renamed Analysis/Winding_Numbers to Winding_Numbers_2; reorganised Analysis/Cauchy_Integral_Theorem by splitting it into Contour_Integration, Winding_Numbers,Cauchy_Integral_Theorem and Cauchy_Integral_Formula.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Cauchy_Integral_Formula.thy Sun Dec 01 19:10:57 2019 +0000
@@ -0,0 +1,2090 @@
+section \<open>Cauchy's Integral Formula\<close>
+
+theory Cauchy_Integral_Formula
+ imports Winding_Numbers
+begin
+
+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
+
+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
+
+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)
+
+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)
+
+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
+
+
+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
+
+
+subsection\<open>Existence of all higher derivatives\<close>
+
+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
+
+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>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 integral formula\<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
+
+subsection \<open>Generalised Cauchy's integral theorem\<close>
+
+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 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
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Sun Dec 01 19:10:57 2019 +0000
@@ -1,15 +1,17 @@
-section \<open>Complex Path Integrals and Cauchy's Integral Theorem\<close>
+section \<open>Cauchy's Integral Theorem\<close>
-text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2015)\<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
+ Contour_Integration
Weierstrass_Theorems
Retracts
begin
+subsection \<open>Misc\<close>
+
+(*TODO: move. Not used in HOL/Analysis.*)
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)"
@@ -20,6 +22,7 @@
using leibniz_rule_field_differentiable[OF assms(1-3) _ assms(4)]
by (auto simp: holomorphic_on_def)
+(*TODO: move. Not used in HOL/Analysis.*)
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
@@ -36,1968 +39,12 @@
finally show ?thesis .
qed
+(*TODO: move. Not used in HOL/Analysis.*)
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:
@@ -3362,7 +1409,6 @@
qed
qed
-
lemma
assumes "open S" "path p" "path_image p \<subseteq> S"
shows contour_integral_nearby_ends:
@@ -3453,1189 +1499,6 @@
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:
@@ -4826,3022 +1689,7 @@
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/Complex_Transcendental.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Sun Dec 01 19:10:57 2019 +0000
@@ -4053,4 +4053,6 @@
apply (auto simp: Re_complex_div_eq_0 exp_of_nat_mult [symmetric] mult_ac exp_Euler)
done
+
+
end
--- a/src/HOL/Analysis/Conformal_Mappings.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Conformal_Mappings.thy Sun Dec 01 19:10:57 2019 +0000
@@ -5,12 +5,10 @@
text\<open>Also Cauchy's residue theorem by Wenda Li (2016)\<close>
theory Conformal_Mappings
-imports Cauchy_Integral_Theorem
-
+imports Cauchy_Integral_Formula
begin
-(* FIXME mv to Cauchy_Integral_Theorem.thy *)
-subsection\<open>Cauchy's inequality and more versions of Liouville\<close>
+subsection\<open>Liouville's theorem\<close>
lemma Cauchy_higher_deriv_bound:
assumes holf: "f holomorphic_on (ball z r)"
@@ -55,6 +53,7 @@
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"
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Contour_Integration.thy Sun Dec 01 19:10:57 2019 +0000
@@ -0,0 +1,2681 @@
+section \<open>Contour Integration\<close>
+
+theory Contour_Integration
+ imports Henstock_Kurzweil_Integration Path_Connected Complex_Transcendental
+begin
+
+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])
+
+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
+
+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
+
+
+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
+
+
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Homeomorphism.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Homeomorphism.thy Sun Dec 01 19:10:57 2019 +0000
@@ -2184,7 +2184,6 @@
qed
qed
-
corollary covering_space_lift_stronger:
fixes p :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
and f :: "'c::real_normed_vector \<Rightarrow> 'b"
@@ -2252,4 +2251,36 @@
by (metis that covering_space_lift_strong [OF cov _ \<open>z \<in> U\<close> U contf fim])
qed
+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
+
end
--- a/src/HOL/Analysis/Path_Connected.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Path_Connected.thy Sun Dec 01 19:10:57 2019 +0000
@@ -4003,4 +4003,5 @@
shows "\<exists>g. homeomorphism S T f g"
using assms injective_into_1d_eq_homeomorphism by blast
+
end
--- a/src/HOL/Analysis/Retracts.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Retracts.thy Sun Dec 01 19:10:57 2019 +0000
@@ -2591,4 +2591,51 @@
shows "connected(-S)"
using assms path_connected_complement_homeomorphic_interval path_connected_imp_connected by blast
+
+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
+
end
--- a/src/HOL/Analysis/Winding_Numbers.thy Wed Nov 27 16:54:33 2019 +0000
+++ b/src/HOL/Analysis/Winding_Numbers.thy Sun Dec 01 19:10:57 2019 +0000
@@ -1,1211 +1,1330 @@
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
+theory Winding_Numbers
+ imports Cauchy_Integral_Theorem
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
+text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
+
+subsection \<open>Basic Winding Numbers\<close>
-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)
+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"
-subsection\<open>Winding number for a triangle\<close>
+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 wn_triangle1:
- assumes "0 \<in> interior(convex hull {a,b,c})"
- shows "\<not> (Im(a/b) \<le> 0 \<and> 0 \<le> Im(b/c))"
+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 -
- { 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)
+ 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
- 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
+ then show ?thesis
+ unfolding winding_number_def by (rule someI2_ex) (blast intro: \<open>0<e\<close>)
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))"
+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 [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)
+ 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
-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))"
+(*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 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]
+ 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
-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"
+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 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)"
+ 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_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)
+ 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
- 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)"
+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 [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)
+ 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 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)
+ 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
-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"
+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 -
- 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
+ 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 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"
+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 -
- 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
+ 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
-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"
+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]: "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}"
+ 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
- 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)
+ 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
- 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
+ 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
-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"
+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 -
- 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)
+ 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 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
+ 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
-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
+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)
-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
+subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number is constant on a connected region\<close>
-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")
+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 -
- 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 .
+ 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 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 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 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"
+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 -
- 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)
+ 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
-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>)"
+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 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
+ 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
- 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 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
- apply (simp add: pathfinish_def pathstart_def)
- using \<open>g 0 = p 0\<close> \<open>g 1 = p 1\<close> by auto
+ by (auto intro: winding_number_unique [OF \<gamma>] simp add: winding_number_prop_def wnot)
qed
finally show ?thesis .
qed
-subsection\<^marker>\<open>tag unimportant\<close> \<open>The winding number defines a continuous logarithm for the path itself\<close>
+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
+
+subsection \<open>The real part of winding numbers\<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)"
+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 -
- let ?q = "\<lambda>t. 2 * of_real pi * \<i> * winding_number(subpath 0 t p) \<zeta> + Ln(pathstart p - \<zeta>)"
+ 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
- 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
+ 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 -
- 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)
+ 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
- show "path ?q"
- unfolding path_def
- by (intro continuous_intros) (simp add: continuous_on_eq_continuous_within *)
+ 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
- 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
+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
-subsection \<open>Winding number equality is the same as path/loop homotopy in C - {0}\<close>
+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
+
-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
+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
-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)
+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
+
+subsection \<open>Invariance of winding numbers under homotopy\<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_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)
+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
-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)
+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>Winding numbers of some simple paths\<close>
+
+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
- assume ?rhs
- then show ?lhs
- by (simp add: winding_number_homotopic_paths)
+ 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
-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)
+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)
+
+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
-end
-
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Winding_Numbers_2.thy Sun Dec 01 19:10:57 2019 +0000
@@ -0,0 +1,1211 @@
+section \<open>More Winding Numbers\<close>
+
+text\<open>By John Harrison et al. Ported from HOL Light by L C Paulson (2017)\<close>
+
+theory Winding_Numbers_2
+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
+