + (winding_number pg z
+ + winding_number (cp +++ (reversepath pg)) z)"
+ proof (subst add_left_cancel,rule winding_number_join)
+ show "path pg" and "path (cp +++ reversepath pg)"
+ and "pathfinish pg = pathstart (cp +++ reversepath pg)"
+ by (auto simp add: valid_path_imp_path)
+ show "z \<notin> path_image pg" using pg(4) z by blast
+ show "z \<notin> path_image (cp +++ reversepath pg)" using z
+ by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
+ not_in_path_image_join path_image_reversepath singletonD)
+ qed
+ also have "... = winding_number g z + (winding_number pg z
+ + (winding_number cp z + winding_number (reversepath pg) z))"
+ apply (auto intro!:winding_number_join simp: valid_path_imp_path)
+ apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
+ by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
+ also have "... = winding_number g z + winding_number cp z"
+ apply (subst winding_number_reversepath)
+ apply (auto simp: valid_path_imp_path)
+ by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
+ finally have "winding_number g' z = winding_number g z + winding_number cp z"
+ unfolding g'_def .
+ moreover have "winding_number g z + winding_number cp z = 0"
+ using winding z \<open>n=winding_number g p\<close> by auto
+ ultimately show "winding_number g' z = 0" unfolding g'_def by auto
+ qed
+ show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
+ using h by fastforce
+ qed
+ moreover have "contour_integral g' f = contour_integral g f
+ - winding_number g p * contour_integral p_circ f"
+ proof -
+ have "contour_integral g' f = contour_integral g f
+ + contour_integral (pg +++ cp +++ reversepath pg) f"
+ unfolding g'_def
+ apply (subst contour_integral_join)
+ by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
+ intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
+ contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral pg f
+ + contour_integral (cp +++ reversepath pg) f"
+ apply (subst contour_integral_join)
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral pg f
+ + contour_integral cp f + contour_integral (reversepath pg) f"
+ apply (subst contour_integral_join)
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f + contour_integral cp f"
+ using contour_integral_reversepath
+ by (auto simp add:contour_integrable_reversepath)
+ also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
+ using \<open>n=winding_number g p\<close> by auto
+ finally show ?thesis .
+ qed
+ moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
+ proof -
+ have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
+ using "2.prems"(8) that
+ apply blast
+ apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
+ by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
+ have "winding_number g' p' = winding_number g p'
+ + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
+ apply (subst winding_number_join)
+ apply (simp_all add: valid_path_imp_path)
+ apply (intro not_in_path_image_join)
+ by auto
+ also have "... = winding_number g p' + winding_number pg p'
+ + winding_number (cp +++ reversepath pg) p'"
+ apply (subst winding_number_join)
+ apply (simp_all add: valid_path_imp_path)
+ apply (intro not_in_path_image_join)
+ by auto
+ also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
+ + winding_number (reversepath pg) p'"
+ apply (subst winding_number_join)
+ by (simp_all add: valid_path_imp_path)
+ also have "... = winding_number g p' + winding_number cp p'"
+ apply (subst winding_number_reversepath)
+ by (simp_all add: valid_path_imp_path)
+ also have "... = winding_number g p'" using that by auto
+ finally show ?thesis .
+ qed
+ ultimately show ?case unfolding p_circ_def
+ apply (subst (asm) sum.cong[OF refl,
+ of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
+ by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
+qed
+
+lemma Cauchy_theorem_singularities:
+ assumes "open s" "connected s" "finite pts" and
+ holo:"f holomorphic_on s-pts" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ "path_image g \<subseteq> s-pts" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
+ avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ (is "?L=?R")
+proof -
+ define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
+ define pts1 where "pts1 \<equiv> pts \<inter> s"
+ define pts2 where "pts2 \<equiv> pts - pts1"
+ have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
+ unfolding pts1_def pts2_def by auto
+ have "contour_integral g f = (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
+ proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
+ have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
+ then show "connected (s - pts1)"
+ using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
+ next
+ show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
+ show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
+ show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
+ show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
+ by (simp add: avoid pts1_def)
+ qed
+ moreover have "sum circ pts2=0"
+ proof -
+ have "winding_number g p=0" when "p\<in>pts2" for p
+ using \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
+ thus ?thesis unfolding circ_def
+ apply (intro sum.neutral)
+ by auto
+ qed
+ moreover have "?R=sum circ pts1 + sum circ pts2"
+ unfolding circ_def
+ using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
+ by blast
+ ultimately show ?thesis
+ apply (fold circ_def)
+ by auto
+qed
+
+theorem Residue_theorem:
+ fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
+ and g::"real \<Rightarrow> complex"
+ assumes "open s" "connected s" "finite pts" and
+ holo:"f holomorphic_on s-pts" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ "path_image g \<subseteq> s-pts" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0"
+ shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
+proof -
+ define c where "c \<equiv> 2 * pi * \<i>"
+ obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
+ using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
+ have "contour_integral g f
+ = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
+ using Cauchy_theorem_singularities[OF assms avoid] .
+ also have "... = (\<Sum>p\<in>pts. c * winding_number g p * residue f p)"
+ proof (intro sum.cong)
+ show "pts = pts" by simp
+ next
+ fix x assume "x \<in> pts"
+ show "winding_number g x * contour_integral (circlepath x (h x)) f
+ = c * winding_number g x * residue f x"
+ proof (cases "x\<in>s")
+ case False
+ then have "winding_number g x=0" using homo by auto
+ thus ?thesis by auto
+ next
+ case True
+ have "contour_integral (circlepath x (h x)) f = c* residue f x"
+ using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
+ apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
+ by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
+ then show ?thesis by auto
+ qed
+ qed
+ also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
+ by (simp add: sum_distrib_left algebra_simps)
+ finally show ?thesis unfolding c_def .
+qed
+
+subsection \<open>Non-essential singular points\<close>
+
+definition\<^marker>\<open>tag important\<close> is_pole ::
+ "('a::topological_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool" where
+ "is_pole f a = (LIM x (at a). f x :> at_infinity)"
+
+lemma is_pole_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
+ shows "is_pole f a \<longleftrightarrow> is_pole g b"
+ unfolding is_pole_def using assms by (intro filterlim_cong,auto)
+
+lemma is_pole_transform:
+ assumes "is_pole f a" "eventually (\<lambda>x. f x = g x) (at a)" "a=b"
+ shows "is_pole g b"
+ using is_pole_cong assms by auto
+
+lemma is_pole_tendsto:
+ fixes f::"('a::topological_space \<Rightarrow> 'b::real_normed_div_algebra)"
+ shows "is_pole f x \<Longrightarrow> ((inverse o f) \<longlongrightarrow> 0) (at x)"
+unfolding is_pole_def
+by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)
+
+lemma is_pole_inverse_holomorphic:
+ assumes "open s"
+ and f_holo:"f holomorphic_on (s-{z})"
+ and pole:"is_pole f z"
+ and non_z:"\<forall>x\<in>s-{z}. f x\<noteq>0"
+ shows "(\<lambda>x. if x=z then 0 else inverse (f x)) holomorphic_on s"
+proof -
+ define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
+ have "isCont g z" unfolding isCont_def using is_pole_tendsto[OF pole]
+ apply (subst Lim_cong_at[where b=z and y=0 and g="inverse \<circ> f"])
+ by (simp_all add:g_def)
+ moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
+ hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
+ by (auto elim!:continuous_on_inverse simp add:non_z)
+ hence "continuous_on (s-{z}) g" unfolding g_def
+ apply (subst continuous_on_cong[where t="s-{z}" and g="inverse o f"])
+ by auto
+ ultimately have "continuous_on s g" using open_delete[OF \<open>open s\<close>] \<open>open s\<close>
+ by (auto simp add:continuous_on_eq_continuous_at)
+ moreover have "(inverse o f) holomorphic_on (s-{z})"
+ unfolding comp_def using f_holo
+ by (auto elim!:holomorphic_on_inverse simp add:non_z)
+ hence "g holomorphic_on (s-{z})"
+ apply (subst holomorphic_cong[where t="s-{z}" and g="inverse o f"])
+ by (auto simp add:g_def)
+ ultimately show ?thesis unfolding g_def using \<open>open s\<close>
+ by (auto elim!: no_isolated_singularity)
+qed
+
+lemma not_is_pole_holomorphic:
+ assumes "open A" "x \<in> A" "f holomorphic_on A"
+ shows "\<not>is_pole f x"
+proof -
+ have "continuous_on A f" by (intro holomorphic_on_imp_continuous_on) fact
+ with assms have "isCont f x" by (simp add: continuous_on_eq_continuous_at)
+ hence "f \<midarrow>x\<rightarrow> f x" by (simp add: isCont_def)
+ thus "\<not>is_pole f x" unfolding is_pole_def
+ using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
+qed
+
+lemma is_pole_inverse_power: "n > 0 \<Longrightarrow> is_pole (\<lambda>z::complex. 1 / (z - a) ^ n) a"
+ unfolding is_pole_def inverse_eq_divide [symmetric]
+ by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
+ (auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)
+
+lemma is_pole_inverse: "is_pole (\<lambda>z::complex. 1 / (z - a)) a"
+ using is_pole_inverse_power[of 1 a] by simp
+
+lemma is_pole_divide:
+ fixes f :: "'a :: t2_space \<Rightarrow> 'b :: real_normed_field"
+ assumes "isCont f z" "filterlim g (at 0) (at z)" "f z \<noteq> 0"
+ shows "is_pole (\<lambda>z. f z / g z) z"
+proof -
+ have "filterlim (\<lambda>z. f z * inverse (g z)) at_infinity (at z)"
+ by (intro tendsto_mult_filterlim_at_infinity[of _ "f z"]
+ filterlim_compose[OF filterlim_inverse_at_infinity])+
+ (insert assms, auto simp: isCont_def)
+ thus ?thesis by (simp add: field_split_simps is_pole_def)
+qed
+
+lemma is_pole_basic:
+ assumes "f holomorphic_on A" "open A" "z \<in> A" "f z \<noteq> 0" "n > 0"
+ shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z"
+proof (rule is_pole_divide)
+ have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
+ with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
+ have "filterlim (\<lambda>w. (w - z) ^ n) (nhds 0) (at z)"
+ using assms by (auto intro!: tendsto_eq_intros)
+ thus "filterlim (\<lambda>w. (w - z) ^ n) (at 0) (at z)"
+ by (intro filterlim_atI tendsto_eq_intros)
+ (insert assms, auto simp: eventually_at_filter)
+qed fact+
+
+lemma is_pole_basic':
+ assumes "f holomorphic_on A" "open A" "0 \<in> A" "f 0 \<noteq> 0" "n > 0"
+ shows "is_pole (\<lambda>w. f w / w ^ n) 0"
+ using is_pole_basic[of f A 0] assms by simp
+
+text \<open>The proposition
+ \<^term>\<open>\<exists>x. ((f::complex\<Rightarrow>complex) \<longlongrightarrow> x) (at z) \<or> is_pole f z\<close>
+can be interpreted as the complex function \<^term>\<open>f\<close> has a non-essential singularity at \<^term>\<open>z\<close>
+(i.e. the singularity is either removable or a pole).\<close>
+definition not_essential::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "not_essential f z = (\<exists>x. f\<midarrow>z\<rightarrow>x \<or> is_pole f z)"
+
+definition isolated_singularity_at::"[complex \<Rightarrow> complex, complex] \<Rightarrow> bool" where
+ "isolated_singularity_at f z = (\<exists>r>0. f analytic_on ball z r-{z})"
+
+named_theorems singularity_intros "introduction rules for singularities"
+
+lemma holomorphic_factor_unique:
+ fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int
+ assumes "r>0" "g z\<noteq>0" "h z\<noteq>0"
+ and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0"
+ and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
+ shows "n=m"
+proof -
+ have [simp]:"at z within ball z r \<noteq> bot" using \<open>r>0\<close>
+ by (auto simp add:at_within_ball_bot_iff)
+ have False when "n>m"
+ proof -
+ have "(h \<longlongrightarrow> 0) (at z within ball z r)"
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"])
+ have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w"
+ using \<open>n>m\<close> asm \<open>r>0\<close>
+ apply (auto simp add:field_simps powr_diff)
+ by force
+ then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
+ \<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto
+ next
+ define F where "F \<equiv> at z within ball z r"
+ define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)"
+ have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto
+ moreover have "continuous F f'" unfolding f'_def F_def continuous_def
+ apply (subst Lim_ident_at)
+ using \<open>n>m\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
+ ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
+ by (simp add: continuous_within)
+ moreover have "(g \<longlongrightarrow> g z) F"
+ using holomorphic_on_imp_continuous_on[OF g_holo,unfolded continuous_on_def] \<open>r>0\<close>
+ unfolding F_def by auto
+ ultimately show " ((\<lambda>w. f' w * g w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
+ qed
+ moreover have "(h \<longlongrightarrow> h z) (at z within ball z r)"
+ using holomorphic_on_imp_continuous_on[OF h_holo]
+ by (auto simp add:continuous_on_def \<open>r>0\<close>)
+ ultimately have "h z=0" by (auto intro!: tendsto_unique)
+ thus False using \<open>h z\<noteq>0\<close> by auto
+ qed
+ moreover have False when "m>n"
+ proof -
+ have "(g \<longlongrightarrow> 0) (at z within ball z r)"
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"])
+ have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm
+ apply (auto simp add:field_simps powr_diff)
+ by force
+ then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
+ \<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto
+ next
+ define F where "F \<equiv> at z within ball z r"
+ define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)"
+ have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto
+ moreover have "continuous F f'" unfolding f'_def F_def continuous_def
+ apply (subst Lim_ident_at)
+ using \<open>m>n\<close> by (auto intro!:tendsto_powr_complex_0 tendsto_eq_intros)
+ ultimately have "(f' \<longlongrightarrow> 0) F" unfolding F_def
+ by (simp add: continuous_within)
+ moreover have "(h \<longlongrightarrow> h z) F"
+ using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] \<open>r>0\<close>
+ unfolding F_def by auto
+ ultimately show " ((\<lambda>w. f' w * h w) \<longlongrightarrow> 0) F" using tendsto_mult by fastforce
+ qed
+ moreover have "(g \<longlongrightarrow> g z) (at z within ball z r)"
+ using holomorphic_on_imp_continuous_on[OF g_holo]
+ by (auto simp add:continuous_on_def \<open>r>0\<close>)
+ ultimately have "g z=0" by (auto intro!: tendsto_unique)
+ thus False using \<open>g z\<noteq>0\<close> by auto
+ qed
+ ultimately show "n=m" by fastforce
+qed
+
+lemma holomorphic_factor_puncture:
+ assumes f_iso:"isolated_singularity_at f z"
+ and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close>
+ and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close>
+ shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)"
+proof -
+ define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r"
+ proof (rule ex_ex1I[OF that])
+ fix n1 n2 :: int
+ assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2"
+ define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0"
+ obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0"
+ and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
+ obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0"
+ and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
+ define r where "r \<equiv> min r1 r2"
+ have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto
+ moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0
+ \<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0"
+ using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def
+ by fastforce
+ ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close>
+ apply (elim holomorphic_factor_unique)
+ by (auto simp add:r_def)
+ qed
+
+ have P_exist:"\<exists> n g r. P h n g r" when
+ "\<exists>z'. (h \<longlongrightarrow> z') (at z)" "isolated_singularity_at h z" "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
+ for h
+ proof -
+ from that(2) obtain r where "r>0" "h analytic_on ball z r - {z}"
+ unfolding isolated_singularity_at_def by auto
+ obtain z' where "(h \<longlongrightarrow> z') (at z)" using \<open>\<exists>z'. (h \<longlongrightarrow> z') (at z)\<close> by auto
+ define h' where "h'=(\<lambda>x. if x=z then z' else h x)"
+ have "h' holomorphic_on ball z r"
+ apply (rule no_isolated_singularity'[of "{z}"])
+ subgoal by (metis LIM_equal Lim_at_imp_Lim_at_within \<open>h \<midarrow>z\<rightarrow> z'\<close> empty_iff h'_def insert_iff)
+ subgoal using \<open>h analytic_on ball z r - {z}\<close> analytic_imp_holomorphic h'_def holomorphic_transform
+ by fastforce
+ by auto
+ have ?thesis when "z'=0"
+ proof -
+ have "h' z=0" using that unfolding h'_def by auto
+ moreover have "\<not> h' constant_on ball z r"
+ using \<open>\<exists>\<^sub>Fw in (at z). h w\<noteq>0\<close> unfolding constant_on_def frequently_def eventually_at h'_def
+ apply simp
+ by (metis \<open>0 < r\<close> centre_in_ball dist_commute mem_ball that)
+ moreover note \<open>h' holomorphic_on ball z r\<close>
+ ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1 \<subseteq> ball z r" and
+ g:"g holomorphic_on ball z r1"
+ "\<And>w. w \<in> ball z r1 \<Longrightarrow> h' w = (w - z) ^ n * g w"
+ "\<And>w. w \<in> ball z r1 \<Longrightarrow> g w \<noteq> 0"
+ using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
+ OF \<open>h' holomorphic_on ball z r\<close> \<open>r>0\<close> \<open>h' z=0\<close> \<open>\<not> h' constant_on ball z r\<close>]
+ by (auto simp add:dist_commute)
+ define rr where "rr=r1/2"
+ have "P h' n g rr"
+ unfolding P_def rr_def
+ using \<open>n>0\<close> \<open>r1>0\<close> g by (auto simp add:powr_nat)
+ then have "P h n g rr"
+ unfolding h'_def P_def by auto
+ then show ?thesis unfolding P_def by blast
+ qed
+ moreover have ?thesis when "z'\<noteq>0"
+ proof -
+ have "h' z\<noteq>0" using that unfolding h'_def by auto
+ obtain r1 where "r1>0" "cball z r1 \<subseteq> ball z r" "\<forall>x\<in>cball z r1. h' x\<noteq>0"
+ proof -
+ have "isCont h' z" "h' z\<noteq>0"
+ by (auto simp add: Lim_cong_within \<open>h \<midarrow>z\<rightarrow> z'\<close> \<open>z'\<noteq>0\<close> continuous_at h'_def)
+ then obtain r2 where r2:"r2>0" "\<forall>x\<in>ball z r2. h' x\<noteq>0"
+ using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
+ define r1 where "r1=min r2 r / 2"
+ have "0 < r1" "cball z r1 \<subseteq> ball z r"
+ using \<open>r2>0\<close> \<open>r>0\<close> unfolding r1_def by auto
+ moreover have "\<forall>x\<in>cball z r1. h' x \<noteq> 0"
+ using r2 unfolding r1_def by simp
+ ultimately show ?thesis using that by auto
+ qed
+ then have "P h' 0 h' r1" using \<open>h' holomorphic_on ball z r\<close> unfolding P_def by auto
+ then have "P h 0 h' r1" unfolding P_def h'_def by auto
+ then show ?thesis unfolding P_def by blast
+ qed
+ ultimately show ?thesis by auto
+ qed
+
+ have ?thesis when "\<exists>x. (f \<longlongrightarrow> x) (at z)"
+ apply (rule_tac imp_unique[unfolded P_def])
+ using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
+ moreover have ?thesis when "is_pole f z"
+ proof (rule imp_unique[unfolded P_def])
+ obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "\<forall>x\<in>ball z e-{z}. f x\<noteq>0"
+ proof -
+ have "\<forall>\<^sub>F z in at z. f z \<noteq> 0"
+ using \<open>is_pole f z\<close> filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
+ by auto
+ then obtain e1 where e1:"e1>0" "\<forall>x\<in>ball z e1-{z}. f x\<noteq>0"
+ using that eventually_at[of "\<lambda>x. f x\<noteq>0" z UNIV,simplified] by (auto simp add:dist_commute)
+ obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
+ define e where "e=min e1 e2"
+ show ?thesis
+ apply (rule that[of e])
+ using e1 e2 unfolding e_def by auto
+ qed
+
+ define h where "h \<equiv> \<lambda>x. inverse (f x)"
+
+ have "\<exists>n g r. P h n g r"
+ proof -
+ have "h \<midarrow>z\<rightarrow> 0"
+ using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
+ moreover have "\<exists>\<^sub>Fw in (at z). h w\<noteq>0"
+ using non_zero
+ apply (elim frequently_rev_mp)
+ unfolding h_def eventually_at by (auto intro:exI[where x=1])
+ moreover have "isolated_singularity_at h z"
+ unfolding isolated_singularity_at_def h_def
+ apply (rule exI[where x=e])
+ using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open
+ holomorphic_on_inverse open_delete)
+ ultimately show ?thesis
+ using P_exist[of h] by auto
+ qed
+ then obtain n g r
+ where "0 < r" and
+ g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and
+ g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ unfolding P_def by auto
+ have "P f (-n) (inverse o g) r"
+ proof -
+ have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w
+ using g_fac[rule_format,of w] that unfolding h_def
+ apply (auto simp add:powr_minus )
+ by (metis inverse_inverse_eq inverse_mult_distrib)
+ then show ?thesis
+ unfolding P_def comp_def
+ using \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros)
+ qed
+ then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
+ unfolding P_def by blast
+ qed
+ ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger
+qed
+
+lemma not_essential_transform:
+ assumes "not_essential g z"
+ assumes "\<forall>\<^sub>F w in (at z). g w = f w"
+ shows "not_essential f z"
+ using assms unfolding not_essential_def
+ by (simp add: filterlim_cong is_pole_cong)
+
+lemma isolated_singularity_at_transform:
+ assumes "isolated_singularity_at g z"
+ assumes "\<forall>\<^sub>F w in (at z). g w = f w"
+ shows "isolated_singularity_at f z"
+proof -
+ obtain r1 where "r1>0" and r1:"g analytic_on ball z r1 - {z}"
+ using assms(1) unfolding isolated_singularity_at_def by auto
+ obtain r2 where "r2>0" and r2:" \<forall>x. x \<noteq> z \<and> dist x z < r2 \<longrightarrow> g x = f x"
+ using assms(2) unfolding eventually_at by auto
+ define r3 where "r3=min r1 r2"
+ have "r3>0" unfolding r3_def using \<open>r1>0\<close> \<open>r2>0\<close> by auto
+ moreover have "f analytic_on ball z r3 - {z}"
+ proof -
+ have "g holomorphic_on ball z r3 - {z}"
+ using r1 unfolding r3_def by (subst (asm) analytic_on_open,auto)
+ then have "f holomorphic_on ball z r3 - {z}"
+ using r2 unfolding r3_def
+ by (auto simp add:dist_commute elim!:holomorphic_transform)
+ then show ?thesis by (subst analytic_on_open,auto)
+ qed
+ ultimately show ?thesis unfolding isolated_singularity_at_def by auto
+qed
+
+lemma not_essential_powr[singularity_intros]:
+ assumes "LIM w (at z). f w :> (at x)"
+ shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z"
+proof -
+ define fp where "fp=(\<lambda>w. (f w) powr (of_int n))"
+ have ?thesis when "n>0"
+ proof -
+ have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n"
+ using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def
+ apply (elim Lim_transform_within[where d=1],simp)
+ by (metis less_le powr_0 powr_of_int that zero_less_nat_eq zero_power)
+ then show ?thesis unfolding not_essential_def fp_def by auto
+ qed
+ moreover have ?thesis when "n=0"
+ proof -
+ have "fp \<midarrow>z\<rightarrow> 1 "
+ apply (subst tendsto_cong[where g="\<lambda>_.1"])
+ using that filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def by auto
+ then show ?thesis unfolding fp_def not_essential_def by auto
+ qed
+ moreover have ?thesis when "n<0"
+ proof (cases "x=0")
+ case True
+ have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
+ apply (subst filterlim_inverse_at_iff[symmetric],simp)
+ apply (rule filterlim_atI)
+ subgoal using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ subgoal using filterlim_at_within_not_equal[OF assms,of 0]
+ by (eventually_elim,insert that,auto)
+ done
+ then have "LIM w (at z). fp w :> at_infinity"
+ proof (elim filterlim_mono_eventually)
+ show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x"
+ using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
+ apply eventually_elim
+ using powr_of_int that by auto
+ qed auto
+ then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
+ next
+ case False
+ let ?xx= "inverse (x ^ (nat (-n)))"
+ have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx"
+ using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
+ then have "fp \<midarrow>z\<rightarrow>?xx"
+ apply (elim Lim_transform_within[where d=1],simp)
+ unfolding fp_def by (metis inverse_zero nat_mono_iff nat_zero_as_int neg_0_less_iff_less
+ not_le power_eq_0_iff powr_0 powr_of_int that)
+ then show ?thesis unfolding fp_def not_essential_def by auto
+ qed
+ ultimately show ?thesis by linarith
+qed
+
+lemma isolated_singularity_at_powr[singularity_intros]:
+ assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+ shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z"
+proof -
+ obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
+ using assms(1) unfolding isolated_singularity_at_def by auto
+ then have r1:"f holomorphic_on ball z r1 - {z}"
+ using analytic_on_open[of "ball z r1-{z}" f] by blast
+ obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
+ using assms(2) unfolding eventually_at by auto
+ define r3 where "r3=min r1 r2"
+ have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
+ apply (rule holomorphic_on_powr_of_int)
+ subgoal unfolding r3_def using r1 by auto
+ subgoal unfolding r3_def using r2 by (auto simp add:dist_commute)
+ done
+ moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith
+ ultimately show ?thesis unfolding isolated_singularity_at_def
+ apply (subst (asm) analytic_on_open[symmetric])
+ by auto
+qed
+
+lemma non_zero_neighbour:
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+proof -
+ obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
+ have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0"
+ using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
+ moreover have "(w - z) powr of_int fn \<noteq>0"
+ unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto
+ ultimately show ?thesis by auto
+ qed
+ then show ?thesis using \<open>fr>0\<close> unfolding eventually_at by auto
+qed
+
+lemma non_zero_neighbour_pole:
+ assumes "is_pole f z"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
+ using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
+ unfolding is_pole_def by auto
+
+lemma non_zero_neighbour_alt:
+ assumes holo: "f holomorphic_on S"
+ and "open S" "connected S" "z \<in> S" "\<beta> \<in> S" "f \<beta> \<noteq> 0"
+ shows "\<forall>\<^sub>F w in (at z). f w\<noteq>0 \<and> w\<in>S"
+proof (cases "f z = 0")
+ case True
+ from isolated_zeros[OF holo \<open>open S\<close> \<open>connected S\<close> \<open>z \<in> S\<close> True \<open>\<beta> \<in> S\<close> \<open>f \<beta> \<noteq> 0\<close>]
+ obtain r where "0 < r" "ball z r \<subseteq> S" "\<forall>w \<in> ball z r - {z}.f w \<noteq> 0" by metis
+ then show ?thesis unfolding eventually_at
+ apply (rule_tac x=r in exI)
+ by (auto simp add:dist_commute)
+next
+ case False
+ obtain r1 where r1:"r1>0" "\<forall>y. dist z y < r1 \<longrightarrow> f y \<noteq> 0"
+ using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
+ holo holomorphic_on_imp_continuous_on by blast
+ obtain r2 where r2:"r2>0" "ball z r2 \<subseteq> S"
+ using assms(2) assms(4) openE by blast
+ show ?thesis unfolding eventually_at
+ apply (rule_tac x="min r1 r2" in exI)
+ using r1 r2 by (auto simp add:dist_commute)
+qed
+
+lemma not_essential_times[singularity_intros]:
+ assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ shows "not_essential (\<lambda>w. f w * g w) z"
+proof -
+ define fg where "fg = (\<lambda>w. f w * g w)"
+ have ?thesis when "\<not> ((\<exists>\<^sub>Fw in (at z). f w\<noteq>0) \<and> (\<exists>\<^sub>Fw in (at z). g w\<noteq>0))"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). fg w=0"
+ using that[unfolded frequently_def, simplified] unfolding fg_def
+ by (auto elim: eventually_rev_mp)
+ from tendsto_cong[OF this] have "fg \<midarrow>z\<rightarrow>0" by auto
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" and g_nconst:"\<exists>\<^sub>Fw in (at z). g w\<noteq>0"
+ proof -
+ obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
+ obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
+ and gr: "gp holomorphic_on cball z gr"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
+
+ define r1 where "r1=(min fr gr)"
+ have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ when "w\<in>ball z r1 - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that unfolding r1_def by auto
+ moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ using gr(2)[rule_format, of w] that unfolding r1_def by auto
+ ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
+ unfolding fg_def by (auto simp add:powr_add)
+ qed
+
+ have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
+ using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close>
+ by (meson open_ball ball_subset_cball centre_in_ball
+ continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
+ holomorphic_on_subset)+
+ have ?thesis when "fn+gn>0"
+ proof -
+ have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0"
+ using that by (auto intro!:tendsto_eq_intros)
+ then have "fg \<midarrow>z\<rightarrow> 0"
+ apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
+ by (metis (no_types, hide_lams) Diff_iff cball_trivial dist_commute dist_self
+ eq_iff_diff_eq_0 fg_times less_le linorder_not_le mem_ball mem_cball powr_of_int
+ that)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when "fn+gn=0"
+ proof -
+ have "(\<lambda>w. fp w * gp w) \<midarrow>z\<rightarrow>fp z*gp z"
+ using that by (auto intro!:tendsto_eq_intros)
+ then have "fg \<midarrow>z\<rightarrow> fp z*gp z"
+ apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
+ apply (subst fg_times)
+ by (auto simp add:dist_commute that)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ moreover have ?thesis when "fn+gn<0"
+ proof -
+ have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
+ apply (rule filterlim_divide_at_infinity)
+ apply (insert that, auto intro!:tendsto_eq_intros filterlim_atI)
+ using eventually_at_topological by blast
+ then have "is_pole fg z" unfolding is_pole_def
+ apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>],simp)
+ apply (subst fg_times,simp add:dist_commute)
+ apply (subst powr_of_int)
+ using that by (auto simp add:field_split_simps)
+ then show ?thesis unfolding not_essential_def fg_def by auto
+ qed
+ ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma not_essential_inverse[singularity_intros]:
+ assumes f_ness:"not_essential f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ shows "not_essential (\<lambda>w. inverse (f w)) z"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). f w=0"
+ using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
+ then have "\<forall>\<^sub>Fw in (at z). vf w=0"
+ unfolding vf_def by auto
+ from tendsto_cong[OF this] have "vf \<midarrow>z\<rightarrow>0" unfolding vf_def by auto
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "is_pole f z"
+ proof -
+ have "vf \<midarrow>z\<rightarrow>0"
+ using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "\<exists>x. f\<midarrow>z\<rightarrow>x " and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ proof -
+ from that obtain fz where fz:"f\<midarrow>z\<rightarrow>fz" by auto
+ have ?thesis when "fz=0"
+ proof -
+ have "(\<lambda>w. inverse (vf w)) \<midarrow>z\<rightarrow>0"
+ using fz that unfolding vf_def by auto
+ moreover have "\<forall>\<^sub>F w in at z. inverse (vf w) \<noteq> 0"
+ using non_zero_neighbour[OF f_iso f_ness f_nconst]
+ unfolding vf_def by auto
+ ultimately have "is_pole vf z"
+ using filterlim_inverse_at_iff[of vf "at z"] unfolding filterlim_at is_pole_def by auto
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ moreover have ?thesis when "fz\<noteq>0"
+ proof -
+ have "vf \<midarrow>z\<rightarrow>inverse fz"
+ using fz that unfolding vf_def by (auto intro:tendsto_eq_intros)
+ then show ?thesis unfolding not_essential_def vf_def by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ ultimately show ?thesis using f_ness unfolding not_essential_def by auto
+qed
+
+lemma isolated_singularity_at_inverse[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ shows "isolated_singularity_at (\<lambda>w. inverse (f w)) z"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ have ?thesis when "\<not>(\<exists>\<^sub>Fw in (at z). f w\<noteq>0)"
+ proof -
+ have "\<forall>\<^sub>Fw in (at z). f w=0"
+ using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
+ then have "\<forall>\<^sub>Fw in (at z). vf w=0"
+ unfolding vf_def by auto
+ then obtain d1 where "d1>0" and d1:"\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> vf x = 0"
+ unfolding eventually_at by auto
+ then have "vf holomorphic_on ball z d1-{z}"
+ apply (rule_tac holomorphic_transform[of "\<lambda>_. 0"])
+ by (auto simp add:dist_commute)
+ then have "vf analytic_on ball z d1 - {z}"
+ by (simp add: analytic_on_open open_delete)
+ then show ?thesis using \<open>d1>0\<close> unfolding isolated_singularity_at_def vf_def by auto
+ qed
+ moreover have ?thesis when f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ proof -
+ have "\<forall>\<^sub>F w in at z. f w \<noteq> 0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
+ then obtain d1 where d1:"d1>0" "\<forall>x. x \<noteq> z \<and> dist x z < d1 \<longrightarrow> f x \<noteq> 0"
+ unfolding eventually_at by auto
+ obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
+ using f_iso unfolding isolated_singularity_at_def by auto
+ define d3 where "d3=min d1 d2"
+ have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
+ moreover have "vf analytic_on ball z d3 - {z}"
+ unfolding vf_def
+ apply (rule analytic_on_inverse)
+ subgoal using d2 unfolding d3_def by (elim analytic_on_subset) auto
+ subgoal for w using d1 unfolding d3_def by (auto simp add:dist_commute)
+ done
+ ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma not_essential_divide[singularity_intros]:
+ assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ shows "not_essential (\<lambda>w. f w / g w) z"
+proof -
+ have "not_essential (\<lambda>w. f w * inverse (g w)) z"
+ apply (rule not_essential_times[where g="\<lambda>w. inverse (g w)"])
+ using assms by (auto intro: isolated_singularity_at_inverse not_essential_inverse)
+ then show ?thesis by (simp add:field_simps)
+qed
+
+lemma
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ shows isolated_singularity_at_times[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. f w * g w) z" and
+ isolated_singularity_at_add[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. f w + g w) z"
+proof -
+ obtain d1 d2 where "d1>0" "d2>0"
+ and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
+ using f_iso g_iso unfolding isolated_singularity_at_def by auto
+ define d3 where "d3=min d1 d2"
+ have "d3>0" unfolding d3_def using \<open>d1>0\<close> \<open>d2>0\<close> by auto
+
+ have "(\<lambda>w. f w * g w) analytic_on ball z d3 - {z}"
+ apply (rule analytic_on_mult)
+ using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
+ then show "isolated_singularity_at (\<lambda>w. f w * g w) z"
+ using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
+ have "(\<lambda>w. f w + g w) analytic_on ball z d3 - {z}"
+ apply (rule analytic_on_add)
+ using d1 d2 unfolding d3_def by (auto elim:analytic_on_subset)
+ then show "isolated_singularity_at (\<lambda>w. f w + g w) z"
+ using \<open>d3>0\<close> unfolding isolated_singularity_at_def by auto
+qed
+
+lemma isolated_singularity_at_uminus[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ shows "isolated_singularity_at (\<lambda>w. - f w) z"
+ using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast
+
+lemma isolated_singularity_at_id[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. w) z"
+ unfolding isolated_singularity_at_def by (simp add: gt_ex)
+
+lemma isolated_singularity_at_minus[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ shows "isolated_singularity_at (\<lambda>w. f w - g w) z"
+ using isolated_singularity_at_uminus[THEN isolated_singularity_at_add[OF f_iso,of "\<lambda>w. - g w"]
+ ,OF g_iso] by simp
+
+lemma isolated_singularity_at_divide[singularity_intros]:
+ assumes f_iso:"isolated_singularity_at f z"
+ and g_iso:"isolated_singularity_at g z"
+ and g_ness:"not_essential g z"
+ shows "isolated_singularity_at (\<lambda>w. f w / g w) z"
+ using isolated_singularity_at_inverse[THEN isolated_singularity_at_times[OF f_iso,
+ of "\<lambda>w. inverse (g w)"],OF g_iso g_ness] by (simp add:field_simps)
+
+lemma isolated_singularity_at_const[singularity_intros]:
+ "isolated_singularity_at (\<lambda>w. c) z"
+ unfolding isolated_singularity_at_def by (simp add: gt_ex)
+
+lemma isolated_singularity_at_holomorphic:
+ assumes "f holomorphic_on s-{z}" "open s" "z\<in>s"
+ shows "isolated_singularity_at f z"
+ using assms unfolding isolated_singularity_at_def
+ by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
+
+subsubsection \<open>The order of non-essential singularities (i.e. removable singularities or poles)\<close>
+
+
+definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
+ "zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n)
+ \<and> h w \<noteq>0)))"
+
+definition\<^marker>\<open>tag important\<close> zor_poly
+ ::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where
+ "zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
+ \<and> h w \<noteq>0))"
+
+lemma zorder_exist:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))"
+proof -
+ define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have "\<exists>!n. \<exists>g r. P n g r"
+ using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
+ then have "\<exists>g r. P n g r"
+ unfolding n_def P_def zorder_def
+ by (drule_tac theI',argo)
+ then have "\<exists>r. P n g r"
+ unfolding P_def zor_poly_def g_def n_def
+ by (drule_tac someI_ex,argo)
+ then obtain r1 where "P n g r1" by auto
+ then show ?thesis unfolding P_def by auto
+qed
+
+lemma
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z"
+ and f_ness:"not_essential f z"
+ and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
+ shows zorder_inverse: "zorder (\<lambda>w. inverse (f w)) z = - zorder f z"
+ and zor_poly_inverse: "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. inverse (f w)) z w
+ = inverse (zor_poly f z w)"
+proof -
+ define vf where "vf = (\<lambda>w. inverse (f w))"
+ define fn vfn where
+ "fn = zorder f z" and "vfn = zorder vf z"
+ define fp vfp where
+ "fp = zor_poly f z" and "vfp = zor_poly vf z"
+
+ obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
+ by auto
+ have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
+ and fr_nz: "inverse (fp w)\<noteq>0"
+ when "w\<in>ball z fr - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that by auto
+ then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0"
+ unfolding vf_def by (auto simp add:powr_minus)
+ qed
+ obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
+ "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)"
+ proof -
+ have "isolated_singularity_at vf z"
+ using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
+ moreover have "not_essential vf z"
+ using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
+ moreover have "\<exists>\<^sub>F w in at z. vf w \<noteq> 0"
+ using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
+ ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
+ qed
+
+
+ define r1 where "r1 = min fr vfr"
+ have "r1>0" using \<open>fr>0\<close> \<open>vfr>0\<close> unfolding r1_def by simp
+ show "vfn = - fn"
+ apply (rule holomorphic_factor_unique[of r1 vfp z "\<lambda>w. inverse (fp w)" vf])
+ subgoal using \<open>r1>0\<close> by simp
+ subgoal by simp
+ subgoal by simp
+ subgoal
+ proof (rule ballI)
+ fix w assume "w \<in> ball z r1 - {z}"
+ then have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" unfolding r1_def by auto
+ from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)]
+ show "vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0
+ \<and> vf w = inverse (fp w) * (w - z) powr of_int (- fn) \<and> inverse (fp w) \<noteq> 0" by auto
+ qed
+ subgoal using vfr(1) unfolding r1_def by (auto intro!:holomorphic_intros)
+ subgoal using fr unfolding r1_def by (auto intro!:holomorphic_intros)
+ done
+
+ have "vfp w = inverse (fp w)" when "w\<in>ball z r1-{z}" for w
+ proof -
+ have "w \<in> ball z fr - {z}" "w \<in> cball z vfr - {z}" "w\<noteq>z" using that unfolding r1_def by auto
+ from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] \<open>vfn = - fn\<close> \<open>w\<noteq>z\<close>
+ show ?thesis by auto
+ qed
+ then show "\<forall>\<^sub>Fw in (at z). vfp w = inverse (fp w)"
+ unfolding eventually_at using \<open>r1>0\<close>
+ apply (rule_tac x=r1 in exI)
+ by (auto simp add:dist_commute)
+qed
+
+lemma
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ and f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
+ shows zorder_times:"zorder (\<lambda>w. f w * g w) z = zorder f z + zorder g z" and
+ zor_poly_times:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w * g w) z w
+ = zor_poly f z w *zor_poly g z w"
+proof -
+ define fg where "fg = (\<lambda>w. f w * g w)"
+ define fn gn fgn where
+ "fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
+ define fp gp fgp where
+ "fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
+ have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
+ using fg_nconst by (auto elim!:frequently_elim1)
+ obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
+ and fr: "fp holomorphic_on cball z fr"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
+ obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
+ and gr: "gp holomorphic_on cball z gr"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
+ define r1 where "r1=min fr gr"
+ have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ when "w\<in>ball z r1 - {z}" for w
+ proof -
+ have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ using fr(2)[rule_format,of w] that unfolding r1_def by auto
+ moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ using gr(2)[rule_format, of w] that unfolding r1_def by auto
+ ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
+ unfolding fg_def by (auto simp add:powr_add)
+ qed
+
+ obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0"
+ and fgr: "fgp holomorphic_on cball z fgr"
+ "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0"
+ proof -
+ have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))"
+ apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
+ subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
+ subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
+ subgoal unfolding fg_def using fg_nconst .
+ done
+ then show ?thesis using that by blast
+ qed
+ define r2 where "r2 = min fgr r1"
+ have "r2>0" using \<open>r1>0\<close> \<open>fgr>0\<close> unfolding r2_def by simp
+ show "fgn = fn + gn "
+ apply (rule holomorphic_factor_unique[of r2 fgp z "\<lambda>w. fp w * gp w" fg])
+ subgoal using \<open>r2>0\<close> by simp
+ subgoal by simp
+ subgoal by simp
+ subgoal
+ proof (rule ballI)
+ fix w assume "w \<in> ball z r2 - {z}"
+ then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto
+ from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
+ show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0
+ \<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
+ qed
+ subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
+ subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
+ done
+
+ have "fgp w = fp w *gp w" when "w\<in>ball z r2-{z}" for w
+ proof -
+ have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" "w\<noteq>z" using that unfolding r2_def by auto
+ from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] \<open>fgn = fn + gn\<close> \<open>w\<noteq>z\<close>
+ show ?thesis by auto
+ qed
+ then show "\<forall>\<^sub>Fw in (at z). fgp w = fp w * gp w"
+ using \<open>r2>0\<close> unfolding eventually_at by (auto simp add:dist_commute)
+qed
+
+lemma
+ fixes f g::"complex \<Rightarrow> complex" and z::complex
+ assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
+ and f_ness:"not_essential f z" and g_ness:"not_essential g z"
+ and fg_nconst: "\<exists>\<^sub>Fw in (at z). f w * g w\<noteq> 0"
+ shows zorder_divide:"zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z" and
+ zor_poly_divide:"\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w
+ = zor_poly f z w / zor_poly g z w"
+proof -
+ have f_nconst:"\<exists>\<^sub>Fw in (at z). f w \<noteq> 0" and g_nconst:"\<exists>\<^sub>Fw in (at z).g w\<noteq> 0"
+ using fg_nconst by (auto elim!:frequently_elim1)
+ define vg where "vg=(\<lambda>w. inverse (g w))"
+ have "zorder (\<lambda>w. f w * vg w) z = zorder f z + zorder vg z"
+ apply (rule zorder_times[OF f_iso _ f_ness,of vg])
+ subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
+ subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
+ subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
+ done
+ then show "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
+ using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
+ by (auto simp add:field_simps)
+
+ have "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
+ apply (rule zor_poly_times[OF f_iso _ f_ness,of vg])
+ subgoal unfolding vg_def using isolated_singularity_at_inverse[OF g_iso g_ness] .
+ subgoal unfolding vg_def using not_essential_inverse[OF g_ness g_iso] .
+ subgoal unfolding vg_def using fg_nconst by (auto elim!:frequently_elim1)
+ done
+ then show "\<forall>\<^sub>Fw in (at z). zor_poly (\<lambda>w. f w / g w) z w = zor_poly f z w / zor_poly g z w"
+ using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
+ apply eventually_elim
+ by (auto simp add:field_simps)
+qed
+
+lemma zorder_exist_zero:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes holo: "f holomorphic_on s" and
+ "open s" "connected s" "z\<in>s"
+ and non_const: "\<exists>w\<in>s. f w \<noteq> 0"
+ shows "(if f z=0 then n > 0 else n=0) \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))"
+proof -
+ obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ proof -
+ have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ proof (rule zorder_exist[of f z,folded g_def n_def])
+ show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using holo assms(4,6)
+ by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
+ show "not_essential f z" unfolding not_essential_def
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
+ by fastforce
+ have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w\<in>s"
+ proof -
+ obtain w where "w\<in>s" "f w\<noteq>0" using non_const by auto
+ then show ?thesis
+ by (rule non_zero_neighbour_alt[OF holo \<open>open s\<close> \<open>connected s\<close> \<open>z\<in>s\<close>])
+ qed
+ then show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ apply (elim eventually_frequentlyE)
+ by auto
+ qed
+ then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ by auto
+ obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
+ using assms(4,6) open_contains_cball_eq by blast
+ define r3 where "r3=min r1 r2"
+ have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
+ moreover have "g holomorphic_on cball z r3"
+ using r1(1) unfolding r3_def by auto
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ using r1(2) unfolding r3_def by auto
+ ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
+ qed
+
+ have if_0:"if f z=0 then n > 0 else n=0"
+ proof -
+ have "f\<midarrow> z \<rightarrow> f z"
+ by (metis assms(4,6,7) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
+ then have "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z"
+ apply (elim Lim_transform_within_open[where s="ball z r"])
+ using r by auto
+ moreover have "g \<midarrow>z\<rightarrow>g z"
+ by (metis (mono_tags, lifting) open_ball at_within_open_subset
+ ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
+ ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
+ apply (rule_tac tendsto_divide)
+ using \<open>g z\<noteq>0\<close> by auto
+ then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z"
+ apply (elim Lim_transform_within_open[where s="ball z r"])
+ using r by auto
+
+ have ?thesis when "n\<ge>0" "f z=0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
+ using powr_tendsto
+ apply (elim Lim_transform_within[where d=r])
+ by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp
+ moreover have False when "n=0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 1"
+ using \<open>n=0\<close> by auto
+ then show False using * using LIM_unique zero_neq_one by blast
+ qed
+ ultimately show ?thesis using that by fastforce
+ qed
+ moreover have ?thesis when "n\<ge>0" "f z\<noteq>0"
+ proof -
+ have False when "n>0"
+ proof -
+ have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
+ using powr_tendsto
+ apply (elim Lim_transform_within[where d=r])
+ by (auto simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0"
+ using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros)
+ ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast
+ qed
+ then show ?thesis using that by force
+ qed
+ moreover have False when "n<0"
+ proof -
+ have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z"
+ "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
+ subgoal using powr_tendsto powr_of_int that
+ by (elim Lim_transform_within_open[where s=UNIV],auto)
+ subgoal using that by (auto intro!:tendsto_eq_intros)
+ done
+ from tendsto_mult[OF this,simplified]
+ have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" .
+ then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0"
+ by (elim Lim_transform_within_open[where s=UNIV],auto)
+ then show False using LIM_const_eq by fastforce
+ qed
+ ultimately show ?thesis by fastforce
+ qed
+ moreover have "f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0" when "w\<in>cball z r" for w
+ proof (cases "w=z")
+ case True
+ then have "f \<midarrow>z\<rightarrow>f w"
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on by fastforce
+ then have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow>f w"
+ proof (elim Lim_transform_within[OF _ \<open>r>0\<close>])
+ fix x assume "0 < dist x z" "dist x z < r"
+ then have "x \<in> cball z r - {z}" "x\<noteq>z"
+ unfolding cball_def by (auto simp add: dist_commute)
+ then have "f x = g x * (x - z) powr of_int n"
+ using r(4)[rule_format,of x] by simp
+ also have "... = g x * (x - z) ^ nat n"
+ apply (subst powr_of_int)
+ using if_0 \<open>x\<noteq>z\<close> by (auto split:if_splits)
+ finally show "f x = g x * (x - z) ^ nat n" .
+ qed
+ moreover have "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n"
+ using True apply (auto intro!:tendsto_eq_intros)
+ by (metis open_ball at_within_open_subset ball_subset_cball centre_in_ball
+ continuous_on holomorphic_on_imp_continuous_on r(1) r(3) that)
+ ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
+ then show ?thesis using \<open>g z\<noteq>0\<close> True by auto
+ next
+ case False
+ then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
+ using r(4) that by auto
+ then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
+ qed
+ ultimately show ?thesis using r by auto
+qed
+
+lemma zorder_exist_pole:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n\<equiv>zorder f z" and "g\<equiv>zor_poly f z"
+ assumes holo: "f holomorphic_on s-{z}" and
+ "open s" "z\<in>s"
+ and "is_pole f z"
+ shows "n < 0 \<and> g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> cball z r \<subseteq> s \<and> g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))"
+proof -
+ obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ proof -
+ have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ proof (rule zorder_exist[of f z,folded g_def n_def])
+ show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using holo assms(4,5)
+ by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
+ show "not_essential f z" unfolding not_essential_def
+ using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
+ by fastforce
+ from non_zero_neighbour_pole[OF \<open>is_pole f z\<close>] show "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ apply (elim eventually_frequentlyE)
+ by auto
+ qed
+ then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ by auto
+ obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
+ using assms(4,5) open_contains_cball_eq by metis
+ define r3 where "r3=min r1 r2"
+ have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
+ moreover have "g holomorphic_on cball z r3"
+ using r1(1) unfolding r3_def by auto
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ using r1(2) unfolding r3_def by auto
+ ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
+ qed
+
+ have "n<0"
+ proof (rule ccontr)
+ assume " \<not> n < 0"
+ define c where "c=(if n=0 then g z else 0)"
+ have [simp]:"g \<midarrow>z\<rightarrow> g z"
+ by (metis open_ball at_within_open ball_subset_cball centre_in_ball
+ continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
+ have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
+ unfolding eventually_at_topological
+ apply (rule_tac exI[where x="ball z r"])
+ using r powr_of_int \<open>\<not> n < 0\<close> by auto
+ moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow>c"
+ proof (cases "n=0")
+ case True
+ then show ?thesis unfolding c_def by simp
+ next
+ case False
+ then have "(\<lambda>x. (x - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>\<not> n < 0\<close>
+ by (auto intro!:tendsto_eq_intros)
+ from tendsto_mult[OF _ this,of g "g z",simplified]
+ show ?thesis unfolding c_def using False by simp
+ qed
+ ultimately have "f \<midarrow>z\<rightarrow>c" using tendsto_cong by fast
+ then show False using \<open>is_pole f z\<close> at_neq_bot not_tendsto_and_filterlim_at_infinity
+ unfolding is_pole_def by blast
+ qed
+ moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0"
+ using r(4) \<open>n<0\<close> powr_of_int
+ by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
+ ultimately show ?thesis using r(1-3) \<open>g z\<noteq>0\<close> by auto
+qed
+
+lemma zorder_eqI:
+ assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
+ assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n"
+ shows "zorder f z = n"
+proof -
+ have "continuous_on s g" by (rule holomorphic_on_imp_continuous_on) fact
+ moreover have "open (-{0::complex})" by auto
+ ultimately have "open ((g -` (-{0})) \<inter> s)"
+ unfolding continuous_on_open_vimage[OF \<open>open s\<close>] by blast
+ moreover from assms have "z \<in> (g -` (-{0})) \<inter> s" by auto
+ ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))"
+ unfolding open_contains_cball by blast
+
+ let ?gg= "(\<lambda>w. g w * (w - z) powr n)"
+ define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ have "P n g r"
+ unfolding P_def using r assms(3,4,5) by auto
+ then have "\<exists>g r. P n g r" by auto
+ moreover have unique: "\<exists>!n. \<exists>g r. P n g r" unfolding P_def
+ proof (rule holomorphic_factor_puncture)
+ have "ball z r-{z} \<subseteq> s" using r using ball_subset_cball by blast
+ then have "?gg holomorphic_on ball z r-{z}"
+ using \<open>g holomorphic_on s\<close> r by (auto intro!: holomorphic_intros)
+ then have "f holomorphic_on ball z r - {z}"
+ apply (elim holomorphic_transform)
+ using fg_eq \<open>ball z r-{z} \<subseteq> s\<close> by auto
+ then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
+ using analytic_on_open open_delete r(1) by blast
+ next
+ have "not_essential ?gg z"
+ proof (intro singularity_intros)
+ show "not_essential g z"
+ by (meson \<open>continuous_on s g\<close> assms(1) assms(2) continuous_on_eq_continuous_at
+ isCont_def not_essential_def)
+ show " \<forall>\<^sub>F w in at z. w - z \<noteq> 0" by (simp add: eventually_at_filter)
+ then show "LIM w at z. w - z :> at 0"
+ unfolding filterlim_at by (auto intro:tendsto_eq_intros)
+ show "isolated_singularity_at g z"
+ by (meson Diff_subset open_ball analytic_on_holomorphic
+ assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
+ qed
+ then show "not_essential f z"
+ apply (elim not_essential_transform)
+ unfolding eventually_at using assms(1,2) assms(5)[symmetric]
+ by (metis dist_commute mem_ball openE subsetCE)
+ show "\<exists>\<^sub>F w in at z. f w \<noteq> 0" unfolding frequently_at
+ proof (rule,rule)
+ fix d::real assume "0 < d"
+ define z' where "z'=z+min d r / 2"
+ have "z' \<noteq> z" " dist z' z < d "
+ unfolding z'_def using \<open>d>0\<close> \<open>r>0\<close>
+ by (auto simp add:dist_norm)
+ moreover have "f z' \<noteq> 0"
+ proof (subst fg_eq[OF _ \<open>z'\<noteq>z\<close>])
+ have "z' \<in> cball z r" unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm)
+ then show " z' \<in> s" using r(2) by blast
+ show "g z' * (z' - z) powr of_int n \<noteq> 0"
+ using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> calculation(1) by auto
+ qed
+ ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto
+ qed
+ qed
+ ultimately have "(THE n. \<exists>g r. P n g r) = n"
+ by (rule_tac the1_equality)
+ then show ?thesis unfolding zorder_def P_def by blast
+qed
+
+lemma residue_pole_order:
+ fixes f::"complex \<Rightarrow> complex" and z::complex
+ defines "n \<equiv> nat (- zorder f z)" and "h \<equiv> zor_poly f z"
+ assumes f_iso:"isolated_singularity_at f z"
+ and pole:"is_pole f z"
+ shows "residue f z = ((deriv ^^ (n - 1)) h z / fact (n-1))"
+proof -
+ define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
+ obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
+ obtain r where "0 < n" "0 < r" and r_cball:"cball z r \<subseteq> ball z e" and h_holo: "h holomorphic_on cball z r"
+ and h_divide:"(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
+ proof -
+ obtain r where r:"zorder f z < 0" "h z \<noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
+ "(\<forall>w\<in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
+ using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>,folded n_def h_def] by auto
+ have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
+ moreover have "(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
+ using \<open>h z\<noteq>0\<close> r(6) by blast
+ ultimately show ?thesis using r(3,4,5) that by blast
+ qed
+ have r_nonzero:"\<And>w. w \<in> ball z r - {z} \<Longrightarrow> f w \<noteq> 0"
+ using h_divide by simp
+ define c where "c \<equiv> 2 * pi * \<i>"
+ define der_f where "der_f \<equiv> ((deriv ^^ (n - 1)) h z / fact (n-1))"
+ define h' where "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
+ have "(h' has_contour_integral c / fact (n - 1) * (deriv ^^ (n - 1)) h z) (circlepath z r)"
+ unfolding h'_def
+ proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath[of z r h z "n-1",
+ folded c_def Suc_pred'[OF \<open>n>0\<close>]])
+ show "continuous_on (cball z r) h" using holomorphic_on_imp_continuous_on h_holo by simp
+ show "h holomorphic_on ball z r" using h_holo by auto
+ show " z \<in> ball z r" using \<open>r>0\<close> by auto
+ qed
+ then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def by auto
+ then have "(f has_contour_integral c * der_f) (circlepath z r)"
+ proof (elim has_contour_integral_eq)
+ fix x assume "x \<in> path_image (circlepath z r)"
+ hence "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
+ then show "h' x = f x" using h_divide unfolding h'_def by auto
+ qed
+ moreover have "(f has_contour_integral c * residue f z) (circlepath z r)"
+ using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
+ unfolding c_def by simp
+ ultimately have "c * der_f = c * residue f z" using has_contour_integral_unique by blast
+ hence "der_f = residue f z" unfolding c_def by auto
+ thus ?thesis unfolding der_f_def by auto
+qed
+
+lemma simple_zeroI:
+ assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
+ assumes "\<And>w. w \<in> s \<Longrightarrow> f w = g w * (w - z)"
+ shows "zorder f z = 1"
+ using assms(1-4) by (rule zorder_eqI) (use assms(5) in auto)
+
+lemma higher_deriv_power:
+ shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w =
+ pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
+proof (induction j arbitrary: w)
+ case 0
+ thus ?case by auto
+next
+ case (Suc j w)
+ have "(deriv ^^ Suc j) (\<lambda>w. (w - z) ^ n) w = deriv ((deriv ^^ j) (\<lambda>w. (w - z) ^ n)) w"
+ by simp
+ also have "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) =
+ (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
+ using Suc by (intro Suc.IH ext)
+ also {
+ have "(\<dots> has_field_derivative of_nat (n - j) *
+ pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
+ using Suc.prems by (auto intro!: derivative_eq_intros)
+ also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
+ pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
+ by (cases "Suc j \<le> n", subst pochhammer_rec)
+ (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
+ finally have "deriv (\<lambda>w. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
+ \<dots> * (w - z) ^ (n - Suc j)"
+ by (rule DERIV_imp_deriv)
+ }
+ finally show ?case .
+qed
+
+lemma zorder_zero_eqI:
+ assumes f_holo:"f holomorphic_on s" and "open s" "z \<in> s"
+ assumes zero: "\<And>i. i < nat n \<Longrightarrow> (deriv ^^ i) f z = 0"
+ assumes nz: "(deriv ^^ nat n) f z \<noteq> 0" and "n\<ge>0"
+ shows "zorder f z = n"
+proof -
+ obtain r where [simp]:"r>0" and "ball z r \<subseteq> s"
+ using \<open>open s\<close> \<open>z\<in>s\<close> openE by blast
+ have nz':"\<exists>w\<in>ball z r. f w \<noteq> 0"
+ proof (rule ccontr)
+ assume "\<not> (\<exists>w\<in>ball z r. f w \<noteq> 0)"
+ then have "eventually (\<lambda>u. f u = 0) (nhds z)"
+ using \<open>r>0\<close> unfolding eventually_nhds
+ apply (rule_tac x="ball z r" in exI)
+ by auto
+ then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (\<lambda>_. 0) z"
+ by (intro higher_deriv_cong_ev) auto
+ also have "(deriv ^^ nat n) (\<lambda>_. 0) z = 0"
+ by (induction n) simp_all
+ finally show False using nz by contradiction
+ qed
+
+ define zn g where "zn = zorder f z" and "g = zor_poly f z"
+ obtain e where e_if:"if f z = 0 then 0 < zn else zn = 0" and
+ [simp]:"e>0" and "cball z e \<subseteq> ball z r" and
+ g_holo:"g holomorphic_on cball z e" and
+ e_fac:"(\<forall>w\<in>cball z e. f w = g w * (w - z) ^ nat zn \<and> g w \<noteq> 0)"
+ proof -
+ have "f holomorphic_on ball z r"
+ using f_holo \<open>ball z r \<subseteq> s\<close> by auto
+ from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
+ show ?thesis by blast
+ qed
+ from this(1,2,5) have "zn\<ge>0" "g z\<noteq>0"
+ subgoal by (auto split:if_splits)
+ subgoal using \<open>0 < e\<close> ball_subset_cball centre_in_ball e_fac by blast
+ done
+
+ define A where "A = (\<lambda>i. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
+ have deriv_A:"(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)" for i
+ proof -
+ have "eventually (\<lambda>w. w \<in> ball z e) (nhds z)"
+ using \<open>cball z e \<subseteq> ball z r\<close> \<open>e>0\<close> by (intro eventually_nhds_in_open) auto
+ hence "eventually (\<lambda>w. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
+ apply eventually_elim
+ by (use e_fac in auto)
+ hence "(deriv ^^ i) f z = (deriv ^^ i) (\<lambda>w. (w - z) ^ nat zn * g w) z"
+ by (intro higher_deriv_cong_ev) auto
+ also have "\<dots> = (\<Sum>j=0..i. of_nat (i choose j) *
+ (deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
+ using g_holo \<open>e>0\<close>
+ by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
+ also have "\<dots> = (\<Sum>j=0..i. if j = nat zn then
+ of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
+ proof (intro sum.cong refl, goal_cases)
+ case (1 j)
+ have "(deriv ^^ j) (\<lambda>w. (w - z) ^ nat zn) z =
+ pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
+ by (subst higher_deriv_power) auto
+ also have "\<dots> = (if j = nat zn then fact j else 0)"
+ by (auto simp: not_less pochhammer_0_left pochhammer_fact)
+ also have "of_nat (i choose j) * \<dots> * (deriv ^^ (i - j)) g z =
+ (if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
+ * (deriv ^^ (i - nat zn)) g z else 0)"
+ by simp
+ finally show ?case .
+ qed
+ also have "\<dots> = (if i \<ge> zn then A i else 0)"
+ by (auto simp: A_def)
+ finally show "(deriv ^^ i) f z = \<dots>" .
+ qed
+
+ have False when "n<zn"
+ proof -
+ have "(deriv ^^ nat n) f z = 0"
+ using deriv_A[of "nat n"] that \<open>n\<ge>0\<close> by auto
+ with nz show False by auto
+ qed
+ moreover have "n\<le>zn"
+ proof -
+ have "g z \<noteq> 0" using e_fac[rule_format,of z] \<open>e>0\<close> by simp
+ then have "(deriv ^^ nat zn) f z \<noteq> 0"
+ using deriv_A[of "nat zn"] by(auto simp add:A_def)
+ then have "nat zn \<ge> nat n" using zero[of "nat zn"] by linarith
+ moreover have "zn\<ge>0" using e_if by (auto split:if_splits)
+ ultimately show ?thesis using nat_le_eq_zle by blast
+ qed
+ ultimately show ?thesis unfolding zn_def by fastforce
+qed
+
+lemma
+ assumes "eventually (\<lambda>z. f z = g z) (at z)" "z = z'"
+ shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
+proof -
+ define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
+ \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))"
+ have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h
+ proof -
+ have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g
+ proof -
+ from that(1) obtain r1 where r1_P:"P f n h r1" by auto
+ from that(2) obtain r2 where "r2>0" and r2_dist:"\<forall>x. x \<noteq> z \<and> dist x z \<le> r2 \<longrightarrow> f x = g x"
+ unfolding eventually_at_le by auto
+ define r where "r=min r1 r2"
+ have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto
+ moreover have "h holomorphic_on cball z r"
+ using r1_P unfolding P_def r_def by auto
+ moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
+ proof -
+ have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0"
+ using r1_P that unfolding P_def r_def by auto
+ moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
+ by (simp add: dist_commute)
+ ultimately show ?thesis by simp
+ qed
+ ultimately show ?thesis unfolding P_def by auto
+ qed
+ from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
+ by (simp add: eq_commute)
+ show ?thesis
+ by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
+ qed
+ then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
+ using \<open>z=z'\<close> unfolding P_def zorder_def zor_poly_def by auto
+qed
+
+lemma zorder_nonzero_div_power:
+ assumes "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" "n > 0"
+ shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
+ apply (rule zorder_eqI[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>f holomorphic_on s\<close> \<open>f z\<noteq>0\<close>])
+ apply (subst powr_of_int)
+ using \<open>n>0\<close> by (auto simp add:field_simps)
+
+lemma zor_poly_eq:
+ assumes "isolated_singularity_at f z" "not_essential f z" "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
+proof -
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
+ using zorder_exist[OF assms] by blast
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
+ by (auto simp: field_simps powr_minus)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_zero_eq:
+ assumes "f holomorphic_on s" "open s" "connected s" "z \<in> s" "\<exists>w\<in>s. f w \<noteq> 0"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
+proof -
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
+ using zorder_exist_zero[OF assms] by auto
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
+ by (auto simp: field_simps powr_minus)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_pole_eq:
+ assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
+proof -
+ obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
+ using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
+ obtain r where r:"r>0"
+ "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
+ using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>] by auto
+ then have *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
+ by (auto simp: field_simps)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r eventually_at_ball'[of r z UNIV] by auto
+ thus ?thesis by eventually_elim (insert *, auto)
+qed
+
+lemma zor_poly_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
+ assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ from zorder_exist[OF assms(2-4)] obtain r where
+ r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
+ "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n"
+ unfolding n_def by blast
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
+ by eventually_elim (insert r, auto simp: field_simps powr_minus)
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+lemma zor_poly_zero_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes "f holomorphic_on A" "open A" "connected A" "z0 \<in> A" "\<exists>z\<in>A. f z \<noteq> 0"
+ assumes lim: "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ from zorder_exist_zero[OF assms(2-6)] obtain r where
+ r: "r > 0" "cball z0 r \<subseteq> A" "zor_poly f z0 holomorphic_on cball z0 r"
+ "\<And>w. w \<in> cball z0 r \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) ^ nat n"
+ unfolding n_def by blast
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ hence "eventually (\<lambda>w. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
+ by eventually_elim (insert r, auto simp: field_simps)
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w / (w - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) / (g x - z0) ^ nat n) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+lemma zor_poly_pole_eqI:
+ fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
+ defines "n \<equiv> zorder f z0"
+ assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
+ assumes lim: "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> c) F"
+ assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
+ shows "zor_poly f z0 z0 = c"
+proof -
+ obtain r where r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
+ proof -
+ have "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
+ using non_zero_neighbour_pole[OF \<open>is_pole f z0\<close>] by (auto elim:eventually_frequentlyE)
+ moreover have "not_essential f z0" unfolding not_essential_def using \<open>is_pole f z0\<close> by simp
+ ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
+ qed
+ from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
+ using eventually_at_ball'[of r z0 UNIV] by auto
+ have "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
+ using zor_poly_pole_eq[OF f_iso \<open>is_pole f z0\<close>] unfolding n_def .
+ moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
+ using r by (intro holomorphic_on_imp_continuous_on) auto
+ with r(1,2) have "isCont (zor_poly f z0) z0"
+ by (auto simp: continuous_on_eq_continuous_at)
+ hence "(zor_poly f z0 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ unfolding isCont_def .
+ ultimately have "((\<lambda>w. f w * (w - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ by (blast intro: Lim_transform_eventually)
+ hence "((\<lambda>x. f (g x) * (g x - z0) ^ nat (-n)) \<longlongrightarrow> zor_poly f z0 z0) F"
+ by (rule filterlim_compose[OF _ g])
+ from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
+qed
+
+lemma residue_simple_pole:
+ assumes "isolated_singularity_at f z0"
+ assumes "is_pole f z0" "zorder f z0 = - 1"
+ shows "residue f z0 = zor_poly f z0 z0"
+ using assms by (subst residue_pole_order) simp_all
+
+lemma residue_simple_pole_limit:
+ assumes "isolated_singularity_at f z0"
+ assumes "is_pole f z0" "zorder f z0 = - 1"
+ assumes "((\<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
+ assumes "filterlim g (at z0) F" "F \<noteq> bot"
+ shows "residue f z0 = c"
+proof -
+ have "residue f z0 = zor_poly f z0 z0"
+ by (rule residue_simple_pole assms)+
+ also have "\<dots> = c"
+ apply (rule zor_poly_pole_eqI)
+ using assms by auto
+ finally show ?thesis .
+qed
+
+lemma lhopital_complex_simple:
+ assumes "(f has_field_derivative f') (at z)"
+ assumes "(g has_field_derivative g') (at z)"
+ assumes "f z = 0" "g z = 0" "g' \<noteq> 0" "f' / g' = c"
+ shows "((\<lambda>w. f w / g w) \<longlongrightarrow> c) (at z)"
+proof -
+ have "eventually (\<lambda>w. w \<noteq> z) (at z)"
+ by (auto simp: eventually_at_filter)
+ hence "eventually (\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z)) = f w / g w) (at z)"
+ by eventually_elim (simp add: assms field_split_simps)
+ moreover have "((\<lambda>w. ((f w - f z) / (w - z)) / ((g w - g z) / (w - z))) \<longlongrightarrow> f' / g') (at z)"
+ by (intro tendsto_divide has_field_derivativeD assms)
+ ultimately have "((\<lambda>w. f w / g w) \<longlongrightarrow> f' / g') (at z)"
+ by (blast intro: Lim_transform_eventually)
+ with assms show ?thesis by simp
+qed
+
+lemma
+ assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
+ and "open s" "connected s" "z \<in> s"
+ assumes g_deriv:"(g has_field_derivative g') (at z)"
+ assumes "f z \<noteq> 0" "g z = 0" "g' \<noteq> 0"
+ shows porder_simple_pole_deriv: "zorder (\<lambda>w. f w / g w) z = - 1"
+ and residue_simple_pole_deriv: "residue (\<lambda>w. f w / g w) z = f z / g'"
+proof -
+ have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
+ using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
+ by (meson Diff_subset holomorphic_on_subset)+
+ have [simp]:"not_essential f z" "not_essential g z"
+ unfolding not_essential_def using f_holo g_holo assms(3,5)
+ by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
+ have g_nconst:"\<exists>\<^sub>F w in at z. g w \<noteq>0 "
+ proof (rule ccontr)
+ assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
+ then have "\<forall>\<^sub>F w in nhds z. g w = 0"
+ unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
+ by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
+ then have "deriv g z = deriv (\<lambda>_. 0) z"
+ by (intro deriv_cong_ev) auto
+ then have "deriv g z = 0" by auto
+ then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
+ then show False using \<open>g'\<noteq>0\<close> by auto
+ qed
+
+ have "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
+ proof -
+ have "\<forall>\<^sub>F w in at z. f w \<noteq>0 \<and> w\<in>s"
+ apply (rule non_zero_neighbour_alt)
+ using assms by auto
+ with g_nconst have "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
+ by (elim frequently_rev_mp eventually_rev_mp,auto)
+ then show ?thesis using zorder_divide[of f z g] by auto
+ qed
+ moreover have "zorder f z=0"
+ apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
+ using \<open>f z\<noteq>0\<close> by auto
+ moreover have "zorder g z=1"
+ apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
+ subgoal using assms(8) by auto
+ subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
+ subgoal by simp
+ done
+ ultimately show "zorder (\<lambda>w. f w / g w) z = - 1" by auto
+
+ show "residue (\<lambda>w. f w / g w) z = f z / g'"
+ proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
+ show "zorder (\<lambda>w. f w / g w) z = - 1" by fact
+ show "isolated_singularity_at (\<lambda>w. f w / g w) z"
+ by (auto intro: singularity_intros)
+ show "is_pole (\<lambda>w. f w / g w) z"
+ proof (rule is_pole_divide)
+ have "\<forall>\<^sub>F x in at z. g x \<noteq> 0"
+ apply (rule non_zero_neighbour)
+ using g_nconst by auto
+ moreover have "g \<midarrow>z\<rightarrow> 0"
+ using DERIV_isCont assms(8) continuous_at g_deriv by force
+ ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
+ show "isCont f z"
+ using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
+ by auto
+ show "f z \<noteq> 0" by fact
+ qed
+ show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
+ have "((\<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
+ proof (rule lhopital_complex_simple)
+ show "((\<lambda>w. f w * (w - z)) has_field_derivative f z) (at z)"
+ using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
+ show "(g has_field_derivative g') (at z)" by fact
+ qed (insert assms, auto)
+ then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
+ by (simp add: field_split_simps)
+ qed
+qed
+
+subsection \<open>The argument principle\<close>
+
+theorem argument_principle:
+ fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
+ defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
+ assumes "open s" and
+ "connected s" and
+ f_holo:"f holomorphic_on s-poles" and
+ h_holo:"h holomorphic_on s" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ path_img:"path_image g \<subseteq> s - pz" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
+ finite:"finite pz" and
+ poles:"\<forall>p\<in>poles. is_pole f p"
+ shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
+ (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
+ (is "?L=?R")
+proof -
+ define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
+ define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
+ define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
+ define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
+
+ have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
+ proof -
+ obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
+ using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
+ have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
+ proof -
+ define po where "po \<equiv> zorder f p"
+ define pp where "pp \<equiv> zor_poly f p"
+ define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
+ define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
+ obtain r where "pp p\<noteq>0" "r>0" and
+ "r<e1" and
+ pp_holo:"pp holomorphic_on cball p r" and
+ pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
+ proof -
+ have "isolated_singularity_at f p"
+ proof -
+ have "f holomorphic_on ball p e1 - {p}"
+ apply (intro holomorphic_on_subset[OF f_holo])
+ using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
+ then show ?thesis unfolding isolated_singularity_at_def
+ using \<open>e1>0\<close> analytic_on_open open_delete by blast
+ qed
+ moreover have "not_essential f p"
+ proof (cases "is_pole f p")
+ case True
+ then show ?thesis unfolding not_essential_def by auto
+ next
+ case False
+ then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
+ moreover have "open (s-poles)"
+ using \<open>open s\<close>
+ apply (elim open_Diff)
+ apply (rule finite_imp_closed)
+ using finite unfolding pz_def by simp
+ ultimately have "isCont f p"
+ using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
+ by auto
+ then show ?thesis unfolding isCont_def not_essential_def by auto
+ qed
+ moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
+ proof (rule ccontr)
+ assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
+ then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
+ then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
+ unfolding eventually_at by (auto simp add:dist_commute)
+ then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
+ moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
+ ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
+ then have "infinite pz"
+ unfolding pz_def infinite_super by auto
+ then show False using \<open>finite pz\<close> by auto
+ qed
+ ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
+ "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
+ using zorder_exist[of f p,folded po_def pp_def] by auto
+ define r1 where "r1=min r e1 / 2"
+ have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
+ moreover have "r1>0" "pp holomorphic_on cball p r1"
+ "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
+ unfolding r1_def using \<open>e1>0\<close> r by auto
+ ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
+ qed
+
+ define e2 where "e2 \<equiv> r/2"
+ have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
+ define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
+ define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
+ have "((\<lambda>w. prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
+ proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
+ have "ball p r \<subseteq> s"
+ using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
+ then have "cball p e2 \<subseteq> s"
+ using \<open>r>0\<close> unfolding e2_def by auto
+ then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
+ using h_holo by (auto intro!: holomorphic_intros)
+ then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
+ using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
+ unfolding prin_def by (auto simp add: mult.assoc)
+ have "anal holomorphic_on ball p r" unfolding anal_def
+ using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
+ by (auto intro!: holomorphic_intros)
+ then show "(anal has_contour_integral 0) (circlepath p e2)"
+ using e2_def \<open>r>0\<close>
+ by (auto elim!: Cauchy_theorem_disc_simple)
+ qed
+ then have "cont ff' p e2" unfolding cont_def po_def
+ proof (elim has_contour_integral_eq)
+ fix w assume "w \<in> path_image (circlepath p e2)"
+ then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
+ define wp where "wp \<equiv> w-p"
+ have "wp\<noteq>0" and "pp w \<noteq>0"
+ unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
+ moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
+ proof (rule DERIV_imp_deriv)
+ have "(pp has_field_derivative (deriv pp w)) (at w)"
+ using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
+ by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
+ then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
+ + deriv pp w * (w - p) powr of_int po) (at w)"
+ unfolding f'_def using \<open>w\<noteq>p\<close>
+ by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
+ qed
+ ultimately show "prin w + anal w = ff' w"
+ unfolding ff'_def prin_def anal_def
+ apply simp
+ apply (unfold f'_def)
+ apply (fold wp_def)
+ apply (auto simp add:field_simps)
+ by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
+ qed
+ then have "cont ff p e2" unfolding cont_def
+ proof (elim has_contour_integral_eq)
+ fix w assume "w \<in> path_image (circlepath p e2)"
+ then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
+ have "deriv f' w = deriv f w"
+ proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
+ show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
+ by (auto intro!: holomorphic_intros)
+ next
+ have "ball p e1 - {p} \<subseteq> s - poles"
+ using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
+ by auto
+ then have "ball p r - {p} \<subseteq> s - poles"
+ apply (elim dual_order.trans)
+ using \<open>r<e1\<close> by auto
+ then show "f holomorphic_on ball p r - {p}" using f_holo
+ by auto
+ next
+ show "open (ball p r - {p})" by auto
+ show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
+ next
+ fix x assume "x \<in> ball p r - {p}"
+ then show "f' x = f x"
+ using pp_po unfolding f'_def by auto
+ qed
+ moreover have " f' w = f w "
+ using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
+ unfolding f'_def by auto
+ ultimately show "ff' w = ff w"
+ unfolding ff'_def ff_def by simp
+ qed
+ moreover have "cball p e2 \<subseteq> ball p e1"
+ using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
+ ultimately show ?thesis using \<open>e2>0\<close> by auto
+ qed
+ then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
+ by auto
+ define e4 where "e4 \<equiv> if p\<in>pz then e2 else e1"
+ have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
+ moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
+ moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
+ by (auto simp add: e2 e4_def)
+ ultimately show ?thesis by auto
+ qed
+ then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
+ \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
+ by metis
+ define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
+ define w where "w \<equiv> \<lambda>p. winding_number g p"
+ have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
+ proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
+ path_img homo])
+ have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
+ then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
+ by (auto intro!: holomorphic_intros simp add:pz_def)
+ next
+ show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
+ using get_e using avoid_def by blast
+ qed
+ also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
+ proof (rule sum.cong[of pz pz,simplified])
+ fix p assume "p \<in> pz"
+ show "w p * ci p = c * w p * h p * (zorder f p)"
+ proof (cases "p\<in>s")
+ assume "p \<in> s"
+ have "ci p = c * h p * (zorder f p)" unfolding ci_def
+ apply (rule contour_integral_unique)
+ using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
+ thus ?thesis by auto
+ next
+ assume "p\<notin>s"
+ then have "w p=0" using homo unfolding w_def by auto
+ then show ?thesis by auto
+ qed
+ qed
+ also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
+ unfolding sum_distrib_left by (simp add:algebra_simps)
+ finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
+ then show ?thesis unfolding ff_def c_def w_def by simp
+qed
+
+subsection \<open>Rouche's theorem \<close>
+
+theorem Rouche_theorem:
+ fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
+ defines "fg\<equiv>(\<lambda>p. f p + g p)"
+ defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
+ assumes
+ "open s" and "connected s" and
+ "finite zeros_fg" and
+ "finite zeros_f" and
+ f_holo:"f holomorphic_on s" and
+ g_holo:"g holomorphic_on s" and
+ "valid_path \<gamma>" and
+ loop:"pathfinish \<gamma> = pathstart \<gamma>" and
+ path_img:"path_image \<gamma> \<subseteq> s " and
+ path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
+ shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
+ = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
+proof -
+ have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
+ proof -
+ have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
+ proof -
+ have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
+ moreover have "f z = - g z" using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
+ then have "cmod (f z) = cmod (g z)" by auto
+ ultimately show False by auto
+ qed
+ then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
+ qed
+ have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
+ proof -
+ have False when "z\<in>path_image \<gamma>" and "f z =0" for z
+ proof -
+ have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
+ then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
+ then show False by auto
+ qed
+ then show ?thesis unfolding zeros_f_def using path_img by auto
+ qed
+ define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
+ define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
+ define h where "h \<equiv> \<lambda>p. g p / f p + 1"
+ obtain spikes
+ where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
+ using \<open>valid_path \<gamma>\<close>
+ by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
+ have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
+ proof -
+ have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
+ proof -
+ have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
+ proof -
+ have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
+ apply (cases "cmod (f p) = 0")
+ by (auto simp add: norm_divide)
+ then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
+ qed
+ then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
+ by (simp add: image_subset_iff path_image_compose)
+ moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
+ ultimately show "?thesis"
+ using convex_in_outside[of "ball 1 1" 0] outside_mono by blast
+ qed
+ have valid_h:"valid_path (h \<circ> \<gamma>)"
+ proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
+ show "h holomorphic_on s - zeros_f"
+ unfolding h_def using f_holo g_holo
+ by (auto intro!: holomorphic_intros simp add:zeros_f_def)
+ next
+ show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
+ by auto
+ qed
+ have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
+ proof -
+ have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
+ then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
+ using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
+ unfolding c_def by auto
+ moreover have "winding_number (h o \<gamma>) 0 = 0"
+ proof -
+ have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
+ moreover have "path (h o \<gamma>)"
+ using valid_h by (simp add: valid_path_imp_path)
+ moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
+ by (simp add: loop pathfinish_compose pathstart_compose)
+ ultimately show ?thesis using winding_number_zero_in_outside by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
+ when "x\<in>{0..1} - spikes" for x
+ proof (rule vector_derivative_chain_at_general)
+ show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
+ next
+ define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ define t where "t \<equiv> \<gamma> x"
+ have "f t\<noteq>0" unfolding zeros_f_def t_def
+ by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
+ moreover have "t\<in>s"
+ using contra_subsetD path_image_def path_fg t_def that by fastforce
+ ultimately have "(h has_field_derivative der t) (at t)"
+ unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
+ by (auto intro!: holomorphic_derivI derivative_eq_intros)
+ then show "h field_differentiable at (\<gamma> x)"
+ unfolding t_def field_differentiable_def by blast
+ qed
+ then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
+ = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
+ unfolding has_contour_integral
+ apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
+ by auto
+ ultimately show ?thesis by auto
+ qed
+ then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
+ using contour_integral_unique by simp
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
+ + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
+ proof -
+ have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
+ proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
+ show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
+ by auto
+ then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
+ using f_holo
+ by (auto intro!: holomorphic_intros simp add:zeros_f_def)
+ qed
+ moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
+ using h_contour
+ by (simp add: has_contour_integral_integrable)
+ ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
+ contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
+ using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
+ by auto
+ moreover have "deriv fg p / fg p = deriv f p / f p + deriv h p / h p"
+ when "p\<in> path_image \<gamma>" for p
+ proof -
+ have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
+ by auto
+ have "h p\<noteq>0"
+ proof (rule ccontr)
+ assume "\<not> h p \<noteq> 0"
+ then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
+ then have "cmod (g p/f p) = 1" by auto
+ moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
+ apply (cases "cmod (f p) = 0")
+ by (auto simp add: norm_divide)
+ ultimately show False by auto
+ qed
+ have der_fg:"deriv fg p = deriv f p + deriv g p" unfolding fg_def
+ using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _ \<open>open s\<close>] path_img that
+ by auto
+ have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ proof -
+ define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
+ have "p\<in>s" using path_img that by auto
+ then have "(h has_field_derivative der p) (at p)"
+ unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
+ by (auto intro!: derivative_eq_intros holomorphic_derivI)
+ then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
+ qed
+ show ?thesis
+ apply (simp only:der_fg der_h)
+ apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
+ by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
+ qed
+ then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
+ = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
+ by (elim contour_integral_eq)
+ ultimately show ?thesis by auto
+ qed
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
+ unfolding c_def zeros_fg_def w_def
+ proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
+ , of _ "{}" "\<lambda>_. 1",simplified])
+ show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
+ show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
+ show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
+ qed
+ moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
+ unfolding c_def zeros_f_def w_def
+ proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
+ , of _ "{}" "\<lambda>_. 1",simplified])
+ show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
+ show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
+ show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
+ qed
+ ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
+ by auto
+ then show ?thesis unfolding c_def using w_def by auto
+qed
+
+
+subsection \<open>Poles and residues of some well-known functions\<close>
+
+(* TODO: add more material here for other functions *)
+lemma is_pole_Gamma: "is_pole Gamma (-of_nat n)"
+ unfolding is_pole_def using Gamma_poles .
+
+lemma Gamme_residue:
+ "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
+proof (rule residue_simple')
+ show "open (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
+ by (intro open_Compl closed_subset_Ints) auto
+ show "Gamma holomorphic_on (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) - {- of_nat n})"
+ by (rule holomorphic_Gamma) auto
+ show "(\<lambda>w. Gamma w * (w - (-of_nat n))) \<midarrow>(-of_nat n)\<rightarrow> (- 1) ^ n / fact n"
+ using Gamma_residues[of n] by simp
+qed auto
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Great_Picard.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,1862 @@
+section \<open>The Great Picard Theorem and its Applications\<close>
+
+text\<open>Ported from HOL Light (cauchy.ml) by L C Paulson, 2017\<close>
+
+theory Great_Picard
+ imports Conformal_Mappings
+
+begin
+
+subsection\<open>Schottky's theorem\<close>
+
+lemma Schottky_lemma0:
+ assumes holf: "f holomorphic_on S" and cons: "contractible S" and "a \<in> S"
+ and f: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> -1"
+ obtains g where "g holomorphic_on S"
+ "norm(g a) \<le> 1 + norm(f a) / 3"
+ "\<And>z. z \<in> S \<Longrightarrow> f z = cos(of_real pi * g z)"
+proof -
+ obtain g where holg: "g holomorphic_on S" and g: "norm(g a) \<le> pi + norm(f a)"
+ and f_eq_cos: "\<And>z. z \<in> S \<Longrightarrow> f z = cos(g z)"
+ using contractible_imp_holomorphic_arccos_bounded [OF assms]
+ by blast
+ show ?thesis
+ proof
+ show "(\<lambda>z. g z / pi) holomorphic_on S"
+ by (auto intro: holomorphic_intros holg)
+ have "3 \<le> pi"
+ using pi_approx by force
+ have "3 * norm(g a) \<le> 3 * (pi + norm(f a))"
+ using g by auto
+ also have "... \<le> pi * 3 + pi * cmod (f a)"
+ using \<open>3 \<le> pi\<close> by (simp add: mult_right_mono algebra_simps)
+ finally show "cmod (g a / complex_of_real pi) \<le> 1 + cmod (f a) / 3"
+ by (simp add: field_simps norm_divide)
+ show "\<And>z. z \<in> S \<Longrightarrow> f z = cos (complex_of_real pi * (g z / complex_of_real pi))"
+ by (simp add: f_eq_cos)
+ qed
+qed
+
+
+lemma Schottky_lemma1:
+ fixes n::nat
+ assumes "0 < n"
+ shows "0 < n + sqrt(real n ^ 2 - 1)"
+proof -
+ have "(n-1)^2 \<le> n^2 - 1"
+ using assms by (simp add: algebra_simps power2_eq_square)
+ then have "real (n - 1) \<le> sqrt (real (n\<^sup>2 - 1))"
+ by (metis of_nat_le_iff of_nat_power real_le_rsqrt)
+ then have "n-1 \<le> sqrt(real n ^ 2 - 1)"
+ by (simp add: Suc_leI assms of_nat_diff)
+ then show ?thesis
+ using assms by linarith
+qed
+
+
+lemma Schottky_lemma2:
+ fixes x::real
+ assumes "0 \<le> x"
+ obtains n where "0 < n" "\<bar>x - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
+proof -
+ obtain n0::nat where "0 < n0" "ln(n0 + sqrt(real n0 ^ 2 - 1)) / pi \<le> x"
+ proof
+ show "ln(real 1 + sqrt(real 1 ^ 2 - 1))/pi \<le> x"
+ by (auto simp: assms)
+ qed auto
+ moreover
+ obtain M::nat where "\<And>n. \<lbrakk>0 < n; ln(n + sqrt(real n ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> n \<le> M"
+ proof
+ fix n::nat
+ assume "0 < n" "ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"
+ then have "ln (n + sqrt ((real n)\<^sup>2 - 1)) \<le> x * pi"
+ by (simp add: field_split_simps)
+ then have *: "exp (ln (n + sqrt ((real n)\<^sup>2 - 1))) \<le> exp (x * pi)"
+ by blast
+ have 0: "0 \<le> sqrt ((real n)\<^sup>2 - 1)"
+ using \<open>0 < n\<close> by auto
+ have "n + sqrt ((real n)\<^sup>2 - 1) = exp (ln (n + sqrt ((real n)\<^sup>2 - 1)))"
+ by (simp add: Suc_leI \<open>0 < n\<close> add_pos_nonneg real_of_nat_ge_one_iff)
+ also have "... \<le> exp (x * pi)"
+ using "*" by blast
+ finally have "real n \<le> exp (x * pi)"
+ using 0 by linarith
+ then show "n \<le> nat (ceiling (exp(x * pi)))"
+ by linarith
+ qed
+ ultimately obtain n where
+ "0 < n" and le_x: "ln(n + sqrt(real n ^ 2 - 1)) / pi \<le> x"
+ and le_n: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> n"
+ using bounded_Max_nat [of "\<lambda>n. 0<n \<and> ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi \<le> x"] by metis
+ define a where "a \<equiv> ln(n + sqrt(real n ^ 2 - 1)) / pi"
+ define b where "b \<equiv> ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / pi"
+ have le_xa: "a \<le> x"
+ and le_na: "\<And>k. \<lbrakk>0 < k; ln(k + sqrt(real k ^ 2 - 1)) / pi \<le> x\<rbrakk> \<Longrightarrow> k \<le> n"
+ using le_x le_n by (auto simp: a_def)
+ moreover have "x < b"
+ using le_n [of "Suc n"] by (force simp: b_def)
+ moreover have "b - a < 1"
+ proof -
+ have "ln (1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) - ln (real n + sqrt ((real n)\<^sup>2 - 1)) =
+ ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1)))"
+ by (simp add: \<open>0 < n\<close> Schottky_lemma1 add_pos_nonneg ln_div [symmetric])
+ also have "... \<le> 3"
+ proof (cases "n = 1")
+ case True
+ have "sqrt 3 \<le> 2"
+ by (simp add: real_le_lsqrt)
+ then have "(2 + sqrt 3) \<le> 4"
+ by simp
+ also have "... \<le> exp 3"
+ using exp_ge_add_one_self [of "3::real"] by simp
+ finally have "ln (2 + sqrt 3) \<le> 3"
+ by (metis add_nonneg_nonneg add_pos_nonneg dbl_def dbl_inc_def dbl_inc_simps(3)
+ dbl_simps(3) exp_gt_zero ln_exp ln_le_cancel_iff real_sqrt_ge_0_iff zero_le_one zero_less_one)
+ then show ?thesis
+ by (simp add: True)
+ next
+ case False with \<open>0 < n\<close> have "1 < n" "2 \<le> n"
+ by linarith+
+ then have 1: "1 \<le> real n * real n"
+ by (metis less_imp_le_nat one_le_power power2_eq_square real_of_nat_ge_one_iff)
+ have *: "4 + (m+2) * 2 \<le> (m+2) * ((m+2) * 3)" for m::nat
+ by simp
+ have "4 + n * 2 \<le> n * (n * 3)"
+ using * [of "n-2"] \<open>2 \<le> n\<close>
+ by (metis le_add_diff_inverse2)
+ then have **: "4 + real n * 2 \<le> real n * (real n * 3)"
+ by (metis (mono_tags, hide_lams) of_nat_le_iff of_nat_add of_nat_mult of_nat_numeral)
+ have "sqrt ((1 + real n)\<^sup>2 - 1) \<le> 2 * sqrt ((real n)\<^sup>2 - 1)"
+ by (auto simp: real_le_lsqrt power2_eq_square algebra_simps 1 **)
+ then
+ have "((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> 2"
+ using Schottky_lemma1 \<open>0 < n\<close> by (simp add: field_split_simps)
+ then have "ln ((1 + real n + sqrt ((1 + real n)\<^sup>2 - 1)) / (real n + sqrt ((real n)\<^sup>2 - 1))) \<le> ln 2"
+ apply (subst ln_le_cancel_iff)
+ using Schottky_lemma1 \<open>0 < n\<close> by auto (force simp: field_split_simps)
+ also have "... \<le> 3"
+ using ln_add_one_self_le_self [of 1] by auto
+ finally show ?thesis .
+ qed
+ also have "... < pi"
+ using pi_approx by simp
+ finally show ?thesis
+ by (simp add: a_def b_def field_split_simps)
+ qed
+ ultimately have "\<bar>x - a\<bar> < 1/2 \<or> \<bar>x - b\<bar> < 1/2"
+ by (auto simp: abs_if)
+ then show thesis
+ proof
+ assume "\<bar>x - a\<bar> < 1 / 2"
+ then show ?thesis
+ by (rule_tac n=n in that) (auto simp: a_def \<open>0 < n\<close>)
+ next
+ assume "\<bar>x - b\<bar> < 1 / 2"
+ then show ?thesis
+ by (rule_tac n="Suc n" in that) (auto simp: b_def \<open>0 < n\<close>)
+ qed
+qed
+
+
+lemma Schottky_lemma3:
+ fixes z::complex
+ assumes "z \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)})
+ \<union> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
+ shows "cos(pi * cos(pi * z)) = 1 \<or> cos(pi * cos(pi * z)) = -1"
+proof -
+ have sqrt2 [simp]: "complex_of_real (sqrt x) * complex_of_real (sqrt x) = x" if "x \<ge> 0" for x::real
+ by (metis abs_of_nonneg of_real_mult real_sqrt_mult_self that)
+ have 1: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) -
+ (ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
+ inverse
+ (exp (\<i> * (of_int m * complex_of_real pi) -
+ (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
+ if "n > 0" for m n
+ proof -
+ have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> inverse e^2 - 2 * n*inverse e + 1 = 0" for n e::complex
+ by (auto simp: field_simps power2_eq_square)
+ have [simp]: "1 \<le> real n * real n"
+ by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
+ show ?thesis
+ apply (simp add: eeq)
+ using Schottky_lemma1 [OF that]
+ apply (auto simp: eeq exp_diff exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
+ apply (rule_tac x="int n" in exI)
+ apply (auto simp: power2_eq_square algebra_simps)
+ apply (rule_tac x="- int n" in exI)
+ apply (auto simp: power2_eq_square algebra_simps)
+ done
+ qed
+ have 2: "\<exists>k. exp (\<i> * (of_int m * complex_of_real pi) +
+ (ln (real n + sqrt ((real n)\<^sup>2 - 1)))) +
+ inverse
+ (exp (\<i> * (of_int m * complex_of_real pi) +
+ (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
+ if "n > 0" for m n
+ proof -
+ have eeq: "e \<noteq> 0 \<Longrightarrow> e + inverse e = n*2 \<longleftrightarrow> e^2 - 2 * n*e + 1 = 0" for n e::complex
+ by (auto simp: field_simps power2_eq_square)
+ have [simp]: "1 \<le> real n * real n"
+ by (metis One_nat_def add.commute nat_less_real_le of_nat_1 of_nat_Suc one_le_power power2_eq_square that)
+ show ?thesis
+ apply (simp add: eeq)
+ using Schottky_lemma1 [OF that]
+ apply (auto simp: exp_add exp_Euler exp_of_real algebra_simps sin_int_times_real cos_int_times_real)
+ apply (rule_tac x="int n" in exI)
+ apply (auto simp: power2_eq_square algebra_simps)
+ apply (rule_tac x="- int n" in exI)
+ apply (auto simp: power2_eq_square algebra_simps)
+ done
+ qed
+ have "\<exists>x. cos (complex_of_real pi * z) = of_int x"
+ using assms
+ apply safe
+ apply (auto simp: Ints_def cos_exp_eq exp_minus Complex_eq)
+ apply (auto simp: algebra_simps dest: 1 2)
+ done
+ then have "sin(pi * cos(pi * z)) ^ 2 = 0"
+ by (simp add: Complex_Transcendental.sin_eq_0)
+ then have "1 - cos(pi * cos(pi * z)) ^ 2 = 0"
+ by (simp add: sin_squared_eq)
+ then show ?thesis
+ using power2_eq_1_iff by auto
+qed
+
+
+theorem Schottky:
+ assumes holf: "f holomorphic_on cball 0 1"
+ and nof0: "norm(f 0) \<le> r"
+ and not01: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not>(f z = 0 \<or> f z = 1)"
+ and "0 < t" "t < 1" "norm z \<le> t"
+ shows "norm(f z) \<le> exp(pi * exp(pi * (2 + 2 * r + 12 * t / (1 - t))))"
+proof -
+ obtain h where holf: "h holomorphic_on cball 0 1"
+ and nh0: "norm (h 0) \<le> 1 + norm(2 * f 0 - 1) / 3"
+ and h: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 = cos(of_real pi * h z)"
+ proof (rule Schottky_lemma0 [of "\<lambda>z. 2 * f z - 1" "cball 0 1" 0])
+ show "(\<lambda>z. 2 * f z - 1) holomorphic_on cball 0 1"
+ by (intro holomorphic_intros holf)
+ show "contractible (cball (0::complex) 1)"
+ by (auto simp: convex_imp_contractible)
+ show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> 2 * f z - 1 \<noteq> 1 \<and> 2 * f z - 1 \<noteq> - 1"
+ using not01 by force
+ qed auto
+ obtain g where holg: "g holomorphic_on cball 0 1"
+ and ng0: "norm(g 0) \<le> 1 + norm(h 0) / 3"
+ and g: "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z = cos(of_real pi * g z)"
+ proof (rule Schottky_lemma0 [OF holf convex_imp_contractible, of 0])
+ show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> h z \<noteq> 1 \<and> h z \<noteq> - 1"
+ using h not01 by fastforce+
+ qed auto
+ have g0_2_f0: "norm(g 0) \<le> 2 + norm(f 0)"
+ proof -
+ have "cmod (2 * f 0 - 1) \<le> cmod (2 * f 0) + 1"
+ by (metis norm_one norm_triangle_ineq4)
+ also have "... \<le> 2 + cmod (f 0) * 3"
+ by simp
+ finally have "1 + norm(2 * f 0 - 1) / 3 \<le> (2 + norm(f 0) - 1) * 3"
+ apply (simp add: field_split_simps)
+ using norm_ge_zero [of "f 0 * 2 - 1"]
+ by linarith
+ with nh0 have "norm(h 0) \<le> (2 + norm(f 0) - 1) * 3"
+ by linarith
+ then have "1 + norm(h 0) / 3 \<le> 2 + norm(f 0)"
+ by simp
+ with ng0 show ?thesis
+ by auto
+ qed
+ have "z \<in> ball 0 1"
+ using assms by auto
+ have norm_g_12: "norm(g z - g 0) \<le> (12 * t) / (1 - t)"
+ proof -
+ obtain g' where g': "\<And>x. x \<in> cball 0 1 \<Longrightarrow> (g has_field_derivative g' x) (at x within cball 0 1)"
+ using holg [unfolded holomorphic_on_def field_differentiable_def] by metis
+ have int_g': "(g' has_contour_integral g z - g 0) (linepath 0 z)"
+ using contour_integral_primitive [OF g' valid_path_linepath, of 0 z]
+ using \<open>z \<in> ball 0 1\<close> segment_bound1 by fastforce
+ have "cmod (g' w) \<le> 12 / (1 - t)" if "w \<in> closed_segment 0 z" for w
+ proof -
+ have w: "w \<in> ball 0 1"
+ using segment_bound [OF that] \<open>z \<in> ball 0 1\<close> by simp
+ have ttt: "\<And>z. z \<in> frontier (cball 0 1) \<Longrightarrow> 1 - t \<le> dist w z"
+ using \<open>norm z \<le> t\<close> segment_bound1 [OF \<open>w \<in> closed_segment 0 z\<close>]
+ apply (simp add: dist_complex_def)
+ by (metis diff_left_mono dist_commute dist_complex_def norm_triangle_ineq2 order_trans)
+ have *: "\<lbrakk>\<And>b. (\<exists>w \<in> T \<union> U. w \<in> ball b 1); \<And>x. x \<in> D \<Longrightarrow> g x \<notin> T \<union> U\<rbrakk> \<Longrightarrow> \<nexists>b. ball b 1 \<subseteq> g ` D" for T U D
+ by force
+ have "\<nexists>b. ball b 1 \<subseteq> g ` cball 0 1"
+ proof (rule *)
+ show "(\<exists>w \<in> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
+ (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)}). w \<in> ball b 1)" for b
+ proof -
+ obtain m where m: "m \<in> \<int>" "\<bar>Re b - m\<bar> \<le> 1/2"
+ by (metis Ints_of_int abs_minus_commute of_int_round_abs_le)
+ show ?thesis
+ proof (cases "0::real" "Im b" rule: le_cases)
+ case le
+ then obtain n where "0 < n" and n: "\<bar>Im b - ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
+ using Schottky_lemma2 [of "Im b"] by blast
+ have "dist b (Complex m (Im b)) \<le> 1/2"
+ by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
+ moreover
+ have "dist (Complex m (Im b)) (Complex m (ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1/2"
+ using n by (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
+ ultimately have "dist b (Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1"
+ by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
+ with le m \<open>0 < n\<close> show ?thesis
+ apply (rule_tac x = "Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)" in bexI)
+ apply (simp_all del: Complex_eq greaterThan_0)
+ by blast
+ next
+ case ge
+ then obtain n where "0 < n" and n: "\<bar>- Im b - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1/2"
+ using Schottky_lemma2 [of "- Im b"] by auto
+ have "dist b (Complex m (Im b)) \<le> 1/2"
+ by (metis cancel_comm_monoid_add_class.diff_cancel cmod_eq_Re complex.sel(1) complex.sel(2) dist_norm m(2) minus_complex.code)
+ moreover
+ have "dist (Complex m (- ln (n + sqrt ((real n)\<^sup>2 - 1)) / pi)) (Complex m (Im b)) < 1/2"
+ using n
+ apply (simp add: complex_norm cmod_eq_Re complex_diff dist_norm del: Complex_eq)
+ by (metis add.commute add_uminus_conv_diff)
+ ultimately have "dist b (Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)) < 1"
+ by (simp add: dist_triangle_lt [of b "Complex m (Im b)"] dist_commute)
+ with ge m \<open>0 < n\<close> show ?thesis
+ apply (rule_tac x = "Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)" in bexI)
+ apply (simp_all del: Complex_eq greaterThan_0)
+ by blast
+ qed
+ qed
+ show "g v \<notin> (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (ln(n + sqrt(real n ^ 2 - 1)) / pi)}) \<union>
+ (\<Union>m \<in> Ints. \<Union>n \<in> {0<..}. {Complex m (-ln(n + sqrt(real n ^ 2 - 1)) / pi)})"
+ if "v \<in> cball 0 1" for v
+ using not01 [OF that]
+ by (force simp: g [OF that, symmetric] h [OF that, symmetric] dest!: Schottky_lemma3 [of "g v"])
+ qed
+ then have 12: "(1 - t) * cmod (deriv g w) / 12 < 1"
+ using Bloch_general [OF holg _ ttt, of 1] w by force
+ have "g field_differentiable at w within cball 0 1"
+ using holg w by (simp add: holomorphic_on_def)
+ then have "g field_differentiable at w within ball 0 1"
+ using ball_subset_cball field_differentiable_within_subset by blast
+ with w have der_gw: "(g has_field_derivative deriv g w) (at w)"
+ by (simp add: field_differentiable_within_open [of _ "ball 0 1"] field_differentiable_derivI)
+ with DERIV_unique [OF der_gw] g' w have "deriv g w = g' w"
+ by (metis open_ball at_within_open ball_subset_cball has_field_derivative_subset subsetCE)
+ then show "cmod (g' w) \<le> 12 / (1 - t)"
+ using g' 12 \<open>t < 1\<close> by (simp add: field_simps)
+ qed
+ then have "cmod (g z - g 0) \<le> 12 / (1 - t) * cmod z"
+ using has_contour_integral_bound_linepath [OF int_g', of "12/(1 - t)"] assms
+ by simp
+ with \<open>cmod z \<le> t\<close> \<open>t < 1\<close> show ?thesis
+ by (simp add: field_split_simps)
+ qed
+ have fz: "f z = (1 + cos(of_real pi * h z)) / 2"
+ using h \<open>z \<in> ball 0 1\<close> by (auto simp: field_simps)
+ have "cmod (f z) \<le> exp (cmod (complex_of_real pi * h z))"
+ by (simp add: fz mult.commute norm_cos_plus1_le)
+ also have "... \<le> exp (pi * exp (pi * (2 + 2 * r + 12 * t / (1 - t))))"
+ proof (simp add: norm_mult)
+ have "cmod (g z - g 0) \<le> 12 * t / (1 - t)"
+ using norm_g_12 \<open>t < 1\<close> by (simp add: norm_mult)
+ then have "cmod (g z) - cmod (g 0) \<le> 12 * t / (1 - t)"
+ using norm_triangle_ineq2 order_trans by blast
+ then have *: "cmod (g z) \<le> 2 + 2 * r + 12 * t / (1 - t)"
+ using g0_2_f0 norm_ge_zero [of "f 0"] nof0
+ by linarith
+ have "cmod (h z) \<le> exp (cmod (complex_of_real pi * g z))"
+ using \<open>z \<in> ball 0 1\<close> by (simp add: g norm_cos_le)
+ also have "... \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))"
+ using \<open>t < 1\<close> nof0 * by (simp add: norm_mult)
+ finally show "cmod (h z) \<le> exp (pi * (2 + 2 * r + 12 * t / (1 - t)))" .
+ qed
+ finally show ?thesis .
+qed
+
+
+subsection\<open>The Little Picard Theorem\<close>
+
+theorem Landau_Picard:
+ obtains R
+ where "\<And>z. 0 < R z"
+ "\<And>f. \<lbrakk>f holomorphic_on cball 0 (R(f 0));
+ \<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1\<rbrakk> \<Longrightarrow> norm(deriv f 0) < 1"
+proof -
+ define R where "R \<equiv> \<lambda>z. 3 * exp(pi * exp(pi*(2 + 2 * cmod z + 12)))"
+ show ?thesis
+ proof
+ show Rpos: "\<And>z. 0 < R z"
+ by (auto simp: R_def)
+ show "norm(deriv f 0) < 1"
+ if holf: "f holomorphic_on cball 0 (R(f 0))"
+ and Rf: "\<And>z. norm z \<le> R(f 0) \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1" for f
+ proof -
+ let ?r = "R(f 0)"
+ define g where "g \<equiv> f \<circ> (\<lambda>z. of_real ?r * z)"
+ have "0 < ?r"
+ using Rpos by blast
+ have holg: "g holomorphic_on cball 0 1"
+ unfolding g_def
+ apply (intro holomorphic_intros holomorphic_on_compose holomorphic_on_subset [OF holf])
+ using Rpos by (auto simp: less_imp_le norm_mult)
+ have *: "norm(g z) \<le> exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12 * t / (1 - t))))"
+ if "0 < t" "t < 1" "norm z \<le> t" for t z
+ proof (rule Schottky [OF holg])
+ show "cmod (g 0) \<le> cmod (f 0)"
+ by (simp add: g_def)
+ show "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not> (g z = 0 \<or> g z = 1)"
+ using Rpos by (simp add: g_def less_imp_le norm_mult Rf)
+ qed (auto simp: that)
+ have C1: "g holomorphic_on ball 0 (1 / 2)"
+ by (rule holomorphic_on_subset [OF holg]) auto
+ have C2: "continuous_on (cball 0 (1 / 2)) g"
+ by (meson cball_divide_subset_numeral holg holomorphic_on_imp_continuous_on holomorphic_on_subset)
+ have C3: "cmod (g z) \<le> R (f 0) / 3" if "cmod (0 - z) = 1/2" for z
+ proof -
+ have "norm(g z) \<le> exp(pi * exp(pi * (2 + 2 * norm (f 0) + 12)))"
+ using * [of "1/2"] that by simp
+ also have "... = ?r / 3"
+ by (simp add: R_def)
+ finally show ?thesis .
+ qed
+ then have cmod_g'_le: "cmod (deriv g 0) * 3 \<le> R (f 0) * 2"
+ using Cauchy_inequality [OF C1 C2 _ C3, of 1] by simp
+ have holf': "f holomorphic_on ball 0 (R(f 0))"
+ by (rule holomorphic_on_subset [OF holf]) auto
+ then have fd0: "f field_differentiable at 0"
+ by (rule holomorphic_on_imp_differentiable_at [OF _ open_ball])
+ (auto simp: Rpos [of "f 0"])
+ have g_eq: "deriv g 0 = of_real ?r * deriv f 0"
+ apply (rule DERIV_imp_deriv)
+ apply (simp add: g_def)
+ by (metis DERIV_chain DERIV_cmult_Id fd0 field_differentiable_derivI mult.commute mult_zero_right)
+ show ?thesis
+ using cmod_g'_le Rpos [of "f 0"] by (simp add: g_eq norm_mult)
+ qed
+ qed
+qed
+
+lemma little_Picard_01:
+ assumes holf: "f holomorphic_on UNIV" and f01: "\<And>z. f z \<noteq> 0 \<and> f z \<noteq> 1"
+ obtains c where "f = (\<lambda>x. c)"
+proof -
+ obtain R
+ where Rpos: "\<And>z. 0 < R z"
+ and R: "\<And>h. \<lbrakk>h holomorphic_on cball 0 (R(h 0));
+ \<And>z. norm z \<le> R(h 0) \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1\<rbrakk> \<Longrightarrow> norm(deriv h 0) < 1"
+ using Landau_Picard by metis
+ have contf: "continuous_on UNIV f"
+ by (simp add: holf holomorphic_on_imp_continuous_on)
+ show ?thesis
+ proof (cases "\<forall>x. deriv f x = 0")
+ case True
+ obtain c where "\<And>x. f(x) = c"
+ apply (rule DERIV_zero_connected_constant [OF connected_UNIV open_UNIV finite.emptyI contf])
+ apply (metis True DiffE holf holomorphic_derivI open_UNIV, auto)
+ done
+ then show ?thesis
+ using that by auto
+ next
+ case False
+ then obtain w where w: "deriv f w \<noteq> 0" by auto
+ define fw where "fw \<equiv> (f \<circ> (\<lambda>z. w + z / deriv f w))"
+ have norm_let1: "norm(deriv fw 0) < 1"
+ proof (rule R)
+ show "fw holomorphic_on cball 0 (R (fw 0))"
+ unfolding fw_def
+ by (intro holomorphic_intros holomorphic_on_compose w holomorphic_on_subset [OF holf] subset_UNIV)
+ show "fw z \<noteq> 0 \<and> fw z \<noteq> 1" if "cmod z \<le> R (fw 0)" for z
+ using f01 by (simp add: fw_def)
+ qed
+ have "(fw has_field_derivative deriv f w * inverse (deriv f w)) (at 0)"
+ apply (simp add: fw_def)
+ apply (rule DERIV_chain)
+ using holf holomorphic_derivI apply force
+ apply (intro derivative_eq_intros w)
+ apply (auto simp: field_simps)
+ done
+ then show ?thesis
+ using norm_let1 w by (simp add: DERIV_imp_deriv)
+ qed
+qed
+
+
+theorem little_Picard:
+ assumes holf: "f holomorphic_on UNIV"
+ and "a \<noteq> b" "range f \<inter> {a,b} = {}"
+ obtains c where "f = (\<lambda>x. c)"
+proof -
+ let ?g = "\<lambda>x. 1/(b - a)*(f x - b) + 1"
+ obtain c where "?g = (\<lambda>x. c)"
+ proof (rule little_Picard_01)
+ show "?g holomorphic_on UNIV"
+ by (intro holomorphic_intros holf)
+ show "\<And>z. ?g z \<noteq> 0 \<and> ?g z \<noteq> 1"
+ using assms by (auto simp: field_simps)
+ qed auto
+ then have "?g x = c" for x
+ by meson
+ then have "f x = c * (b-a) + a" for x
+ using assms by (auto simp: field_simps)
+ then show ?thesis
+ using that by blast
+qed
+
+
+text\<open>A couple of little applications of Little Picard\<close>
+
+lemma holomorphic_periodic_fixpoint:
+ assumes holf: "f holomorphic_on UNIV"
+ and "p \<noteq> 0" and per: "\<And>z. f(z + p) = f z"
+ obtains x where "f x = x"
+proof -
+ have False if non: "\<And>x. f x \<noteq> x"
+ proof -
+ obtain c where "(\<lambda>z. f z - z) = (\<lambda>z. c)"
+ proof (rule little_Picard)
+ show "(\<lambda>z. f z - z) holomorphic_on UNIV"
+ by (simp add: holf holomorphic_on_diff)
+ show "range (\<lambda>z. f z - z) \<inter> {p,0} = {}"
+ using assms non by auto (metis add.commute diff_eq_eq)
+ qed (auto simp: assms)
+ with per show False
+ by (metis add.commute add_cancel_left_left \<open>p \<noteq> 0\<close> diff_add_cancel)
+ qed
+ then show ?thesis
+ using that by blast
+qed
+
+
+lemma holomorphic_involution_point:
+ assumes holfU: "f holomorphic_on UNIV" and non: "\<And>a. f \<noteq> (\<lambda>x. a + x)"
+ obtains x where "f(f x) = x"
+proof -
+ { assume non_ff [simp]: "\<And>x. f(f x) \<noteq> x"
+ then have non_fp [simp]: "f z \<noteq> z" for z
+ by metis
+ have holf: "f holomorphic_on X" for X
+ using assms holomorphic_on_subset by blast
+ obtain c where c: "(\<lambda>x. (f(f x) - x)/(f x - x)) = (\<lambda>x. c)"
+ proof (rule little_Picard_01)
+ show "(\<lambda>x. (f(f x) - x)/(f x - x)) holomorphic_on UNIV"
+ apply (intro holomorphic_intros holf holomorphic_on_compose [unfolded o_def, OF holf])
+ using non_fp by auto
+ qed auto
+ then obtain "c \<noteq> 0" "c \<noteq> 1"
+ by (metis (no_types, hide_lams) non_ff diff_zero divide_eq_0_iff right_inverse_eq)
+ have eq: "f(f x) - c * f x = x*(1 - c)" for x
+ using fun_cong [OF c, of x] by (simp add: field_simps)
+ have df_times_dff: "deriv f z * (deriv f (f z) - c) = 1 * (1 - c)" for z
+ proof (rule DERIV_unique)
+ show "((\<lambda>x. f (f x) - c * f x) has_field_derivative
+ deriv f z * (deriv f (f z) - c)) (at z)"
+ apply (intro derivative_eq_intros)
+ apply (rule DERIV_chain [unfolded o_def, of f])
+ apply (auto simp: algebra_simps intro!: holomorphic_derivI [OF holfU])
+ done
+ show "((\<lambda>x. f (f x) - c * f x) has_field_derivative 1 * (1 - c)) (at z)"
+ by (simp add: eq mult_commute_abs)
+ qed
+ { fix z::complex
+ obtain k where k: "deriv f \<circ> f = (\<lambda>x. k)"
+ proof (rule little_Picard)
+ show "(deriv f \<circ> f) holomorphic_on UNIV"
+ by (meson holfU holomorphic_deriv holomorphic_on_compose holomorphic_on_subset open_UNIV subset_UNIV)
+ obtain "deriv f (f x) \<noteq> 0" "deriv f (f x) \<noteq> c" for x
+ using df_times_dff \<open>c \<noteq> 1\<close> eq_iff_diff_eq_0
+ by (metis lambda_one mult_zero_left mult_zero_right)
+ then show "range (deriv f \<circ> f) \<inter> {0,c} = {}"
+ by force
+ qed (use \<open>c \<noteq> 0\<close> in auto)
+ have "\<not> f constant_on UNIV"
+ by (meson UNIV_I non_ff constant_on_def)
+ with holf open_mapping_thm have "open(range f)"
+ by blast
+ obtain l where l: "\<And>x. f x - k * x = l"
+ proof (rule DERIV_zero_connected_constant [of UNIV "{}" "\<lambda>x. f x - k * x"], simp_all)
+ have "deriv f w - k = 0" for w
+ proof (rule analytic_continuation [OF _ open_UNIV connected_UNIV subset_UNIV, of "\<lambda>z. deriv f z - k" "f z" "range f" w])
+ show "(\<lambda>z. deriv f z - k) holomorphic_on UNIV"
+ by (intro holomorphic_intros holf open_UNIV)
+ show "f z islimpt range f"
+ by (metis (no_types, lifting) IntI UNIV_I \<open>open (range f)\<close> image_eqI inf.absorb_iff2 inf_aci(1) islimpt_UNIV islimpt_eq_acc_point open_Int top_greatest)
+ show "\<And>z. z \<in> range f \<Longrightarrow> deriv f z - k = 0"
+ by (metis comp_def diff_self image_iff k)
+ qed auto
+ moreover
+ have "((\<lambda>x. f x - k * x) has_field_derivative deriv f x - k) (at x)" for x
+ by (metis DERIV_cmult_Id Deriv.field_differentiable_diff UNIV_I field_differentiable_derivI holf holomorphic_on_def)
+ ultimately
+ show "\<forall>x. ((\<lambda>x. f x - k * x) has_field_derivative 0) (at x)"
+ by auto
+ show "continuous_on UNIV (\<lambda>x. f x - k * x)"
+ by (simp add: continuous_on_diff holf holomorphic_on_imp_continuous_on)
+ qed (auto simp: connected_UNIV)
+ have False
+ proof (cases "k=1")
+ case True
+ then have "\<exists>x. k * x + l \<noteq> a + x" for a
+ using l non [of a] ext [of f "(+) a"]
+ by (metis add.commute diff_eq_eq)
+ with True show ?thesis by auto
+ next
+ case False
+ have "\<And>x. (1 - k) * x \<noteq> f 0"
+ using l [of 0] apply (simp add: algebra_simps)
+ by (metis diff_add_cancel l mult.commute non_fp)
+ then show False
+ by (metis False eq_iff_diff_eq_0 mult.commute nonzero_mult_div_cancel_right times_divide_eq_right)
+ qed
+ }
+ }
+ then show thesis
+ using that by blast
+qed
+
+
+subsection\<open>The Arzelà --Ascoli theorem\<close>
+
+lemma subsequence_diagonalization_lemma:
+ fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"
+ assumes sub: "\<And>i r. \<exists>k. strict_mono (k :: nat \<Rightarrow> nat) \<and> P i (r \<circ> k)"
+ and P_P: "\<And>i r::nat \<Rightarrow> 'a. \<And>k1 k2 N.
+ \<lbrakk>P i (r \<circ> k1); \<And>j. N \<le> j \<Longrightarrow> \<exists>j'. j \<le> j' \<and> k2 j = k1 j'\<rbrakk> \<Longrightarrow> P i (r \<circ> k2)"
+ obtains k where "strict_mono (k :: nat \<Rightarrow> nat)" "\<And>i. P i (r \<circ> k)"
+proof -
+ obtain kk where "\<And>i r. strict_mono (kk i r :: nat \<Rightarrow> nat) \<and> P i (r \<circ> (kk i r))"
+ using sub by metis
+ then have sub_kk: "\<And>i r. strict_mono (kk i r)" and P_kk: "\<And>i r. P i (r \<circ> (kk i r))"
+ by auto
+ define rr where "rr \<equiv> rec_nat (kk 0 r) (\<lambda>n x. x \<circ> kk (Suc n) (r \<circ> x))"
+ then have [simp]: "rr 0 = kk 0 r" "\<And>n. rr(Suc n) = rr n \<circ> kk (Suc n) (r \<circ> rr n)"
+ by auto
+ show thesis
+ proof
+ have sub_rr: "strict_mono (rr i)" for i
+ using sub_kk by (induction i) (auto simp: strict_mono_def o_def)
+ have P_rr: "P i (r \<circ> rr i)" for i
+ using P_kk by (induction i) (auto simp: o_def)
+ have "i \<le> i+d \<Longrightarrow> rr i n \<le> rr (i+d) n" for d i n
+ proof (induction d)
+ case 0 then show ?case
+ by simp
+ next
+ case (Suc d) then show ?case
+ apply simp
+ using seq_suble [OF sub_kk] order_trans strict_mono_less_eq [OF sub_rr] by blast
+ qed
+ then have "\<And>i j n. i \<le> j \<Longrightarrow> rr i n \<le> rr j n"
+ by (metis le_iff_add)
+ show "strict_mono (\<lambda>n. rr n n)"
+ apply (simp add: strict_mono_Suc_iff)
+ by (meson lessI less_le_trans seq_suble strict_monoD sub_kk sub_rr)
+ have "\<exists>j. i \<le> j \<and> rr (n+d) i = rr n j" for d n i
+ apply (induction d arbitrary: i, auto)
+ by (meson order_trans seq_suble sub_kk)
+ then have "\<And>m n i. n \<le> m \<Longrightarrow> \<exists>j. i \<le> j \<and> rr m i = rr n j"
+ by (metis le_iff_add)
+ then show "P i (r \<circ> (\<lambda>n. rr n n))" for i
+ by (meson P_rr P_P)
+ qed
+qed
+
+lemma function_convergent_subsequence:
+ fixes f :: "[nat,'a] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
+ assumes "countable S" and M: "\<And>n::nat. \<And>x. x \<in> S \<Longrightarrow> norm(f n x) \<le> M"
+ obtains k where "strict_mono (k::nat\<Rightarrow>nat)" "\<And>x. x \<in> S \<Longrightarrow> \<exists>l. (\<lambda>n. f (k n) x) \<longlonglongrightarrow> l"
+proof (cases "S = {}")
+ case True
+ then show ?thesis
+ using strict_mono_id that by fastforce
+next
+ case False
+ with \<open>countable S\<close> obtain \<sigma> :: "nat \<Rightarrow> 'a" where \<sigma>: "S = range \<sigma>"
+ using uncountable_def by blast
+ obtain k where "strict_mono k" and k: "\<And>i. \<exists>l. (\<lambda>n. (f \<circ> k) n (\<sigma> i)) \<longlonglongrightarrow> l"
+ proof (rule subsequence_diagonalization_lemma
+ [of "\<lambda>i r. \<exists>l. ((\<lambda>n. (f \<circ> r) n (\<sigma> i)) \<longlongrightarrow> l) sequentially" id])
+ show "\<exists>k::nat\<Rightarrow>nat. strict_mono k \<and> (\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k)) n (\<sigma> i)) \<longlonglongrightarrow> l)" for i r
+ proof -
+ have "f (r n) (\<sigma> i) \<in> cball 0 M" for n
+ by (simp add: \<sigma> M)
+ then show ?thesis
+ using compact_def [of "cball (0::'b) M"] apply simp
+ apply (drule_tac x="(\<lambda>n. f (r n) (\<sigma> i))" in spec)
+ apply (force simp: o_def)
+ done
+ qed
+ show "\<And>i r k1 k2 N.
+ \<lbrakk>\<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k1)) n (\<sigma> i)) \<longlonglongrightarrow> l; \<And>j. N \<le> j \<Longrightarrow> \<exists>j'\<ge>j. k2 j = k1 j'\<rbrakk>
+ \<Longrightarrow> \<exists>l. (\<lambda>n. (f \<circ> (r \<circ> k2)) n (\<sigma> i)) \<longlonglongrightarrow> l"
+ apply (simp add: lim_sequentially)
+ apply (erule ex_forward all_forward imp_forward)+
+ apply auto
+ by (metis (no_types, hide_lams) le_cases order_trans)
+ qed auto
+ with \<sigma> that show ?thesis
+ by force
+qed
+
+
+theorem Arzela_Ascoli:
+ fixes \<F> :: "[nat,'a::euclidean_space] \<Rightarrow> 'b::{real_normed_vector,heine_borel}"
+ assumes "compact S"
+ and M: "\<And>n x. x \<in> S \<Longrightarrow> norm(\<F> n x) \<le> M"
+ and equicont:
+ "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
+ \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n y. y \<in> S \<and> norm(x - y) < d \<longrightarrow> norm(\<F> n x - \<F> n y) < e)"
+ obtains g k where "continuous_on S g" "strict_mono (k :: nat \<Rightarrow> nat)"
+ "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(k n) x - g x) < e"
+proof -
+ have UEQ: "\<And>e. 0 < e \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e)"
+ apply (rule compact_uniformly_equicontinuous [OF \<open>compact S\<close>, of "range \<F>"])
+ using equicont by (force simp: dist_commute dist_norm)+
+ have "continuous_on S g"
+ if "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<and> x \<in> S \<longrightarrow> norm(\<F>(r n) x - g x) < e"
+ for g:: "'a \<Rightarrow> 'b" and r :: "nat \<Rightarrow> nat"
+ proof (rule uniform_limit_theorem [of _ "\<F> \<circ> r"])
+ show "\<forall>\<^sub>F n in sequentially. continuous_on S ((\<F> \<circ> r) n)"
+ apply (simp add: eventually_sequentially)
+ apply (rule_tac x=0 in exI)
+ using UEQ apply (force simp: continuous_on_iff)
+ done
+ show "uniform_limit S (\<F> \<circ> r) g sequentially"
+ apply (simp add: uniform_limit_iff eventually_sequentially)
+ by (metis dist_norm that)
+ qed auto
+ moreover
+ obtain R where "countable R" "R \<subseteq> S" and SR: "S \<subseteq> closure R"
+ by (metis separable that)
+ obtain k where "strict_mono k" and k: "\<And>x. x \<in> R \<Longrightarrow> \<exists>l. (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> l"
+ apply (rule function_convergent_subsequence [OF \<open>countable R\<close> M])
+ using \<open>R \<subseteq> S\<close> apply force+
+ done
+ then have Cauchy: "Cauchy ((\<lambda>n. \<F> (k n) x))" if "x \<in> R" for x
+ using convergent_eq_Cauchy that by blast
+ have "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> x \<in> S \<longrightarrow> dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
+ if "0 < e" for e
+ proof -
+ obtain d where "0 < d"
+ and d: "\<And>n. \<forall>x \<in> S. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (\<F> n x') (\<F> n x) < e/3"
+ by (metis UEQ \<open>0 < e\<close> divide_pos_pos zero_less_numeral)
+ obtain T where "T \<subseteq> R" and "finite T" and T: "S \<subseteq> (\<Union>c\<in>T. ball c d)"
+ proof (rule compactE_image [OF \<open>compact S\<close>, of R "(\<lambda>x. ball x d)"])
+ have "closure R \<subseteq> (\<Union>c\<in>R. ball c d)"
+ apply clarsimp
+ using \<open>0 < d\<close> closure_approachable by blast
+ with SR show "S \<subseteq> (\<Union>c\<in>R. ball c d)"
+ by auto
+ qed auto
+ have "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (\<F> (k m) x) (\<F> (k n) x) < e/3" if "x \<in> R" for x
+ using Cauchy \<open>0 < e\<close> that unfolding Cauchy_def
+ by (metis less_divide_eq_numeral1(1) mult_zero_left)
+ then obtain MF where MF: "\<And>x m n. \<lbrakk>x \<in> R; m \<ge> MF x; n \<ge> MF x\<rbrakk> \<Longrightarrow> norm (\<F> (k m) x - \<F> (k n) x) < e/3"
+ using dist_norm by metis
+ have "dist ((\<F> \<circ> k) m x) ((\<F> \<circ> k) n x) < e"
+ if m: "Max (MF ` T) \<le> m" and n: "Max (MF ` T) \<le> n" "x \<in> S" for m n x
+ proof -
+ obtain t where "t \<in> T" and t: "x \<in> ball t d"
+ using \<open>x \<in> S\<close> T by auto
+ have "norm(\<F> (k m) t - \<F> (k m) x) < e / 3"
+ by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> d dist_norm mem_ball subset_iff t \<open>x \<in> S\<close>)
+ moreover
+ have "norm(\<F> (k n) t - \<F> (k n) x) < e / 3"
+ by (metis \<open>R \<subseteq> S\<close> \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> subsetD d dist_norm mem_ball t \<open>x \<in> S\<close>)
+ moreover
+ have "norm(\<F> (k m) t - \<F> (k n) t) < e / 3"
+ proof (rule MF)
+ show "t \<in> R"
+ using \<open>T \<subseteq> R\<close> \<open>t \<in> T\<close> by blast
+ show "MF t \<le> m" "MF t \<le> n"
+ by (meson Max_ge \<open>finite T\<close> \<open>t \<in> T\<close> finite_imageI imageI le_trans m n)+
+ qed
+ ultimately
+ show ?thesis
+ unfolding dist_norm [symmetric] o_def
+ by (metis dist_triangle_third dist_commute)
+ qed
+ then show ?thesis
+ by force
+ qed
+ then have "\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> S. norm(\<F>(k n) x - g x) < e"
+ using uniformly_convergent_eq_cauchy [of "\<lambda>x. x \<in> S" "\<F> \<circ> k"]
+ apply (simp add: o_def dist_norm)
+ by meson
+ ultimately show thesis
+ by (metis that \<open>strict_mono k\<close>)
+qed
+
+
+
+subsubsection\<^marker>\<open>tag important\<close>\<open>Montel's theorem\<close>
+
+text\<open>a sequence of holomorphic functions uniformly bounded
+on compact subsets of an open set S has a subsequence that converges to a
+holomorphic function, and converges \emph{uniformly} on compact subsets of S.\<close>
+
+
+theorem Montel:
+ fixes \<F> :: "[nat,complex] \<Rightarrow> complex"
+ assumes "open S"
+ and \<H>: "\<And>h. h \<in> \<H> \<Longrightarrow> h holomorphic_on S"
+ and bounded: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K. norm(h z) \<le> B"
+ and rng_f: "range \<F> \<subseteq> \<H>"
+ obtains g r
+ where "g holomorphic_on S" "strict_mono (r :: nat \<Rightarrow> nat)"
+ "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>n. \<F> (r n) x) \<longlongrightarrow> g x) sequentially"
+ "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) g sequentially"
+proof -
+ obtain K where comK: "\<And>n. compact(K n)" and KS: "\<And>n::nat. K n \<subseteq> S"
+ and subK: "\<And>X. \<lbrakk>compact X; X \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. X \<subseteq> K n"
+ using open_Union_compact_subsets [OF \<open>open S\<close>] by metis
+ then have "\<And>i. \<exists>B. \<forall>h \<in> \<H>. \<forall> z \<in> K i. norm(h z) \<le> B"
+ by (simp add: bounded)
+ then obtain B where B: "\<And>i h z. \<lbrakk>h \<in> \<H>; z \<in> K i\<rbrakk> \<Longrightarrow> norm(h z) \<le> B i"
+ by metis
+ have *: "\<exists>r g. strict_mono (r::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r) n x - g x) < e)"
+ if "\<And>n. \<F> n \<in> \<H>" for \<F> i
+ proof -
+ obtain g k where "continuous_on (K i) g" "strict_mono (k::nat\<Rightarrow>nat)"
+ "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm(\<F>(k n) x - g x) < e"
+ proof (rule Arzela_Ascoli [of "K i" "\<F>" "B i"])
+ show "\<exists>d>0. \<forall>n y. y \<in> K i \<and> cmod (z - y) < d \<longrightarrow> cmod (\<F> n z - \<F> n y) < e"
+ if z: "z \<in> K i" and "0 < e" for z e
+ proof -
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ using z KS [of i] \<open>open S\<close> by (force simp: open_contains_cball)
+ have "cball z (2 / 3 * r) \<subseteq> cball z r"
+ using \<open>0 < r\<close> by (simp add: cball_subset_cball_iff)
+ then have z23S: "cball z (2 / 3 * r) \<subseteq> S"
+ using r by blast
+ obtain M where "0 < M" and M: "\<And>n w. dist z w \<le> 2/3 * r \<Longrightarrow> norm(\<F> n w) \<le> M"
+ proof -
+ obtain N where N: "\<forall>n\<ge>N. cball z (2/3 * r) \<subseteq> K n"
+ using subK compact_cball [of z "(2 / 3 * r)"] z23S by force
+ have "cmod (\<F> n w) \<le> \<bar>B N\<bar> + 1" if "dist z w \<le> 2 / 3 * r" for n w
+ proof -
+ have "w \<in> K N"
+ using N mem_cball that by blast
+ then have "cmod (\<F> n w) \<le> B N"
+ using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
+ also have "... \<le> \<bar>B N\<bar> + 1"
+ by simp
+ finally show ?thesis .
+ qed
+ then show ?thesis
+ by (rule_tac M="\<bar>B N\<bar> + 1" in that) auto
+ qed
+ have "cmod (\<F> n z - \<F> n y) < e"
+ if "y \<in> K i" and y_near_z: "cmod (z - y) < r/3" "cmod (z - y) < e * r / (6 * M)"
+ for n y
+ proof -
+ have "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral
+ (2 * pi) * \<i> * winding_number (circlepath z (2 / 3 * r)) \<xi> * \<F> n \<xi>)
+ (circlepath z (2 / 3 * r))"
+ if "dist \<xi> z < (2 / 3 * r)" for \<xi>
+ proof (rule Cauchy_integral_formula_convex_simple)
+ have "\<F> n holomorphic_on S"
+ by (simp add: \<H> \<open>\<And>n. \<F> n \<in> \<H>\<close>)
+ with z23S show "\<F> n holomorphic_on cball z (2 / 3 * r)"
+ using holomorphic_on_subset by blast
+ qed (use that \<open>0 < r\<close> in \<open>auto simp: dist_commute\<close>)
+ then have *: "((\<lambda>w. \<F> n w / (w - \<xi>)) has_contour_integral (2 * pi) * \<i> * \<F> n \<xi>)
+ (circlepath z (2 / 3 * r))"
+ if "dist \<xi> z < (2 / 3 * r)" for \<xi>
+ using that by (simp add: winding_number_circlepath dist_norm)
+ have y: "((\<lambda>w. \<F> n w / (w - y)) has_contour_integral (2 * pi) * \<i> * \<F> n y)
+ (circlepath z (2 / 3 * r))"
+ apply (rule *)
+ using that \<open>0 < r\<close> by (simp only: dist_norm norm_minus_commute)
+ have z: "((\<lambda>w. \<F> n w / (w - z)) has_contour_integral (2 * pi) * \<i> * \<F> n z)
+ (circlepath z (2 / 3 * r))"
+ apply (rule *)
+ using \<open>0 < r\<close> by simp
+ have le_er: "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) \<le> e / r"
+ if "cmod (x - z) = r/3 + r/3" for x
+ proof -
+ have "\<not> (cmod (x - y) < r/3)"
+ using y_near_z(1) that \<open>M > 0\<close> \<open>r > 0\<close>
+ by (metis (full_types) norm_diff_triangle_less norm_minus_commute order_less_irrefl)
+ then have r4_le_xy: "r/4 \<le> cmod (x - y)"
+ using \<open>r > 0\<close> by simp
+ then have neq: "x \<noteq> y" "x \<noteq> z"
+ using that \<open>r > 0\<close> by (auto simp: field_split_simps norm_minus_commute)
+ have leM: "cmod (\<F> n x) \<le> M"
+ by (simp add: M dist_commute dist_norm that)
+ have "cmod (\<F> n x / (x - y) - \<F> n x / (x - z)) = cmod (\<F> n x) * cmod (1 / (x - y) - 1 / (x - z))"
+ by (metis (no_types, lifting) divide_inverse mult.left_neutral norm_mult right_diff_distrib')
+ also have "... = cmod (\<F> n x) * cmod ((y - z) / ((x - y) * (x - z)))"
+ using neq by (simp add: field_split_simps)
+ also have "... = cmod (\<F> n x) * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
+ by (simp add: norm_mult norm_divide that)
+ also have "... \<le> M * (cmod (y - z) / (cmod(x - y) * (2/3 * r)))"
+ apply (rule mult_mono)
+ apply (rule leM)
+ using \<open>r > 0\<close> \<open>M > 0\<close> neq by auto
+ also have "... < M * ((e * r / (6 * M)) / (cmod(x - y) * (2/3 * r)))"
+ unfolding mult_less_cancel_left
+ using y_near_z(2) \<open>M > 0\<close> \<open>r > 0\<close> neq
+ apply (simp add: field_simps mult_less_0_iff norm_minus_commute)
+ done
+ also have "... \<le> e/r"
+ using \<open>e > 0\<close> \<open>r > 0\<close> r4_le_xy by (simp add: field_split_simps)
+ finally show ?thesis by simp
+ qed
+ have "(2 * pi) * cmod (\<F> n y - \<F> n z) = cmod ((2 * pi) * \<i> * \<F> n y - (2 * pi) * \<i> * \<F> n z)"
+ by (simp add: right_diff_distrib [symmetric] norm_mult)
+ also have "cmod ((2 * pi) * \<i> * \<F> n y - (2 * pi) * \<i> * \<F> n z) \<le> e / r * (2 * pi * (2 / 3 * r))"
+ apply (rule has_contour_integral_bound_circlepath [OF has_contour_integral_diff [OF y z], of "e/r"])
+ using \<open>e > 0\<close> \<open>r > 0\<close> le_er by auto
+ also have "... = (2 * pi) * e * ((2 / 3))"
+ using \<open>r > 0\<close> by (simp add: field_split_simps)
+ finally have "cmod (\<F> n y - \<F> n z) \<le> e * (2 / 3)"
+ by simp
+ also have "... < e"
+ using \<open>e > 0\<close> by simp
+ finally show ?thesis by (simp add: norm_minus_commute)
+ qed
+ then show ?thesis
+ apply (rule_tac x="min (r/3) ((e * r)/(6 * M))" in exI)
+ using \<open>0 < e\<close> \<open>0 < r\<close> \<open>0 < M\<close> by simp
+ qed
+ show "\<And>n x. x \<in> K i \<Longrightarrow> cmod (\<F> n x) \<le> B i"
+ using B \<open>\<And>n. \<F> n \<in> \<H>\<close> by blast
+ next
+ fix g :: "complex \<Rightarrow> complex" and k :: "nat \<Rightarrow> nat"
+ assume *: "\<And>(g::complex\<Rightarrow>complex) (k::nat\<Rightarrow>nat). continuous_on (K i) g \<Longrightarrow>
+ strict_mono k \<Longrightarrow>
+ (\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod (\<F> (k n) x - g x) < e) \<Longrightarrow> thesis"
+ "continuous_on (K i) g"
+ "strict_mono k"
+ "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. N \<le> n \<and> x \<in> K i \<longrightarrow> cmod (\<F> (k n) x - g x) < e"
+ show ?thesis
+ by (rule *(1)[OF *(2,3)], drule *(4)) auto
+ qed (use comK in simp_all)
+ then show ?thesis
+ by auto
+ qed
+ have "\<exists>k g. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<forall>e > 0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> r \<circ> k) n x - g x) < e)"
+ for i r
+ apply (rule *)
+ using rng_f by auto
+ then have **: "\<And>i r. \<exists>k. strict_mono (k::nat\<Rightarrow>nat) \<and> (\<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> K i. norm((\<F> \<circ> (r \<circ> k)) n x - g x) < e)"
+ by (force simp: o_assoc)
+ obtain k :: "nat \<Rightarrow> nat" where "strict_mono k"
+ and "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> (id \<circ> k)) n x - g x) < e"
+ (* TODO: clean up this mess *)
+ apply (rule subsequence_diagonalization_lemma [OF **, of id id])
+ apply (erule ex_forward all_forward imp_forward)+
+ apply force
+ apply (erule exE)
+ apply (rename_tac i r k1 k2 N g e Na)
+ apply (rule_tac x="max N Na" in exI)
+ apply fastforce+
+ done
+ then have lt_e: "\<And>i. \<exists>g. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - g x) < e"
+ by simp
+ have "\<exists>l. \<forall>e>0. \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - l) < e" if "z \<in> S" for z
+ proof -
+ obtain G where G: "\<And>i e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K i. cmod ((\<F> \<circ> k) n x - G i x) < e"
+ using lt_e by metis
+ obtain N where "\<And>n. n \<ge> N \<Longrightarrow> z \<in> K n"
+ using subK [of "{z}"] that \<open>z \<in> S\<close> by auto
+ moreover have "\<And>e. e > 0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> k) n x - G N x) < e"
+ using G by auto
+ ultimately show ?thesis
+ by (metis comp_apply order_refl)
+ qed
+ then obtain g where g: "\<And>z e. \<lbrakk>z \<in> S; e > 0\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. norm(\<F> (k n) z - g z) < e"
+ by metis
+ show ?thesis
+ proof
+ show g_lim: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (k n) x) \<longlonglongrightarrow> g x"
+ by (simp add: lim_sequentially g dist_norm)
+ have dg_le_e: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>T. cmod (\<F> (k n) x - g x) < e"
+ if T: "compact T" "T \<subseteq> S" and "0 < e" for T e
+ proof -
+ obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> T \<subseteq> K n"
+ using subK [OF T] by blast
+ obtain h where h: "\<And>e. e>0 \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>x\<in>K N. cmod ((\<F> \<circ> k) n x - h x) < e"
+ using lt_e by blast
+ have geq: "g w = h w" if "w \<in> T" for w
+ apply (rule LIMSEQ_unique [of "\<lambda>n. \<F>(k n) w"])
+ using \<open>T \<subseteq> S\<close> g_lim that apply blast
+ using h N that by (force simp: lim_sequentially dist_norm)
+ show ?thesis
+ using T h N \<open>0 < e\<close> by (fastforce simp add: geq)
+ qed
+ then show "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk>
+ \<Longrightarrow> uniform_limit K (\<F> \<circ> k) g sequentially"
+ by (simp add: uniform_limit_iff dist_norm eventually_sequentially)
+ show "g holomorphic_on S"
+ proof (rule holomorphic_uniform_sequence [OF \<open>open S\<close> \<H>])
+ show "\<And>n. (\<F> \<circ> k) n \<in> \<H>"
+ by (simp add: range_subsetD rng_f)
+ show "\<exists>d>0. cball z d \<subseteq> S \<and> uniform_limit (cball z d) (\<lambda>n. (\<F> \<circ> k) n) g sequentially"
+ if "z \<in> S" for z
+ proof -
+ obtain d where d: "d>0" "cball z d \<subseteq> S"
+ using \<open>open S\<close> \<open>z \<in> S\<close> open_contains_cball by blast
+ then have "uniform_limit (cball z d) (\<F> \<circ> k) g sequentially"
+ using dg_le_e compact_cball by (auto simp: uniform_limit_iff eventually_sequentially dist_norm)
+ with d show ?thesis by blast
+ qed
+ qed
+ qed (auto simp: \<open>strict_mono k\<close>)
+qed
+
+
+
+subsection\<open>Some simple but useful cases of Hurwitz's theorem\<close>
+
+proposition Hurwitz_no_zeros:
+ assumes S: "open S" "connected S"
+ and holf: "\<And>n::nat. \<F> n holomorphic_on S"
+ and holg: "g holomorphic_on S"
+ and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
+ and nonconst: "\<not> g constant_on S"
+ and nz: "\<And>n z. z \<in> S \<Longrightarrow> \<F> n z \<noteq> 0"
+ and "z0 \<in> S"
+ shows "g z0 \<noteq> 0"
+proof
+ assume g0: "g z0 = 0"
+ obtain h r m
+ where "0 < m" "0 < r" and subS: "ball z0 r \<subseteq> S"
+ and holh: "h holomorphic_on ball z0 r"
+ and geq: "\<And>w. w \<in> ball z0 r \<Longrightarrow> g w = (w - z0)^m * h w"
+ and hnz: "\<And>w. w \<in> ball z0 r \<Longrightarrow> h w \<noteq> 0"
+ by (blast intro: holomorphic_factor_zero_nonconstant [OF holg S \<open>z0 \<in> S\<close> g0 nonconst])
+ then have holf0: "\<F> n holomorphic_on ball z0 r" for n
+ by (meson holf holomorphic_on_subset)
+ have *: "((\<lambda>z. deriv (\<F> n) z / \<F> n z) has_contour_integral 0) (circlepath z0 (r/2))" for n
+ proof (rule Cauchy_theorem_disc_simple [of _ z0 r])
+ show "(\<lambda>z. deriv (\<F> n) z / \<F> n z) holomorphic_on ball z0 r"
+ apply (intro holomorphic_intros holomorphic_deriv holf holf0 open_ball nz)
+ using \<open>ball z0 r \<subseteq> S\<close> by blast
+ qed (use \<open>0 < r\<close> in auto)
+ have hol_dg: "deriv g holomorphic_on S"
+ by (simp add: \<open>open S\<close> holg holomorphic_deriv)
+ have "continuous_on (sphere z0 (r/2)) (deriv g)"
+ apply (intro holomorphic_on_imp_continuous_on holomorphic_on_subset [OF hol_dg])
+ using \<open>0 < r\<close> subS by auto
+ then have "compact (deriv g ` (sphere z0 (r/2)))"
+ by (rule compact_continuous_image [OF _ compact_sphere])
+ then have bo_dg: "bounded (deriv g ` (sphere z0 (r/2)))"
+ using compact_imp_bounded by blast
+ have "continuous_on (sphere z0 (r/2)) (cmod \<circ> g)"
+ apply (intro continuous_intros holomorphic_on_imp_continuous_on holomorphic_on_subset [OF holg])
+ using \<open>0 < r\<close> subS by auto
+ then have "compact ((cmod \<circ> g) ` sphere z0 (r/2))"
+ by (rule compact_continuous_image [OF _ compact_sphere])
+ moreover have "(cmod \<circ> g) ` sphere z0 (r/2) \<noteq> {}"
+ using \<open>0 < r\<close> by auto
+ ultimately obtain b where b: "b \<in> (cmod \<circ> g) ` sphere z0 (r/2)"
+ "\<And>t. t \<in> (cmod \<circ> g) ` sphere z0 (r/2) \<Longrightarrow> b \<le> t"
+ using compact_attains_inf [of "(norm \<circ> g) ` (sphere z0 (r/2))"] by blast
+ have "(\<lambda>n. contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv (\<F> n) z / \<F> n z)) \<longlonglongrightarrow>
+ contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
+ proof (rule contour_integral_uniform_limit_circlepath)
+ show "\<forall>\<^sub>F n in sequentially. (\<lambda>z. deriv (\<F> n) z / \<F> n z) contour_integrable_on circlepath z0 (r/2)"
+ using * contour_integrable_on_def eventually_sequentiallyI by meson
+ show "uniform_limit (sphere z0 (r/2)) (\<lambda>n z. deriv (\<F> n) z / \<F> n z) (\<lambda>z. deriv g z / g z) sequentially"
+ proof (rule uniform_lim_divide [OF _ _ bo_dg])
+ show "uniform_limit (sphere z0 (r/2)) (\<lambda>a. deriv (\<F> a)) (deriv g) sequentially"
+ proof (rule uniform_limitI)
+ fix e::real
+ assume "0 < e"
+ have *: "dist (deriv (\<F> n) w) (deriv g w) < e"
+ if e8: "\<And>x. dist z0 x \<le> 3 * r / 4 \<Longrightarrow> dist (\<F> n x) (g x) * 8 < r * e"
+ and w: "dist w z0 = r/2" for n w
+ proof -
+ have "ball w (r/4) \<subseteq> ball z0 r" "cball w (r/4) \<subseteq> ball z0 r"
+ using \<open>0 < r\<close> by (simp_all add: ball_subset_ball_iff cball_subset_ball_iff w)
+ with subS have wr4_sub: "ball w (r/4) \<subseteq> S" "cball w (r/4) \<subseteq> S" by force+
+ moreover
+ have "(\<lambda>z. \<F> n z - g z) holomorphic_on S"
+ by (intro holomorphic_intros holf holg)
+ ultimately have hol: "(\<lambda>z. \<F> n z - g z) holomorphic_on ball w (r/4)"
+ and cont: "continuous_on (cball w (r / 4)) (\<lambda>z. \<F> n z - g z)"
+ using holomorphic_on_subset by (blast intro: holomorphic_on_imp_continuous_on)+
+ have "w \<in> S"
+ using \<open>0 < r\<close> wr4_sub by auto
+ have "\<And>y. dist w y < r / 4 \<Longrightarrow> dist z0 y \<le> 3 * r / 4"
+ apply (rule dist_triangle_le [where z=w])
+ using w by (simp add: dist_commute)
+ with e8 have in_ball: "\<And>y. y \<in> ball w (r/4) \<Longrightarrow> \<F> n y - g y \<in> ball 0 (r/4 * e/2)"
+ by (simp add: dist_norm [symmetric])
+ have "\<F> n field_differentiable at w"
+ by (metis holomorphic_on_imp_differentiable_at \<open>w \<in> S\<close> holf \<open>open S\<close>)
+ moreover
+ have "g field_differentiable at w"
+ using \<open>w \<in> S\<close> \<open>open S\<close> holg holomorphic_on_imp_differentiable_at by auto
+ moreover
+ have "cmod (deriv (\<lambda>w. \<F> n w - g w) w) * 2 \<le> e"
+ apply (rule Cauchy_higher_deriv_bound [OF hol cont in_ball, of 1, simplified])
+ using \<open>r > 0\<close> by auto
+ ultimately have "dist (deriv (\<F> n) w) (deriv g w) \<le> e/2"
+ by (simp add: dist_norm)
+ then show ?thesis
+ using \<open>e > 0\<close> by auto
+ qed
+ have "cball z0 (3 * r / 4) \<subseteq> ball z0 r"
+ by (simp add: cball_subset_ball_iff \<open>0 < r\<close>)
+ with subS have "uniform_limit (cball z0 (3 * r/4)) \<F> g sequentially"
+ by (force intro: ul_g)
+ then have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>cball z0 (3 * r / 4). dist (\<F> n x) (g x) < r / 4 * e / 2"
+ using \<open>0 < e\<close> \<open>0 < r\<close> by (force simp: intro!: uniform_limitD)
+ then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> sphere z0 (r/2). dist (deriv (\<F> n) x) (deriv g x) < e"
+ apply (simp add: eventually_sequentially)
+ apply (elim ex_forward all_forward imp_forward asm_rl)
+ using * apply (force simp: dist_commute)
+ done
+ qed
+ show "uniform_limit (sphere z0 (r/2)) \<F> g sequentially"
+ proof (rule uniform_limitI)
+ fix e::real
+ assume "0 < e"
+ have "sphere z0 (r/2) \<subseteq> ball z0 r"
+ using \<open>0 < r\<close> by auto
+ with subS have "uniform_limit (sphere z0 (r/2)) \<F> g sequentially"
+ by (force intro: ul_g)
+ then show "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> sphere z0 (r/2). dist (\<F> n x) (g x) < e"
+ apply (rule uniform_limitD)
+ using \<open>0 < e\<close> by force
+ qed
+ show "b > 0" "\<And>x. x \<in> sphere z0 (r/2) \<Longrightarrow> b \<le> cmod (g x)"
+ using b \<open>0 < r\<close> by (fastforce simp: geq hnz)+
+ qed
+ qed (use \<open>0 < r\<close> in auto)
+ then have "(\<lambda>n. 0) \<longlonglongrightarrow> contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z)"
+ by (simp add: contour_integral_unique [OF *])
+ then have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) = 0"
+ by (simp add: LIMSEQ_const_iff)
+ moreover
+ have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. deriv g z / g z) =
+ contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z)"
+ proof (rule contour_integral_eq, use \<open>0 < r\<close> in simp)
+ fix w
+ assume w: "dist z0 w * 2 = r"
+ then have w_inb: "w \<in> ball z0 r"
+ using \<open>0 < r\<close> by auto
+ have h_der: "(h has_field_derivative deriv h w) (at w)"
+ using holh holomorphic_derivI w_inb by blast
+ have "deriv g w = ((of_nat m * h w + deriv h w * (w - z0)) * (w - z0) ^ m) / (w - z0)"
+ if "r = dist z0 w * 2" "w \<noteq> z0"
+ proof -
+ have "((\<lambda>w. (w - z0) ^ m * h w) has_field_derivative
+ (m * h w + deriv h w * (w - z0)) * (w - z0) ^ m / (w - z0)) (at w)"
+ apply (rule derivative_eq_intros h_der refl)+
+ using that \<open>m > 0\<close> \<open>0 < r\<close> apply (simp add: divide_simps distrib_right)
+ apply (metis Suc_pred mult.commute power_Suc)
+ done
+ then show ?thesis
+ apply (rule DERIV_imp_deriv [OF has_field_derivative_transform_within_open [where S = "ball z0 r"]])
+ using that \<open>m > 0\<close> \<open>0 < r\<close>
+ apply (simp_all add: hnz geq)
+ done
+ qed
+ with \<open>0 < r\<close> \<open>0 < m\<close> w w_inb show "deriv g w / g w = of_nat m / (w - z0) + deriv h w / h w"
+ by (auto simp: geq field_split_simps hnz)
+ qed
+ moreover
+ have "contour_integral (circlepath z0 (r/2)) (\<lambda>z. m / (z - z0) + deriv h z / h z) =
+ 2 * of_real pi * \<i> * m + 0"
+ proof (rule contour_integral_unique [OF has_contour_integral_add])
+ show "((\<lambda>x. m / (x - z0)) has_contour_integral 2 * of_real pi * \<i> * m) (circlepath z0 (r/2))"
+ by (force simp: \<open>0 < r\<close> intro: Cauchy_integral_circlepath_simple)
+ show "((\<lambda>x. deriv h x / h x) has_contour_integral 0) (circlepath z0 (r/2))"
+ apply (rule Cauchy_theorem_disc_simple [of _ z0 r])
+ using hnz holh holomorphic_deriv holomorphic_on_divide \<open>0 < r\<close>
+ apply force+
+ done
+ qed
+ ultimately show False using \<open>0 < m\<close> by auto
+qed
+
+corollary Hurwitz_injective:
+ assumes S: "open S" "connected S"
+ and holf: "\<And>n::nat. \<F> n holomorphic_on S"
+ and holg: "g holomorphic_on S"
+ and ul_g: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"
+ and nonconst: "\<not> g constant_on S"
+ and inj: "\<And>n. inj_on (\<F> n) S"
+ shows "inj_on g S"
+proof -
+ have False if z12: "z1 \<in> S" "z2 \<in> S" "z1 \<noteq> z2" "g z2 = g z1" for z1 z2
+ proof -
+ obtain z0 where "z0 \<in> S" and z0: "g z0 \<noteq> g z2"
+ using constant_on_def nonconst by blast
+ have "(\<lambda>z. g z - g z1) holomorphic_on S"
+ by (intro holomorphic_intros holg)
+ then obtain r where "0 < r" "ball z2 r \<subseteq> S" "\<And>z. dist z2 z < r \<and> z \<noteq> z2 \<Longrightarrow> g z \<noteq> g z1"
+ apply (rule isolated_zeros [of "\<lambda>z. g z - g z1" S z2 z0])
+ using S \<open>z0 \<in> S\<close> z0 z12 by auto
+ have "g z2 - g z1 \<noteq> 0"
+ proof (rule Hurwitz_no_zeros [of "S - {z1}" "\<lambda>n z. \<F> n z - \<F> n z1" "\<lambda>z. g z - g z1"])
+ show "open (S - {z1})"
+ by (simp add: S open_delete)
+ show "connected (S - {z1})"
+ by (simp add: connected_open_delete [OF S])
+ show "\<And>n. (\<lambda>z. \<F> n z - \<F> n z1) holomorphic_on S - {z1}"
+ by (intro holomorphic_intros holomorphic_on_subset [OF holf]) blast
+ show "(\<lambda>z. g z - g z1) holomorphic_on S - {z1}"
+ by (intro holomorphic_intros holomorphic_on_subset [OF holg]) blast
+ show "uniform_limit K (\<lambda>n z. \<F> n z - \<F> n z1) (\<lambda>z. g z - g z1) sequentially"
+ if "compact K" "K \<subseteq> S - {z1}" for K
+ proof (rule uniform_limitI)
+ fix e::real
+ assume "e > 0"
+ have "uniform_limit K \<F> g sequentially"
+ using that ul_g by fastforce
+ then have K: "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> K. dist (\<F> n x) (g x) < e/2"
+ using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
+ have "uniform_limit {z1} \<F> g sequentially"
+ by (simp add: ul_g z12)
+ then have "\<forall>\<^sub>F n in sequentially. \<forall>x \<in> {z1}. dist (\<F> n x) (g x) < e/2"
+ using \<open>0 < e\<close> by (force simp: intro!: uniform_limitD)
+ then have z1: "\<forall>\<^sub>F n in sequentially. dist (\<F> n z1) (g z1) < e/2"
+ by simp
+ have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e/2 + e/2"
+ apply (rule eventually_mono [OF eventually_conj [OF K z1]])
+ apply (simp add: dist_norm algebra_simps del: divide_const_simps)
+ by (metis add.commute dist_commute dist_norm dist_triangle_add_half)
+ have "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e/2 + e/2"
+ using eventually_conj [OF K z1]
+ apply (rule eventually_mono)
+ by (metis (no_types, hide_lams) diff_add_eq diff_diff_eq2 dist_commute dist_norm dist_triangle_add_half field_sum_of_halves)
+ then
+ show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>K. dist (\<F> n x - \<F> n z1) (g x - g z1) < e"
+ by simp
+ qed
+ show "\<not> (\<lambda>z. g z - g z1) constant_on S - {z1}"
+ unfolding constant_on_def
+ by (metis Diff_iff \<open>z0 \<in> S\<close> empty_iff insert_iff right_minus_eq z0 z12)
+ show "\<And>n z. z \<in> S - {z1} \<Longrightarrow> \<F> n z - \<F> n z1 \<noteq> 0"
+ by (metis DiffD1 DiffD2 eq_iff_diff_eq_0 inj inj_onD insertI1 \<open>z1 \<in> S\<close>)
+ show "z2 \<in> S - {z1}"
+ using \<open>z2 \<in> S\<close> \<open>z1 \<noteq> z2\<close> by auto
+ qed
+ with z12 show False by auto
+ qed
+ then show ?thesis by (auto simp: inj_on_def)
+qed
+
+
+
+subsection\<open>The Great Picard theorem\<close>
+
+lemma GPicard1:
+ assumes S: "open S" "connected S" and "w \<in> S" "0 < r" "Y \<subseteq> X"
+ and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
+ and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
+ and r: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> r"
+ obtains B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S" "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
+proof -
+ obtain e where "e > 0" and e: "cball w e \<subseteq> S"
+ using assms open_contains_cball_eq by blast
+ show ?thesis
+ proof
+ show "0 < exp(pi * exp(pi * (2 + 2 * r + 12)))"
+ by simp
+ show "ball w (e / 2) \<subseteq> S"
+ using e ball_divide_subset_numeral ball_subset_cball by blast
+ show "cmod (h z) \<le> exp (pi * exp (pi * (2 + 2 * r + 12)))"
+ if "h \<in> Y" "z \<in> ball w (e / 2)" for h z
+ proof -
+ have "h \<in> X"
+ using \<open>Y \<subseteq> X\<close> \<open>h \<in> Y\<close> by blast
+ with holX have "h holomorphic_on S"
+ by auto
+ then have "h holomorphic_on cball w e"
+ by (metis e holomorphic_on_subset)
+ then have hol_h_o: "(h \<circ> (\<lambda>z. (w + of_real e * z))) holomorphic_on cball 0 1"
+ apply (intro holomorphic_intros holomorphic_on_compose)
+ apply (erule holomorphic_on_subset)
+ using that \<open>e > 0\<close> by (auto simp: dist_norm norm_mult)
+ have norm_le_r: "cmod ((h \<circ> (\<lambda>z. w + complex_of_real e * z)) 0) \<le> r"
+ by (auto simp: r \<open>h \<in> Y\<close>)
+ have le12: "norm (of_real(inverse e) * (z - w)) \<le> 1/2"
+ using that \<open>e > 0\<close> by (simp add: inverse_eq_divide dist_norm norm_minus_commute norm_divide)
+ have non01: "\<And>z::complex. cmod z \<le> 1 \<Longrightarrow> h (w + e * z) \<noteq> 0 \<and> h (w + e * z) \<noteq> 1"
+ apply (rule X01 [OF \<open>h \<in> X\<close>])
+ apply (rule subsetD [OF e])
+ using \<open>0 < e\<close> by (auto simp: dist_norm norm_mult)
+ have "cmod (h z) \<le> cmod (h (w + of_real e * (inverse e * (z - w))))"
+ using \<open>0 < e\<close> by (simp add: field_split_simps)
+ also have "... \<le> exp (pi * exp (pi * (14 + 2 * r)))"
+ using r [OF \<open>h \<in> Y\<close>] Schottky [OF hol_h_o norm_le_r _ _ _ le12] non01 by auto
+ finally
+ show ?thesis by simp
+ qed
+ qed (use \<open>e > 0\<close> in auto)
+qed
+
+lemma GPicard2:
+ assumes "S \<subseteq> T" "connected T" "S \<noteq> {}" "open S" "\<And>x. \<lbrakk>x islimpt S; x \<in> T\<rbrakk> \<Longrightarrow> x \<in> S"
+ shows "S = T"
+ by (metis assms open_subset connected_clopen closedin_limpt)
+
+
+lemma GPicard3:
+ assumes S: "open S" "connected S" "w \<in> S" and "Y \<subseteq> X"
+ and holX: "\<And>h. h \<in> X \<Longrightarrow> h holomorphic_on S"
+ and X01: "\<And>h z. \<lbrakk>h \<in> X; z \<in> S\<rbrakk> \<Longrightarrow> h z \<noteq> 0 \<and> h z \<noteq> 1"
+ and no_hw_le1: "\<And>h. h \<in> Y \<Longrightarrow> norm(h w) \<le> 1"
+ and "compact K" "K \<subseteq> S"
+ obtains B where "\<And>h z. \<lbrakk>h \<in> Y; z \<in> K\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
+proof -
+ define U where "U \<equiv> {z \<in> S. \<exists>B Z. 0 < B \<and> open Z \<and> z \<in> Z \<and> Z \<subseteq> S \<and>
+ (\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B)}"
+ then have "U \<subseteq> S" by blast
+ have "U = S"
+ proof (rule GPicard2 [OF \<open>U \<subseteq> S\<close> \<open>connected S\<close>])
+ show "U \<noteq> {}"
+ proof -
+ obtain B Z where "0 < B" "open Z" "w \<in> Z" "Z \<subseteq> S"
+ and "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
+ apply (rule GPicard1 [OF S zero_less_one \<open>Y \<subseteq> X\<close> holX])
+ using no_hw_le1 X01 by force+
+ then show ?thesis
+ unfolding U_def using \<open>w \<in> S\<close> by blast
+ qed
+ show "open U"
+ unfolding open_subopen [of U] by (auto simp: U_def)
+ fix v
+ assume v: "v islimpt U" "v \<in> S"
+ have "\<not> (\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v))"
+ proof
+ assume "\<forall>r>0. \<exists>h\<in>Y. r < cmod (h v)"
+ then have "\<forall>n. \<exists>h\<in>Y. Suc n < cmod (h v)"
+ by simp
+ then obtain \<F> where FY: "\<And>n. \<F> n \<in> Y" and ltF: "\<And>n. Suc n < cmod (\<F> n v)"
+ by metis
+ define \<G> where "\<G> \<equiv> \<lambda>n z. inverse(\<F> n z)"
+ have hol\<G>: "\<G> n holomorphic_on S" for n
+ apply (simp add: \<G>_def)
+ using FY X01 \<open>Y \<subseteq> X\<close> holX apply (blast intro: holomorphic_on_inverse)
+ done
+ have \<G>not0: "\<G> n z \<noteq> 0" and \<G>not1: "\<G> n z \<noteq> 1" if "z \<in> S" for n z
+ using FY X01 \<open>Y \<subseteq> X\<close> that by (force simp: \<G>_def)+
+ have \<G>_le1: "cmod (\<G> n v) \<le> 1" for n
+ using less_le_trans linear ltF
+ by (fastforce simp add: \<G>_def norm_inverse inverse_le_1_iff)
+ define W where "W \<equiv> {h. h holomorphic_on S \<and> (\<forall>z \<in> S. h z \<noteq> 0 \<and> h z \<noteq> 1)}"
+ obtain B Z where "0 < B" "open Z" "v \<in> Z" "Z \<subseteq> S"
+ and B: "\<And>h z. \<lbrakk>h \<in> range \<G>; z \<in> Z\<rbrakk> \<Longrightarrow> norm(h z) \<le> B"
+ apply (rule GPicard1 [OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> zero_less_one, of "range \<G>" W])
+ using hol\<G> \<G>not0 \<G>not1 \<G>_le1 by (force simp: W_def)+
+ then obtain e where "e > 0" and e: "ball v e \<subseteq> Z"
+ by (meson open_contains_ball)
+ obtain h j where holh: "h holomorphic_on ball v e" and "strict_mono j"
+ and lim: "\<And>x. x \<in> ball v e \<Longrightarrow> (\<lambda>n. \<G> (j n) x) \<longlonglongrightarrow> h x"
+ and ulim: "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk>
+ \<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
+ proof (rule Montel)
+ show "\<And>h. h \<in> range \<G> \<Longrightarrow> h holomorphic_on ball v e"
+ by (metis \<open>Z \<subseteq> S\<close> e hol\<G> holomorphic_on_subset imageE)
+ show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>range \<G>. \<forall>z\<in>K. cmod (h z) \<le> B"
+ using B e by blast
+ qed auto
+ have "h v = 0"
+ proof (rule LIMSEQ_unique)
+ show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> h v"
+ using \<open>e > 0\<close> lim by simp
+ have lt_Fj: "real x \<le> cmod (\<F> (j x) v)" for x
+ by (metis of_nat_Suc ltF \<open>strict_mono j\<close> add.commute less_eq_real_def less_le_trans nat_le_real_less seq_suble)
+ show "(\<lambda>n. \<G> (j n) v) \<longlonglongrightarrow> 0"
+ proof (rule Lim_null_comparison [OF eventually_sequentiallyI lim_inverse_n])
+ show "cmod (\<G> (j x) v) \<le> inverse (real x)" if "1 \<le> x" for x
+ using that by (simp add: \<G>_def norm_inverse_le_norm [OF lt_Fj])
+ qed
+ qed
+ have "h v \<noteq> 0"
+ proof (rule Hurwitz_no_zeros [of "ball v e" "\<G> \<circ> j" h])
+ show "\<And>n. (\<G> \<circ> j) n holomorphic_on ball v e"
+ using \<open>Z \<subseteq> S\<close> e hol\<G> by force
+ show "\<And>n z. z \<in> ball v e \<Longrightarrow> (\<G> \<circ> j) n z \<noteq> 0"
+ using \<G>not0 \<open>Z \<subseteq> S\<close> e by fastforce
+ show "\<not> h constant_on ball v e"
+ proof (clarsimp simp: constant_on_def)
+ fix c
+ have False if "\<And>z. dist v z < e \<Longrightarrow> h z = c"
+ proof -
+ have "h v = c"
+ by (simp add: \<open>0 < e\<close> that)
+ obtain y where "y \<in> U" "y \<noteq> v" and y: "dist y v < e"
+ using v \<open>e > 0\<close> by (auto simp: islimpt_approachable)
+ then obtain C T where "y \<in> S" "C > 0" "open T" "y \<in> T" "T \<subseteq> S"
+ and "\<And>h z'. \<lbrakk>h \<in> Y; z' \<in> T\<rbrakk> \<Longrightarrow> cmod (h z') \<le> C"
+ using \<open>y \<in> U\<close> by (auto simp: U_def)
+ then have le_C: "\<And>n. cmod (\<F> n y) \<le> C"
+ using FY by blast
+ have "\<forall>\<^sub>F n in sequentially. dist (\<G> (j n) y) (h y) < inverse C"
+ using uniform_limitD [OF ulim [of "{y}"], of "inverse C"] \<open>C > 0\<close> y
+ by (simp add: dist_commute)
+ then obtain n where "dist (\<G> (j n) y) (h y) < inverse C"
+ by (meson eventually_at_top_linorder order_refl)
+ moreover
+ have "h y = h v"
+ by (metis \<open>h v = c\<close> dist_commute that y)
+ ultimately have "norm (\<G> (j n) y) < inverse C"
+ by (simp add: \<open>h v = 0\<close>)
+ then have "C < norm (\<F> (j n) y)"
+ apply (simp add: \<G>_def)
+ by (metis FY X01 \<open>0 < C\<close> \<open>y \<in> S\<close> \<open>Y \<subseteq> X\<close> inverse_less_iff_less norm_inverse subsetD zero_less_norm_iff)
+ show False
+ using \<open>C < cmod (\<F> (j n) y)\<close> le_C not_less by blast
+ qed
+ then show "\<exists>x\<in>ball v e. h x \<noteq> c" by force
+ qed
+ show "h holomorphic_on ball v e"
+ by (simp add: holh)
+ show "\<And>K. \<lbrakk>compact K; K \<subseteq> ball v e\<rbrakk> \<Longrightarrow> uniform_limit K (\<G> \<circ> j) h sequentially"
+ by (simp add: ulim)
+ qed (use \<open>e > 0\<close> in auto)
+ with \<open>h v = 0\<close> show False by blast
+ qed
+ then show "v \<in> U"
+ apply (clarsimp simp add: U_def v)
+ apply (rule GPicard1[OF \<open>open S\<close> \<open>connected S\<close> \<open>v \<in> S\<close> _ \<open>Y \<subseteq> X\<close> holX])
+ using X01 no_hw_le1 apply (meson | force simp: not_less)+
+ done
+ qed
+ have "\<And>x. x \<in> K \<longrightarrow> x \<in> U"
+ using \<open>U = S\<close> \<open>K \<subseteq> S\<close> by blast
+ then have "\<And>x. x \<in> K \<longrightarrow> (\<exists>B Z. 0 < B \<and> open Z \<and> x \<in> Z \<and>
+ (\<forall>h z'. h \<in> Y \<and> z' \<in> Z \<longrightarrow> norm(h z') \<le> B))"
+ unfolding U_def by blast
+ then obtain F Z where F: "\<And>x. x \<in> K \<Longrightarrow> open (Z x) \<and> x \<in> Z x \<and>
+ (\<forall>h z'. h \<in> Y \<and> z' \<in> Z x \<longrightarrow> norm(h z') \<le> F x)"
+ by metis
+ then obtain L where "L \<subseteq> K" "finite L" and L: "K \<subseteq> (\<Union>c \<in> L. Z c)"
+ by (auto intro: compactE_image [OF \<open>compact K\<close>, of K Z])
+ then have *: "\<And>x h z'. \<lbrakk>x \<in> L; h \<in> Y \<and> z' \<in> Z x\<rbrakk> \<Longrightarrow> cmod (h z') \<le> F x"
+ using F by blast
+ have "\<exists>B. \<forall>h z. h \<in> Y \<and> z \<in> K \<longrightarrow> norm(h z) \<le> B"
+ proof (cases "L = {}")
+ case True with L show ?thesis by simp
+ next
+ case False
+ with \<open>finite L\<close> show ?thesis
+ apply (rule_tac x = "Max (F ` L)" in exI)
+ apply (simp add: linorder_class.Max_ge_iff)
+ using * F by (metis L UN_E subsetD)
+ qed
+ with that show ?thesis by metis
+qed
+
+
+lemma GPicard4:
+ assumes "0 < k" and holf: "f holomorphic_on (ball 0 k - {0})"
+ and AE: "\<And>e. \<lbrakk>0 < e; e < k\<rbrakk> \<Longrightarrow> \<exists>d. 0 < d \<and> d < e \<and> (\<forall>z \<in> sphere 0 d. norm(f z) \<le> B)"
+ obtains \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> norm(f z) \<le> B"
+proof -
+ obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < k/2" and \<epsilon>: "\<And>z. norm z = \<epsilon> \<Longrightarrow> norm(f z) \<le> B"
+ using AE [of "k/2"] \<open>0 < k\<close> by auto
+ show ?thesis
+ proof
+ show "\<epsilon> < k"
+ using \<open>0 < k\<close> \<open>\<epsilon> < k/2\<close> by auto
+ show "cmod (f \<xi>) \<le> B" if \<xi>: "\<xi> \<in> ball 0 \<epsilon> - {0}" for \<xi>
+ proof -
+ obtain d where "0 < d" "d < norm \<xi>" and d: "\<And>z. norm z = d \<Longrightarrow> norm(f z) \<le> B"
+ using AE [of "norm \<xi>"] \<open>\<epsilon> < k\<close> \<xi> by auto
+ have [simp]: "closure (cball 0 \<epsilon> - ball 0 d) = cball 0 \<epsilon> - ball 0 d"
+ by (blast intro!: closure_closed)
+ have [simp]: "interior (cball 0 \<epsilon> - ball 0 d) = ball 0 \<epsilon> - cball (0::complex) d"
+ using \<open>0 < \<epsilon>\<close> \<open>0 < d\<close> by (simp add: interior_diff)
+ have *: "norm(f w) \<le> B" if "w \<in> cball 0 \<epsilon> - ball 0 d" for w
+ proof (rule maximum_modulus_frontier [of f "cball 0 \<epsilon> - ball 0 d"])
+ show "f holomorphic_on interior (cball 0 \<epsilon> - ball 0 d)"
+ apply (rule holomorphic_on_subset [OF holf])
+ using \<open>\<epsilon> < k\<close> \<open>0 < d\<close> that by auto
+ show "continuous_on (closure (cball 0 \<epsilon> - ball 0 d)) f"
+ apply (rule holomorphic_on_imp_continuous_on)
+ apply (rule holomorphic_on_subset [OF holf])
+ using \<open>0 < d\<close> \<open>\<epsilon> < k\<close> by auto
+ show "\<And>z. z \<in> frontier (cball 0 \<epsilon> - ball 0 d) \<Longrightarrow> cmod (f z) \<le> B"
+ apply (simp add: frontier_def)
+ using \<epsilon> d less_eq_real_def by blast
+ qed (use that in auto)
+ show ?thesis
+ using * \<open>d < cmod \<xi>\<close> that by auto
+ qed
+ qed (use \<open>0 < \<epsilon>\<close> in auto)
+qed
+
+
+lemma GPicard5:
+ assumes holf: "f holomorphic_on (ball 0 1 - {0})"
+ and f01: "\<And>z. z \<in> ball 0 1 - {0} \<Longrightarrow> f z \<noteq> 0 \<and> f z \<noteq> 1"
+ obtains e B where "0 < e" "e < 1" "0 < B"
+ "(\<forall>z \<in> ball 0 e - {0}. norm(f z) \<le> B) \<or>
+ (\<forall>z \<in> ball 0 e - {0}. norm(f z) \<ge> B)"
+proof -
+ have [simp]: "1 + of_nat n \<noteq> (0::complex)" for n
+ using of_nat_eq_0_iff by fastforce
+ have [simp]: "cmod (1 + of_nat n) = 1 + of_nat n" for n
+ by (metis norm_of_nat of_nat_Suc)
+ have *: "(\<lambda>x::complex. x / of_nat (Suc n)) ` (ball 0 1 - {0}) \<subseteq> ball 0 1 - {0}" for n
+ by (auto simp: norm_divide field_split_simps split: if_split_asm)
+ define h where "h \<equiv> \<lambda>n z::complex. f (z / (Suc n))"
+ have holh: "(h n) holomorphic_on ball 0 1 - {0}" for n
+ unfolding h_def
+ proof (rule holomorphic_on_compose_gen [unfolded o_def, OF _ holf *])
+ show "(\<lambda>x. x / of_nat (Suc n)) holomorphic_on ball 0 1 - {0}"
+ by (intro holomorphic_intros) auto
+ qed
+ have h01: "\<And>n z. z \<in> ball 0 1 - {0} \<Longrightarrow> h n z \<noteq> 0 \<and> h n z \<noteq> 1"
+ unfolding h_def
+ apply (rule f01)
+ using * by force
+ obtain w where w: "w \<in> ball 0 1 - {0::complex}"
+ by (rule_tac w = "1/2" in that) auto
+ consider "infinite {n. norm(h n w) \<le> 1}" | "infinite {n. 1 \<le> norm(h n w)}"
+ by (metis (mono_tags, lifting) infinite_nat_iff_unbounded_le le_cases mem_Collect_eq)
+ then show ?thesis
+ proof cases
+ case 1
+ with infinite_enumerate obtain r :: "nat \<Rightarrow> nat"
+ where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<le> 1}"
+ by blast
+ obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (h \<circ> r)\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
+ proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
+ show "range (h \<circ> r) \<subseteq>
+ {g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 1)}"
+ apply clarsimp
+ apply (intro conjI holomorphic_intros holomorphic_on_compose holh)
+ using h01 apply auto
+ done
+ show "connected (ball 0 1 - {0::complex})"
+ by (simp add: connected_open_delete)
+ qed (use r in auto)
+ have normf_le_B: "cmod(f z) \<le> B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
+ proof -
+ have *: "\<And>w. norm w = 1/2 \<Longrightarrow> cmod((f (w / (1 + of_nat (r n))))) \<le> B"
+ using B by (auto simp: h_def o_def)
+ have half: "norm (z * (1 + of_nat (r n))) = 1/2"
+ by (simp add: norm_mult divide_simps that)
+ show ?thesis
+ using * [OF half] by simp
+ qed
+ obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod(f z) \<le> B"
+ proof (rule GPicard4 [OF zero_less_one holf, of B])
+ fix e::real
+ assume "0 < e" "e < 1"
+ obtain n where "(1/e - 2) / 2 < real n"
+ using reals_Archimedean2 by blast
+ also have "... \<le> r n"
+ using \<open>strict_mono r\<close> by (simp add: seq_suble)
+ finally have "(1/e - 2) / 2 < real (r n)" .
+ with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
+ by (simp add: field_simps)
+ show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod (f z) \<le> B)"
+ apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
+ using normf_le_B by (simp add: e)
+ qed blast
+ then have \<epsilon>: "cmod (f z) \<le> \<bar>B\<bar> + 1" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
+ using that by fastforce
+ have "0 < \<bar>B\<bar> + 1"
+ by simp
+ then show ?thesis
+ apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
+ using \<epsilon> by auto
+ next
+ case 2
+ with infinite_enumerate obtain r :: "nat \<Rightarrow> nat"
+ where "strict_mono r" and r: "\<And>n. r n \<in> {n. norm(h n w) \<ge> 1}"
+ by blast
+ obtain B where B: "\<And>j z. \<lbrakk>norm z = 1/2; j \<in> range (\<lambda>n. inverse \<circ> h (r n))\<rbrakk> \<Longrightarrow> norm(j z) \<le> B"
+ proof (rule GPicard3 [OF _ _ w, where K = "sphere 0 (1/2)"])
+ show "range (\<lambda>n. inverse \<circ> h (r n)) \<subseteq>
+ {g. g holomorphic_on ball 0 1 - {0} \<and> (\<forall>z\<in>ball 0 1 - {0}. g z \<noteq> 0 \<and> g z \<noteq> 1)}"
+ apply clarsimp
+ apply (intro conjI holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holh] holomorphic_on_compose)
+ using h01 apply auto
+ done
+ show "connected (ball 0 1 - {0::complex})"
+ by (simp add: connected_open_delete)
+ show "\<And>j. j \<in> range (\<lambda>n. inverse \<circ> h (r n)) \<Longrightarrow> cmod (j w) \<le> 1"
+ using r norm_inverse_le_norm by fastforce
+ qed (use r in auto)
+ have norm_if_le_B: "cmod(inverse (f z)) \<le> B" if "norm z = 1 / (2 * (1 + of_nat (r n)))" for z n
+ proof -
+ have *: "inverse (cmod((f (z / (1 + of_nat (r n)))))) \<le> B" if "norm z = 1/2" for z
+ using B [OF that] by (force simp: norm_inverse h_def)
+ have half: "norm (z * (1 + of_nat (r n))) = 1/2"
+ by (simp add: norm_mult divide_simps that)
+ show ?thesis
+ using * [OF half] by (simp add: norm_inverse)
+ qed
+ have hol_if: "(inverse \<circ> f) holomorphic_on (ball 0 1 - {0})"
+ by (metis (no_types, lifting) holf comp_apply f01 holomorphic_on_inverse holomorphic_transform)
+ obtain \<epsilon> where "0 < \<epsilon>" "\<epsilon> < 1" and leB: "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod((inverse \<circ> f) z) \<le> B"
+ proof (rule GPicard4 [OF zero_less_one hol_if, of B])
+ fix e::real
+ assume "0 < e" "e < 1"
+ obtain n where "(1/e - 2) / 2 < real n"
+ using reals_Archimedean2 by blast
+ also have "... \<le> r n"
+ using \<open>strict_mono r\<close> by (simp add: seq_suble)
+ finally have "(1/e - 2) / 2 < real (r n)" .
+ with \<open>0 < e\<close> have e: "e > 1 / (2 + 2 * real (r n))"
+ by (simp add: field_simps)
+ show "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod ((inverse \<circ> f) z) \<le> B)"
+ apply (rule_tac x="1 / (2 * (1 + of_nat (r n)))" in exI)
+ using norm_if_le_B by (simp add: e)
+ qed blast
+ have \<epsilon>: "cmod (f z) \<ge> inverse B" and "B > 0" if "cmod z < \<epsilon>" "z \<noteq> 0" for z
+ proof -
+ have "inverse (cmod (f z)) \<le> B"
+ using leB that by (simp add: norm_inverse)
+ moreover
+ have "f z \<noteq> 0"
+ using \<open>\<epsilon> < 1\<close> f01 that by auto
+ ultimately show "cmod (f z) \<ge> inverse B"
+ by (simp add: norm_inverse inverse_le_imp_le)
+ show "B > 0"
+ using \<open>f z \<noteq> 0\<close> \<open>inverse (cmod (f z)) \<le> B\<close> not_le order.trans by fastforce
+ qed
+ then have "B > 0"
+ by (metis \<open>0 < \<epsilon>\<close> dense leI order.asym vector_choose_size)
+ then have "inverse B > 0"
+ by (simp add: field_split_simps)
+ then show ?thesis
+ apply (rule that [OF \<open>0 < \<epsilon>\<close> \<open>\<epsilon> < 1\<close>])
+ using \<epsilon> by auto
+ qed
+qed
+
+
+lemma GPicard6:
+ assumes "open M" "z \<in> M" "a \<noteq> 0" and holf: "f holomorphic_on (M - {z})"
+ and f0a: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> 0 \<and> f w \<noteq> a"
+ obtains r where "0 < r" "ball z r \<subseteq> M"
+ "bounded(f ` (ball z r - {z})) \<or>
+ bounded((inverse \<circ> f) ` (ball z r - {z}))"
+proof -
+ obtain r where "0 < r" and r: "ball z r \<subseteq> M"
+ using assms openE by blast
+ let ?g = "\<lambda>w. f (z + of_real r * w) / a"
+ obtain e B where "0 < e" "e < 1" "0 < B"
+ and B: "(\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<le> B) \<or> (\<forall>z \<in> ball 0 e - {0}. norm(?g z) \<ge> B)"
+ proof (rule GPicard5)
+ show "?g holomorphic_on ball 0 1 - {0}"
+ apply (intro holomorphic_intros holomorphic_on_compose_gen [unfolded o_def, OF _ holf])
+ using \<open>0 < r\<close> \<open>a \<noteq> 0\<close> r
+ by (auto simp: dist_norm norm_mult subset_eq)
+ show "\<And>w. w \<in> ball 0 1 - {0} \<Longrightarrow> f (z + of_real r * w) / a \<noteq> 0 \<and> f (z + of_real r * w) / a \<noteq> 1"
+ apply (simp add: field_split_simps \<open>a \<noteq> 0\<close>)
+ apply (rule f0a)
+ using \<open>0 < r\<close> r by (auto simp: dist_norm norm_mult subset_eq)
+ qed
+ show ?thesis
+ proof
+ show "0 < e*r"
+ by (simp add: \<open>0 < e\<close> \<open>0 < r\<close>)
+ have "ball z (e * r) \<subseteq> ball z r"
+ by (simp add: \<open>0 < r\<close> \<open>e < 1\<close> order.strict_implies_order subset_ball)
+ then show "ball z (e * r) \<subseteq> M"
+ using r by blast
+ consider "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<le> B" | "\<And>z. z \<in> ball 0 e - {0} \<Longrightarrow> norm(?g z) \<ge> B"
+ using B by blast
+ then show "bounded (f ` (ball z (e * r) - {z})) \<or>
+ bounded ((inverse \<circ> f) ` (ball z (e * r) - {z}))"
+ proof cases
+ case 1
+ have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<le> B * norm a" for w
+ using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 1 [of "(w - z) / r"]
+ by (simp add: norm_divide dist_norm field_split_simps)
+ then show ?thesis
+ by (force simp: intro!: boundedI)
+ next
+ case 2
+ have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> cmod (f w) \<ge> B * norm a" for w
+ using \<open>a \<noteq> 0\<close> \<open>0 < r\<close> 2 [of "(w - z) / r"]
+ by (simp add: norm_divide dist_norm field_split_simps)
+ then have "\<lbrakk>dist z w < e * r; w \<noteq> z\<rbrakk> \<Longrightarrow> inverse (cmod (f w)) \<le> inverse (B * norm a)" for w
+ by (metis \<open>0 < B\<close> \<open>a \<noteq> 0\<close> mult_pos_pos norm_inverse norm_inverse_le_norm zero_less_norm_iff)
+ then show ?thesis
+ by (force simp: norm_inverse intro!: boundedI)
+ qed
+ qed
+qed
+
+
+theorem great_Picard:
+ assumes "open M" "z \<in> M" "a \<noteq> b" and holf: "f holomorphic_on (M - {z})"
+ and fab: "\<And>w. w \<in> M - {z} \<Longrightarrow> f w \<noteq> a \<and> f w \<noteq> b"
+ obtains l where "(f \<longlongrightarrow> l) (at z) \<or> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
+proof -
+ obtain r where "0 < r" and zrM: "ball z r \<subseteq> M"
+ and r: "bounded((\<lambda>z. f z - a) ` (ball z r - {z})) \<or>
+ bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
+ proof (rule GPicard6 [OF \<open>open M\<close> \<open>z \<in> M\<close>])
+ show "b - a \<noteq> 0"
+ using assms by auto
+ show "(\<lambda>z. f z - a) holomorphic_on M - {z}"
+ by (intro holomorphic_intros holf)
+ qed (use fab in auto)
+ have holfb: "f holomorphic_on ball z r - {z}"
+ apply (rule holomorphic_on_subset [OF holf])
+ using zrM by auto
+ have holfb_i: "(\<lambda>z. inverse(f z - a)) holomorphic_on ball z r - {z}"
+ apply (intro holomorphic_intros holfb)
+ using fab zrM by fastforce
+ show ?thesis
+ using r
+ proof
+ assume "bounded ((\<lambda>z. f z - a) ` (ball z r - {z}))"
+ then obtain B where B: "\<And>w. w \<in> (\<lambda>z. f z - a) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
+ by (force simp: bounded_iff)
+ have "\<forall>\<^sub>F w in at z. cmod (f w - a) \<le> B"
+ apply (simp add: eventually_at)
+ apply (rule_tac x=r in exI)
+ using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
+ then have "\<exists>B. \<forall>\<^sub>F w in at z. cmod (f w) \<le> B"
+ apply (rule_tac x="B + norm a" in exI)
+ apply (erule eventually_mono)
+ by (metis add.commute add_le_cancel_right norm_triangle_sub order.trans)
+ then obtain g where holg: "g holomorphic_on ball z r" and gf: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = f w"
+ using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb] by auto
+ then have "g \<midarrow>z\<rightarrow> g z"
+ apply (simp add: continuous_at [symmetric])
+ using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
+ then have "(f \<longlongrightarrow> g z) (at z)"
+ apply (rule Lim_transform_within_open [of g "g z" z UNIV "ball z r"])
+ using \<open>0 < r\<close> by (auto simp: gf)
+ then show ?thesis
+ using that by blast
+ next
+ assume "bounded((inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}))"
+ then obtain B where B: "\<And>w. w \<in> (inverse \<circ> (\<lambda>z. f z - a)) ` (ball z r - {z}) \<Longrightarrow> norm w \<le> B"
+ by (force simp: bounded_iff)
+ have "\<forall>\<^sub>F w in at z. cmod (inverse (f w - a)) \<le> B"
+ apply (simp add: eventually_at)
+ apply (rule_tac x=r in exI)
+ using \<open>0 < r\<close> by (auto simp: dist_commute intro!: B)
+ then have "\<exists>B. \<forall>\<^sub>F z in at z. cmod (inverse (f z - a)) \<le> B"
+ by blast
+ then obtain g where holg: "g holomorphic_on ball z r" and gf: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w = inverse (f w - a)"
+ using \<open>0 < r\<close> holomorphic_on_extend_bounded [OF holfb_i] by auto
+ then have gz: "g \<midarrow>z\<rightarrow> g z"
+ apply (simp add: continuous_at [symmetric])
+ using \<open>0 < r\<close> centre_in_ball field_differentiable_imp_continuous_at holomorphic_on_imp_differentiable_at by blast
+ have gnz: "\<And>w. w \<in> ball z r - {z} \<Longrightarrow> g w \<noteq> 0"
+ using gf fab zrM by fastforce
+ show ?thesis
+ proof (cases "g z = 0")
+ case True
+ have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
+ by (auto simp: field_simps)
+ have "(inverse \<circ> f) \<midarrow>z\<rightarrow> 0"
+ proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
+ show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> 0"
+ using True by (auto simp: intro!: tendsto_eq_intros gz)
+ show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
+ using * gf gnz by simp
+ qed (use \<open>0 < r\<close> in auto)
+ with that show ?thesis by blast
+ next
+ case False
+ show ?thesis
+ proof (cases "1 + a * g z = 0")
+ case True
+ have "(f \<longlongrightarrow> 0) (at z)"
+ proof (rule Lim_transform_within_open [of "\<lambda>w. (1 + a * g w) / g w" _ _ _ "ball z r"])
+ show "(\<lambda>w. (1 + a * g w) / g w) \<midarrow>z\<rightarrow> 0"
+ apply (rule tendsto_eq_intros refl gz \<open>g z \<noteq> 0\<close>)+
+ by (simp add: True)
+ show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> (1 + a * g x) / g x = f x"
+ using fab fab zrM by (fastforce simp add: gf field_split_simps)
+ qed (use \<open>0 < r\<close> in auto)
+ then show ?thesis
+ using that by blast
+ next
+ case False
+ have *: "\<lbrakk>g \<noteq> 0; inverse g = f - a\<rbrakk> \<Longrightarrow> g / (1 + a * g) = inverse f" for f g::complex
+ by (auto simp: field_simps)
+ have "(inverse \<circ> f) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
+ proof (rule Lim_transform_within_open [of "\<lambda>w. g w / (1 + a * g w)" _ _ UNIV "ball z r"])
+ show "(\<lambda>w. g w / (1 + a * g w)) \<midarrow>z\<rightarrow> g z / (1 + a * g z)"
+ using False by (auto simp: False intro!: tendsto_eq_intros gz)
+ show "\<And>x. \<lbrakk>x \<in> ball z r; x \<noteq> z\<rbrakk> \<Longrightarrow> g x / (1 + a * g x) = (inverse \<circ> f) x"
+ using * gf gnz by simp
+ qed (use \<open>0 < r\<close> in auto)
+ with that show ?thesis by blast
+ qed
+ qed
+ qed
+qed
+
+
+corollary great_Picard_alt:
+ assumes M: "open M" "z \<in> M" and holf: "f holomorphic_on (M - {z})"
+ and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
+ obtains a where "- {a} \<subseteq> f ` (M - {z})"
+ apply (simp add: subset_iff image_iff)
+ by (metis great_Picard [OF M _ holf] non)
+
+
+corollary great_Picard_infinite:
+ assumes M: "open M" "z \<in> M" and holf: "f holomorphic_on (M - {z})"
+ and non: "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
+ obtains a where "\<And>w. w \<noteq> a \<Longrightarrow> infinite {x. x \<in> M - {z} \<and> f x = w}"
+proof -
+ have False if "a \<noteq> b" and ab: "finite {x. x \<in> M - {z} \<and> f x = a}" "finite {x. x \<in> M - {z} \<and> f x = b}" for a b
+ proof -
+ have finab: "finite {x. x \<in> M - {z} \<and> f x \<in> {a,b}}"
+ using finite_UnI [OF ab] unfolding mem_Collect_eq insert_iff empty_iff
+ by (simp add: conj_disj_distribL)
+ obtain r where "0 < r" and zrM: "ball z r \<subseteq> M" and r: "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a,b}\<rbrakk> \<Longrightarrow> x \<notin> ball z r"
+ proof -
+ obtain e where "e > 0" and e: "ball z e \<subseteq> M"
+ using assms openE by blast
+ show ?thesis
+ proof (cases "{x \<in> M - {z}. f x \<in> {a, b}} = {}")
+ case True
+ then show ?thesis
+ apply (rule_tac r=e in that)
+ using e \<open>e > 0\<close> by auto
+ next
+ case False
+ let ?r = "min e (Min (dist z ` {x \<in> M - {z}. f x \<in> {a,b}}))"
+ show ?thesis
+ proof
+ show "0 < ?r"
+ using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> by auto
+ have "ball z ?r \<subseteq> ball z e"
+ by (simp add: subset_ball)
+ with e show "ball z ?r \<subseteq> M" by blast
+ show "\<And>x. \<lbrakk>x \<in> M - {z}; f x \<in> {a, b}\<rbrakk> \<Longrightarrow> x \<notin> ball z ?r"
+ using min_less_iff_conj Min_gr_iff finab False \<open>0 < e\<close> by auto
+ qed
+ qed
+ qed
+ have holfb: "f holomorphic_on (ball z r - {z})"
+ apply (rule holomorphic_on_subset [OF holf])
+ using zrM by auto
+ show ?thesis
+ apply (rule great_Picard [OF open_ball _ \<open>a \<noteq> b\<close> holfb])
+ using non \<open>0 < r\<close> r zrM by auto
+ qed
+ with that show thesis
+ by meson
+qed
+
+theorem Casorati_Weierstrass:
+ assumes "open M" "z \<in> M" "f holomorphic_on (M - {z})"
+ and "\<And>l. \<not> (f \<longlongrightarrow> l) (at z)" "\<And>l. \<not> ((inverse \<circ> f) \<longlongrightarrow> l) (at z)"
+ shows "closure(f ` (M - {z})) = UNIV"
+proof -
+ obtain a where a: "- {a} \<subseteq> f ` (M - {z})"
+ using great_Picard_alt [OF assms] .
+ have "UNIV = closure(- {a})"
+ by (simp add: closure_interior)
+ also have "... \<subseteq> closure(f ` (M - {z}))"
+ by (simp add: a closure_mono)
+ finally show ?thesis
+ by blast
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Riemann_Mapping.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,1489 @@
+(* Title: HOL/Analysis/Riemann_Mapping.thy
+ Authors: LC Paulson, based on material from HOL Light
+*)
+
+section \<open>Moebius functions, Equivalents of Simply Connected Sets, Riemann Mapping Theorem\<close>
+
+theory Riemann_Mapping
+imports Great_Picard
+begin
+
+subsection\<open>Moebius functions are biholomorphisms of the unit disc\<close>
+
+definition\<^marker>\<open>tag important\<close> Moebius_function :: "[real,complex,complex] \<Rightarrow> complex" where
+ "Moebius_function \<equiv> \<lambda>t w z. exp(\<i> * of_real t) * (z - w) / (1 - cnj w * z)"
+
+lemma Moebius_function_simple:
+ "Moebius_function 0 w z = (z - w) / (1 - cnj w * z)"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_eq_zero:
+ "Moebius_function t w w = 0"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_of_zero:
+ "Moebius_function t w 0 = - exp(\<i> * of_real t) * w"
+ by (simp add: Moebius_function_def)
+
+lemma Moebius_function_norm_lt_1:
+ assumes w1: "norm w < 1" and z1: "norm z < 1"
+ shows "norm (Moebius_function t w z) < 1"
+proof -
+ have "1 - cnj w * z \<noteq> 0"
+ by (metis complex_cnj_cnj complex_mod_sqrt_Re_mult_cnj mult.commute mult_less_cancel_right1 norm_ge_zero norm_mult norm_one order.asym right_minus_eq w1 z1)
+ then have VV: "1 - w * cnj z \<noteq> 0"
+ by (metis complex_cnj_cnj complex_cnj_mult complex_cnj_one right_minus_eq)
+ then have "1 - norm (Moebius_function t w z) ^ 2 =
+ ((1 - norm w ^ 2) / (norm (1 - cnj w * z) ^ 2)) * (1 - norm z ^ 2)"
+ apply (cases w)
+ apply (cases z)
+ apply (simp add: Moebius_function_def divide_simps norm_divide norm_mult)
+ apply (simp add: complex_norm complex_diff complex_mult one_complex.code complex_cnj)
+ apply (auto simp: algebra_simps power2_eq_square)
+ done
+ then have "1 - (cmod (Moebius_function t w z))\<^sup>2 = (1 - cmod (w * w)) / (cmod (1 - cnj w * z))\<^sup>2 * (1 - cmod (z * z))"
+ by (simp add: norm_mult power2_eq_square)
+ moreover have "0 < 1 - cmod (z * z)"
+ by (metis (no_types) z1 diff_gt_0_iff_gt mult.left_neutral norm_mult_less)
+ ultimately have "0 < 1 - norm (Moebius_function t w z) ^ 2"
+ using \<open>1 - cnj w * z \<noteq> 0\<close> w1 norm_mult_less by fastforce
+ then show ?thesis
+ using linorder_not_less by fastforce
+qed
+
+lemma Moebius_function_holomorphic:
+ assumes "norm w < 1"
+ shows "Moebius_function t w holomorphic_on ball 0 1"
+proof -
+ have *: "1 - z * w \<noteq> 0" if "norm z < 1" for z
+ proof -
+ have "norm (1::complex) \<noteq> norm (z * w)"
+ using assms that norm_mult_less by fastforce
+ then show ?thesis by auto
+ qed
+ show ?thesis
+ apply (simp add: Moebius_function_def)
+ apply (intro holomorphic_intros)
+ using assms *
+ by (metis complex_cnj_cnj complex_cnj_mult complex_cnj_one complex_mod_cnj mem_ball_0 mult.commute right_minus_eq)
+qed
+
+lemma Moebius_function_compose:
+ assumes meq: "-w1 = w2" and "norm w1 < 1" "norm z < 1"
+ shows "Moebius_function 0 w1 (Moebius_function 0 w2 z) = z"
+proof -
+ have "norm w2 < 1"
+ using assms by auto
+ then have "-w1 = z" if "cnj w2 * z = 1"
+ by (metis assms(3) complex_mod_cnj less_irrefl mult.right_neutral norm_mult_less norm_one that)
+ moreover have "z=0" if "1 - cnj w2 * z = cnj w1 * (z - w2)"
+ proof -
+ have "w2 * cnj w2 = 1"
+ using that meq by (auto simp: algebra_simps)
+ then show "z = 0"
+ by (metis (no_types) \<open>cmod w2 < 1\<close> complex_mod_cnj less_irrefl mult.right_neutral norm_mult_less norm_one)
+ qed
+ moreover have "z - w2 - w1 * (1 - cnj w2 * z) = z * (1 - cnj w2 * z - cnj w1 * (z - w2))"
+ using meq by (fastforce simp: algebra_simps)
+ ultimately
+ show ?thesis
+ by (simp add: Moebius_function_def divide_simps norm_divide norm_mult)
+qed
+
+lemma ball_biholomorphism_exists:
+ assumes "a \<in> ball 0 1"
+ obtains f g where "f a = 0"
+ "f holomorphic_on ball 0 1" "f ` ball 0 1 \<subseteq> ball 0 1"
+ "g holomorphic_on ball 0 1" "g ` ball 0 1 \<subseteq> ball 0 1"
+ "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z"
+ "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z"
+proof
+ show "Moebius_function 0 a holomorphic_on ball 0 1" "Moebius_function 0 (-a) holomorphic_on ball 0 1"
+ using Moebius_function_holomorphic assms mem_ball_0 by auto
+ show "Moebius_function 0 a a = 0"
+ by (simp add: Moebius_function_eq_zero)
+ show "Moebius_function 0 a ` ball 0 1 \<subseteq> ball 0 1"
+ "Moebius_function 0 (- a) ` ball 0 1 \<subseteq> ball 0 1"
+ using Moebius_function_norm_lt_1 assms by auto
+ show "Moebius_function 0 a (Moebius_function 0 (- a) z) = z"
+ "Moebius_function 0 (- a) (Moebius_function 0 a z) = z" if "z \<in> ball 0 1" for z
+ using Moebius_function_compose assms that by auto
+qed
+
+
+subsection\<open>A big chain of equivalents of simple connectedness for an open set\<close>
+
+lemma biholomorphic_to_disc_aux:
+ assumes "open S" "connected S" "0 \<in> S" and S01: "S \<subseteq> ball 0 1"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0; inj_on f S\<rbrakk>
+ \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z)"
+proof -
+ define F where "F \<equiv> {h. h holomorphic_on S \<and> h ` S \<subseteq> ball 0 1 \<and> h 0 = 0 \<and> inj_on h S}"
+ have idF: "id \<in> F"
+ using S01 by (auto simp: F_def)
+ then have "F \<noteq> {}"
+ by blast
+ have imF_ne: "((\<lambda>h. norm(deriv h 0)) ` F) \<noteq> {}"
+ using idF by auto
+ have holF: "\<And>h. h \<in> F \<Longrightarrow> h holomorphic_on S"
+ by (auto simp: F_def)
+ obtain f where "f \<in> F" and normf: "\<And>h. h \<in> F \<Longrightarrow> norm(deriv h 0) \<le> norm(deriv f 0)"
+ proof -
+ obtain r where "r > 0" and r: "ball 0 r \<subseteq> S"
+ using \<open>open S\<close> \<open>0 \<in> S\<close> openE by auto
+ have bdd: "bdd_above ((\<lambda>h. norm(deriv h 0)) ` F)"
+ proof (intro bdd_aboveI exI ballI, clarify)
+ show "norm (deriv f 0) \<le> 1 / r" if "f \<in> F" for f
+ proof -
+ have r01: "(*) (complex_of_real r) ` ball 0 1 \<subseteq> S"
+ using that \<open>r > 0\<close> by (auto simp: norm_mult r [THEN subsetD])
+ then have "f holomorphic_on (*) (complex_of_real r) ` ball 0 1"
+ using holomorphic_on_subset [OF holF] by (simp add: that)
+ then have holf: "f \<circ> (\<lambda>z. (r * z)) holomorphic_on (ball 0 1)"
+ by (intro holomorphic_intros holomorphic_on_compose)
+ have f0: "(f \<circ> (*) (complex_of_real r)) 0 = 0"
+ using F_def that by auto
+ have "f ` S \<subseteq> ball 0 1"
+ using F_def that by blast
+ with r01 have fr1: "\<And>z. norm z < 1 \<Longrightarrow> norm ((f \<circ> (*)(of_real r))z) < 1"
+ by force
+ have *: "((\<lambda>w. f (r * w)) has_field_derivative deriv f (r * z) * r) (at z)"
+ if "z \<in> ball 0 1" for z::complex
+ proof (rule DERIV_chain' [where g=f])
+ show "(f has_field_derivative deriv f (of_real r * z)) (at (of_real r * z))"
+ apply (rule holomorphic_derivI [OF holF \<open>open S\<close>])
+ apply (rule \<open>f \<in> F\<close>)
+ by (meson imageI r01 subset_iff that)
+ qed simp
+ have df0: "((\<lambda>w. f (r * w)) has_field_derivative deriv f 0 * r) (at 0)"
+ using * [of 0] by simp
+ have deq: "deriv (\<lambda>x. f (complex_of_real r * x)) 0 = deriv f 0 * complex_of_real r"
+ using DERIV_imp_deriv df0 by blast
+ have "norm (deriv (f \<circ> (*) (complex_of_real r)) 0) \<le> 1"
+ by (auto intro: Schwarz_Lemma [OF holf f0 fr1, of 0])
+ with \<open>r > 0\<close> show ?thesis
+ by (simp add: deq norm_mult divide_simps o_def)
+ qed
+ qed
+ define l where "l \<equiv> SUP h\<in>F. norm (deriv h 0)"
+ have eql: "norm (deriv f 0) = l" if le: "l \<le> norm (deriv f 0)" and "f \<in> F" for f
+ apply (rule order_antisym [OF _ le])
+ using \<open>f \<in> F\<close> bdd cSUP_upper by (fastforce simp: l_def)
+ obtain \<F> where \<F>in: "\<And>n. \<F> n \<in> F" and \<F>lim: "(\<lambda>n. norm (deriv (\<F> n) 0)) \<longlonglongrightarrow> l"
+ proof -
+ have "\<exists>f. f \<in> F \<and> \<bar>norm (deriv f 0) - l\<bar> < 1 / (Suc n)" for n
+ proof -
+ obtain f where "f \<in> F" and f: "l < norm (deriv f 0) + 1/(Suc n)"
+ using cSup_least [OF imF_ne, of "l - 1/(Suc n)"] by (fastforce simp add: l_def)
+ then have "\<bar>norm (deriv f 0) - l\<bar> < 1 / (Suc n)"
+ by (fastforce simp add: abs_if not_less eql)
+ with \<open>f \<in> F\<close> show ?thesis
+ by blast
+ qed
+ then obtain \<F> where fF: "\<And>n. (\<F> n) \<in> F"
+ and fless: "\<And>n. \<bar>norm (deriv (\<F> n) 0) - l\<bar> < 1 / (Suc n)"
+ by metis
+ have "(\<lambda>n. norm (deriv (\<F> n) 0)) \<longlonglongrightarrow> l"
+ proof (rule metric_LIMSEQ_I)
+ fix e::real
+ assume "e > 0"
+ then obtain N::nat where N: "e > 1/(Suc N)"
+ using nat_approx_posE by blast
+ show "\<exists>N. \<forall>n\<ge>N. dist (norm (deriv (\<F> n) 0)) l < e"
+ proof (intro exI allI impI)
+ fix n assume "N \<le> n"
+ have "dist (norm (deriv (\<F> n) 0)) l < 1 / (Suc n)"
+ using fless by (simp add: dist_norm)
+ also have "... < e"
+ using N \<open>N \<le> n\<close> inverse_of_nat_le le_less_trans by blast
+ finally show "dist (norm (deriv (\<F> n) 0)) l < e" .
+ qed
+ qed
+ with fF show ?thesis
+ using that by blast
+ qed
+ have "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>B. \<forall>h\<in>F. \<forall>z\<in>K. norm (h z) \<le> B"
+ by (rule_tac x=1 in exI) (force simp: F_def)
+ moreover have "range \<F> \<subseteq> F"
+ using \<open>\<And>n. \<F> n \<in> F\<close> by blast
+ ultimately obtain f and r :: "nat \<Rightarrow> nat"
+ where holf: "f holomorphic_on S" and r: "strict_mono r"
+ and limf: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<F> (r n) x) \<longlonglongrightarrow> f x"
+ and ulimf: "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K (\<F> \<circ> r) f sequentially"
+ using Montel [of S F \<F>, OF \<open>open S\<close> holF] by auto+
+ have der: "\<And>n x. x \<in> S \<Longrightarrow> ((\<F> \<circ> r) n has_field_derivative ((\<lambda>n. deriv (\<F> n)) \<circ> r) n x) (at x)"
+ using \<open>\<And>n. \<F> n \<in> F\<close> \<open>open S\<close> holF holomorphic_derivI by fastforce
+ have ulim: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<F> \<circ> r) f sequentially"
+ by (meson ulimf \<open>open S\<close> compact_cball open_contains_cball)
+ obtain f' :: "complex\<Rightarrow>complex" where f': "(f has_field_derivative f' 0) (at 0)"
+ and tof'0: "(\<lambda>n. ((\<lambda>n. deriv (\<F> n)) \<circ> r) n 0) \<longlonglongrightarrow> f' 0"
+ using has_complex_derivative_uniform_sequence [OF \<open>open S\<close> der ulim] \<open>0 \<in> S\<close> by metis
+ then have derf0: "deriv f 0 = f' 0"
+ by (simp add: DERIV_imp_deriv)
+ have "f field_differentiable (at 0)"
+ using field_differentiable_def f' by blast
+ have "(\<lambda>x. (norm (deriv (\<F> (r x)) 0))) \<longlonglongrightarrow> norm (deriv f 0)"
+ using isCont_tendsto_compose [OF continuous_norm [OF continuous_ident] tof'0] derf0 by auto
+ with LIMSEQ_subseq_LIMSEQ [OF \<F>lim r] have no_df0: "norm(deriv f 0) = l"
+ by (force simp: o_def intro: tendsto_unique)
+ have nonconstf: "\<not> f constant_on S"
+ proof -
+ have False if "\<And>x. x \<in> S \<Longrightarrow> f x = c" for c
+ proof -
+ have "deriv f 0 = 0"
+ by (metis that \<open>open S\<close> \<open>0 \<in> S\<close> DERIV_imp_deriv [OF has_field_derivative_transform_within_open [OF DERIV_const]])
+ with no_df0 have "l = 0"
+ by auto
+ with eql [OF _ idF] show False by auto
+ qed
+ then show ?thesis
+ by (meson constant_on_def)
+ qed
+ show ?thesis
+ proof
+ show "f \<in> F"
+ unfolding F_def
+ proof (intro CollectI conjI holf)
+ have "norm(f z) \<le> 1" if "z \<in> S" for z
+ proof (intro Lim_norm_ubound [OF _ limf] always_eventually allI that)
+ fix n
+ have "\<F> (r n) \<in> F"
+ by (simp add: \<F>in)
+ then show "norm (\<F> (r n) z) \<le> 1"
+ using that by (auto simp: F_def)
+ qed simp
+ then have fless1: "norm(f z) < 1" if "z \<in> S" for z
+ using maximum_modulus_principle [OF holf \<open>open S\<close> \<open>connected S\<close> \<open>open S\<close>] nonconstf that
+ by fastforce
+ then show "f ` S \<subseteq> ball 0 1"
+ by auto
+ have "(\<lambda>n. \<F> (r n) 0) \<longlonglongrightarrow> 0"
+ using \<F>in by (auto simp: F_def)
+ then show "f 0 = 0"
+ using tendsto_unique [OF _ limf ] \<open>0 \<in> S\<close> trivial_limit_sequentially by blast
+ show "inj_on f S"
+ proof (rule Hurwitz_injective [OF \<open>open S\<close> \<open>connected S\<close> _ holf])
+ show "\<And>n. (\<F> \<circ> r) n holomorphic_on S"
+ by (simp add: \<F>in holF)
+ show "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K(\<F> \<circ> r) f sequentially"
+ by (metis ulimf)
+ show "\<not> f constant_on S"
+ using nonconstf by auto
+ show "\<And>n. inj_on ((\<F> \<circ> r) n) S"
+ using \<F>in by (auto simp: F_def)
+ qed
+ qed
+ show "\<And>h. h \<in> F \<Longrightarrow> norm (deriv h 0) \<le> norm (deriv f 0)"
+ by (metis eql le_cases no_df0)
+ qed
+ qed
+ have holf: "f holomorphic_on S" and injf: "inj_on f S" and f01: "f ` S \<subseteq> ball 0 1"
+ using \<open>f \<in> F\<close> by (auto simp: F_def)
+ obtain g where holg: "g holomorphic_on (f ` S)"
+ and derg: "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1"
+ and gf: "\<And>z. z \<in> S \<Longrightarrow> g(f z) = z"
+ using holomorphic_has_inverse [OF holf \<open>open S\<close> injf] by metis
+ have "ball 0 1 \<subseteq> f ` S"
+ proof
+ fix a::complex
+ assume a: "a \<in> ball 0 1"
+ have False if "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> a"
+ proof -
+ obtain h k where "h a = 0"
+ and holh: "h holomorphic_on ball 0 1" and h01: "h ` ball 0 1 \<subseteq> ball 0 1"
+ and holk: "k holomorphic_on ball 0 1" and k01: "k ` ball 0 1 \<subseteq> ball 0 1"
+ and hk: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> h (k z) = z"
+ and kh: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> k (h z) = z"
+ using ball_biholomorphism_exists [OF a] by blast
+ have nf1: "\<And>z. z \<in> S \<Longrightarrow> norm(f z) < 1"
+ using \<open>f \<in> F\<close> by (auto simp: F_def)
+ have 1: "h \<circ> f holomorphic_on S"
+ using F_def \<open>f \<in> F\<close> holh holomorphic_on_compose holomorphic_on_subset by blast
+ have 2: "\<And>z. z \<in> S \<Longrightarrow> (h \<circ> f) z \<noteq> 0"
+ by (metis \<open>h a = 0\<close> a comp_eq_dest_lhs nf1 kh mem_ball_0 that)
+ have 3: "inj_on (h \<circ> f) S"
+ by (metis (no_types, lifting) F_def \<open>f \<in> F\<close> comp_inj_on inj_on_inverseI injf kh mem_Collect_eq subset_inj_on)
+ obtain \<psi> where hol\<psi>: "\<psi> holomorphic_on ((h \<circ> f) ` S)"
+ and \<psi>2: "\<And>z. z \<in> S \<Longrightarrow> \<psi>(h (f z)) ^ 2 = h(f z)"
+ proof (rule exE [OF prev [OF 1 2 3]], safe)
+ fix \<theta>
+ assume hol\<theta>: "\<theta> holomorphic_on S" and \<theta>2: "(\<forall>z\<in>S. (h \<circ> f) z = (\<theta> z)\<^sup>2)"
+ show thesis
+ proof
+ show "(\<theta> \<circ> g \<circ> k) holomorphic_on (h \<circ> f) ` S"
+ proof (intro holomorphic_on_compose)
+ show "k holomorphic_on (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF holk])
+ using f01 h01 by force
+ show "g holomorphic_on k ` (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF holg])
+ by (auto simp: kh nf1)
+ show "\<theta> holomorphic_on g ` k ` (h \<circ> f) ` S"
+ apply (rule holomorphic_on_subset [OF hol\<theta>])
+ by (auto simp: gf kh nf1)
+ qed
+ show "((\<theta> \<circ> g \<circ> k) (h (f z)))\<^sup>2 = h (f z)" if "z \<in> S" for z
+ proof -
+ have "f z \<in> ball 0 1"
+ by (simp add: nf1 that)
+ then have "(\<theta> (g (k (h (f z)))))\<^sup>2 = (\<theta> (g (f z)))\<^sup>2"
+ by (metis kh)
+ also have "... = h (f z)"
+ using \<theta>2 gf that by auto
+ finally show ?thesis
+ by (simp add: o_def)
+ qed
+ qed
+ qed
+ have norm\<psi>1: "norm(\<psi> (h (f z))) < 1" if "z \<in> S" for z
+ proof -
+ have "norm (\<psi> (h (f z)) ^ 2) < 1"
+ by (metis (no_types) that DIM_complex \<psi>2 h01 image_subset_iff mem_ball_0 nf1)
+ then show ?thesis
+ by (metis le_less_trans mult_less_cancel_left2 norm_ge_zero norm_power not_le power2_eq_square)
+ qed
+ then have \<psi>01: "\<psi> (h (f 0)) \<in> ball 0 1"
+ by (simp add: \<open>0 \<in> S\<close>)
+ obtain p q where p0: "p (\<psi> (h (f 0))) = 0"
+ and holp: "p holomorphic_on ball 0 1" and p01: "p ` ball 0 1 \<subseteq> ball 0 1"
+ and holq: "q holomorphic_on ball 0 1" and q01: "q ` ball 0 1 \<subseteq> ball 0 1"
+ and pq: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> p (q z) = z"
+ and qp: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> q (p z) = z"
+ using ball_biholomorphism_exists [OF \<psi>01] by metis
+ have "p \<circ> \<psi> \<circ> h \<circ> f \<in> F"
+ unfolding F_def
+ proof (intro CollectI conjI holf)
+ show "p \<circ> \<psi> \<circ> h \<circ> f holomorphic_on S"
+ proof (intro holomorphic_on_compose holf)
+ show "h holomorphic_on f ` S"
+ apply (rule holomorphic_on_subset [OF holh])
+ using f01 by force
+ show "\<psi> holomorphic_on h ` f ` S"
+ apply (rule holomorphic_on_subset [OF hol\<psi>])
+ by auto
+ show "p holomorphic_on \<psi> ` h ` f ` S"
+ apply (rule holomorphic_on_subset [OF holp])
+ by (auto simp: norm\<psi>1)
+ qed
+ show "(p \<circ> \<psi> \<circ> h \<circ> f) ` S \<subseteq> ball 0 1"
+ apply clarsimp
+ by (meson norm\<psi>1 p01 image_subset_iff mem_ball_0)
+ show "(p \<circ> \<psi> \<circ> h \<circ> f) 0 = 0"
+ by (simp add: \<open>p (\<psi> (h (f 0))) = 0\<close>)
+ show "inj_on (p \<circ> \<psi> \<circ> h \<circ> f) S"
+ unfolding inj_on_def o_def
+ by (metis \<psi>2 dist_0_norm gf kh mem_ball nf1 norm\<psi>1 qp)
+ qed
+ then have le_norm_df0: "norm (deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) \<le> norm (deriv f 0)"
+ by (rule normf)
+ have 1: "k \<circ> power2 \<circ> q holomorphic_on ball 0 1"
+ proof (intro holomorphic_on_compose holq)
+ show "power2 holomorphic_on q ` ball 0 1"
+ using holomorphic_on_subset holomorphic_on_power
+ by (blast intro: holomorphic_on_ident)
+ show "k holomorphic_on power2 ` q ` ball 0 1"
+ apply (rule holomorphic_on_subset [OF holk])
+ using q01 by (auto simp: norm_power abs_square_less_1)
+ qed
+ have 2: "(k \<circ> power2 \<circ> q) 0 = 0"
+ using p0 F_def \<open>f \<in> F\<close> \<psi>01 \<psi>2 \<open>0 \<in> S\<close> kh qp by force
+ have 3: "norm ((k \<circ> power2 \<circ> q) z) < 1" if "norm z < 1" for z
+ proof -
+ have "norm ((power2 \<circ> q) z) < 1"
+ using that q01 by (force simp: norm_power abs_square_less_1)
+ with k01 show ?thesis
+ by fastforce
+ qed
+ have False if c: "\<forall>z. norm z < 1 \<longrightarrow> (k \<circ> power2 \<circ> q) z = c * z" and "norm c = 1" for c
+ proof -
+ have "c \<noteq> 0" using that by auto
+ have "norm (p(1/2)) < 1" "norm (p(-1/2)) < 1"
+ using p01 by force+
+ then have "(k \<circ> power2 \<circ> q) (p(1/2)) = c * p(1/2)" "(k \<circ> power2 \<circ> q) (p(-1/2)) = c * p(-1/2)"
+ using c by force+
+ then have "p (1/2) = p (- (1/2))"
+ by (auto simp: \<open>c \<noteq> 0\<close> qp o_def)
+ then have "q (p (1/2)) = q (p (- (1/2)))"
+ by simp
+ then have "1/2 = - (1/2::complex)"
+ by (auto simp: qp)
+ then show False
+ by simp
+ qed
+ moreover
+ have False if "norm (deriv (k \<circ> power2 \<circ> q) 0) \<noteq> 1" "norm (deriv (k \<circ> power2 \<circ> q) 0) \<le> 1"
+ and le: "\<And>\<xi>. norm \<xi> < 1 \<Longrightarrow> norm ((k \<circ> power2 \<circ> q) \<xi>) \<le> norm \<xi>"
+ proof -
+ have "norm (deriv (k \<circ> power2 \<circ> q) 0) < 1"
+ using that by simp
+ moreover have eq: "deriv f 0 = deriv (k \<circ> (\<lambda>z. z ^ 2) \<circ> q) 0 * deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0"
+ proof (intro DERIV_imp_deriv has_field_derivative_transform_within_open [OF DERIV_chain])
+ show "(k \<circ> power2 \<circ> q has_field_derivative deriv (k \<circ> power2 \<circ> q) 0) (at ((p \<circ> \<psi> \<circ> h \<circ> f) 0))"
+ using "1" holomorphic_derivI p0 by auto
+ show "(p \<circ> \<psi> \<circ> h \<circ> f has_field_derivative deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) (at 0)"
+ using \<open>p \<circ> \<psi> \<circ> h \<circ> f \<in> F\<close> \<open>open S\<close> \<open>0 \<in> S\<close> holF holomorphic_derivI by blast
+ show "\<And>x. x \<in> S \<Longrightarrow> (k \<circ> power2 \<circ> q \<circ> (p \<circ> \<psi> \<circ> h \<circ> f)) x = f x"
+ using \<psi>2 f01 kh norm\<psi>1 qp by auto
+ qed (use assms in simp_all)
+ ultimately have "cmod (deriv (p \<circ> \<psi> \<circ> h \<circ> f) 0) \<le> 0"
+ using le_norm_df0
+ by (metis linorder_not_le mult.commute mult_less_cancel_left2 norm_mult)
+ moreover have "1 \<le> norm (deriv f 0)"
+ using normf [of id] by (simp add: idF)
+ ultimately show False
+ by (simp add: eq)
+ qed
+ ultimately show ?thesis
+ using Schwarz_Lemma [OF 1 2 3] norm_one by blast
+ qed
+ then show "a \<in> f ` S"
+ by blast
+ qed
+ then have "f ` S = ball 0 1"
+ using F_def \<open>f \<in> F\<close> by blast
+ then show ?thesis
+ apply (rule_tac x=f in exI)
+ apply (rule_tac x=g in exI)
+ using holf holg derg gf by safe force+
+qed
+
+
+locale SC_Chain =
+ fixes S :: "complex set"
+ assumes openS: "open S"
+begin
+
+lemma winding_number_zero:
+ assumes "simply_connected S"
+ shows "connected S \<and>
+ (\<forall>\<gamma> z. path \<gamma> \<and> path_image \<gamma> \<subseteq> S \<and>
+ pathfinish \<gamma> = pathstart \<gamma> \<and> z \<notin> S \<longrightarrow> winding_number \<gamma> z = 0)"
+ using assms
+ by (auto simp: simply_connected_imp_connected simply_connected_imp_winding_number_zero)
+
+lemma contour_integral_zero:
+ assumes "valid_path g" "path_image g \<subseteq> S" "pathfinish g = pathstart g" "f holomorphic_on S"
+ "\<And>\<gamma> z. \<lbrakk>path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> S\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
+ shows "(f has_contour_integral 0) g"
+ using assms by (meson Cauchy_theorem_global openS valid_path_imp_path)
+
+lemma global_primitive:
+ assumes "connected S" and holf: "f holomorphic_on S"
+ and prev: "\<And>\<gamma> f. \<lbrakk>valid_path \<gamma>; path_image \<gamma> \<subseteq> S; pathfinish \<gamma> = pathstart \<gamma>; f holomorphic_on S\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) \<gamma>"
+ shows "\<exists>h. \<forall>z \<in> S. (h has_field_derivative f z) (at z)"
+proof (cases "S = {}")
+case True then show ?thesis
+ by simp
+next
+ case False
+ then obtain a where "a \<in> S"
+ by blast
+ show ?thesis
+ proof (intro exI ballI)
+ fix x assume "x \<in> S"
+ then obtain d where "d > 0" and d: "cball x d \<subseteq> S"
+ using openS open_contains_cball_eq by blast
+ let ?g = "\<lambda>z. (SOME g. polynomial_function g \<and> path_image g \<subseteq> S \<and> pathstart g = a \<and> pathfinish g = z)"
+ show "((\<lambda>z. contour_integral (?g z) f) has_field_derivative f x)
+ (at x)"
+ proof (simp add: has_field_derivative_def has_derivative_at2 bounded_linear_mult_right, rule Lim_transform)
+ show "(\<lambda>y. inverse(norm(y - x)) *\<^sub>R (contour_integral(linepath x y) f - f x * (y - x))) \<midarrow>x\<rightarrow> 0"
+ proof (clarsimp simp add: Lim_at)
+ fix e::real assume "e > 0"
+ moreover have "continuous (at x) f"
+ using openS \<open>x \<in> S\<close> holf continuous_on_eq_continuous_at holomorphic_on_imp_continuous_on by auto
+ ultimately obtain d1 where "d1 > 0"
+ and d1: "\<And>x'. dist x' x < d1 \<Longrightarrow> dist (f x') (f x) < e/2"
+ unfolding continuous_at_eps_delta
+ by (metis less_divide_eq_numeral1(1) mult_zero_left)
+ obtain d2 where "d2 > 0" and d2: "ball x d2 \<subseteq> S"
+ using openS \<open>x \<in> S\<close> open_contains_ball_eq by blast
+ have "inverse (norm (y - x)) * norm (contour_integral (linepath x y) f - f x * (y - x)) < e"
+ if "0 < d1" "0 < d2" "y \<noteq> x" "dist y x < d1" "dist y x < d2" for y
+ proof -
+ have "f contour_integrable_on linepath x y"
+ proof (rule contour_integrable_continuous_linepath [OF continuous_on_subset])
+ show "continuous_on S f"
+ by (simp add: holf holomorphic_on_imp_continuous_on)
+ have "closed_segment x y \<subseteq> ball x d2"
+ by (meson dist_commute_lessI dist_in_closed_segment le_less_trans mem_ball subsetI that(5))
+ with d2 show "closed_segment x y \<subseteq> S"
+ by blast
+ qed
+ then obtain z where z: "(f has_contour_integral z) (linepath x y)"
+ by (force simp: contour_integrable_on_def)
+ have con: "((\<lambda>w. f x) has_contour_integral f x * (y - x)) (linepath x y)"
+ using has_contour_integral_const_linepath [of "f x" y x] by metis
+ have "norm (z - f x * (y - x)) \<le> (e/2) * norm (y - x)"
+ proof (rule has_contour_integral_bound_linepath)
+ show "((\<lambda>w. f w - f x) has_contour_integral z - f x * (y - x)) (linepath x y)"
+ by (rule has_contour_integral_diff [OF z con])
+ show "\<And>w. w \<in> closed_segment x y \<Longrightarrow> norm (f w - f x) \<le> e/2"
+ by (metis d1 dist_norm less_le_trans not_less not_less_iff_gr_or_eq segment_bound1 that(4))
+ qed (use \<open>e > 0\<close> in auto)
+ with \<open>e > 0\<close> have "inverse (norm (y - x)) * norm (z - f x * (y - x)) \<le> e/2"
+ by (simp add: field_split_simps)
+ also have "... < e"
+ using \<open>e > 0\<close> by simp
+ finally show ?thesis
+ by (simp add: contour_integral_unique [OF z])
+ qed
+ with \<open>d1 > 0\<close> \<open>d2 > 0\<close>
+ show "\<exists>d>0. \<forall>z. z \<noteq> x \<and> dist z x < d \<longrightarrow>
+ inverse (norm (z - x)) * norm (contour_integral (linepath x z) f - f x * (z - x)) < e"
+ by (rule_tac x="min d1 d2" in exI) auto
+ qed
+ next
+ have *: "(1 / norm (y - x)) *\<^sub>R (contour_integral (?g y) f -
+ (contour_integral (?g x) f + f x * (y - x))) =
+ (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R norm (y - x)"
+ if "0 < d" "y \<noteq> x" and yx: "dist y x < d" for y
+ proof -
+ have "y \<in> S"
+ by (metis subsetD d dist_commute less_eq_real_def mem_cball yx)
+ have gxy: "polynomial_function (?g x) \<and> path_image (?g x) \<subseteq> S \<and> pathstart (?g x) = a \<and> pathfinish (?g x) = x"
+ "polynomial_function (?g y) \<and> path_image (?g y) \<subseteq> S \<and> pathstart (?g y) = a \<and> pathfinish (?g y) = y"
+ using someI_ex [OF connected_open_polynomial_connected [OF openS \<open>connected S\<close> \<open>a \<in> S\<close>]] \<open>x \<in> S\<close> \<open>y \<in> S\<close>
+ by meson+
+ then have vp: "valid_path (?g x)" "valid_path (?g y)"
+ by (simp_all add: valid_path_polynomial_function)
+ have f0: "(f has_contour_integral 0) ((?g x) +++ linepath x y +++ reversepath (?g y))"
+ proof (rule prev)
+ show "valid_path ((?g x) +++ linepath x y +++ reversepath (?g y))"
+ using gxy vp by (auto simp: valid_path_join)
+ have "closed_segment x y \<subseteq> cball x d"
+ using yx by (auto simp: dist_commute dest!: dist_in_closed_segment)
+ then have "closed_segment x y \<subseteq> S"
+ using d by blast
+ then show "path_image ((?g x) +++ linepath x y +++ reversepath (?g y)) \<subseteq> S"
+ using gxy by (auto simp: path_image_join)
+ qed (use gxy holf in auto)
+ then have fintxy: "f contour_integrable_on linepath x y"
+ by (metis (no_types, lifting) contour_integrable_joinD1 contour_integrable_joinD2 gxy(2) has_contour_integral_integrable pathfinish_linepath pathstart_reversepath valid_path_imp_reverse valid_path_join valid_path_linepath vp(2))
+ have fintgx: "f contour_integrable_on (?g x)" "f contour_integrable_on (?g y)"
+ using openS contour_integrable_holomorphic_simple gxy holf vp by blast+
+ show ?thesis
+ apply (clarsimp simp add: divide_simps)
+ using contour_integral_unique [OF f0]
+ apply (simp add: fintxy gxy contour_integrable_reversepath contour_integral_reversepath fintgx vp)
+ apply (simp add: algebra_simps)
+ done
+ qed
+ show "(\<lambda>z. (1 / norm (z - x)) *\<^sub>R
+ (contour_integral (?g z) f - (contour_integral (?g x) f + f x * (z - x))) -
+ (contour_integral (linepath x z) f - f x * (z - x)) /\<^sub>R norm (z - x))
+ \<midarrow>x\<rightarrow> 0"
+ apply (rule tendsto_eventually)
+ apply (simp add: eventually_at)
+ apply (rule_tac x=d in exI)
+ using \<open>d > 0\<close> * by simp
+ qed
+ qed
+qed
+
+lemma holomorphic_log:
+ assumes "connected S" and holf: "f holomorphic_on S" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f. f holomorphic_on S \<Longrightarrow> \<exists>h. \<forall>z \<in> S. (h has_field_derivative f z) (at z)"
+ shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))"
+proof -
+ have "(\<lambda>z. deriv f z / f z) holomorphic_on S"
+ by (simp add: openS holf holomorphic_deriv holomorphic_on_divide nz)
+ then obtain g where g: "\<And>z. z \<in> S \<Longrightarrow> (g has_field_derivative deriv f z / f z) (at z)"
+ using prev [of "\<lambda>z. deriv f z / f z"] by metis
+ have hfd: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>z. exp (g z) / f z) has_field_derivative 0) (at x)"
+ apply (rule derivative_eq_intros g| simp)+
+ apply (subst DERIV_deriv_iff_field_differentiable)
+ using openS holf holomorphic_on_imp_differentiable_at nz apply auto
+ done
+ obtain c where c: "\<And>x. x \<in> S \<Longrightarrow> exp (g x) / f x = c"
+ proof (rule DERIV_zero_connected_constant[OF \<open>connected S\<close> openS finite.emptyI])
+ show "continuous_on S (\<lambda>z. exp (g z) / f z)"
+ by (metis (full_types) openS g continuous_on_divide continuous_on_exp holf holomorphic_on_imp_continuous_on holomorphic_on_open nz)
+ then show "\<forall>x\<in>S - {}. ((\<lambda>z. exp (g z) / f z) has_field_derivative 0) (at x)"
+ using hfd by (blast intro: DERIV_zero_connected_constant [OF \<open>connected S\<close> openS finite.emptyI, of "\<lambda>z. exp(g z) / f z"])
+ qed auto
+ show ?thesis
+ proof (intro exI ballI conjI)
+ show "(\<lambda>z. Ln(inverse c) + g z) holomorphic_on S"
+ apply (intro holomorphic_intros)
+ using openS g holomorphic_on_open by blast
+ fix z :: complex
+ assume "z \<in> S"
+ then have "exp (g z) / c = f z"
+ by (metis c divide_divide_eq_right exp_not_eq_zero nonzero_mult_div_cancel_left)
+ moreover have "1 / c \<noteq> 0"
+ using \<open>z \<in> S\<close> c nz by fastforce
+ ultimately show "f z = exp (Ln (inverse c) + g z)"
+ by (simp add: exp_add inverse_eq_divide)
+ qed
+qed
+
+lemma holomorphic_sqrt:
+ assumes holf: "f holomorphic_on S" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))"
+ shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+proof -
+ obtain g where holg: "g holomorphic_on S" and g: "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)"
+ using prev [of f] holf nz by metis
+ show ?thesis
+ proof (intro exI ballI conjI)
+ show "(\<lambda>z. exp(g z/2)) holomorphic_on S"
+ by (intro holomorphic_intros) (auto simp: holg)
+ show "\<And>z. z \<in> S \<Longrightarrow> f z = (exp (g z/2))\<^sup>2"
+ by (metis (no_types) g exp_double nonzero_mult_div_cancel_left times_divide_eq_right zero_neq_numeral)
+ qed
+qed
+
+lemma biholomorphic_to_disc:
+ assumes "connected S" and S: "S \<noteq> {}" "S \<noteq> UNIV"
+ and prev: "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z)"
+proof -
+ obtain a b where "a \<in> S" "b \<notin> S"
+ using S by blast
+ then obtain \<delta> where "\<delta> > 0" and \<delta>: "ball a \<delta> \<subseteq> S"
+ using openS openE by blast
+ obtain g where holg: "g holomorphic_on S" and eqg: "\<And>z. z \<in> S \<Longrightarrow> z - b = (g z)\<^sup>2"
+ proof (rule exE [OF prev [of "\<lambda>z. z - b"]])
+ show "(\<lambda>z. z - b) holomorphic_on S"
+ by (intro holomorphic_intros)
+ qed (use \<open>b \<notin> S\<close> in auto)
+ have "\<not> g constant_on S"
+ proof -
+ have "(a + \<delta>/2) \<in> ball a \<delta>" "a + (\<delta>/2) \<noteq> a"
+ using \<open>\<delta> > 0\<close> by (simp_all add: dist_norm)
+ then show ?thesis
+ unfolding constant_on_def
+ using eqg [of a] eqg [of "a + \<delta>/2"] \<open>a \<in> S\<close> \<delta>
+ by (metis diff_add_cancel subset_eq)
+ qed
+ then have "open (g ` ball a \<delta>)"
+ using open_mapping_thm [of g S "ball a \<delta>", OF holg openS \<open>connected S\<close>] \<delta> by blast
+ then obtain r where "r > 0" and r: "ball (g a) r \<subseteq> (g ` ball a \<delta>)"
+ by (metis \<open>0 < \<delta>\<close> centre_in_ball imageI openE)
+ have g_not_r: "g z \<notin> ball (-(g a)) r" if "z \<in> S" for z
+ proof
+ assume "g z \<in> ball (-(g a)) r"
+ then have "- g z \<in> ball (g a) r"
+ by (metis add.inverse_inverse dist_minus mem_ball)
+ with r have "- g z \<in> (g ` ball a \<delta>)"
+ by blast
+ then obtain w where w: "- g z = g w" "dist a w < \<delta>"
+ by auto
+ then have "w \<in> ball a \<delta>"
+ by simp
+ then have "w \<in> S"
+ using \<delta> by blast
+ then have "w = z"
+ by (metis diff_add_cancel eqg power_minus_Bit0 that w(1))
+ then have "g z = 0"
+ using \<open>- g z = g w\<close> by auto
+ with eqg [OF that] have "z = b"
+ by auto
+ with that \<open>b \<notin> S\<close> show False
+ by simp
+ qed
+ then have nz: "\<And>z. z \<in> S \<Longrightarrow> g z + g a \<noteq> 0"
+ by (metis \<open>0 < r\<close> add.commute add_diff_cancel_left' centre_in_ball diff_0)
+ let ?f = "\<lambda>z. (r/3) / (g z + g a) - (r/3) / (g a + g a)"
+ obtain h where holh: "h holomorphic_on S" and "h a = 0" and h01: "h ` S \<subseteq> ball 0 1" and "inj_on h S"
+ proof
+ show "?f holomorphic_on S"
+ by (intro holomorphic_intros holg nz)
+ have 3: "\<lbrakk>norm x \<le> 1/3; norm y \<le> 1/3\<rbrakk> \<Longrightarrow> norm(x - y) < 1" for x y::complex
+ using norm_triangle_ineq4 [of x y] by simp
+ have "norm ((r/3) / (g z + g a) - (r/3) / (g a + g a)) < 1" if "z \<in> S" for z
+ apply (rule 3)
+ unfolding norm_divide
+ using \<open>r > 0\<close> g_not_r [OF \<open>z \<in> S\<close>] g_not_r [OF \<open>a \<in> S\<close>]
+ by (simp_all add: field_split_simps dist_commute dist_norm)
+ then show "?f ` S \<subseteq> ball 0 1"
+ by auto
+ show "inj_on ?f S"
+ using \<open>r > 0\<close> eqg apply (clarsimp simp: inj_on_def)
+ by (metis diff_add_cancel)
+ qed auto
+ obtain k where holk: "k holomorphic_on (h ` S)"
+ and derk: "\<And>z. z \<in> S \<Longrightarrow> deriv h z * deriv k (h z) = 1"
+ and kh: "\<And>z. z \<in> S \<Longrightarrow> k(h z) = z"
+ using holomorphic_has_inverse [OF holh openS \<open>inj_on h S\<close>] by metis
+
+ have 1: "open (h ` S)"
+ by (simp add: \<open>inj_on h S\<close> holh openS open_mapping_thm3)
+ have 2: "connected (h ` S)"
+ by (simp add: connected_continuous_image \<open>connected S\<close> holh holomorphic_on_imp_continuous_on)
+ have 3: "0 \<in> h ` S"
+ using \<open>a \<in> S\<close> \<open>h a = 0\<close> by auto
+ have 4: "\<exists>g. g holomorphic_on h ` S \<and> (\<forall>z\<in>h ` S. f z = (g z)\<^sup>2)"
+ if holf: "f holomorphic_on h ` S" and nz: "\<And>z. z \<in> h ` S \<Longrightarrow> f z \<noteq> 0" "inj_on f (h ` S)" for f
+ proof -
+ obtain g where holg: "g holomorphic_on S" and eqg: "\<And>z. z \<in> S \<Longrightarrow> (f \<circ> h) z = (g z)\<^sup>2"
+ proof -
+ have "f \<circ> h holomorphic_on S"
+ by (simp add: holh holomorphic_on_compose holf)
+ moreover have "\<forall>z\<in>S. (f \<circ> h) z \<noteq> 0"
+ by (simp add: nz)
+ ultimately show thesis
+ using prev that by blast
+ qed
+ show ?thesis
+ proof (intro exI conjI)
+ show "g \<circ> k holomorphic_on h ` S"
+ proof -
+ have "k ` h ` S \<subseteq> S"
+ by (simp add: \<open>\<And>z. z \<in> S \<Longrightarrow> k (h z) = z\<close> image_subset_iff)
+ then show ?thesis
+ by (meson holg holk holomorphic_on_compose holomorphic_on_subset)
+ qed
+ show "\<forall>z\<in>h ` S. f z = ((g \<circ> k) z)\<^sup>2"
+ using eqg kh by auto
+ qed
+ qed
+ obtain f g where f: "f holomorphic_on h ` S" and g: "g holomorphic_on ball 0 1"
+ and gf: "\<forall>z\<in>h ` S. f z \<in> ball 0 1 \<and> g (f z) = z" and fg:"\<forall>z\<in>ball 0 1. g z \<in> h ` S \<and> f (g z) = z"
+ using biholomorphic_to_disc_aux [OF 1 2 3 h01 4] by blast
+ show ?thesis
+ proof (intro exI conjI)
+ show "f \<circ> h holomorphic_on S"
+ by (simp add: f holh holomorphic_on_compose)
+ show "k \<circ> g holomorphic_on ball 0 1"
+ by (metis holomorphic_on_subset image_subset_iff fg holk g holomorphic_on_compose)
+ qed (use fg gf kh in auto)
+qed
+
+lemma homeomorphic_to_disc:
+ assumes S: "S \<noteq> {}"
+ and prev: "S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))" (is "_ \<or> ?P")
+ shows "S homeomorphic ball (0::complex) 1"
+ using prev
+proof
+ assume "S = UNIV" then show ?thesis
+ using homeomorphic_ball01_UNIV homeomorphic_sym by blast
+next
+ assume ?P
+ then show ?thesis
+ unfolding homeomorphic_minimal
+ using holomorphic_on_imp_continuous_on by blast
+qed
+
+lemma homeomorphic_to_disc_imp_simply_connected:
+ assumes "S = {} \<or> S homeomorphic ball (0::complex) 1"
+ shows "simply_connected S"
+ using assms homeomorphic_simply_connected_eq convex_imp_simply_connected by auto
+
+end
+
+proposition
+ assumes "open S"
+ shows simply_connected_eq_winding_number_zero:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>g z. path g \<and> path_image g \<subseteq> S \<and>
+ pathfinish g = pathstart g \<and> (z \<notin> S)
+ \<longrightarrow> winding_number g z = 0)" (is "?wn0")
+ and simply_connected_eq_contour_integral_zero:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>g f. valid_path g \<and> path_image g \<subseteq> S \<and>
+ pathfinish g = pathstart g \<and> f holomorphic_on S
+ \<longrightarrow> (f has_contour_integral 0) g)" (is "?ci0")
+ and simply_connected_eq_global_primitive:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<longrightarrow>
+ (\<exists>h. \<forall>z. z \<in> S \<longrightarrow> (h has_field_derivative f z) (at z)))" (is "?gp")
+ and simply_connected_eq_holomorphic_log:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = exp(g z))))" (is "?log")
+ and simply_connected_eq_holomorphic_sqrt:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f. f holomorphic_on S \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. g holomorphic_on S \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)))" (is "?sqrt")
+ and simply_connected_eq_biholomorphic_to_disc:
+ "simply_connected S \<longleftrightarrow>
+ S = {} \<or> S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))" (is "?bih")
+ and simply_connected_eq_homeomorphic_to_disc:
+ "simply_connected S \<longleftrightarrow> S = {} \<or> S homeomorphic ball (0::complex) 1" (is "?disc")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?wn0 \<and> ?ci0 \<and> ?gp \<and> ?log \<and> ?sqrt \<and> ?bih \<and> ?disc"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<delta>; \<delta> \<Longrightarrow> \<zeta>; \<zeta> \<Longrightarrow> \<eta>; \<eta> \<Longrightarrow> \<theta>; \<theta> \<Longrightarrow> \<xi>; \<xi> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>) \<and> (\<alpha> \<longleftrightarrow> \<delta>) \<and> (\<alpha> \<longleftrightarrow> \<zeta>) \<and>
+ (\<alpha> \<longleftrightarrow> \<eta>) \<and> (\<alpha> \<longleftrightarrow> \<theta>) \<and> (\<alpha> \<longleftrightarrow> \<xi>)" for \<alpha> \<beta> \<gamma> \<delta> \<zeta> \<eta> \<theta> \<xi>
+ by blast
+ show ?thesis
+ apply (rule *)
+ using winding_number_zero apply metis
+ using contour_integral_zero apply metis
+ using global_primitive apply metis
+ using holomorphic_log apply metis
+ using holomorphic_sqrt apply simp
+ using biholomorphic_to_disc apply blast
+ using homeomorphic_to_disc apply blast
+ using homeomorphic_to_disc_imp_simply_connected apply blast
+ done
+qed
+ then show ?wn0 ?ci0 ?gp ?log ?sqrt ?bih ?disc
+ by safe
+qed
+
+corollary contractible_eq_simply_connected_2d:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> (contractible S \<longleftrightarrow> simply_connected S)"
+ apply safe
+ apply (simp add: contractible_imp_simply_connected)
+ using convex_imp_contractible homeomorphic_contractible_eq simply_connected_eq_homeomorphic_to_disc by auto
+
+subsection\<open>A further chain of equivalences about components of the complement of a simply connected set\<close>
+
+text\<open>(following 1.35 in Burckel'S book)\<close>
+
+context SC_Chain
+begin
+
+lemma frontier_properties:
+ assumes "simply_connected S"
+ shows "if bounded S then connected(frontier S)
+ else \<forall>C \<in> components(frontier S). \<not> bounded C"
+proof -
+ have "S = {} \<or> S homeomorphic ball (0::complex) 1"
+ using simply_connected_eq_homeomorphic_to_disc assms openS by blast
+ then show ?thesis
+ proof
+ assume "S = {}"
+ then have "bounded S"
+ by simp
+ with \<open>S = {}\<close> show ?thesis
+ by simp
+ next
+ assume S01: "S homeomorphic ball (0::complex) 1"
+ then obtain g f
+ where gim: "g ` S = ball 0 1" and fg: "\<And>x. x \<in> S \<Longrightarrow> f(g x) = x"
+ and fim: "f ` ball 0 1 = S" and gf: "\<And>y. cmod y < 1 \<Longrightarrow> g(f y) = y"
+ and contg: "continuous_on S g" and contf: "continuous_on (ball 0 1) f"
+ by (fastforce simp: homeomorphism_def homeomorphic_def)
+ define D where "D \<equiv> \<lambda>n. ball (0::complex) (1 - 1/(of_nat n + 2))"
+ define A where "A \<equiv> \<lambda>n. {z::complex. 1 - 1/(of_nat n + 2) < norm z \<and> norm z < 1}"
+ define X where "X \<equiv> \<lambda>n::nat. closure(f ` A n)"
+ have D01: "D n \<subseteq> ball 0 1" for n
+ by (simp add: D_def ball_subset_ball_iff)
+ have A01: "A n \<subseteq> ball 0 1" for n
+ by (auto simp: A_def)
+ have cloX: "closed(X n)" for n
+ by (simp add: X_def)
+ have Xsubclo: "X n \<subseteq> closure S" for n
+ unfolding X_def by (metis A01 closure_mono fim image_mono)
+ have connX: "connected(X n)" for n
+ unfolding X_def
+ apply (rule connected_imp_connected_closure)
+ apply (rule connected_continuous_image)
+ apply (simp add: continuous_on_subset [OF contf A01])
+ using connected_annulus [of _ "0::complex"] by (simp add: A_def)
+ have nestX: "X n \<subseteq> X m" if "m \<le> n" for m n
+ proof -
+ have "1 - 1 / (real m + 2) \<le> 1 - 1 / (real n + 2)"
+ using that by (auto simp: field_simps)
+ then show ?thesis
+ by (auto simp: X_def A_def intro!: closure_mono)
+ qed
+ have "closure S - S \<subseteq> (\<Inter>n. X n)"
+ proof
+ fix x
+ assume "x \<in> closure S - S"
+ then have "x \<in> closure S" "x \<notin> S" by auto
+ show "x \<in> (\<Inter>n. X n)"
+ proof
+ fix n
+ have "ball 0 1 = closure (D n) \<union> A n"
+ by (auto simp: D_def A_def le_less_trans)
+ with fim have Seq: "S = f ` (closure (D n)) \<union> f ` (A n)"
+ by (simp add: image_Un)
+ have "continuous_on (closure (D n)) f"
+ by (simp add: D_def cball_subset_ball_iff continuous_on_subset [OF contf])
+ moreover have "compact (closure (D n))"
+ by (simp add: D_def)
+ ultimately have clo_fim: "closed (f ` closure (D n))"
+ using compact_continuous_image compact_imp_closed by blast
+ have *: "(f ` cball 0 (1 - 1 / (real n + 2))) \<subseteq> S"
+ by (force simp: D_def Seq)
+ show "x \<in> X n"
+ using \<open>x \<in> closure S\<close> unfolding X_def Seq
+ using \<open>x \<notin> S\<close> * D_def clo_fim by auto
+ qed
+ qed
+ moreover have "(\<Inter>n. X n) \<subseteq> closure S - S"
+ proof -
+ have "(\<Inter>n. X n) \<subseteq> closure S"
+ proof -
+ have "(\<Inter>n. X n) \<subseteq> X 0"
+ by blast
+ also have "... \<subseteq> closure S"
+ apply (simp add: X_def fim [symmetric])
+ apply (rule closure_mono)
+ by (auto simp: A_def)
+ finally show "(\<Inter>n. X n) \<subseteq> closure S" .
+ qed
+ moreover have "(\<Inter>n. X n) \<inter> S \<subseteq> {}"
+ proof (clarify, clarsimp simp: X_def fim [symmetric])
+ fix x assume x [rule_format]: "\<forall>n. f x \<in> closure (f ` A n)" and "cmod x < 1"
+ then obtain n where n: "1 / (1 - norm x) < of_nat n"
+ using reals_Archimedean2 by blast
+ with \<open>cmod x < 1\<close> gr0I have XX: "1 / of_nat n < 1 - norm x" and "n > 0"
+ by (fastforce simp: field_split_simps algebra_simps)+
+ have "f x \<in> f ` (D n)"
+ using n \<open>cmod x < 1\<close> by (auto simp: field_split_simps algebra_simps D_def)
+ moreover have " f ` D n \<inter> closure (f ` A n) = {}"
+ proof -
+ have op_fDn: "open(f ` (D n))"
+ proof (rule invariance_of_domain)
+ show "continuous_on (D n) f"
+ by (rule continuous_on_subset [OF contf D01])
+ show "open (D n)"
+ by (simp add: D_def)
+ show "inj_on f (D n)"
+ unfolding inj_on_def using D01 by (metis gf mem_ball_0 subsetCE)
+ qed
+ have injf: "inj_on f (ball 0 1)"
+ by (metis mem_ball_0 inj_on_def gf)
+ have "D n \<union> A n \<subseteq> ball 0 1"
+ using D01 A01 by simp
+ moreover have "D n \<inter> A n = {}"
+ by (auto simp: D_def A_def)
+ ultimately have "f ` D n \<inter> f ` A n = {}"
+ by (metis A01 D01 image_is_empty inj_on_image_Int injf)
+ then show ?thesis
+ by (simp add: open_Int_closure_eq_empty [OF op_fDn])
+ qed
+ ultimately show False
+ using x [of n] by blast
+ qed
+ ultimately
+ show "(\<Inter>n. X n) \<subseteq> closure S - S"
+ using closure_subset disjoint_iff_not_equal by blast
+ qed
+ ultimately have "closure S - S = (\<Inter>n. X n)" by blast
+ then have frontierS: "frontier S = (\<Inter>n. X n)"
+ by (simp add: frontier_def openS interior_open)
+ show ?thesis
+ proof (cases "bounded S")
+ case True
+ have bouX: "bounded (X n)" for n
+ apply (simp add: X_def)
+ apply (rule bounded_closure)
+ by (metis A01 fim image_mono bounded_subset [OF True])
+ have compaX: "compact (X n)" for n
+ apply (simp add: compact_eq_bounded_closed bouX)
+ apply (auto simp: X_def)
+ done
+ have "connected (\<Inter>n. X n)"
+ by (metis nestX compaX connX connected_nest)
+ then show ?thesis
+ by (simp add: True \<open>frontier S = (\<Inter>n. X n)\<close>)
+ next
+ case False
+ have unboundedX: "\<not> bounded(X n)" for n
+ proof
+ assume bXn: "bounded(X n)"
+ have "continuous_on (cball 0 (1 - 1 / (2 + real n))) f"
+ by (simp add: cball_subset_ball_iff continuous_on_subset [OF contf])
+ then have "bounded (f ` cball 0 (1 - 1 / (2 + real n)))"
+ by (simp add: compact_imp_bounded [OF compact_continuous_image])
+ moreover have "bounded (f ` A n)"
+ by (auto simp: X_def closure_subset image_subset_iff bounded_subset [OF bXn])
+ ultimately have "bounded (f ` (cball 0 (1 - 1/(2 + real n)) \<union> A n))"
+ by (simp add: image_Un)
+ then have "bounded (f ` ball 0 1)"
+ apply (rule bounded_subset)
+ apply (auto simp: A_def algebra_simps)
+ done
+ then show False
+ using False by (simp add: fim [symmetric])
+ qed
+ have clo_INTX: "closed(\<Inter>(range X))"
+ by (metis cloX closed_INT)
+ then have lcX: "locally compact (\<Inter>(range X))"
+ by (metis closed_imp_locally_compact)
+ have False if C: "C \<in> components (frontier S)" and boC: "bounded C" for C
+ proof -
+ have "closed C"
+ by (metis C closed_components frontier_closed)
+ then have "compact C"
+ by (metis boC compact_eq_bounded_closed)
+ have Cco: "C \<in> components (\<Inter>(range X))"
+ by (metis frontierS C)
+ obtain K where "C \<subseteq> K" "compact K"
+ and Ksub: "K \<subseteq> \<Inter>(range X)" and clo: "closed(\<Inter>(range X) - K)"
+ proof (cases "{k. C \<subseteq> k \<and> compact k \<and> openin (top_of_set (\<Inter>(range X))) k} = {}")
+ case True
+ then show ?thesis
+ using Sura_Bura [OF lcX Cco \<open>compact C\<close>] boC
+ by (simp add: True)
+ next
+ case False
+ then obtain L where "compact L" "C \<subseteq> L" and K: "openin (top_of_set (\<Inter>x. X x)) L"
+ by blast
+ show ?thesis
+ proof
+ show "L \<subseteq> \<Inter>(range X)"
+ by (metis K openin_imp_subset)
+ show "closed (\<Inter>(range X) - L)"
+ by (metis closedin_diff closedin_self closedin_closed_trans [OF _ clo_INTX] K)
+ qed (use \<open>compact L\<close> \<open>C \<subseteq> L\<close> in auto)
+ qed
+ obtain U V where "open U" and "compact (closure U)" and "open V" "K \<subseteq> U"
+ and V: "\<Inter>(range X) - K \<subseteq> V" and "U \<inter> V = {}"
+ using separation_normal_compact [OF \<open>compact K\<close> clo] by blast
+ then have "U \<inter> (\<Inter> (range X) - K) = {}"
+ by blast
+ have "(closure U - U) \<inter> (\<Inter>n. X n \<inter> closure U) \<noteq> {}"
+ proof (rule compact_imp_fip)
+ show "compact (closure U - U)"
+ by (metis \<open>compact (closure U)\<close> \<open>open U\<close> compact_diff)
+ show "\<And>T. T \<in> range (\<lambda>n. X n \<inter> closure U) \<Longrightarrow> closed T"
+ by clarify (metis cloX closed_Int closed_closure)
+ show "(closure U - U) \<inter> \<Inter>\<F> \<noteq> {}"
+ if "finite \<F>" and \<F>: "\<F> \<subseteq> range (\<lambda>n. X n \<inter> closure U)" for \<F>
+ proof
+ assume empty: "(closure U - U) \<inter> \<Inter>\<F> = {}"
+ obtain J where "finite J" and J: "\<F> = (\<lambda>n. X n \<inter> closure U) ` J"
+ using finite_subset_image [OF \<open>finite \<F>\<close> \<F>] by auto
+ show False
+ proof (cases "J = {}")
+ case True
+ with J empty have "closed U"
+ by (simp add: closure_subset_eq)
+ have "C \<noteq> {}"
+ using C in_components_nonempty by blast
+ then have "U \<noteq> {}"
+ using \<open>K \<subseteq> U\<close> \<open>C \<subseteq> K\<close> by blast
+ moreover have "U \<noteq> UNIV"
+ using \<open>compact (closure U)\<close> by auto
+ ultimately show False
+ using \<open>open U\<close> \<open>closed U\<close> clopen by blast
+ next
+ case False
+ define j where "j \<equiv> Max J"
+ have "j \<in> J"
+ by (simp add: False \<open>finite J\<close> j_def)
+ have jmax: "\<And>m. m \<in> J \<Longrightarrow> m \<le> j"
+ by (simp add: j_def \<open>finite J\<close>)
+ have "\<Inter> ((\<lambda>n. X n \<inter> closure U) ` J) = X j \<inter> closure U"
+ using False jmax nestX \<open>j \<in> J\<close> by auto
+ then have "X j \<inter> closure U = X j \<inter> U"
+ apply safe
+ using DiffI J empty apply auto[1]
+ using closure_subset by blast
+ then have "openin (top_of_set (X j)) (X j \<inter> closure U)"
+ by (simp add: openin_open_Int \<open>open U\<close>)
+ moreover have "closedin (top_of_set (X j)) (X j \<inter> closure U)"
+ by (simp add: closedin_closed_Int)
+ moreover have "X j \<inter> closure U \<noteq> X j"
+ by (metis unboundedX \<open>compact (closure U)\<close> bounded_subset compact_eq_bounded_closed inf.order_iff)
+ moreover have "X j \<inter> closure U \<noteq> {}"
+ proof -
+ have "C \<noteq> {}"
+ using C in_components_nonempty by blast
+ moreover have "C \<subseteq> X j \<inter> closure U"
+ using \<open>C \<subseteq> K\<close> \<open>K \<subseteq> U\<close> Ksub closure_subset by blast
+ ultimately show ?thesis by blast
+ qed
+ ultimately show False
+ using connX [of j] by (force simp: connected_clopen)
+ qed
+ qed
+ qed
+ moreover have "(\<Inter>n. X n \<inter> closure U) = (\<Inter>n. X n) \<inter> closure U"
+ by blast
+ moreover have "x \<in> U" if "\<And>n. x \<in> X n" "x \<in> closure U" for x
+ proof -
+ have "x \<notin> V"
+ using \<open>U \<inter> V = {}\<close> \<open>open V\<close> closure_iff_nhds_not_empty that(2) by blast
+ then show ?thesis
+ by (metis (no_types) Diff_iff INT_I V \<open>K \<subseteq> U\<close> contra_subsetD that(1))
+ qed
+ ultimately show False
+ by (auto simp: open_Int_closure_eq_empty [OF \<open>open V\<close>, of U])
+ qed
+ then show ?thesis
+ by (auto simp: False)
+ qed
+ qed
+qed
+
+
+lemma unbounded_complement_components:
+ assumes C: "C \<in> components (- S)" and S: "connected S"
+ and prev: "if bounded S then connected(frontier S)
+ else \<forall>C \<in> components(frontier S). \<not> bounded C"
+ shows "\<not> bounded C"
+proof (cases "bounded S")
+ case True
+ with prev have "S \<noteq> UNIV" and confr: "connected(frontier S)"
+ by auto
+ obtain w where C_ccsw: "C = connected_component_set (- S) w" and "w \<notin> S"
+ using C by (auto simp: components_def)
+ show ?thesis
+ proof (cases "S = {}")
+ case True with C show ?thesis by auto
+ next
+ case False
+ show ?thesis
+ proof
+ assume "bounded C"
+ then have "outside C \<noteq> {}"
+ using outside_bounded_nonempty by metis
+ then obtain z where z: "\<not> bounded (connected_component_set (- C) z)" and "z \<notin> C"
+ by (auto simp: outside_def)
+ have clo_ccs: "closed (connected_component_set (- S) x)" for x
+ by (simp add: closed_Compl closed_connected_component openS)
+ have "connected_component_set (- S) w = connected_component_set (- S) z"
+ proof (rule joinable_connected_component_eq [OF confr])
+ show "frontier S \<subseteq> - S"
+ using openS by (auto simp: frontier_def interior_open)
+ have False if "connected_component_set (- S) w \<inter> frontier (- S) = {}"
+ proof -
+ have "C \<inter> frontier S = {}"
+ using that by (simp add: C_ccsw)
+ then show False
+ by (metis C_ccsw ComplI Compl_eq_Compl_iff Diff_subset False \<open>w \<notin> S\<close> clo_ccs closure_closed compl_bot_eq connected_component_eq_UNIV connected_component_eq_empty empty_subsetI frontier_complement frontier_def frontier_not_empty frontier_of_connected_component_subset le_inf_iff subset_antisym)
+ qed
+ then show "connected_component_set (- S) w \<inter> frontier S \<noteq> {}"
+ by auto
+ have *: "\<lbrakk>frontier C \<subseteq> C; frontier C \<subseteq> F; frontier C \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> F \<noteq> {}" for C F::"complex set"
+ by blast
+ have "connected_component_set (- S) z \<inter> frontier (- S) \<noteq> {}"
+ proof (rule *)
+ show "frontier (connected_component_set (- S) z) \<subseteq> connected_component_set (- S) z"
+ by (auto simp: closed_Compl closed_connected_component frontier_def openS)
+ show "frontier (connected_component_set (- S) z) \<subseteq> frontier (- S)"
+ using frontier_of_connected_component_subset by fastforce
+ have "\<not> bounded (-S)"
+ by (simp add: True cobounded_imp_unbounded)
+ then have "connected_component_set (- S) z \<noteq> {}"
+ apply (simp only: connected_component_eq_empty)
+ using confr openS \<open>bounded C\<close> \<open>w \<notin> S\<close>
+ apply (simp add: frontier_def interior_open C_ccsw)
+ by (metis ComplI Compl_eq_Diff_UNIV connected_UNIV closed_closure closure_subset connected_component_eq_self
+ connected_diff_open_from_closed subset_UNIV)
+ then show "frontier (connected_component_set (- S) z) \<noteq> {}"
+ apply (simp add: frontier_eq_empty connected_component_eq_UNIV)
+ apply (metis False compl_top_eq double_compl)
+ done
+ qed
+ then show "connected_component_set (- S) z \<inter> frontier S \<noteq> {}"
+ by auto
+ qed
+ then show False
+ by (metis C_ccsw Compl_iff \<open>w \<notin> S\<close> \<open>z \<notin> C\<close> connected_component_eq_empty connected_component_idemp)
+ qed
+ qed
+next
+ case False
+ obtain w where C_ccsw: "C = connected_component_set (- S) w" and "w \<notin> S"
+ using C by (auto simp: components_def)
+ have "frontier (connected_component_set (- S) w) \<subseteq> connected_component_set (- S) w"
+ by (simp add: closed_Compl closed_connected_component frontier_subset_eq openS)
+ moreover have "frontier (connected_component_set (- S) w) \<subseteq> frontier S"
+ using frontier_complement frontier_of_connected_component_subset by blast
+ moreover have "frontier (connected_component_set (- S) w) \<noteq> {}"
+ by (metis C C_ccsw False bounded_empty compl_top_eq connected_component_eq_UNIV double_compl frontier_not_empty in_components_nonempty)
+ ultimately obtain z where zin: "z \<in> frontier S" and z: "z \<in> connected_component_set (- S) w"
+ by blast
+ have *: "connected_component_set (frontier S) z \<in> components(frontier S)"
+ by (simp add: \<open>z \<in> frontier S\<close> componentsI)
+ with prev False have "\<not> bounded (connected_component_set (frontier S) z)"
+ by simp
+ moreover have "connected_component (- S) w = connected_component (- S) z"
+ using connected_component_eq [OF z] by force
+ ultimately show ?thesis
+ by (metis C_ccsw * zin bounded_subset closed_Compl closure_closed connected_component_maximal
+ connected_component_refl connected_connected_component frontier_closures in_components_subset le_inf_iff mem_Collect_eq openS)
+qed
+
+lemma empty_inside:
+ assumes "connected S" "\<And>C. C \<in> components (- S) \<Longrightarrow> \<not> bounded C"
+ shows "inside S = {}"
+ using assms by (auto simp: components_def inside_def)
+
+lemma empty_inside_imp_simply_connected:
+ "\<lbrakk>connected S; inside S = {}\<rbrakk> \<Longrightarrow> simply_connected S"
+ by (metis ComplI inside_Un_outside openS outside_mono simply_connected_eq_winding_number_zero subsetCE sup_bot.left_neutral winding_number_zero_in_outside)
+
+end
+
+proposition
+ fixes S :: "complex set"
+ assumes "open S"
+ shows simply_connected_eq_frontier_properties:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (if bounded S then connected(frontier S)
+ else (\<forall>C \<in> components(frontier S). \<not>bounded C))" (is "?fp")
+ and simply_connected_eq_unbounded_complement_components:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and> (\<forall>C \<in> components(- S). \<not>bounded C)" (is "?ucc")
+ and simply_connected_eq_empty_inside:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and> inside S = {}" (is "?ei")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?fp \<and> ?ucc \<and> ?ei"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<delta>; \<delta> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>) \<and> (\<alpha> \<longleftrightarrow> \<delta>)" for \<alpha> \<beta> \<gamma> \<delta>
+ by blast
+ show ?thesis
+ apply (rule *)
+ using frontier_properties simply_connected_imp_connected apply blast
+apply clarify
+ using unbounded_complement_components simply_connected_imp_connected apply blast
+ using empty_inside apply blast
+ using empty_inside_imp_simply_connected apply blast
+ done
+qed
+ then show ?fp ?ucc ?ei
+ by safe
+qed
+
+lemma simply_connected_iff_simple:
+ fixes S :: "complex set"
+ assumes "open S" "bounded S"
+ shows "simply_connected S \<longleftrightarrow> connected S \<and> connected(- S)"
+ apply (simp add: simply_connected_eq_unbounded_complement_components assms, safe)
+ apply (metis DIM_complex assms(2) cobounded_has_bounded_component double_compl order_refl)
+ by (meson assms inside_bounded_complement_connected_empty simply_connected_eq_empty_inside simply_connected_eq_unbounded_complement_components)
+
+subsection\<open>Further equivalences based on continuous logs and sqrts\<close>
+
+context SC_Chain
+begin
+
+lemma continuous_log:
+ fixes f :: "complex\<Rightarrow>complex"
+ assumes S: "simply_connected S"
+ and contf: "continuous_on S f" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ shows "\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp(g z))"
+proof -
+ consider "S = {}" | "S homeomorphic ball (0::complex) 1"
+ using simply_connected_eq_homeomorphic_to_disc [OF openS] S by metis
+ then show ?thesis
+ proof cases
+ case 1 then show ?thesis
+ by simp
+ next
+ case 2
+ then obtain h k :: "complex\<Rightarrow>complex"
+ where kh: "\<And>x. x \<in> S \<Longrightarrow> k(h x) = x" and him: "h ` S = ball 0 1"
+ and conth: "continuous_on S h"
+ and hk: "\<And>y. y \<in> ball 0 1 \<Longrightarrow> h(k y) = y" and kim: "k ` ball 0 1 = S"
+ and contk: "continuous_on (ball 0 1) k"
+ unfolding homeomorphism_def homeomorphic_def by metis
+ obtain g where contg: "continuous_on (ball 0 1) g"
+ and expg: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> (f \<circ> k) z = exp (g z)"
+ proof (rule continuous_logarithm_on_ball)
+ show "continuous_on (ball 0 1) (f \<circ> k)"
+ apply (rule continuous_on_compose [OF contk])
+ using kim continuous_on_subset [OF contf]
+ by blast
+ show "\<And>z. z \<in> ball 0 1 \<Longrightarrow> (f \<circ> k) z \<noteq> 0"
+ using kim nz by auto
+ qed auto
+ then show ?thesis
+ by (metis comp_apply conth continuous_on_compose him imageI kh)
+ qed
+qed
+
+lemma continuous_sqrt:
+ fixes f :: "complex\<Rightarrow>complex"
+ assumes contf: "continuous_on S f" and nz: "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"
+ and prev: "\<And>f::complex\<Rightarrow>complex.
+ \<lbrakk>continuous_on S f; \<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0\<rbrakk>
+ \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp(g z))"
+ shows "\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+proof -
+ obtain g where contg: "continuous_on S g" and geq: "\<And>z. z \<in> S \<Longrightarrow> f z = exp(g z)"
+ using contf nz prev by metis
+ show ?thesis
+proof (intro exI ballI conjI)
+ show "continuous_on S (\<lambda>z. exp(g z/2))"
+ by (intro continuous_intros) (auto simp: contg)
+ show "\<And>z. z \<in> S \<Longrightarrow> f z = (exp (g z/2))\<^sup>2"
+ by (metis (no_types, lifting) divide_inverse exp_double geq mult.left_commute mult.right_neutral right_inverse zero_neq_numeral)
+ qed
+qed
+
+lemma continuous_sqrt_imp_simply_connected:
+ assumes "connected S"
+ and prev: "\<And>f::complex\<Rightarrow>complex. \<lbrakk>continuous_on S f; \<forall>z \<in> S. f z \<noteq> 0\<rbrakk>
+ \<Longrightarrow> \<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)"
+ shows "simply_connected S"
+proof (clarsimp simp add: simply_connected_eq_holomorphic_sqrt [OF openS] \<open>connected S\<close>)
+ fix f
+ assume "f holomorphic_on S" and nz: "\<forall>z\<in>S. f z \<noteq> 0"
+ then obtain g where contg: "continuous_on S g" and geq: "\<And>z. z \<in> S \<Longrightarrow> f z = (g z)\<^sup>2"
+ by (metis holomorphic_on_imp_continuous_on prev)
+ show "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
+ proof (intro exI ballI conjI)
+ show "g holomorphic_on S"
+ proof (clarsimp simp add: holomorphic_on_open [OF openS])
+ fix z
+ assume "z \<in> S"
+ with nz geq have "g z \<noteq> 0"
+ by auto
+ obtain \<delta> where "0 < \<delta>" "\<And>w. \<lbrakk>w \<in> S; dist w z < \<delta>\<rbrakk> \<Longrightarrow> dist (g w) (g z) < cmod (g z)"
+ using contg [unfolded continuous_on_iff] by (metis \<open>g z \<noteq> 0\<close> \<open>z \<in> S\<close> zero_less_norm_iff)
+ then have \<delta>: "\<And>w. \<lbrakk>w \<in> S; w \<in> ball z \<delta>\<rbrakk> \<Longrightarrow> g w + g z \<noteq> 0"
+ apply (clarsimp simp: dist_norm)
+ by (metis \<open>g z \<noteq> 0\<close> add_diff_cancel_left' diff_0_right norm_eq_zero norm_increases_online norm_minus_commute norm_not_less_zero not_less_iff_gr_or_eq)
+ have *: "(\<lambda>x. (f x - f z) / (x - z) / (g x + g z)) \<midarrow>z\<rightarrow> deriv f z / (g z + g z)"
+ apply (intro tendsto_intros)
+ using SC_Chain.openS SC_Chain_axioms \<open>f holomorphic_on S\<close> \<open>z \<in> S\<close> has_field_derivativeD holomorphic_derivI apply fastforce
+ using \<open>z \<in> S\<close> contg continuous_on_eq_continuous_at isCont_def openS apply blast
+ by (simp add: \<open>g z \<noteq> 0\<close>)
+ then have "(g has_field_derivative deriv f z / (g z + g z)) (at z)"
+ unfolding has_field_derivative_iff
+ proof (rule Lim_transform_within_open)
+ show "open (ball z \<delta> \<inter> S)"
+ by (simp add: openS open_Int)
+ show "z \<in> ball z \<delta> \<inter> S"
+ using \<open>z \<in> S\<close> \<open>0 < \<delta>\<close> by simp
+ show "\<And>x. \<lbrakk>x \<in> ball z \<delta> \<inter> S; x \<noteq> z\<rbrakk>
+ \<Longrightarrow> (f x - f z) / (x - z) / (g x + g z) = (g x - g z) / (x - z)"
+ using \<delta>
+ apply (simp add: geq \<open>z \<in> S\<close> divide_simps)
+ apply (auto simp: algebra_simps power2_eq_square)
+ done
+ qed
+ then show "\<exists>f'. (g has_field_derivative f') (at z)" ..
+ qed
+ qed (use geq in auto)
+qed
+
+end
+
+proposition
+ fixes S :: "complex set"
+ assumes "open S"
+ shows simply_connected_eq_continuous_log:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f::complex\<Rightarrow>complex. continuous_on S f \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = exp (g z))))" (is "?log")
+ and simply_connected_eq_continuous_sqrt:
+ "simply_connected S \<longleftrightarrow>
+ connected S \<and>
+ (\<forall>f::complex\<Rightarrow>complex. continuous_on S f \<and> (\<forall>z \<in> S. f z \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>z \<in> S. f z = (g z)\<^sup>2)))" (is "?sqrt")
+proof -
+ interpret SC_Chain
+ using assms by (simp add: SC_Chain_def)
+ have "?log \<and> ?sqrt"
+proof -
+ have *: "\<lbrakk>\<alpha> \<Longrightarrow> \<beta>; \<beta> \<Longrightarrow> \<gamma>; \<gamma> \<Longrightarrow> \<alpha>\<rbrakk>
+ \<Longrightarrow> (\<alpha> \<longleftrightarrow> \<beta>) \<and> (\<alpha> \<longleftrightarrow> \<gamma>)" for \<alpha> \<beta> \<gamma>
+ by blast
+ show ?thesis
+ apply (rule *)
+ apply (simp add: local.continuous_log winding_number_zero)
+ apply (simp add: continuous_sqrt)
+ apply (simp add: continuous_sqrt_imp_simply_connected)
+ done
+qed
+ then show ?log ?sqrt
+ by safe
+qed
+
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>More Borsukian results\<close>
+
+lemma Borsukian_componentwise_eq:
+ fixes S :: "'a::euclidean_space set"
+ assumes S: "locally connected S \<or> compact S"
+ shows "Borsukian S \<longleftrightarrow> (\<forall>C \<in> components S. Borsukian C)"
+proof -
+ have *: "ANR(-{0::complex})"
+ by (simp add: ANR_delete open_Compl open_imp_ANR)
+ show ?thesis
+ using cohomotopically_trivial_on_components [OF assms *] by (auto simp: Borsukian_alt)
+qed
+
+lemma Borsukian_componentwise:
+ fixes S :: "'a::euclidean_space set"
+ assumes "locally connected S \<or> compact S" "\<And>C. C \<in> components S \<Longrightarrow> Borsukian C"
+ shows "Borsukian S"
+ by (metis Borsukian_componentwise_eq assms)
+
+lemma simply_connected_eq_Borsukian:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> (simply_connected S \<longleftrightarrow> connected S \<and> Borsukian S)"
+ by (auto simp: simply_connected_eq_continuous_log Borsukian_continuous_logarithm)
+
+lemma Borsukian_eq_simply_connected:
+ fixes S :: "complex set"
+ shows "open S \<Longrightarrow> Borsukian S \<longleftrightarrow> (\<forall>C \<in> components S. simply_connected C)"
+apply (auto simp: Borsukian_componentwise_eq open_imp_locally_connected)
+ using in_components_connected open_components simply_connected_eq_Borsukian apply blast
+ using open_components simply_connected_eq_Borsukian by blast
+
+lemma Borsukian_separation_open_closed:
+ fixes S :: "complex set"
+ assumes S: "open S \<or> closed S" and "bounded S"
+ shows "Borsukian S \<longleftrightarrow> connected(- S)"
+ using S
+proof
+ assume "open S"
+ show ?thesis
+ unfolding Borsukian_eq_simply_connected [OF \<open>open S\<close>]
+ by (meson \<open>open S\<close> \<open>bounded S\<close> bounded_subset in_components_connected in_components_subset nonseparation_by_component_eq open_components simply_connected_iff_simple)
+next
+ assume "closed S"
+ with \<open>bounded S\<close> show ?thesis
+ by (simp add: Borsukian_separation_compact compact_eq_bounded_closed)
+qed
+
+
+subsection\<open>Finally, the Riemann Mapping Theorem\<close>
+
+theorem Riemann_mapping_theorem:
+ "open S \<and> simply_connected S \<longleftrightarrow>
+ S = {} \<or> S = UNIV \<or>
+ (\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and>
+ (\<forall>z \<in> S. f z \<in> ball 0 1 \<and> g(f z) = z) \<and>
+ (\<forall>z \<in> ball 0 1. g z \<in> S \<and> f(g z) = z))"
+ (is "_ = ?rhs")
+proof -
+ have "simply_connected S \<longleftrightarrow> ?rhs" if "open S"
+ by (simp add: simply_connected_eq_biholomorphic_to_disc that)
+ moreover have "open S" if "?rhs"
+ proof -
+ { fix f g
+ assume g: "g holomorphic_on ball 0 1" "\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z"
+ and "\<forall>z\<in>S. cmod (f z) < 1 \<and> g (f z) = z"
+ then have "S = g ` (ball 0 1)"
+ by (force simp:)
+ then have "open S"
+ by (metis open_ball g inj_on_def open_mapping_thm3)
+ }
+ with that show "open S" by auto
+ qed
+ ultimately show ?thesis by metis
+qed
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Winding_Numbers.thy Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,1153 @@
+section \<open>Winding Numbers\<close>
+
+text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2017)\<close>
+
+theory Winding_Numbers
+imports
+ Riemann_Mapping
+begin
+
+lemma simply_connected_inside_simple_path:
+ fixes p :: "real \<Rightarrow> complex"
+ shows "simple_path p \<Longrightarrow> simply_connected(inside(path_image p))"
+ using Jordan_inside_outside connected_simple_path_image inside_simple_curve_imp_closed simply_connected_eq_frontier_properties
+ by fastforce
+
+lemma simply_connected_Int:
+ fixes S :: "complex set"
+ assumes "open S" "open T" "simply_connected S" "simply_connected T" "connected (S \<inter> T)"
+ shows "simply_connected (S \<inter> T)"
+ using assms by (force simp: simply_connected_eq_winding_number_zero open_Int)
+
+subsection\<open>Winding number for a triangle\<close>
+
+lemma wn_triangle1:
+ assumes "0 \<in> interior(convex hull {a,b,c})"
+ shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))"
+proof -
+ { assume 0: "Im(a/b) \<le> 0" "0 \<le> Im(b/c)"
+ have "0 \<notin> interior (convex hull {a,b,c})"
+ proof (cases "a=0 \<or> b=0 \<or> c=0")
+ case True then show ?thesis
+ by (auto simp: not_in_interior_convex_hull_3)
+ next
+ case False
+ then have "b \<noteq> 0" by blast
+ { fix x y::complex and u::real
+ assume eq_f': "Im x * Re b \<le> Im b * Re x" "Im y * Re b \<le> Im b * Re y" "0 \<le> u" "u \<le> 1"
+ then have "((1 - u) * Im x) * Re b \<le> Im b * ((1 - u) * Re x)"
+ by (simp add: mult_left_mono mult.assoc mult.left_commute [of "Im b"])
+ then have "((1 - u) * Im x + u * Im y) * Re b \<le> Im b * ((1 - u) * Re x + u * Re y)"
+ using eq_f' ordered_comm_semiring_class.comm_mult_left_mono
+ by (fastforce simp add: algebra_simps)
+ }
+ with False 0 have "convex hull {a,b,c} \<le> {z. Im z * Re b \<le> Im b * Re z}"
+ apply (simp add: Complex.Im_divide divide_simps complex_neq_0 [symmetric])
+ apply (simp add: algebra_simps)
+ apply (rule hull_minimal)
+ apply (auto simp: algebra_simps convex_alt)
+ done
+ moreover have "0 \<notin> interior({z. Im z * Re b \<le> Im b * Re z})"
+ proof
+ assume "0 \<in> interior {z. Im z * Re b \<le> Im b * Re z}"
+ then obtain e where "e>0" and e: "ball 0 e \<subseteq> {z. Im z * Re b \<le> Im b * Re z}"
+ by (meson mem_interior)
+ define z where "z \<equiv> - sgn (Im b) * (e/3) + sgn (Re b) * (e/3) * \<i>"
+ have "z \<in> ball 0 e"
+ using \<open>e>0\<close>
+ apply (simp add: z_def dist_norm)
+ apply (rule le_less_trans [OF norm_triangle_ineq4])
+ apply (simp add: norm_mult abs_sgn_eq)
+ done
+ then have "z \<in> {z. Im z * Re b \<le> Im b * Re z}"
+ using e by blast
+ then show False
+ using \<open>e>0\<close> \<open>b \<noteq> 0\<close>
+ apply (simp add: z_def dist_norm sgn_if less_eq_real_def mult_less_0_iff complex.expand split: if_split_asm)
+ apply (auto simp: algebra_simps)
+ apply (meson less_asym less_trans mult_pos_pos neg_less_0_iff_less)
+ by (metis less_asym mult_pos_pos neg_less_0_iff_less)
+ qed
+ ultimately show ?thesis
+ using interior_mono by blast
+ qed
+ } with assms show ?thesis by blast
+qed
+
+lemma wn_triangle2_0:
+ assumes "0 \<in> interior(convex hull {a,b,c})"
+ shows
+ "0 < Im((b - a) * cnj (b)) \<and>
+ 0 < Im((c - b) * cnj (c)) \<and>
+ 0 < Im((a - c) * cnj (a))
+ \<or>
+ Im((b - a) * cnj (b)) < 0 \<and>
+ 0 < Im((b - c) * cnj (b)) \<and>
+ 0 < Im((a - b) * cnj (a)) \<and>
+ 0 < Im((c - a) * cnj (c))"
+proof -
+ have [simp]: "{b,c,a} = {a,b,c}" "{c,a,b} = {a,b,c}" by auto
+ show ?thesis
+ using wn_triangle1 [OF assms] wn_triangle1 [of b c a] wn_triangle1 [of c a b] assms
+ by (auto simp: algebra_simps Im_complex_div_gt_0 Im_complex_div_lt_0 not_le not_less)
+qed
+
+lemma wn_triangle2:
+ assumes "z \<in> interior(convex hull {a,b,c})"
+ shows "0 < Im((b - a) * cnj (b - z)) \<and>
+ 0 < Im((c - b) * cnj (c - z)) \<and>
+ 0 < Im((a - c) * cnj (a - z))
+ \<or>
+ Im((b - a) * cnj (b - z)) < 0 \<and>
+ 0 < Im((b - c) * cnj (b - z)) \<and>
+ 0 < Im((a - b) * cnj (a - z)) \<and>
+ 0 < Im((c - a) * cnj (c - z))"
+proof -
+ have 0: "0 \<in> interior(convex hull {a-z, b-z, c-z})"
+ using assms convex_hull_translation [of "-z" "{a,b,c}"]
+ interior_translation [of "-z"]
+ by (simp cong: image_cong_simp)
+ show ?thesis using wn_triangle2_0 [OF 0]
+ by simp
+qed
+
+lemma wn_triangle3:
+ assumes z: "z \<in> interior(convex hull {a,b,c})"
+ and "0 < Im((b-a) * cnj (b-z))"
+ "0 < Im((c-b) * cnj (c-z))"
+ "0 < Im((a-c) * cnj (a-z))"
+ shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1"
+proof -
+ have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
+ using z interior_of_triangle [of a b c]
+ by (auto simp: closed_segment_def)
+ have gt0: "0 < Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z)"
+ using assms
+ by (simp add: winding_number_linepath_pos_lt path_image_join winding_number_join_pos_combined)
+ have lt2: "Re (winding_number (linepath a b +++ linepath b c +++ linepath c a) z) < 2"
+ using winding_number_lt_half_linepath [of _ a b]
+ using winding_number_lt_half_linepath [of _ b c]
+ using winding_number_lt_half_linepath [of _ c a] znot
+ apply (fastforce simp add: winding_number_join path_image_join)
+ done
+ show ?thesis
+ by (rule winding_number_eq_1) (simp_all add: path_image_join gt0 lt2)
+qed
+
+proposition winding_number_triangle:
+ assumes z: "z \<in> interior(convex hull {a,b,c})"
+ shows "winding_number(linepath a b +++ linepath b c +++ linepath c a) z =
+ (if 0 < Im((b - a) * cnj (b - z)) then 1 else -1)"
+proof -
+ have [simp]: "{a,c,b} = {a,b,c}" by auto
+ have znot[simp]: "z \<notin> closed_segment a b" "z \<notin> closed_segment b c" "z \<notin> closed_segment c a"
+ using z interior_of_triangle [of a b c]
+ by (auto simp: closed_segment_def)
+ then have [simp]: "z \<notin> closed_segment b a" "z \<notin> closed_segment c b" "z \<notin> closed_segment a c"
+ using closed_segment_commute by blast+
+ have *: "winding_number (linepath a b +++ linepath b c +++ linepath c a) z =
+ winding_number (reversepath (linepath a c +++ linepath c b +++ linepath b a)) z"
+ by (simp add: reversepath_joinpaths winding_number_join not_in_path_image_join)
+ show ?thesis
+ using wn_triangle2 [OF z] apply (rule disjE)
+ apply (simp add: wn_triangle3 z)
+ apply (simp add: path_image_join winding_number_reversepath * wn_triangle3 z)
+ done
+qed
+
+subsection\<open>Winding numbers for simple closed paths\<close>
+
+lemma winding_number_from_innerpath:
+ assumes "simple_path c1" and c1: "pathstart c1 = a" "pathfinish c1 = b"
+ and "simple_path c2" and c2: "pathstart c2 = a" "pathfinish c2 = b"
+ and "simple_path c" and c: "pathstart c = a" "pathfinish c = b"
+ and c1c2: "path_image c1 \<inter> path_image c2 = {a,b}"
+ and c1c: "path_image c1 \<inter> path_image c = {a,b}"
+ and c2c: "path_image c2 \<inter> path_image c = {a,b}"
+ and ne_12: "path_image c \<inter> inside(path_image c1 \<union> path_image c2) \<noteq> {}"
+ and z: "z \<in> inside(path_image c1 \<union> path_image c)"
+ and wn_d: "winding_number (c1 +++ reversepath c) z = d"
+ and "a \<noteq> b" "d \<noteq> 0"
+ obtains "z \<in> inside(path_image c1 \<union> path_image c2)" "winding_number (c1 +++ reversepath c2) z = d"
+proof -
+ obtain 0: "inside(path_image c1 \<union> path_image c) \<inter> inside(path_image c2 \<union> path_image c) = {}"
+ and 1: "inside(path_image c1 \<union> path_image c) \<union> inside(path_image c2 \<union> path_image c) \<union>
+ (path_image c - {a,b}) = inside(path_image c1 \<union> path_image c2)"
+ by (rule split_inside_simple_closed_curve
+ [OF \<open>simple_path c1\<close> c1 \<open>simple_path c2\<close> c2 \<open>simple_path c\<close> c \<open>a \<noteq> b\<close> c1c2 c1c c2c ne_12])
+ have znot: "z \<notin> path_image c" "z \<notin> path_image c1" "z \<notin> path_image c2"
+ using union_with_outside z 1 by auto
+ have wn_cc2: "winding_number (c +++ reversepath c2) z = 0"
+ apply (rule winding_number_zero_in_outside)
+ apply (simp_all add: \<open>simple_path c2\<close> c c2 \<open>simple_path c\<close> simple_path_imp_path path_image_join)
+ by (metis "0" ComplI UnE disjoint_iff_not_equal sup.commute union_with_inside z znot)
+ show ?thesis
+ proof
+ show "z \<in> inside (path_image c1 \<union> path_image c2)"
+ using "1" z by blast
+ have "winding_number c1 z - winding_number c z = d "
+ using assms znot
+ by (metis wn_d winding_number_join simple_path_imp_path winding_number_reversepath add.commute path_image_reversepath path_reversepath pathstart_reversepath uminus_add_conv_diff)
+ then show "winding_number (c1 +++ reversepath c2) z = d"
+ using wn_cc2 by (simp add: winding_number_join simple_path_imp_path assms znot winding_number_reversepath)
+ qed
+qed
+
+lemma simple_closed_path_wn1:
+ fixes a::complex and e::real
+ assumes "0 < e"
+ and sp_pl: "simple_path(p +++ linepath (a - e) (a + e))"
+ and psp: "pathstart p = a + e"
+ and pfp: "pathfinish p = a - e"
+ and disj: "ball a e \<inter> path_image p = {}"
+obtains z where "z \<in> inside (path_image (p +++ linepath (a - e) (a + e)))"
+ "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1"
+proof -
+ have "arc p" and arc_lp: "arc (linepath (a - e) (a + e))"
+ and pap: "path_image p \<inter> path_image (linepath (a - e) (a + e)) \<subseteq> {pathstart p, a-e}"
+ using simple_path_join_loop_eq [of "linepath (a - e) (a + e)" p] assms by auto
+ have mid_eq_a: "midpoint (a - e) (a + e) = a"
+ by (simp add: midpoint_def)
+ then have "a \<in> path_image(p +++ linepath (a - e) (a + e))"
+ apply (simp add: assms path_image_join)
+ by (metis midpoint_in_closed_segment)
+ have "a \<in> frontier(inside (path_image(p +++ linepath (a - e) (a + e))))"
+ apply (simp add: assms Jordan_inside_outside)
+ apply (simp_all add: assms path_image_join)
+ by (metis mid_eq_a midpoint_in_closed_segment)
+ with \<open>0 < e\<close> obtain c where c: "c \<in> inside (path_image(p +++ linepath (a - e) (a + e)))"
+ and dac: "dist a c < e"
+ by (auto simp: frontier_straddle)
+ then have "c \<notin> path_image(p +++ linepath (a - e) (a + e))"
+ using inside_no_overlap by blast
+ then have "c \<notin> path_image p"
+ "c \<notin> closed_segment (a - of_real e) (a + of_real e)"
+ by (simp_all add: assms path_image_join)
+ with \<open>0 < e\<close> dac have "c \<notin> affine hull {a - of_real e, a + of_real e}"
+ by (simp add: segment_as_ball not_le)
+ with \<open>0 < e\<close> have *: "\<not> collinear {a - e, c,a + e}"
+ using collinear_3_affine_hull [of "a-e" "a+e"] by (auto simp: insert_commute)
+ have 13: "1/3 + 1/3 + 1/3 = (1::real)" by simp
+ have "(1/3) *\<^sub>R (a - of_real e) + (1/3) *\<^sub>R c + (1/3) *\<^sub>R (a + of_real e) \<in> interior(convex hull {a - e, c, a + e})"
+ using interior_convex_hull_3_minimal [OF * DIM_complex]
+ by clarsimp (metis 13 zero_less_divide_1_iff zero_less_numeral)
+ then obtain z where z: "z \<in> interior(convex hull {a - e, c, a + e})" by force
+ have [simp]: "z \<notin> closed_segment (a - e) c"
+ by (metis DIM_complex Diff_iff IntD2 inf_sup_absorb interior_of_triangle z)
+ have [simp]: "z \<notin> closed_segment (a + e) (a - e)"
+ by (metis DIM_complex DiffD2 Un_iff interior_of_triangle z)
+ have [simp]: "z \<notin> closed_segment c (a + e)"
+ by (metis (no_types, lifting) DIM_complex DiffD2 Un_insert_right inf_sup_aci(5) insertCI interior_of_triangle mk_disjoint_insert z)
+ show thesis
+ proof
+ have "norm (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z) = 1"
+ using winding_number_triangle [OF z] by simp
+ have zin: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union> path_image p)"
+ and zeq: "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
+ winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
+ proof (rule winding_number_from_innerpath
+ [of "linepath (a + e) (a - e)" "a+e" "a-e" p
+ "linepath (a + e) c +++ linepath c (a - e)" z
+ "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"])
+ show sp_aec: "simple_path (linepath (a + e) c +++ linepath c (a - e))"
+ proof (rule arc_imp_simple_path [OF arc_join])
+ show "arc (linepath (a + e) c)"
+ by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathstart_in_path_image psp)
+ show "arc (linepath c (a - e))"
+ by (metis \<open>c \<notin> path_image p\<close> arc_linepath pathfinish_in_path_image pfp)
+ show "path_image (linepath (a + e) c) \<inter> path_image (linepath c (a - e)) \<subseteq> {pathstart (linepath c (a - e))}"
+ by clarsimp (metis "*" IntI Int_closed_segment closed_segment_commute singleton_iff)
+ qed auto
+ show "simple_path p"
+ using \<open>arc p\<close> arc_simple_path by blast
+ show sp_ae2: "simple_path (linepath (a + e) (a - e))"
+ using \<open>arc p\<close> arc_distinct_ends pfp psp by fastforce
+ show pa: "pathfinish (linepath (a + e) (a - e)) = a - e"
+ "pathstart (linepath (a + e) c +++ linepath c (a - e)) = a + e"
+ "pathfinish (linepath (a + e) c +++ linepath c (a - e)) = a - e"
+ "pathstart p = a + e" "pathfinish p = a - e"
+ "pathstart (linepath (a + e) (a - e)) = a + e"
+ by (simp_all add: assms)
+ show 1: "path_image (linepath (a + e) (a - e)) \<inter> path_image p = {a + e, a - e}"
+ proof
+ show "path_image (linepath (a + e) (a - e)) \<inter> path_image p \<subseteq> {a + e, a - e}"
+ using pap closed_segment_commute psp segment_convex_hull by fastforce
+ show "{a + e, a - e} \<subseteq> path_image (linepath (a + e) (a - e)) \<inter> path_image p"
+ using pap pathfinish_in_path_image pathstart_in_path_image pfp psp by fastforce
+ qed
+ show 2: "path_image (linepath (a + e) (a - e)) \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) =
+ {a + e, a - e}" (is "?lhs = ?rhs")
+ proof
+ have "\<not> collinear {c, a + e, a - e}"
+ using * by (simp add: insert_commute)
+ then have "convex hull {a + e, a - e} \<inter> convex hull {a + e, c} = {a + e}"
+ "convex hull {a + e, a - e} \<inter> convex hull {c, a - e} = {a - e}"
+ by (metis (full_types) Int_closed_segment insert_commute segment_convex_hull)+
+ then show "?lhs \<subseteq> ?rhs"
+ by (metis Int_Un_distrib equalityD1 insert_is_Un path_image_join path_image_linepath path_join_eq path_linepath segment_convex_hull simple_path_def sp_aec)
+ show "?rhs \<subseteq> ?lhs"
+ using segment_convex_hull by (simp add: path_image_join)
+ qed
+ have "path_image p \<inter> path_image (linepath (a + e) c) \<subseteq> {a + e}"
+ proof (clarsimp simp: path_image_join)
+ fix x
+ assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment (a + e) c"
+ then have "dist x a \<ge> e"
+ by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
+ with x_ac dac \<open>e > 0\<close> show "x = a + e"
+ by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
+ qed
+ moreover
+ have "path_image p \<inter> path_image (linepath c (a - e)) \<subseteq> {a - e}"
+ proof (clarsimp simp: path_image_join)
+ fix x
+ assume "x \<in> path_image p" and x_ac: "x \<in> closed_segment c (a - e)"
+ then have "dist x a \<ge> e"
+ by (metis IntI all_not_in_conv disj dist_commute mem_ball not_less)
+ with x_ac dac \<open>e > 0\<close> show "x = a - e"
+ by (auto simp: norm_minus_commute dist_norm closed_segment_eq_open dest: open_segment_furthest_le [where y=a])
+ qed
+ ultimately
+ have "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) \<subseteq> {a + e, a - e}"
+ by (force simp: path_image_join)
+ then show 3: "path_image p \<inter> path_image (linepath (a + e) c +++ linepath c (a - e)) = {a + e, a - e}"
+ apply (rule equalityI)
+ apply (clarsimp simp: path_image_join)
+ apply (metis pathstart_in_path_image psp pathfinish_in_path_image pfp)
+ done
+ show 4: "path_image (linepath (a + e) c +++ linepath c (a - e)) \<inter>
+ inside (path_image (linepath (a + e) (a - e)) \<union> path_image p) \<noteq> {}"
+ apply (clarsimp simp: path_image_join segment_convex_hull disjoint_iff_not_equal)
+ by (metis (no_types, hide_lams) UnI1 Un_commute c closed_segment_commute ends_in_segment(1) path_image_join
+ path_image_linepath pathstart_linepath pfp segment_convex_hull)
+ show zin_inside: "z \<in> inside (path_image (linepath (a + e) (a - e)) \<union>
+ path_image (linepath (a + e) c +++ linepath c (a - e)))"
+ apply (simp add: path_image_join)
+ by (metis z inside_of_triangle DIM_complex Un_commute closed_segment_commute)
+ show 5: "winding_number
+ (linepath (a + e) (a - e) +++ reversepath (linepath (a + e) c +++ linepath c (a - e))) z =
+ winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
+ by (simp add: reversepath_joinpaths path_image_join winding_number_join)
+ show 6: "winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z \<noteq> 0"
+ by (simp add: winding_number_triangle z)
+ show "winding_number (linepath (a + e) (a - e) +++ reversepath p) z =
+ winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z"
+ by (metis 1 2 3 4 5 6 pa sp_aec sp_ae2 \<open>arc p\<close> \<open>simple_path p\<close> arc_distinct_ends winding_number_from_innerpath zin_inside)
+ qed (use assms \<open>e > 0\<close> in auto)
+ show "z \<in> inside (path_image (p +++ linepath (a - e) (a + e)))"
+ using zin by (simp add: assms path_image_join Un_commute closed_segment_commute)
+ then have "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) =
+ cmod ((winding_number(reversepath (p +++ linepath (a - e) (a + e))) z))"
+ apply (subst winding_number_reversepath)
+ using simple_path_imp_path sp_pl apply blast
+ apply (metis IntI emptyE inside_no_overlap)
+ by (simp add: inside_def)
+ also have "... = cmod (winding_number(linepath (a + e) (a - e) +++ reversepath p) z)"
+ by (simp add: pfp reversepath_joinpaths)
+ also have "... = cmod (winding_number (linepath (a - e) c +++ linepath c (a + e) +++ linepath (a + e) (a - e)) z)"
+ by (simp add: zeq)
+ also have "... = 1"
+ using z by (simp add: interior_of_triangle winding_number_triangle)
+ finally show "cmod (winding_number (p +++ linepath (a - e) (a + e)) z) = 1" .
+ qed
+qed
+
+lemma simple_closed_path_wn2:
+ fixes a::complex and d e::real
+ assumes "0 < d" "0 < e"
+ and sp_pl: "simple_path(p +++ linepath (a - d) (a + e))"
+ and psp: "pathstart p = a + e"
+ and pfp: "pathfinish p = a - d"
+obtains z where "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
+ "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
+proof -
+ have [simp]: "a + of_real x \<in> closed_segment (a - \<alpha>) (a - \<beta>) \<longleftrightarrow> x \<in> closed_segment (-\<alpha>) (-\<beta>)" for x \<alpha> \<beta>::real
+ using closed_segment_translation_eq [of a]
+ by (metis (no_types, hide_lams) add_uminus_conv_diff of_real_minus of_real_closed_segment)
+ have [simp]: "a - of_real x \<in> closed_segment (a + \<alpha>) (a + \<beta>) \<longleftrightarrow> -x \<in> closed_segment \<alpha> \<beta>" for x \<alpha> \<beta>::real
+ by (metis closed_segment_translation_eq diff_conv_add_uminus of_real_closed_segment of_real_minus)
+ have "arc p" and arc_lp: "arc (linepath (a - d) (a + e))" and "path p"
+ and pap: "path_image p \<inter> closed_segment (a - d) (a + e) \<subseteq> {a+e, a-d}"
+ using simple_path_join_loop_eq [of "linepath (a - d) (a + e)" p] assms arc_imp_path by auto
+ have "0 \<in> closed_segment (-d) e"
+ using \<open>0 < d\<close> \<open>0 < e\<close> closed_segment_eq_real_ivl by auto
+ then have "a \<in> path_image (linepath (a - d) (a + e))"
+ using of_real_closed_segment [THEN iffD2]
+ by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
+ then have "a \<notin> path_image p"
+ using \<open>0 < d\<close> \<open>0 < e\<close> pap by auto
+ then obtain k where "0 < k" and k: "ball a k \<inter> (path_image p) = {}"
+ using \<open>0 < e\<close> \<open>path p\<close> not_on_path_ball by blast
+ define kde where "kde \<equiv> (min k (min d e)) / 2"
+ have "0 < kde" "kde < k" "kde < d" "kde < e"
+ using \<open>0 < k\<close> \<open>0 < d\<close> \<open>0 < e\<close> by (auto simp: kde_def)
+ let ?q = "linepath (a + kde) (a + e) +++ p +++ linepath (a - d) (a - kde)"
+ have "- kde \<in> closed_segment (-d) e"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
+ then have a_diff_kde: "a - kde \<in> closed_segment (a - d) (a + e)"
+ using of_real_closed_segment [THEN iffD2]
+ by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
+ then have clsub2: "closed_segment (a - d) (a - kde) \<subseteq> closed_segment (a - d) (a + e)"
+ by (simp add: subset_closed_segment)
+ then have "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a + e, a - d}"
+ using pap by force
+ moreover
+ have "a + e \<notin> path_image p \<inter> closed_segment (a - d) (a - kde)"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
+ ultimately have sub_a_diff_d: "path_image p \<inter> closed_segment (a - d) (a - kde) \<subseteq> {a - d}"
+ by blast
+ have "kde \<in> closed_segment (-d) e"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
+ then have a_diff_kde: "a + kde \<in> closed_segment (a - d) (a + e)"
+ using of_real_closed_segment [THEN iffD2]
+ by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
+ then have clsub1: "closed_segment (a + kde) (a + e) \<subseteq> closed_segment (a - d) (a + e)"
+ by (simp add: subset_closed_segment)
+ then have "closed_segment (a + kde) (a + e) \<inter> path_image p \<subseteq> {a + e, a - d}"
+ using pap by force
+ moreover
+ have "closed_segment (a + kde) (a + e) \<inter> closed_segment (a - d) (a - kde) = {}"
+ proof (clarsimp intro!: equals0I)
+ fix y
+ assume y1: "y \<in> closed_segment (a + kde) (a + e)"
+ and y2: "y \<in> closed_segment (a - d) (a - kde)"
+ obtain u where u: "y = a + of_real u" and "0 < u"
+ using y1 \<open>0 < kde\<close> \<open>kde < e\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
+ apply (rule_tac u = "(1 - u)*kde + u*e" in that)
+ apply (auto simp: scaleR_conv_of_real algebra_simps)
+ by (meson le_less_trans less_add_same_cancel2 less_eq_real_def mult_left_mono)
+ moreover
+ obtain v where v: "y = a + of_real v" and "v \<le> 0"
+ using y2 \<open>0 < kde\<close> \<open>0 < d\<close> \<open>0 < e\<close> apply (clarsimp simp: in_segment)
+ apply (rule_tac v = "- ((1 - u)*d + u*kde)" in that)
+ apply (force simp: scaleR_conv_of_real algebra_simps)
+ by (meson less_eq_real_def neg_le_0_iff_le segment_bound_lemma)
+ ultimately show False
+ by auto
+ qed
+ moreover have "a - d \<notin> closed_segment (a + kde) (a + e)"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>0 < e\<close> by (auto simp: closed_segment_eq_real_ivl)
+ ultimately have sub_a_plus_e:
+ "closed_segment (a + kde) (a + e) \<inter> (path_image p \<union> closed_segment (a - d) (a - kde))
+ \<subseteq> {a + e}"
+ by auto
+ have "kde \<in> closed_segment (-kde) e"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
+ then have a_add_kde: "a + kde \<in> closed_segment (a - kde) (a + e)"
+ using of_real_closed_segment [THEN iffD2]
+ by (force dest: closed_segment_translation_eq [of "a", THEN iffD2] simp del: of_real_closed_segment)
+ have "closed_segment (a - kde) (a + kde) \<inter> closed_segment (a + kde) (a + e) = {a + kde}"
+ by (metis a_add_kde Int_closed_segment)
+ moreover
+ have "path_image p \<inter> closed_segment (a - kde) (a + kde) = {}"
+ proof (rule equals0I, clarify)
+ fix y assume "y \<in> path_image p" "y \<in> closed_segment (a - kde) (a + kde)"
+ with equals0D [OF k, of y] \<open>0 < kde\<close> \<open>kde < k\<close> show False
+ by (auto simp: dist_norm dest: dist_decreases_closed_segment [where c=a])
+ qed
+ moreover
+ have "- kde \<in> closed_segment (-d) kde"
+ using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < e\<close> closed_segment_eq_real_ivl by auto
+ then have a_diff_kde': "a - kde \<in> closed_segment (a - d) (a + kde)"
+ using of_real_closed_segment [THEN iffD2]
+ by (force dest: closed_segment_translation_eq [of a, THEN iffD2] simp del: of_real_closed_segment)
+ then have "closed_segment (a - d) (a - kde) \<inter> closed_segment (a - kde) (a + kde) = {a - kde}"
+ by (metis Int_closed_segment)
+ ultimately
+ have pa_subset_pm_kde: "path_image ?q \<inter> closed_segment (a - kde) (a + kde) \<subseteq> {a - kde, a + kde}"
+ by (auto simp: path_image_join assms)
+ have ge_kde1: "\<exists>y. x = a + y \<and> y \<ge> kde" if "x \<in> closed_segment (a + kde) (a + e)" for x
+ using that \<open>kde < e\<close> mult_le_cancel_left
+ apply (auto simp: in_segment)
+ apply (rule_tac x="(1-u)*kde + u*e" in exI)
+ apply (fastforce simp: algebra_simps scaleR_conv_of_real)
+ done
+ have ge_kde2: "\<exists>y. x = a + y \<and> y \<le> -kde" if "x \<in> closed_segment (a - d) (a - kde)" for x
+ using that \<open>kde < d\<close> affine_ineq
+ apply (auto simp: in_segment)
+ apply (rule_tac x="- ((1-u)*d + u*kde)" in exI)
+ apply (fastforce simp: algebra_simps scaleR_conv_of_real)
+ done
+ have notin_paq: "x \<notin> path_image ?q" if "dist a x < kde" for x
+ using that using \<open>0 < kde\<close> \<open>kde < d\<close> \<open>kde < k\<close>
+ apply (auto simp: path_image_join assms dist_norm dest!: ge_kde1 ge_kde2)
+ by (meson k disjoint_iff_not_equal le_less_trans less_eq_real_def mem_ball that)
+ obtain z where zin: "z \<in> inside (path_image (?q +++ linepath (a - kde) (a + kde)))"
+ and z1: "cmod (winding_number (?q +++ linepath (a - kde) (a + kde)) z) = 1"
+ proof (rule simple_closed_path_wn1 [of kde ?q a])
+ show "simple_path (?q +++ linepath (a - kde) (a + kde))"
+ proof (intro simple_path_join_loop conjI)
+ show "arc ?q"
+ proof (rule arc_join)
+ show "arc (linepath (a + kde) (a + e))"
+ using \<open>kde < e\<close> \<open>arc p\<close> by (force simp: pfp)
+ show "arc (p +++ linepath (a - d) (a - kde))"
+ using \<open>kde < d\<close> \<open>kde < e\<close> \<open>arc p\<close> sub_a_diff_d by (force simp: pfp intro: arc_join)
+ qed (auto simp: psp pfp path_image_join sub_a_plus_e)
+ show "arc (linepath (a - kde) (a + kde))"
+ using \<open>0 < kde\<close> by auto
+ qed (use pa_subset_pm_kde in auto)
+ qed (use \<open>0 < kde\<close> notin_paq in auto)
+ have eq: "path_image (?q +++ linepath (a - kde) (a + kde)) = path_image (p +++ linepath (a - d) (a + e))"
+ (is "?lhs = ?rhs")
+ proof
+ show "?lhs \<subseteq> ?rhs"
+ using clsub1 clsub2 apply (auto simp: path_image_join assms)
+ by (meson subsetCE subset_closed_segment)
+ show "?rhs \<subseteq> ?lhs"
+ apply (simp add: path_image_join assms Un_ac)
+ by (metis Un_closed_segment Un_assoc a_diff_kde a_diff_kde' le_supI2 subset_refl)
+ qed
+ show thesis
+ proof
+ show zzin: "z \<in> inside (path_image (p +++ linepath (a - d) (a + e)))"
+ by (metis eq zin)
+ then have znotin: "z \<notin> path_image p"
+ by (metis ComplD Un_iff inside_Un_outside path_image_join pathfinish_linepath pathstart_reversepath pfp reversepath_linepath)
+ have znotin_de: "z \<notin> closed_segment (a - d) (a + kde)"
+ by (metis ComplD Un_iff Un_closed_segment a_diff_kde inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
+ have "winding_number (linepath (a - d) (a + e)) z =
+ winding_number (linepath (a - d) (a + kde)) z + winding_number (linepath (a + kde) (a + e)) z"
+ apply (rule winding_number_split_linepath)
+ apply (simp add: a_diff_kde)
+ by (metis ComplD Un_iff inside_Un_outside path_image_join path_image_linepath pathstart_linepath pfp zzin)
+ also have "... = winding_number (linepath (a + kde) (a + e)) z +
+ (winding_number (linepath (a - d) (a - kde)) z +
+ winding_number (linepath (a - kde) (a + kde)) z)"
+ by (simp add: winding_number_split_linepath [of "a-kde", symmetric] znotin_de a_diff_kde')
+ finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
+ winding_number p z + winding_number (linepath (a + kde) (a + e)) z +
+ (winding_number (linepath (a - d) (a - kde)) z +
+ winding_number (linepath (a - kde) (a + kde)) z)"
+ by (metis (no_types, lifting) ComplD Un_iff \<open>arc p\<close> add.assoc arc_imp_path eq path_image_join path_join_path_ends path_linepath simple_path_imp_path sp_pl union_with_outside winding_number_join zin)
+ also have "... = winding_number ?q z + winding_number (linepath (a - kde) (a + kde)) z"
+ using \<open>path p\<close> znotin assms zzin clsub1
+ apply (subst winding_number_join, auto)
+ apply (metis (no_types, hide_lams) ComplD Un_iff contra_subsetD inside_Un_outside path_image_join path_image_linepath pathstart_linepath)
+ apply (metis Un_iff Un_closed_segment a_diff_kde' not_in_path_image_join path_image_linepath znotin_de)
+ by (metis Un_iff Un_closed_segment a_diff_kde' path_image_linepath path_linepath pathstart_linepath winding_number_join znotin_de)
+ also have "... = winding_number (?q +++ linepath (a - kde) (a + kde)) z"
+ using \<open>path p\<close> assms zin
+ apply (subst winding_number_join [symmetric], auto)
+ apply (metis ComplD Un_iff path_image_join pathfinish_join pathfinish_linepath pathstart_linepath union_with_outside)
+ by (metis Un_iff Un_closed_segment a_diff_kde' znotin_de)
+ finally have "winding_number (p +++ linepath (a - d) (a + e)) z =
+ winding_number (?q +++ linepath (a - kde) (a + kde)) z" .
+ then show "cmod (winding_number (p +++ linepath (a - d) (a + e)) z) = 1"
+ by (simp add: z1)
+ qed
+qed
+
+lemma simple_closed_path_wn3:
+ fixes p :: "real \<Rightarrow> complex"
+ assumes "simple_path p" and loop: "pathfinish p = pathstart p"
+ obtains z where "z \<in> inside (path_image p)" "cmod (winding_number p z) = 1"
+proof -
+ have ins: "inside(path_image p) \<noteq> {}" "open(inside(path_image p))"
+ "connected(inside(path_image p))"
+ and out: "outside(path_image p) \<noteq> {}" "open(outside(path_image p))"
+ "connected(outside(path_image p))"
+ and bo: "bounded(inside(path_image p))" "\<not> bounded(outside(path_image p))"
+ and ins_out: "inside(path_image p) \<inter> outside(path_image p) = {}"
+ "inside(path_image p) \<union> outside(path_image p) = - path_image p"
+ and fro: "frontier(inside(path_image p)) = path_image p"
+ "frontier(outside(path_image p)) = path_image p"
+ using Jordan_inside_outside [OF assms] by auto
+ obtain a where a: "a \<in> inside(path_image p)"
+ using \<open>inside (path_image p) \<noteq> {}\<close> by blast
+ obtain d::real where "0 < d" and d_fro: "a - d \<in> frontier (inside (path_image p))"
+ and d_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < d\<rbrakk> \<Longrightarrow> (a - \<epsilon>) \<in> inside (path_image p)"
+ apply (rule ray_to_frontier [of "inside (path_image p)" a "-1"])
+ using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
+ apply (auto simp: of_real_def)
+ done
+ obtain e::real where "0 < e" and e_fro: "a + e \<in> frontier (inside (path_image p))"
+ and e_int: "\<And>\<epsilon>. \<lbrakk>0 \<le> \<epsilon>; \<epsilon> < e\<rbrakk> \<Longrightarrow> (a + \<epsilon>) \<in> inside (path_image p)"
+ apply (rule ray_to_frontier [of "inside (path_image p)" a 1])
+ using \<open>bounded (inside (path_image p))\<close> \<open>open (inside (path_image p))\<close> a interior_eq
+ apply (auto simp: of_real_def)
+ done
+ obtain t0 where "0 \<le> t0" "t0 \<le> 1" and pt: "p t0 = a - d"
+ using a d_fro fro by (auto simp: path_image_def)
+ obtain q where "simple_path q" and q_ends: "pathstart q = a - d" "pathfinish q = a - d"
+ and q_eq_p: "path_image q = path_image p"
+ and wn_q_eq_wn_p: "\<And>z. z \<in> inside(path_image p) \<Longrightarrow> winding_number q z = winding_number p z"
+ proof
+ show "simple_path (shiftpath t0 p)"
+ by (simp add: pathstart_shiftpath pathfinish_shiftpath
+ simple_path_shiftpath path_image_shiftpath \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> assms)
+ show "pathstart (shiftpath t0 p) = a - d"
+ using pt by (simp add: \<open>t0 \<le> 1\<close> pathstart_shiftpath)
+ show "pathfinish (shiftpath t0 p) = a - d"
+ by (simp add: \<open>0 \<le> t0\<close> loop pathfinish_shiftpath pt)
+ show "path_image (shiftpath t0 p) = path_image p"
+ by (simp add: \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> loop path_image_shiftpath)
+ show "winding_number (shiftpath t0 p) z = winding_number p z"
+ if "z \<in> inside (path_image p)" for z
+ by (metis ComplD Un_iff \<open>0 \<le> t0\<close> \<open>t0 \<le> 1\<close> \<open>simple_path p\<close> atLeastAtMost_iff inside_Un_outside
+ loop simple_path_imp_path that winding_number_shiftpath)
+ qed
+ have ad_not_ae: "a - d \<noteq> a + e"
+ by (metis \<open>0 < d\<close> \<open>0 < e\<close> add.left_inverse add_left_cancel add_uminus_conv_diff
+ le_add_same_cancel2 less_eq_real_def not_less of_real_add of_real_def of_real_eq_0_iff pt)
+ have ad_ae_q: "{a - d, a + e} \<subseteq> path_image q"
+ using \<open>path_image q = path_image p\<close> d_fro e_fro fro(1) by auto
+ have ada: "open_segment (a - d) a \<subseteq> inside (path_image p)"
+ proof (clarsimp simp: in_segment)
+ fix u::real assume "0 < u" "u < 1"
+ with d_int have "a - (1 - u) * d \<in> inside (path_image p)"
+ by (metis \<open>0 < d\<close> add.commute diff_add_cancel left_diff_distrib' less_add_same_cancel2 less_eq_real_def mult.left_neutral zero_less_mult_iff)
+ then show "(1 - u) *\<^sub>R (a - d) + u *\<^sub>R a \<in> inside (path_image p)"
+ by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
+ qed
+ have aae: "open_segment a (a + e) \<subseteq> inside (path_image p)"
+ proof (clarsimp simp: in_segment)
+ fix u::real assume "0 < u" "u < 1"
+ with e_int have "a + u * e \<in> inside (path_image p)"
+ by (meson \<open>0 < e\<close> less_eq_real_def mult_less_cancel_right2 not_less zero_less_mult_iff)
+ then show "(1 - u) *\<^sub>R a + u *\<^sub>R (a + e) \<in> inside (path_image p)"
+ apply (simp add: algebra_simps)
+ by (simp add: diff_add_eq of_real_def real_vector.scale_right_diff_distrib)
+ qed
+ have "complex_of_real (d * d + (e * e + d * (e + e))) \<noteq> 0"
+ using ad_not_ae
+ by (metis \<open>0 < d\<close> \<open>0 < e\<close> add_strict_left_mono less_add_same_cancel1 not_sum_squares_lt_zero
+ of_real_eq_0_iff zero_less_double_add_iff_zero_less_single_add zero_less_mult_iff)
+ then have a_in_de: "a \<in> open_segment (a - d) (a + e)"
+ using ad_not_ae \<open>0 < d\<close> \<open>0 < e\<close>
+ apply (auto simp: in_segment algebra_simps scaleR_conv_of_real)
+ apply (rule_tac x="d / (d+e)" in exI)
+ apply (auto simp: field_simps)
+ done
+ then have "open_segment (a - d) (a + e) \<subseteq> inside (path_image p)"
+ using ada a aae Un_open_segment [of a "a-d" "a+e"] by blast
+ then have "path_image q \<inter> open_segment (a - d) (a + e) = {}"
+ using inside_no_overlap by (fastforce simp: q_eq_p)
+ with ad_ae_q have paq_Int_cs: "path_image q \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
+ by (simp add: closed_segment_eq_open)
+ obtain t where "0 \<le> t" "t \<le> 1" and qt: "q t = a + e"
+ using a e_fro fro ad_ae_q by (auto simp: path_defs)
+ then have "t \<noteq> 0"
+ by (metis ad_not_ae pathstart_def q_ends(1))
+ then have "t \<noteq> 1"
+ by (metis ad_not_ae pathfinish_def q_ends(2) qt)
+ have q01: "q 0 = a - d" "q 1 = a - d"
+ using q_ends by (auto simp: pathstart_def pathfinish_def)
+ obtain z where zin: "z \<in> inside (path_image (subpath t 0 q +++ linepath (a - d) (a + e)))"
+ and z1: "cmod (winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z) = 1"
+ proof (rule simple_closed_path_wn2 [of d e "subpath t 0 q" a], simp_all add: q01)
+ show "simple_path (subpath t 0 q +++ linepath (a - d) (a + e))"
+ proof (rule simple_path_join_loop, simp_all add: qt q01)
+ have "inj_on q (closed_segment t 0)"
+ using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close>
+ by (fastforce simp: simple_path_def inj_on_def closed_segment_eq_real_ivl)
+ then show "arc (subpath t 0 q)"
+ using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close>
+ by (simp add: arc_subpath_eq simple_path_imp_path)
+ show "arc (linepath (a - d) (a + e))"
+ by (simp add: ad_not_ae)
+ show "path_image (subpath t 0 q) \<inter> closed_segment (a - d) (a + e) \<subseteq> {a + e, a - d}"
+ using qt paq_Int_cs \<open>simple_path q\<close> \<open>0 \<le> t\<close> \<open>t \<le> 1\<close>
+ by (force simp: dest: rev_subsetD [OF _ path_image_subpath_subset] intro: simple_path_imp_path)
+ qed
+ qed (auto simp: \<open>0 < d\<close> \<open>0 < e\<close> qt)
+ have pa01_Un: "path_image (subpath 0 t q) \<union> path_image (subpath 1 t q) = path_image q"
+ unfolding path_image_subpath
+ using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> by (force simp: path_image_def image_iff)
+ with paq_Int_cs have pa_01q:
+ "(path_image (subpath 0 t q) \<union> path_image (subpath 1 t q)) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}"
+ by metis
+ have z_notin_ed: "z \<notin> closed_segment (a + e) (a - d)"
+ using zin q01 by (simp add: path_image_join closed_segment_commute inside_def)
+ have z_notin_0t: "z \<notin> path_image (subpath 0 t q)"
+ by (metis (no_types, hide_lams) IntI Un_upper1 subsetD empty_iff inside_no_overlap path_image_join
+ path_image_subpath_commute pathfinish_subpath pathstart_def pathstart_linepath q_ends(1) qt subpath_trivial zin)
+ have [simp]: "- winding_number (subpath t 0 q) z = winding_number (subpath 0 t q) z"
+ by (metis \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> atLeastAtMost_iff zero_le_one
+ path_image_subpath_commute path_subpath real_eq_0_iff_le_ge_0
+ reversepath_subpath simple_path_imp_path winding_number_reversepath z_notin_0t)
+ obtain z_in_q: "z \<in> inside(path_image q)"
+ and wn_q: "winding_number (subpath 0 t q +++ subpath t 1 q) z = - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
+ proof (rule winding_number_from_innerpath
+ [of "subpath 0 t q" "a-d" "a+e" "subpath 1 t q" "linepath (a - d) (a + e)"
+ z "- winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"],
+ simp_all add: q01 qt pa01_Un reversepath_subpath)
+ show "simple_path (subpath 0 t q)" "simple_path (subpath 1 t q)"
+ by (simp_all add: \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 0\<close> \<open>t \<noteq> 1\<close> simple_path_subpath)
+ show "simple_path (linepath (a - d) (a + e))"
+ using ad_not_ae by blast
+ show "path_image (subpath 0 t q) \<inter> path_image (subpath 1 t q) = {a - d, a + e}" (is "?lhs = ?rhs")
+ proof
+ show "?lhs \<subseteq> ?rhs"
+ using \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close> \<open>t \<noteq> 1\<close> q_ends qt q01
+ by (force simp: pathfinish_def qt simple_path_def path_image_subpath)
+ show "?rhs \<subseteq> ?lhs"
+ using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
+ qed
+ show "path_image (subpath 0 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
+ proof
+ show "?lhs \<subseteq> ?rhs" using paq_Int_cs pa01_Un by fastforce
+ show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
+ qed
+ show "path_image (subpath 1 t q) \<inter> closed_segment (a - d) (a + e) = {a - d, a + e}" (is "?lhs = ?rhs")
+ proof
+ show "?lhs \<subseteq> ?rhs" by (auto simp: pa_01q [symmetric])
+ show "?rhs \<subseteq> ?lhs" using \<open>0 \<le> t\<close> \<open>t \<le> 1\<close> q01 qt by (force simp: path_image_subpath)
+ qed
+ show "closed_segment (a - d) (a + e) \<inter> inside (path_image q) \<noteq> {}"
+ using a a_in_de open_closed_segment pa01_Un q_eq_p by fastforce
+ show "z \<in> inside (path_image (subpath 0 t q) \<union> closed_segment (a - d) (a + e))"
+ by (metis path_image_join path_image_linepath path_image_subpath_commute pathfinish_subpath pathstart_linepath q01(1) zin)
+ show "winding_number (subpath 0 t q +++ linepath (a + e) (a - d)) z =
+ - winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z"
+ using z_notin_ed z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
+ by (simp add: simple_path_imp_path qt q01 path_image_subpath_commute closed_segment_commute winding_number_join winding_number_reversepath [symmetric])
+ show "- d \<noteq> e"
+ using ad_not_ae by auto
+ show "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z \<noteq> 0"
+ using z1 by auto
+ qed
+ show ?thesis
+ proof
+ show "z \<in> inside (path_image p)"
+ using q_eq_p z_in_q by auto
+ then have [simp]: "z \<notin> path_image q"
+ by (metis disjoint_iff_not_equal inside_no_overlap q_eq_p)
+ have [simp]: "z \<notin> path_image (subpath 1 t q)"
+ using inside_def pa01_Un z_in_q by fastforce
+ have "winding_number(subpath 0 t q +++ subpath t 1 q) z = winding_number(subpath 0 1 q) z"
+ using z_notin_0t \<open>0 \<le> t\<close> \<open>simple_path q\<close> \<open>t \<le> 1\<close>
+ by (simp add: simple_path_imp_path qt path_image_subpath_commute winding_number_join winding_number_subpath_combine)
+ with wn_q have "winding_number (subpath t 0 q +++ linepath (a - d) (a + e)) z = - winding_number q z"
+ by auto
+ with z1 have "cmod (winding_number q z) = 1"
+ by simp
+ with z1 wn_q_eq_wn_p show "cmod (winding_number p z) = 1"
+ using z1 wn_q_eq_wn_p by (simp add: \<open>z \<in> inside (path_image p)\<close>)
+ qed
+qed
+
+proposition simple_closed_path_winding_number_inside:
+ assumes "simple_path \<gamma>"
+ obtains "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = 1"
+ | "\<And>z. z \<in> inside(path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = -1"
+proof (cases "pathfinish \<gamma> = pathstart \<gamma>")
+ case True
+ have "path \<gamma>"
+ by (simp add: assms simple_path_imp_path)
+ then have const: "winding_number \<gamma> constant_on inside(path_image \<gamma>)"
+ proof (rule winding_number_constant)
+ show "connected (inside(path_image \<gamma>))"
+ by (simp add: Jordan_inside_outside True assms)
+ qed (use inside_no_overlap True in auto)
+ obtain z where zin: "z \<in> inside (path_image \<gamma>)" and z1: "cmod (winding_number \<gamma> z) = 1"
+ using simple_closed_path_wn3 [of \<gamma>] True assms by blast
+ have "winding_number \<gamma> z \<in> \<int>"
+ using zin integer_winding_number [OF \<open>path \<gamma>\<close> True] inside_def by blast
+ with z1 consider "winding_number \<gamma> z = 1" | "winding_number \<gamma> z = -1"
+ apply (auto simp: Ints_def abs_if split: if_split_asm)
+ by (metis of_int_1 of_int_eq_iff of_int_minus)
+ with that const zin show ?thesis
+ unfolding constant_on_def by metis
+next
+ case False
+ then show ?thesis
+ using inside_simple_curve_imp_closed assms that(2) by blast
+qed
+
+lemma simple_closed_path_abs_winding_number_inside:
+ assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
+ shows "\<bar>Re (winding_number \<gamma> z)\<bar> = 1"
+ by (metis assms norm_minus_cancel norm_one one_complex.simps(1) real_norm_def simple_closed_path_winding_number_inside uminus_complex.simps(1))
+
+lemma simple_closed_path_norm_winding_number_inside:
+ assumes "simple_path \<gamma>" "z \<in> inside(path_image \<gamma>)"
+ shows "norm (winding_number \<gamma> z) = 1"
+proof -
+ have "pathfinish \<gamma> = pathstart \<gamma>"
+ using assms inside_simple_curve_imp_closed by blast
+ with assms integer_winding_number have "winding_number \<gamma> z \<in> \<int>"
+ by (simp add: inside_def simple_path_def)
+ then show ?thesis
+ by (metis assms norm_minus_cancel norm_one simple_closed_path_winding_number_inside)
+qed
+
+lemma simple_closed_path_winding_number_cases:
+ "\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> {-1,0,1}"
+apply (simp add: inside_Un_outside [of "path_image \<gamma>", symmetric, unfolded set_eq_iff Set.Compl_iff] del: inside_Un_outside)
+ apply (rule simple_closed_path_winding_number_inside)
+ using simple_path_def winding_number_zero_in_outside by blast+
+
+lemma simple_closed_path_winding_number_pos:
+ "\<lbrakk>simple_path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>; 0 < Re(winding_number \<gamma> z)\<rbrakk>
+ \<Longrightarrow> winding_number \<gamma> z = 1"
+using simple_closed_path_winding_number_cases
+ by fastforce
+
+subsection \<open>Winding number for rectangular paths\<close>
+
+proposition winding_number_rectpath:
+ assumes "z \<in> box a1 a3"
+ shows "winding_number (rectpath a1 a3) z = 1"
+proof -
+ from assms have less: "Re a1 < Re a3" "Im a1 < Im a3"
+ by (auto simp: in_box_complex_iff)
+ define a2 a4 where "a2 = Complex (Re a3) (Im a1)" and "a4 = Complex (Re a1) (Im a3)"
+ let ?l1 = "linepath a1 a2" and ?l2 = "linepath a2 a3"
+ and ?l3 = "linepath a3 a4" and ?l4 = "linepath a4 a1"
+ from assms and less have "z \<notin> path_image (rectpath a1 a3)"
+ by (auto simp: path_image_rectpath_cbox_minus_box)
+ also have "path_image (rectpath a1 a3) =
+ path_image ?l1 \<union> path_image ?l2 \<union> path_image ?l3 \<union> path_image ?l4"
+ by (simp add: rectpath_def Let_def path_image_join Un_assoc a2_def a4_def)
+ finally have "z \<notin> \<dots>" .
+ moreover have "\<forall>l\<in>{?l1,?l2,?l3,?l4}. Re (winding_number l z) > 0"
+ unfolding ball_simps HOL.simp_thms a2_def a4_def
+ by (intro conjI; (rule winding_number_linepath_pos_lt;
+ (insert assms, auto simp: a2_def a4_def in_box_complex_iff mult_neg_neg))+)
+ ultimately have "Re (winding_number (rectpath a1 a3) z) > 0"
+ by (simp add: winding_number_join path_image_join rectpath_def Let_def a2_def a4_def)
+ thus "winding_number (rectpath a1 a3) z = 1" using assms less
+ by (intro simple_closed_path_winding_number_pos simple_path_rectpath)
+ (auto simp: path_image_rectpath_cbox_minus_box)
+qed
+
+proposition winding_number_rectpath_outside:
+ assumes "Re a1 \<le> Re a3" "Im a1 \<le> Im a3"
+ assumes "z \<notin> cbox a1 a3"
+ shows "winding_number (rectpath a1 a3) z = 0"
+ using assms by (intro winding_number_zero_outside[OF _ _ _ assms(3)]
+ path_image_rectpath_subset_cbox) simp_all
+
+text\<open>A per-function version for continuous logs, a kind of monodromy\<close>
+proposition\<^marker>\<open>tag unimportant\<close> winding_number_compose_exp:
+ assumes "path p"
+ shows "winding_number (exp \<circ> p) 0 = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
+proof -
+ obtain e where "0 < e" and e: "\<And>t. t \<in> {0..1} \<Longrightarrow> e \<le> norm(exp(p t))"
+ proof
+ have "closed (path_image (exp \<circ> p))"
+ by (simp add: assms closed_path_image holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image)
+ then show "0 < setdist {0} (path_image (exp \<circ> p))"
+ by (metis exp_not_eq_zero imageE image_comp infdist_eq_setdist infdist_pos_not_in_closed path_defs(4) path_image_nonempty)
+ next
+ fix t::real
+ assume "t \<in> {0..1}"
+ have "setdist {0} (path_image (exp \<circ> p)) \<le> dist 0 (exp (p t))"
+ apply (rule setdist_le_dist)
+ using \<open>t \<in> {0..1}\<close> path_image_def by fastforce+
+ then show "setdist {0} (path_image (exp \<circ> p)) \<le> cmod (exp (p t))"
+ by simp
+ qed
+ have "bounded (path_image p)"
+ by (simp add: assms bounded_path_image)
+ then obtain B where "0 < B" and B: "path_image p \<subseteq> cball 0 B"
+ by (meson bounded_pos mem_cball_0 subsetI)
+ let ?B = "cball (0::complex) (B+1)"
+ have "uniformly_continuous_on ?B exp"
+ using holomorphic_on_exp holomorphic_on_imp_continuous_on
+ by (force intro: compact_uniformly_continuous)
+ then obtain d where "d > 0"
+ and d: "\<And>x x'. \<lbrakk>x\<in>?B; x'\<in>?B; dist x' x < d\<rbrakk> \<Longrightarrow> norm (exp x' - exp x) < e"
+ using \<open>e > 0\<close> by (auto simp: uniformly_continuous_on_def dist_norm)
+ then have "min 1 d > 0"
+ by force
+ then obtain g where pfg: "polynomial_function g" and "g 0 = p 0" "g 1 = p 1"
+ and gless: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm(g t - p t) < min 1 d"
+ using path_approx_polynomial_function [OF \<open>path p\<close>] \<open>d > 0\<close> \<open>0 < e\<close>
+ unfolding pathfinish_def pathstart_def by meson
+ have "winding_number (exp \<circ> p) 0 = winding_number (exp \<circ> g) 0"
+ proof (rule winding_number_nearby_paths_eq [symmetric])
+ show "path (exp \<circ> p)" "path (exp \<circ> g)"
+ by (simp_all add: pfg assms holomorphic_on_exp holomorphic_on_imp_continuous_on path_continuous_image path_polynomial_function)
+ next
+ fix t :: "real"
+ assume t: "t \<in> {0..1}"
+ with gless have "norm(g t - p t) < 1"
+ using min_less_iff_conj by blast
+ moreover have ptB: "norm (p t) \<le> B"
+ using B t by (force simp: path_image_def)
+ ultimately have "cmod (g t) \<le> B + 1"
+ by (meson add_mono_thms_linordered_field(4) le_less_trans less_imp_le norm_triangle_sub)
+ with ptB gless t have "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < e"
+ by (auto simp: dist_norm d)
+ with e t show "cmod ((exp \<circ> g) t - (exp \<circ> p) t) < cmod ((exp \<circ> p) t - 0)"
+ by fastforce
+ qed (use \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> in \<open>auto simp: pathfinish_def pathstart_def\<close>)
+ also have "... = 1 / (of_real (2 * pi) * \<i>) * contour_integral (exp \<circ> g) (\<lambda>w. 1 / (w - 0))"
+ proof (rule winding_number_valid_path)
+ have "continuous_on (path_image g) (deriv exp)"
+ by (metis DERIV_exp DERIV_imp_deriv continuous_on_cong holomorphic_on_exp holomorphic_on_imp_continuous_on)
+ then show "valid_path (exp \<circ> g)"
+ by (simp add: field_differentiable_within_exp pfg valid_path_compose valid_path_polynomial_function)
+ show "0 \<notin> path_image (exp \<circ> g)"
+ by (auto simp: path_image_def)
+ qed
+ also have "... = 1 / (of_real (2 * pi) * \<i>) * integral {0..1} (\<lambda>x. vector_derivative g (at x))"
+ proof (simp add: contour_integral_integral, rule integral_cong)
+ fix t :: "real"
+ assume t: "t \<in> {0..1}"
+ show "vector_derivative (exp \<circ> g) (at t) / exp (g t) = vector_derivative g (at t)"
+ proof -
+ have "(exp \<circ> g has_vector_derivative vector_derivative (exp \<circ> g) (at t)) (at t)"
+ by (meson DERIV_exp differentiable_def field_vector_diff_chain_at has_vector_derivative_def
+ has_vector_derivative_polynomial_function pfg vector_derivative_works)
+ moreover have "(exp \<circ> g has_vector_derivative vector_derivative g (at t) * exp (g t)) (at t)"
+ apply (rule field_vector_diff_chain_at)
+ apply (metis has_vector_derivative_polynomial_function pfg vector_derivative_at)
+ using DERIV_exp has_field_derivative_def apply blast
+ done
+ ultimately show ?thesis
+ by (simp add: divide_simps, rule vector_derivative_unique_at)
+ qed
+ qed
+ also have "... = (pathfinish p - pathstart p) / (2 * of_real pi * \<i>)"
+ proof -
+ have "((\<lambda>x. vector_derivative g (at x)) has_integral g 1 - g 0) {0..1}"
+ apply (rule fundamental_theorem_of_calculus [OF zero_le_one])
+ by (metis has_vector_derivative_at_within has_vector_derivative_polynomial_function pfg vector_derivative_at)
+ then show ?thesis
+ apply (simp add: pathfinish_def pathstart_def)
+ using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto
+ qed
+ finally show ?thesis .
+qed
+
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
+
+lemma winding_number_as_continuous_log:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ obtains q where "path q"
+ "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
+proof -
+ let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
+ show ?thesis
+ proof
+ have *: "continuous (at t within {0..1}) (\<lambda>x. winding_number (subpath 0 x p) \<zeta>)"
+ if t: "t \<in> {0..1}" for t
+ proof -
+ let ?B = "ball (p t) (norm(p t - \<zeta>))"
+ have "p t \<noteq> \<zeta>"
+ using path_image_def that \<zeta> by blast
+ then have "simply_connected ?B"
+ by (simp add: convex_imp_simply_connected)
+ then have "\<And>f::complex\<Rightarrow>complex. continuous_on ?B f \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> \<noteq> 0)
+ \<longrightarrow> (\<exists>g. continuous_on ?B g \<and> (\<forall>\<zeta> \<in> ?B. f \<zeta> = exp (g \<zeta>)))"
+ by (simp add: simply_connected_eq_continuous_log)
+ moreover have "continuous_on ?B (\<lambda>w. w - \<zeta>)"
+ by (intro continuous_intros)
+ moreover have "(\<forall>z \<in> ?B. z - \<zeta> \<noteq> 0)"
+ by (auto simp: dist_norm)
+ ultimately obtain g where contg: "continuous_on ?B g"
+ and geq: "\<And>z. z \<in> ?B \<Longrightarrow> z - \<zeta> = exp (g z)" by blast
+ obtain d where "0 < d" and d:
+ "\<And>x. \<lbrakk>x \<in> {0..1}; dist x t < d\<rbrakk> \<Longrightarrow> dist (p x) (p t) < cmod (p t - \<zeta>)"
+ using \<open>path p\<close> t unfolding path_def continuous_on_iff
+ by (metis \<open>p t \<noteq> \<zeta>\<close> right_minus_eq zero_less_norm_iff)
+ have "((\<lambda>x. winding_number (\<lambda>w. subpath 0 x p w - \<zeta>) 0 -
+ winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0) \<longlongrightarrow> 0)
+ (at t within {0..1})"
+ proof (rule Lim_transform_within [OF _ \<open>d > 0\<close>])
+ have "continuous (at t within {0..1}) (g o p)"
+ proof (rule continuous_within_compose)
+ show "continuous (at t within {0..1}) p"
+ using \<open>path p\<close> continuous_on_eq_continuous_within path_def that by blast
+ show "continuous (at (p t) within p ` {0..1}) g"
+ by (metis (no_types, lifting) open_ball UNIV_I \<open>p t \<noteq> \<zeta>\<close> centre_in_ball contg continuous_on_eq_continuous_at continuous_within_topological right_minus_eq zero_less_norm_iff)
+ qed
+ with LIM_zero have "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0))) \<longlongrightarrow> 0) (at t within {0..1})"
+ by (auto simp: subpath_def continuous_within o_def)
+ then show "((\<lambda>u. (g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)) \<longlongrightarrow> 0)
+ (at t within {0..1})"
+ by (simp add: tendsto_divide_zero)
+ show "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>) =
+ winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 - winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
+ if "u \<in> {0..1}" "0 < dist u t" "dist u t < d" for u
+ proof -
+ have "closed_segment t u \<subseteq> {0..1}"
+ using closed_segment_eq_real_ivl t that by auto
+ then have piB: "path_image(subpath t u p) \<subseteq> ?B"
+ apply (clarsimp simp add: path_image_subpath_gen)
+ by (metis subsetD le_less_trans \<open>dist u t < d\<close> d dist_commute dist_in_closed_segment)
+ have *: "path (g \<circ> subpath t u p)"
+ apply (rule path_continuous_image)
+ using \<open>path p\<close> t that apply auto[1]
+ using piB contg continuous_on_subset by blast
+ have "(g (subpath t u p 1) - g (subpath t u p 0)) / (2 * of_real pi * \<i>)
+ = winding_number (exp \<circ> g \<circ> subpath t u p) 0"
+ using winding_number_compose_exp [OF *]
+ by (simp add: pathfinish_def pathstart_def o_assoc)
+ also have "... = winding_number (\<lambda>w. subpath t u p w - \<zeta>) 0"
+ proof (rule winding_number_cong)
+ have "exp(g y) = y - \<zeta>" if "y \<in> (subpath t u p) ` {0..1}" for y
+ by (metis that geq path_image_def piB subset_eq)
+ then show "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> (exp \<circ> g \<circ> subpath t u p) x = subpath t u p x - \<zeta>"
+ by auto
+ qed
+ also have "... = winding_number (\<lambda>w. subpath 0 u p w - \<zeta>) 0 -
+ winding_number (\<lambda>w. subpath 0 t p w - \<zeta>) 0"
+ apply (simp add: winding_number_offset [symmetric])
+ using winding_number_subpath_combine [OF \<open>path p\<close> \<zeta>, of 0 t u] \<open>t \<in> {0..1}\<close> \<open>u \<in> {0..1}\<close>
+ by (simp add: add.commute eq_diff_eq)
+ finally show ?thesis .
+ qed
+ qed
+ then show ?thesis
+ by (subst winding_number_offset) (simp add: continuous_within LIM_zero_iff)
+ qed
+ show "path ?q"
+ unfolding path_def
+ by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
+
+ have "\<zeta> \<noteq> p 0"
+ by (metis \<zeta> pathstart_def pathstart_in_path_image)
+ then show "pathfinish ?q - pathstart ?q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ by (simp add: pathfinish_def pathstart_def)
+ show "p t = \<zeta> + exp (?q t)" if "t \<in> {0..1}" for t
+ proof -
+ have "path (subpath 0 t p)"
+ using \<open>path p\<close> that by auto
+ moreover
+ have "\<zeta> \<notin> path_image (subpath 0 t p)"
+ using \<zeta> [unfolded path_image_def] that by (auto simp: path_image_subpath)
+ ultimately show ?thesis
+ using winding_number_exp_2pi [of "subpath 0 t p" \<zeta>] \<open>\<zeta> \<noteq> p 0\<close>
+ by (auto simp: exp_add algebra_simps pathfinish_def pathstart_def subpath_def)
+ qed
+ qed
+qed
+
+subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
+
+lemma winding_number_homotopic_loops_null_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_loops (-{\<zeta>}) p (\<lambda>t. a))"
+ (is "?lhs = ?rhs")
+proof
+ assume [simp]: ?lhs
+ obtain q where "path q"
+ and qeq: "pathfinish q - pathstart q = 2 * of_real pi * \<i> * winding_number p \<zeta>"
+ and peq: "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp(q t)"
+ using winding_number_as_continuous_log [OF assms] by blast
+ have *: "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r)
+ {0..1} (-{\<zeta>}) ((\<lambda>w. \<zeta> + exp w) \<circ> q) ((\<lambda>w. \<zeta> + exp w) \<circ> (\<lambda>t. 0))"
+ proof (rule homotopic_with_compose_continuous_left)
+ show "homotopic_with_canon (\<lambda>f. pathfinish ((\<lambda>w. \<zeta> + exp w) \<circ> f) = pathstart ((\<lambda>w. \<zeta> + exp w) \<circ> f))
+ {0..1} UNIV q (\<lambda>t. 0)"
+ proof (rule homotopic_with_mono, simp_all add: pathfinish_def pathstart_def)
+ have "homotopic_loops UNIV q (\<lambda>t. 0)"
+ by (rule homotopic_loops_linear) (use qeq \<open>path q\<close> in \<open>auto simp: path_defs\<close>)
+ then have "homotopic_with (\<lambda>r. r 1 = r 0) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
+ by (simp add: homotopic_loops_def pathfinish_def pathstart_def)
+ then show "homotopic_with (\<lambda>h. exp (h 1) = exp (h 0)) (top_of_set {0..1}) euclidean q (\<lambda>t. 0)"
+ by (rule homotopic_with_mono) simp
+ qed
+ show "continuous_on UNIV (\<lambda>w. \<zeta> + exp w)"
+ by (rule continuous_intros)+
+ show "range (\<lambda>w. \<zeta> + exp w) \<subseteq> -{\<zeta>}"
+ by auto
+ qed
+ then have "homotopic_with_canon (\<lambda>r. pathfinish r = pathstart r) {0..1} (-{\<zeta>}) p (\<lambda>x. \<zeta> + 1)"
+ by (rule homotopic_with_eq) (auto simp: o_def peq pathfinish_def pathstart_def)
+ then have "homotopic_loops (-{\<zeta>}) p (\<lambda>t. \<zeta> + 1)"
+ by (simp add: homotopic_loops_def)
+ then show ?rhs ..
+next
+ assume ?rhs
+ then obtain a where "homotopic_loops (-{\<zeta>}) p (\<lambda>t. a)" ..
+ then have "winding_number p \<zeta> = winding_number (\<lambda>t. a) \<zeta>" "a \<noteq> \<zeta>"
+ using winding_number_homotopic_loops homotopic_loops_imp_subset by (force simp:)+
+ moreover have "winding_number (\<lambda>t. a) \<zeta> = 0"
+ by (metis winding_number_zero_const \<open>a \<noteq> \<zeta>\<close>)
+ ultimately show ?lhs by metis
+qed
+
+lemma winding_number_homotopic_paths_null_explicit_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p (linepath (pathstart p) (pathstart p))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (auto simp: winding_number_homotopic_loops_null_eq [OF assms])
+ apply (rule homotopic_loops_imp_homotopic_paths_null)
+ apply (simp add: linepath_refl)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ by (metis \<zeta> pathstart_in_path_image winding_number_homotopic_paths winding_number_trivial)
+qed
+
+lemma winding_number_homotopic_paths_null_eq:
+ assumes "path p" and \<zeta>: "\<zeta> \<notin> path_image p"
+ shows "winding_number p \<zeta> = 0 \<longleftrightarrow> (\<exists>a. homotopic_paths (-{\<zeta>}) p (\<lambda>t. a))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ by (auto simp: winding_number_homotopic_paths_null_explicit_eq [OF assms] linepath_refl)
+next
+ assume ?rhs
+ then show ?lhs
+ by (metis \<zeta> homotopic_paths_imp_pathfinish pathfinish_def pathfinish_in_path_image winding_number_homotopic_paths winding_number_zero_const)
+qed
+
+proposition winding_number_homotopic_paths_eq:
+ assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
+ and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
+ and qp: "pathstart q = pathstart p" "pathfinish q = pathfinish p"
+ shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_paths (-{\<zeta>}) p q"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then have "winding_number (p +++ reversepath q) \<zeta> = 0"
+ using assms by (simp add: winding_number_join winding_number_reversepath)
+ moreover
+ have "path (p +++ reversepath q)" "\<zeta> \<notin> path_image (p +++ reversepath q)"
+ using assms by (auto simp: not_in_path_image_join)
+ ultimately obtain a where "homotopic_paths (- {\<zeta>}) (p +++ reversepath q) (linepath a a)"
+ using winding_number_homotopic_paths_null_explicit_eq by blast
+ then show ?rhs
+ using homotopic_paths_imp_pathstart assms
+ by (fastforce simp add: dest: homotopic_paths_imp_homotopic_loops homotopic_paths_loop_parts)
+next
+ assume ?rhs
+ then show ?lhs
+ by (simp add: winding_number_homotopic_paths)
+qed
+
+lemma winding_number_homotopic_loops_eq:
+ assumes "path p" and \<zeta>p: "\<zeta> \<notin> path_image p"
+ and "path q" and \<zeta>q: "\<zeta> \<notin> path_image q"
+ and loops: "pathfinish p = pathstart p" "pathfinish q = pathstart q"
+ shows "winding_number p \<zeta> = winding_number q \<zeta> \<longleftrightarrow> homotopic_loops (-{\<zeta>}) p q"
+ (is "?lhs = ?rhs")
+proof
+ assume L: ?lhs
+ have "pathstart p \<noteq> \<zeta>" "pathstart q \<noteq> \<zeta>"
+ using \<zeta>p \<zeta>q by blast+
+ moreover have "path_connected (-{\<zeta>})"
+ by (simp add: path_connected_punctured_universe)
+ ultimately obtain r where "path r" and rim: "path_image r \<subseteq> -{\<zeta>}"
+ and pas: "pathstart r = pathstart p" and paf: "pathfinish r = pathstart q"
+ by (auto simp: path_connected_def)
+ then have "pathstart r \<noteq> \<zeta>" by blast
+ have "homotopic_loops (- {\<zeta>}) p (r +++ q +++ reversepath r)"
+ proof (rule homotopic_paths_imp_homotopic_loops)
+ show "homotopic_paths (- {\<zeta>}) p (r +++ q +++ reversepath r)"
+ by (metis (mono_tags, hide_lams) \<open>path r\<close> L \<zeta>p \<zeta>q \<open>path p\<close> \<open>path q\<close> homotopic_loops_conjugate loops not_in_path_image_join paf pas path_image_reversepath path_imp_reversepath path_join_eq pathfinish_join pathfinish_reversepath pathstart_join pathstart_reversepath rim subset_Compl_singleton winding_number_homotopic_loops winding_number_homotopic_paths_eq)
+ qed (use loops pas in auto)
+ moreover have "homotopic_loops (- {\<zeta>}) (r +++ q +++ reversepath r) q"
+ using rim \<zeta>q by (auto simp: homotopic_loops_conjugate paf \<open>path q\<close> \<open>path r\<close> loops)
+ ultimately show ?rhs
+ using homotopic_loops_trans by metis
+next
+ assume ?rhs
+ then show ?lhs
+ by (simp add: winding_number_homotopic_loops)
+qed
+
+end
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/document/root.bib Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,3 @@
+
+
+@misc{dummy}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/document/root.tex Sat Nov 30 13:47:33 2019 +0100
@@ -0,0 +1,43 @@
+\documentclass[11pt,a4paper]{article}
+\usepackage{graphicx}
+\usepackage{isabelle}
+\usepackage{isabellesym}
+\usepackage{latexsym}
+\usepackage{textcomp}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[only,bigsqcap]{stmaryrd}
+\usepackage{pdfsetup}
+
+\usepackage{tocloft}
+
+\urlstyle{rm}
+\isabellestyle{literalunderscore}
+\pagestyle{myheadings}
+
+\raggedbottom
+
+\begin{document}
+
+\title{Complex Analysis}
+\maketitle
+
+\tableofcontents
+
+\begin{center}
+ \includegraphics[height=\textheight]{session_graph}
+\end{center}
+
+\newpage
+
+\renewcommand{\setisabellecontext}[1]{\markright{\href{#1.html}{#1.thy}}}
+
+\parindent 0pt\parskip 0.5ex
+\input{session}
+
+\pagestyle{headings}
+\bibliographystyle{abbrv}
+\bibliography{root}
+\nocite{dummy}
+
+\end{document}
--- a/src/HOL/ROOT Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/ROOT Sat Nov 30 13:47:33 2019 +0100
@@ -71,6 +71,15 @@
"root.tex"
"root.bib"
+session "HOL-Complex_Analysis" (main timing) in Complex_Analysis = "HOL-Analysis" +
+ options [document_tags = "theorem%important,corollary%important,proposition%important,class%important,instantiation%important,subsubsection%unimportant,%unimportant",
+ document_variants = "document:manual=-proof,-ML,-unimportant"]
+ theories
+ Complex_Analysis
+ document_files
+ "root.tex"
+ "root.bib"
+
session "HOL-Analysis-ex" in "Analysis/ex" = "HOL-Analysis" +
theories
Approximations