--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex_Analysis/Meromorphic.thy Thu Feb 16 12:21:21 2023 +0000
@@ -0,0 +1,2333 @@
+theory Meromorphic
+ imports Laurent_Convergence Riemann_Mapping
+begin
+
+lemma analytic_at_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
+ shows "f analytic_on {x} \<longleftrightarrow> g analytic_on {y}"
+proof -
+ have "g analytic_on {x}" if "f analytic_on {x}" "eventually (\<lambda>x. f x = g x) (nhds x)" for f g
+ proof -
+ have "(\<lambda>y. f (x + y)) has_fps_expansion fps_expansion f x"
+ by (rule analytic_at_imp_has_fps_expansion) fact
+ also have "?this \<longleftrightarrow> (\<lambda>y. g (x + y)) has_fps_expansion fps_expansion f x"
+ using that by (intro has_fps_expansion_cong refl) (auto simp: nhds_to_0' eventually_filtermap)
+ finally show ?thesis
+ by (rule has_fps_expansion_imp_analytic)
+ qed
+ from this[of f g] this[of g f] show ?thesis using assms
+ by (auto simp: eq_commute)
+qed
+
+definition remove_sings :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
+ "remove_sings f z = (if \<exists>c. f \<midarrow>z\<rightarrow> c then Lim (at z) f else 0)"
+
+lemma remove_sings_eqI [intro]:
+ assumes "f \<midarrow>z\<rightarrow> c"
+ shows "remove_sings f z = c"
+ using assms unfolding remove_sings_def by (auto simp: tendsto_Lim)
+
+lemma remove_sings_at_analytic [simp]:
+ assumes "f analytic_on {z}"
+ shows "remove_sings f z = f z"
+ using assms by (intro remove_sings_eqI) (simp add: analytic_at_imp_isCont isContD)
+
+lemma remove_sings_at_pole [simp]:
+ assumes "is_pole f z"
+ shows "remove_sings f z = 0"
+ using assms unfolding remove_sings_def is_pole_def
+ by (meson at_neq_bot not_tendsto_and_filterlim_at_infinity)
+
+lemma eventually_remove_sings_eq_at:
+ assumes "isolated_singularity_at f z"
+ shows "eventually (\<lambda>w. remove_sings f w = f w) (at z)"
+proof -
+ from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
+ by (auto simp: isolated_singularity_at_def)
+ hence *: "f analytic_on {w}" if "w \<in> ball z r - {z}" for w
+ using r that by (auto intro: analytic_on_subset)
+ have "eventually (\<lambda>w. w \<in> ball z r - {z}) (at z)"
+ using r by (intro eventually_at_in_open) auto
+ thus ?thesis
+ by eventually_elim (auto simp: remove_sings_at_analytic *)
+qed
+
+lemma eventually_remove_sings_eq_nhds:
+ assumes "f analytic_on {z}"
+ shows "eventually (\<lambda>w. remove_sings f w = f w) (nhds z)"
+proof -
+ from assms obtain A where A: "open A" "z \<in> A" "f holomorphic_on A"
+ by (auto simp: analytic_at)
+ have "eventually (\<lambda>z. z \<in> A) (nhds z)"
+ by (intro eventually_nhds_in_open A)
+ thus ?thesis
+ proof eventually_elim
+ case (elim w)
+ from elim have "f analytic_on {w}"
+ using A analytic_at by blast
+ thus ?case by auto
+ qed
+qed
+
+lemma remove_sings_compose:
+ assumes "filtermap g (at z) = at z'"
+ shows "remove_sings (f \<circ> g) z = remove_sings f z'"
+proof (cases "\<exists>c. f \<midarrow>z'\<rightarrow> c")
+ case True
+ then obtain c where c: "f \<midarrow>z'\<rightarrow> c"
+ by auto
+ from c have "remove_sings f z' = c"
+ by blast
+ moreover from c have "remove_sings (f \<circ> g) z = c"
+ using c by (intro remove_sings_eqI) (auto simp: filterlim_def filtermap_compose assms)
+ ultimately show ?thesis
+ by simp
+next
+ case False
+ hence "\<not>(\<exists>c. (f \<circ> g) \<midarrow>z\<rightarrow> c)"
+ by (auto simp: filterlim_def filtermap_compose assms)
+ with False show ?thesis
+ by (auto simp: remove_sings_def)
+qed
+
+lemma remove_sings_cong:
+ assumes "eventually (\<lambda>x. f x = g x) (at z)" "z = z'"
+ shows "remove_sings f z = remove_sings g z'"
+proof (cases "\<exists>c. f \<midarrow>z\<rightarrow> c")
+ case True
+ then obtain c where c: "f \<midarrow>z\<rightarrow> c" by blast
+ hence "remove_sings f z = c"
+ by blast
+ moreover have "f \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z'\<rightarrow> c"
+ using assms by (intro filterlim_cong refl) auto
+ with c have "remove_sings g z' = c"
+ by (intro remove_sings_eqI) auto
+ ultimately show ?thesis
+ by simp
+next
+ case False
+ have "f \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z'\<rightarrow> c" for c
+ using assms by (intro filterlim_cong) auto
+ with False show ?thesis
+ by (auto simp: remove_sings_def)
+qed
+
+
+lemma deriv_remove_sings_at_analytic [simp]:
+ assumes "f analytic_on {z}"
+ shows "deriv (remove_sings f) z = deriv f z"
+ apply (rule deriv_cong_ev)
+ apply (rule eventually_remove_sings_eq_nhds)
+ using assms by auto
+
+lemma isolated_singularity_at_remove_sings [simp, intro]:
+ assumes "isolated_singularity_at f z"
+ shows "isolated_singularity_at (remove_sings f) z"
+ using isolated_singularity_at_cong[OF eventually_remove_sings_eq_at[OF assms] refl] assms
+ by simp
+
+lemma not_essential_remove_sings_iff [simp]:
+ assumes "isolated_singularity_at f z"
+ shows "not_essential (remove_sings f) z \<longleftrightarrow> not_essential f z"
+ using not_essential_cong[OF eventually_remove_sings_eq_at[OF assms(1)] refl]
+ by simp
+
+lemma not_essential_remove_sings [intro]:
+ assumes "isolated_singularity_at f z" "not_essential f z"
+ shows "not_essential (remove_sings f) z"
+ by (subst not_essential_remove_sings_iff) (use assms in auto)
+
+lemma
+ assumes "isolated_singularity_at f z"
+ shows is_pole_remove_sings_iff [simp]:
+ "is_pole (remove_sings f) z \<longleftrightarrow> is_pole f z"
+ and zorder_remove_sings [simp]:
+ "zorder (remove_sings f) z = zorder f z"
+ and zor_poly_remove_sings [simp]:
+ "zor_poly (remove_sings f) z = zor_poly f z"
+ and has_laurent_expansion_remove_sings_iff [simp]:
+ "(\<lambda>w. remove_sings f (z + w)) has_laurent_expansion F \<longleftrightarrow>
+ (\<lambda>w. f (z + w)) has_laurent_expansion F"
+ and tendsto_remove_sings_iff [simp]:
+ "remove_sings f \<midarrow>z\<rightarrow> c \<longleftrightarrow> f \<midarrow>z\<rightarrow> c"
+ by (intro is_pole_cong eventually_remove_sings_eq_at refl zorder_cong
+ zor_poly_cong has_laurent_expansion_cong' tendsto_cong assms)+
+
+lemma get_all_poles_from_remove_sings:
+ fixes f:: "complex \<Rightarrow> complex"
+ defines "ff\<equiv>remove_sings f"
+ assumes f_holo:"f holomorphic_on s - pts" and "finite pts"
+ "pts\<subseteq>s" "open s" and not_ess:"\<forall>x\<in>pts. not_essential f x"
+ obtains pts' where
+ "pts' \<subseteq> pts" "finite pts'" "ff holomorphic_on s - pts'" "\<forall>x\<in>pts'. is_pole ff x"
+proof -
+ define pts' where "pts' = {x\<in>pts. is_pole f x}"
+
+ have "pts' \<subseteq> pts" unfolding pts'_def by auto
+ then have "finite pts'" using \<open>finite pts\<close>
+ using rev_finite_subset by blast
+ then have "open (s - pts')" using \<open>open s\<close>
+ by (simp add: finite_imp_closed open_Diff)
+
+ have isolated:"isolated_singularity_at f z" if "z\<in>pts" for z
+ proof (rule isolated_singularity_at_holomorphic)
+ show "f holomorphic_on (s-(pts-{z})) - {z}"
+ by (metis Diff_insert f_holo insert_Diff that)
+ show " open (s - (pts - {z}))"
+ by (meson assms(3) assms(5) finite_Diff finite_imp_closed open_Diff)
+ show "z \<in> s - (pts - {z})"
+ using assms(4) that by auto
+ qed
+
+ have "ff holomorphic_on s - pts'"
+ proof (rule no_isolated_singularity')
+ show "(ff \<longlongrightarrow> ff z) (at z within s - pts')" if "z \<in> pts-pts'" for z
+ proof -
+ have "at z within s - pts' = at z"
+ apply (rule at_within_open)
+ using \<open>open (s - pts')\<close> that \<open>pts\<subseteq>s\<close> by auto
+ moreover have "ff \<midarrow>z\<rightarrow> ff z"
+ unfolding ff_def
+ proof (subst tendsto_remove_sings_iff)
+ show "isolated_singularity_at f z"
+ apply (rule isolated)
+ using that by auto
+ have "not_essential f z"
+ using not_ess that by auto
+ moreover have "\<not>is_pole f z"
+ using that unfolding pts'_def by auto
+ ultimately have "\<exists>c. f \<midarrow>z\<rightarrow> c"
+ unfolding not_essential_def by auto
+ then show "f \<midarrow>z\<rightarrow> remove_sings f z"
+ using remove_sings_eqI by blast
+ qed
+ ultimately show ?thesis by auto
+ qed
+ have "ff holomorphic_on s - pts"
+ using f_holo
+ proof (elim holomorphic_transform)
+ fix x assume "x \<in> s - pts"
+ then have "f analytic_on {x}"
+ using assms(3) assms(5) f_holo
+ by (meson finite_imp_closed
+ holomorphic_on_imp_analytic_at open_Diff)
+ from remove_sings_at_analytic[OF this]
+ show "f x = ff x" unfolding ff_def by auto
+ qed
+ then show "ff holomorphic_on s - pts' - (pts - pts')"
+ apply (elim holomorphic_on_subset)
+ by blast
+ show "open (s - pts')"
+ by (simp add: \<open>open (s - pts')\<close>)
+ show "finite (pts - pts')"
+ by (simp add: assms(3))
+ qed
+ moreover have "\<forall>x\<in>pts'. is_pole ff x"
+ unfolding pts'_def
+ using ff_def is_pole_remove_sings_iff isolated by blast
+ moreover note \<open>pts' \<subseteq> pts\<close> \<open>finite pts'\<close>
+ ultimately show ?thesis using that by auto
+qed
+
+lemma remove_sings_eq_0_iff:
+ assumes "not_essential f w"
+ shows "remove_sings f w = 0 \<longleftrightarrow> is_pole f w \<or> f \<midarrow>w\<rightarrow> 0"
+proof (cases "is_pole f w")
+ case True
+ then show ?thesis by simp
+next
+ case False
+ then obtain c where c:"f \<midarrow>w\<rightarrow> c"
+ using \<open>not_essential f w\<close> unfolding not_essential_def by auto
+ then show ?thesis
+ using False remove_sings_eqI by auto
+qed
+
+definition meromorphic_on:: "[complex \<Rightarrow> complex, complex set, complex set] \<Rightarrow> bool"
+ ("_ (meromorphic'_on) _ _" [50,50,50]50) where
+ "f meromorphic_on D pts \<equiv>
+ open D \<and> pts \<subseteq> D \<and> (\<forall>z\<in>pts. isolated_singularity_at f z \<and> not_essential f z) \<and>
+ (\<forall>z\<in>D. \<not>(z islimpt pts)) \<and> (f holomorphic_on D-pts)"
+
+lemma meromorphic_imp_holomorphic: "f meromorphic_on D pts \<Longrightarrow> f holomorphic_on (D - pts)"
+ unfolding meromorphic_on_def by auto
+
+lemma meromorphic_imp_closedin_pts:
+ assumes "f meromorphic_on D pts"
+ shows "closedin (top_of_set D) pts"
+ by (meson assms closedin_limpt meromorphic_on_def)
+
+lemma meromorphic_imp_open_diff':
+ assumes "f meromorphic_on D pts" "pts' \<subseteq> pts"
+ shows "open (D - pts')"
+proof -
+ have "D - pts' = D - closure pts'"
+ proof safe
+ fix x assume x: "x \<in> D" "x \<in> closure pts'" "x \<notin> pts'"
+ hence "x islimpt pts'"
+ by (subst islimpt_in_closure) auto
+ hence "x islimpt pts"
+ by (rule islimpt_subset) fact
+ with assms x show False
+ by (auto simp: meromorphic_on_def)
+ qed (use closure_subset in auto)
+ then show ?thesis
+ using assms meromorphic_on_def by auto
+qed
+
+lemma meromorphic_imp_open_diff: "f meromorphic_on D pts \<Longrightarrow> open (D - pts)"
+ by (erule meromorphic_imp_open_diff') auto
+
+lemma meromorphic_pole_subset:
+ assumes merf: "f meromorphic_on D pts"
+ shows "{x\<in>D. is_pole f x} \<subseteq> pts"
+ by (smt (verit) Diff_iff assms mem_Collect_eq meromorphic_imp_open_diff
+ meromorphic_on_def not_is_pole_holomorphic subsetI)
+
+named_theorems meromorphic_intros
+
+lemma meromorphic_on_subset:
+ assumes "f meromorphic_on A pts" "open B" "B \<subseteq> A" "pts' = pts \<inter> B"
+ shows "f meromorphic_on B pts'"
+ unfolding meromorphic_on_def
+proof (intro ballI conjI)
+ fix z assume "z \<in> B"
+ show "\<not>z islimpt pts'"
+ proof
+ assume "z islimpt pts'"
+ hence "z islimpt pts"
+ by (rule islimpt_subset) (use \<open>pts' = _\<close> in auto)
+ thus False using \<open>z \<in> B\<close> \<open>B \<subseteq> A\<close> assms(1)
+ by (auto simp: meromorphic_on_def)
+ qed
+qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma meromorphic_on_superset_pts:
+ assumes "f meromorphic_on A pts" "pts \<subseteq> pts'" "pts' \<subseteq> A" "\<forall>x\<in>A. \<not>x islimpt pts'"
+ shows "f meromorphic_on A pts'"
+ unfolding meromorphic_on_def
+proof (intro conjI ballI impI)
+ fix z assume "z \<in> pts'"
+ from assms(1) have holo: "f holomorphic_on A - pts" and "open A"
+ unfolding meromorphic_on_def by blast+
+ have "open (A - pts)"
+ by (intro meromorphic_imp_open_diff[OF assms(1)])
+
+ show "isolated_singularity_at f z"
+ proof (cases "z \<in> pts")
+ case False
+ thus ?thesis
+ using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close>
+ by (intro isolated_singularity_at_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo])
+ auto
+ qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+ show "not_essential f z"
+ proof (cases "z \<in> pts")
+ case False
+ thus ?thesis
+ using \<open>open (A - pts)\<close> assms \<open>z \<in> pts'\<close>
+ by (intro not_essential_holomorphic[of _ "A - pts"] holomorphic_on_subset[OF holo])
+ auto
+ qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma meromorphic_on_no_singularities: "f meromorphic_on A {} \<longleftrightarrow> f holomorphic_on A \<and> open A"
+ by (auto simp: meromorphic_on_def)
+
+lemma holomorphic_on_imp_meromorphic_on:
+ "f holomorphic_on A \<Longrightarrow> pts \<subseteq> A \<Longrightarrow> open A \<Longrightarrow> \<forall>x\<in>A. \<not>x islimpt pts \<Longrightarrow> f meromorphic_on A pts"
+ by (rule meromorphic_on_superset_pts[where pts = "{}"])
+ (auto simp: meromorphic_on_no_singularities)
+
+lemma meromorphic_on_const [meromorphic_intros]:
+ assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
+ shows "(\<lambda>_. c) meromorphic_on A pts"
+ by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto)
+
+lemma meromorphic_on_ident [meromorphic_intros]:
+ assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
+ shows "(\<lambda>x. x) meromorphic_on A pts"
+ by (rule holomorphic_on_imp_meromorphic_on) (use assms in auto)
+
+lemma meromorphic_on_id [meromorphic_intros]:
+ assumes "open A" "\<forall>x\<in>A. \<not>x islimpt pts" "pts \<subseteq> A"
+ shows "id meromorphic_on A pts"
+ using meromorphic_on_ident assms unfolding id_def .
+
+lemma not_essential_add [singularity_intros]:
+ assumes f_ness: "not_essential f z" and g_ness: "not_essential g z"
+ assumes f_iso: "isolated_singularity_at f z" and g_iso: "isolated_singularity_at g z"
+ shows "not_essential (\<lambda>w. f w + g w) z"
+proof -
+ have "(\<lambda>w. f (z + w) + g (z + w)) has_laurent_expansion laurent_expansion f z + laurent_expansion g z"
+ by (intro not_essential_has_laurent_expansion laurent_expansion_intros assms)
+ hence "not_essential (\<lambda>w. f (z + w) + g (z + w)) 0"
+ using has_laurent_expansion_not_essential_0 by blast
+ thus ?thesis
+ by (simp add: not_essential_shift_0)
+qed
+
+lemma meromorphic_on_uminus [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>z. -f z) meromorphic_on A pts"
+ unfolding meromorphic_on_def
+ by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
+
+lemma meromorphic_on_add [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
+ shows "(\<lambda>z. f z + g z) meromorphic_on A pts"
+ unfolding meromorphic_on_def
+ by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
+
+lemma meromorphic_on_add':
+ assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
+ shows "(\<lambda>z. f z + g z) meromorphic_on A (pts1 \<union> pts2)"
+proof (rule meromorphic_intros)
+ show "f meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(1)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+ show "g meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(2)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+qed
+
+lemma meromorphic_on_add_const [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>z. f z + c) meromorphic_on A pts"
+ unfolding meromorphic_on_def
+ by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
+
+lemma meromorphic_on_minus_const [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>z. f z - c) meromorphic_on A pts"
+ using meromorphic_on_add_const[OF assms,of "-c"] by simp
+
+lemma meromorphic_on_diff [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
+ shows "(\<lambda>z. f z - g z) meromorphic_on A pts"
+ using meromorphic_on_add[OF assms(1) meromorphic_on_uminus[OF assms(2)]] by simp
+
+lemma meromorphic_on_diff':
+ assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
+ shows "(\<lambda>z. f z - g z) meromorphic_on A (pts1 \<union> pts2)"
+proof (rule meromorphic_intros)
+ show "f meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(1)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+ show "g meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(2)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+qed
+
+lemma meromorphic_on_mult [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "g meromorphic_on A pts"
+ shows "(\<lambda>z. f z * g z) meromorphic_on A pts"
+ unfolding meromorphic_on_def
+ by (use assms in \<open>auto simp: meromorphic_on_def intro!: holomorphic_intros singularity_intros\<close>)
+
+lemma meromorphic_on_mult':
+ assumes "f meromorphic_on A pts1" "g meromorphic_on A pts2"
+ shows "(\<lambda>z. f z * g z) meromorphic_on A (pts1 \<union> pts2)"
+proof (rule meromorphic_intros)
+ show "f meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(1)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+ show "g meromorphic_on A (pts1 \<union> pts2)"
+ by (rule meromorphic_on_superset_pts[OF assms(2)])
+ (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un\<close>)
+qed
+
+
+
+lemma meromorphic_on_imp_not_essential:
+ assumes "f meromorphic_on A pts" "z \<in> A"
+ shows "not_essential f z"
+proof (cases "z \<in> pts")
+ case False
+ thus ?thesis
+ using not_essential_holomorphic[of f "A - pts" z] meromorphic_imp_open_diff[OF assms(1)] assms
+ by (auto simp: meromorphic_on_def)
+qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma meromorphic_imp_analytic: "f meromorphic_on D pts \<Longrightarrow> f analytic_on (D - pts)"
+ unfolding meromorphic_on_def
+ apply (subst analytic_on_open)
+ using meromorphic_imp_open_diff meromorphic_on_id apply blast
+ apply auto
+ done
+
+lemma not_islimpt_isolated_zeros:
+ assumes mero: "f meromorphic_on A pts" and "z \<in> A"
+ shows "\<not>z islimpt {w\<in>A. isolated_zero f w}"
+proof
+ assume islimpt: "z islimpt {w\<in>A. isolated_zero f w}"
+ have holo: "f holomorphic_on A - pts" and "open A"
+ using assms by (auto simp: meromorphic_on_def)
+ have open': "open (A - (pts - {z}))"
+ by (intro meromorphic_imp_open_diff'[OF mero]) auto
+ then obtain r where r: "r > 0" "ball z r \<subseteq> A - (pts - {z})"
+ using meromorphic_imp_open_diff[OF mero] \<open>z \<in> A\<close> openE by blast
+
+ have "not_essential f z"
+ using assms by (rule meromorphic_on_imp_not_essential)
+ then consider c where "f \<midarrow>z\<rightarrow> c" | "is_pole f z"
+ unfolding not_essential_def by blast
+ thus False
+ proof cases
+ assume "is_pole f z"
+ hence "eventually (\<lambda>w. f w \<noteq> 0) (at z)"
+ by (rule non_zero_neighbour_pole)
+ hence "\<not>z islimpt {w. f w = 0}"
+ by (simp add: islimpt_conv_frequently_at frequently_def)
+ moreover have "z islimpt {w. f w = 0}"
+ using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def)
+ ultimately show False by contradiction
+ next
+ fix c assume c: "f \<midarrow>z\<rightarrow> c"
+ define g where "g = (\<lambda>w. if w = z then c else f w)"
+ have holo': "g holomorphic_on A - (pts - {z})" unfolding g_def
+ by (intro removable_singularity holomorphic_on_subset[OF holo] open' c) auto
+
+ have eq_zero: "g w = 0" if "w \<in> ball z r" for w
+ proof (rule analytic_continuation[where f = g])
+ show "open (ball z r)" "connected (ball z r)" "{w\<in>ball z r. isolated_zero f w} \<subseteq> ball z r"
+ by auto
+ have "z islimpt {w\<in>A. isolated_zero f w} \<inter> ball z r"
+ using islimpt \<open>r > 0\<close> by (intro islimpt_Int_eventually eventually_at_in_open') auto
+ also have "\<dots> = {w\<in>ball z r. isolated_zero f w}"
+ using r by auto
+ finally show "z islimpt {w\<in>ball z r. isolated_zero f w}"
+ by simp
+ next
+ fix w assume w: "w \<in> {w\<in>ball z r. isolated_zero f w}"
+ show "g w = 0"
+ proof (cases "w = z")
+ case False
+ thus ?thesis using w by (auto simp: g_def isolated_zero_def)
+ next
+ case True
+ have "z islimpt {z. f z = 0}"
+ using islimpt by (rule islimpt_subset) (auto simp: isolated_zero_def)
+ thus ?thesis
+ using w by (simp add: isolated_zero_altdef True)
+ qed
+ qed (use r that in \<open>auto intro!: holomorphic_on_subset[OF holo'] simp: isolated_zero_def\<close>)
+
+ have "infinite ({w\<in>A. isolated_zero f w} \<inter> ball z r)"
+ using islimpt \<open>r > 0\<close> unfolding islimpt_eq_infinite_ball by blast
+ hence "{w\<in>A. isolated_zero f w} \<inter> ball z r \<noteq> {}"
+ by force
+ then obtain z0 where z0: "z0 \<in> A" "isolated_zero f z0" "z0 \<in> ball z r"
+ by blast
+ have "\<forall>\<^sub>F y in at z0. y \<in> ball z r - (if z = z0 then {} else {z}) - {z0}"
+ using r z0 by (intro eventually_at_in_open) auto
+ hence "eventually (\<lambda>w. f w = 0) (at z0)"
+ proof eventually_elim
+ case (elim w)
+ show ?case
+ using eq_zero[of w] elim by (auto simp: g_def split: if_splits)
+ qed
+ hence "eventually (\<lambda>w. f w = 0) (at z0)"
+ by (auto simp: g_def eventually_at_filter elim!: eventually_mono split: if_splits)
+ moreover from z0 have "eventually (\<lambda>w. f w \<noteq> 0) (at z0)"
+ by (simp add: isolated_zero_def)
+ ultimately have "eventually (\<lambda>_. False) (at z0)"
+ by eventually_elim auto
+ thus False
+ by simp
+ qed
+qed
+
+lemma closedin_isolated_zeros:
+ assumes "f meromorphic_on A pts"
+ shows "closedin (top_of_set A) {z\<in>A. isolated_zero f z}"
+ unfolding closedin_limpt using not_islimpt_isolated_zeros[OF assms] by auto
+
+lemma meromorphic_on_deriv':
+ assumes "f meromorphic_on A pts" "open A"
+ assumes "\<And>x. x \<in> A - pts \<Longrightarrow> (f has_field_derivative f' x) (at x)"
+ shows "f' meromorphic_on A pts"
+ unfolding meromorphic_on_def
+proof (intro conjI ballI)
+ have "open (A - pts)"
+ by (intro meromorphic_imp_open_diff[OF assms(1)])
+ thus "f' holomorphic_on A - pts"
+ by (rule derivative_is_holomorphic) (use assms in auto)
+next
+ fix z assume "z \<in> pts"
+ hence "z \<in> A"
+ using assms(1) by (auto simp: meromorphic_on_def)
+ from \<open>z \<in> pts\<close> obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
+ using assms(1) by (auto simp: meromorphic_on_def isolated_singularity_at_def)
+
+ have "open (ball z r \<inter> (A - (pts - {z})))"
+ by (intro open_Int assms meromorphic_imp_open_diff'[OF assms(1)]) auto
+ then obtain r' where r': "r' > 0" "ball z r' \<subseteq> ball z r \<inter> (A - (pts - {z}))"
+ using r \<open>z \<in> A\<close> by (subst (asm) open_contains_ball) fastforce
+
+ have "open (ball z r' - {z})"
+ by auto
+ hence "f' holomorphic_on ball z r' - {z}"
+ by (rule derivative_is_holomorphic[of _ f]) (use r' in \<open>auto intro!: assms(3)\<close>)
+ moreover have "open (ball z r' - {z})"
+ by auto
+ ultimately show "isolated_singularity_at f' z"
+ unfolding isolated_singularity_at_def using \<open>r' > 0\<close>
+ by (auto simp: analytic_on_open intro!: exI[of _ r'])
+next
+ fix z assume z: "z \<in> pts"
+ hence z': "not_essential f z" "z \<in> A"
+ using assms by (auto simp: meromorphic_on_def)
+ from z'(1) show "not_essential f' z"
+ proof (rule not_essential_deriv')
+ show "z \<in> A - (pts - {z})"
+ using \<open>z \<in> A\<close> by blast
+ show "open (A - (pts - {z}))"
+ by (intro meromorphic_imp_open_diff'[OF assms(1)]) auto
+ qed (use assms in auto)
+qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma meromorphic_on_deriv [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "open A"
+ shows "deriv f meromorphic_on A pts"
+proof (intro meromorphic_on_deriv'[OF assms(1)])
+ have *: "open (A - pts)"
+ by (intro meromorphic_imp_open_diff[OF assms(1)])
+ show "(f has_field_derivative deriv f x) (at x)" if "x \<in> A - pts" for x
+ using assms(1) by (intro holomorphic_derivI[OF _ * that]) (auto simp: meromorphic_on_def)
+qed fact
+
+lemma meromorphic_on_imp_analytic_at:
+ assumes "f meromorphic_on A pts" "z \<in> A - pts"
+ shows "f analytic_on {z}"
+ using assms by (metis analytic_at meromorphic_imp_open_diff meromorphic_on_def)
+
+lemma meromorphic_compact_finite_pts:
+ assumes "f meromorphic_on D pts" "compact S" "S \<subseteq> D"
+ shows "finite (S \<inter> pts)"
+proof -
+ { assume "infinite (S \<inter> pts)"
+ then obtain z where "z \<in> S" and z: "z islimpt (S \<inter> pts)"
+ using assms by (metis compact_eq_Bolzano_Weierstrass inf_le1)
+ then have False
+ using assms by (meson in_mono inf_le2 islimpt_subset meromorphic_on_def) }
+ then show ?thesis by metis
+qed
+
+lemma meromorphic_imp_countable:
+ assumes "f meromorphic_on D pts"
+ shows "countable pts"
+proof -
+ obtain K :: "nat \<Rightarrow> complex set" where K: "D = (\<Union>n. K n)" "\<And>n. compact (K n)"
+ using assms unfolding meromorphic_on_def by (metis open_Union_compact_subsets)
+ then have "pts = (\<Union>n. K n \<inter> pts)"
+ using assms meromorphic_on_def by auto
+ moreover have "\<And>n. finite (K n \<inter> pts)"
+ by (metis K(1) K(2) UN_I assms image_iff meromorphic_compact_finite_pts rangeI subset_eq)
+ ultimately show ?thesis
+ by (metis countableI_type countable_UN countable_finite)
+qed
+
+lemma meromorphic_imp_connected_diff':
+ assumes "f meromorphic_on D pts" "connected D" "pts' \<subseteq> pts"
+ shows "connected (D - pts')"
+proof (rule connected_open_diff_countable)
+ show "countable pts'"
+ by (rule countable_subset [OF assms(3)]) (use assms(1) in \<open>auto simp: meromorphic_imp_countable\<close>)
+qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma meromorphic_imp_connected_diff:
+ assumes "f meromorphic_on D pts" "connected D"
+ shows "connected (D - pts)"
+ using meromorphic_imp_connected_diff'[OF assms order.refl] .
+
+lemma meromorphic_on_compose [meromorphic_intros]:
+ assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B"
+ assumes "open B" and "g ` B \<subseteq> A"
+ shows "(\<lambda>x. f (g x)) meromorphic_on B (isolated_points_of (g -` pts \<inter> B))"
+ unfolding meromorphic_on_def
+proof (intro ballI conjI)
+ fix z assume z: "z \<in> isolated_points_of (g -` pts \<inter> B)"
+ hence z': "z \<in> B" "g z \<in> pts"
+ using isolated_points_of_subset by blast+
+ have g': "g analytic_on {z}"
+ using g z' \<open>open B\<close> analytic_at by blast
+
+ show "isolated_singularity_at (\<lambda>x. f (g x)) z"
+ by (rule isolated_singularity_at_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>)
+ show "not_essential (\<lambda>x. f (g x)) z"
+ by (rule not_essential_compose[OF _ g']) (use f z' in \<open>auto simp: meromorphic_on_def\<close>)
+next
+ fix z assume z: "z \<in> B"
+ hence "g z \<in> A"
+ using assms by auto
+ hence "\<not>g z islimpt pts"
+ using f by (auto simp: meromorphic_on_def)
+ hence ev: "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
+ by (auto simp: islimpt_conv_frequently_at frequently_def)
+ have g': "g analytic_on {z}"
+ by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto)
+
+ (* TODO: There's probably a useful lemma somewhere in here to extract... *)
+ have "eventually (\<lambda>w. w \<notin> isolated_points_of (g -` pts \<inter> B)) (at z)"
+ proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
+ case True
+ have "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
+ using ev by (auto simp: islimpt_conv_frequently_at frequently_def)
+ moreover have "g \<midarrow>z\<rightarrow> g z"
+ using analytic_at_imp_isCont[OF g'] isContD by blast
+ hence lim: "filterlim g (at (g z)) (at z)"
+ using True by (auto simp: filterlim_at isolated_zero_def)
+ have "eventually (\<lambda>w. g w \<notin> pts) (at z)"
+ using ev lim by (rule eventually_compose_filterlim)
+ thus ?thesis
+ by eventually_elim (auto simp: isolated_points_of_def)
+ next
+ case False
+ have "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
+ using False by (rule non_isolated_zero) (auto intro!: analytic_intros g')
+ hence "eventually (\<lambda>w. g w = g z \<and> w \<in> B) (nhds z)"
+ using eventually_nhds_in_open[OF \<open>open B\<close> \<open>z \<in> B\<close>]
+ by eventually_elim auto
+ then obtain X where X: "open X" "z \<in> X" "X \<subseteq> B" "\<forall>x\<in>X. g x = g z"
+ unfolding eventually_nhds by blast
+
+ have "z0 \<notin> isolated_points_of (g -` pts \<inter> B)" if "z0 \<in> X" for z0
+ proof (cases "g z \<in> pts")
+ case False
+ with that have "g z0 \<notin> pts"
+ using X by metis
+ thus ?thesis
+ by (auto simp: isolated_points_of_def)
+ next
+ case True
+ have "eventually (\<lambda>w. w \<in> X) (at z0)"
+ by (intro eventually_at_in_open') fact+
+ hence "eventually (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)"
+ by eventually_elim (use X True in fastforce)
+ hence "frequently (\<lambda>w. w \<in> g -` pts \<inter> B) (at z0)"
+ by (meson at_neq_bot eventually_frequently)
+ thus "z0 \<notin> isolated_points_of (g -` pts \<inter> B)"
+ unfolding isolated_points_of_def by (auto simp: frequently_def)
+ qed
+ moreover have "eventually (\<lambda>x. x \<in> X) (at z)"
+ by (intro eventually_at_in_open') fact+
+ ultimately show ?thesis
+ by (auto elim!: eventually_mono)
+ qed
+ thus "\<not>z islimpt isolated_points_of (g -` pts \<inter> B)"
+ by (auto simp: islimpt_conv_frequently_at frequently_def)
+next
+ have "f \<circ> g analytic_on (\<Union>z\<in>B - isolated_points_of (g -` pts \<inter> B). {z})"
+ unfolding analytic_on_UN
+ proof
+ fix z assume z: "z \<in> B - isolated_points_of (g -` pts \<inter> B)"
+ hence "z \<in> B" by blast
+ have g': "g analytic_on {z}"
+ by (rule holomorphic_on_imp_analytic_at[OF g]) (use assms z in auto)
+ show "f \<circ> g analytic_on {z}"
+ proof (cases "g z \<in> pts")
+ case False
+ show ?thesis
+ proof (rule analytic_on_compose)
+ show "f analytic_on g ` {z}" using False z assms
+ by (auto intro!: meromorphic_on_imp_analytic_at[OF f])
+ qed fact
+ next
+ case True
+ show ?thesis
+ proof (cases "isolated_zero (\<lambda>w. g w - g z) z")
+ case False
+ hence "eventually (\<lambda>w. g w - g z = 0) (nhds z)"
+ by (rule non_isolated_zero) (auto intro!: analytic_intros g')
+ hence "f \<circ> g analytic_on {z} \<longleftrightarrow> (\<lambda>_. f (g z)) analytic_on {z}"
+ by (intro analytic_at_cong) (auto elim!: eventually_mono)
+ thus ?thesis
+ by simp
+ next
+ case True
+ hence ev: "eventually (\<lambda>w. g w \<noteq> g z) (at z)"
+ by (auto simp: isolated_zero_def)
+
+ have "\<not>g z islimpt pts"
+ using \<open>g z \<in> pts\<close> f by (auto simp: meromorphic_on_def)
+ hence "eventually (\<lambda>w. w \<notin> pts) (at (g z))"
+ by (auto simp: islimpt_conv_frequently_at frequently_def)
+ moreover have "g \<midarrow>z\<rightarrow> g z"
+ using analytic_at_imp_isCont[OF g'] isContD by blast
+ with ev have "filterlim g (at (g z)) (at z)"
+ by (auto simp: filterlim_at)
+ ultimately have "eventually (\<lambda>w. g w \<notin> pts) (at z)"
+ using eventually_compose_filterlim by blast
+ hence "z \<in> isolated_points_of (g -` pts \<inter> B)"
+ using \<open>g z \<in> pts\<close> \<open>z \<in> B\<close>
+ by (auto simp: isolated_points_of_def elim!: eventually_mono)
+ with z show ?thesis by simp
+ qed
+ qed
+ qed
+ also have "\<dots> = B - isolated_points_of (g -` pts \<inter> B)"
+ by blast
+ finally show "(\<lambda>x. f (g x)) holomorphic_on B - isolated_points_of (g -` pts \<inter> B)"
+ unfolding o_def using analytic_imp_holomorphic by blast
+qed (auto simp: isolated_points_of_def \<open>open B\<close>)
+
+lemma meromorphic_on_compose':
+ assumes f: "f meromorphic_on A pts" and g: "g holomorphic_on B"
+ assumes "open B" and "g ` B \<subseteq> A" and "pts' = (isolated_points_of (g -` pts \<inter> B))"
+ shows "(\<lambda>x. f (g x)) meromorphic_on B pts'"
+ using meromorphic_on_compose[OF assms(1-4)] assms(5) by simp
+
+lemma meromorphic_on_inverse': "inverse meromorphic_on UNIV 0"
+ unfolding meromorphic_on_def
+ by (auto intro!: holomorphic_intros singularity_intros not_essential_inverse
+ isolated_singularity_at_inverse simp: islimpt_finite)
+
+lemma meromorphic_on_inverse [meromorphic_intros]:
+ assumes mero: "f meromorphic_on A pts"
+ shows "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})"
+proof -
+ have "open A"
+ using mero by (auto simp: meromorphic_on_def)
+ have open': "open (A - pts)"
+ by (intro meromorphic_imp_open_diff[OF mero])
+ have holo: "f holomorphic_on A - pts"
+ using assms by (auto simp: meromorphic_on_def)
+ have ana: "f analytic_on A - pts"
+ using open' holo by (simp add: analytic_on_open)
+
+ show ?thesis
+ unfolding meromorphic_on_def
+ proof (intro conjI ballI)
+ fix z assume z: "z \<in> pts \<union> {z\<in>A. isolated_zero f z}"
+ have "isolated_singularity_at f z \<and> not_essential f z"
+ proof (cases "z \<in> pts")
+ case False
+ have "f holomorphic_on A - pts - {z}"
+ by (intro holomorphic_on_subset[OF holo]) auto
+ hence "isolated_singularity_at f z"
+ by (rule isolated_singularity_at_holomorphic)
+ (use z False in \<open>auto intro!: meromorphic_imp_open_diff[OF mero]\<close>)
+ moreover have "not_essential f z"
+ using z False
+ by (intro not_essential_holomorphic[OF holo] meromorphic_imp_open_diff[OF mero]) auto
+ ultimately show ?thesis by blast
+ qed (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+ thus "isolated_singularity_at (\<lambda>z. inverse (f z)) z" "not_essential (\<lambda>z. inverse (f z)) z"
+ by (auto intro!: isolated_singularity_at_inverse not_essential_inverse)
+ next
+ fix z assume "z \<in> A"
+ hence "\<not> z islimpt {z\<in>A. isolated_zero f z}"
+ by (rule not_islimpt_isolated_zeros[OF mero])
+ thus "\<not> z islimpt pts \<union> {z \<in> A. isolated_zero f z}" using \<open>z \<in> A\<close>
+ using mero by (auto simp: islimpt_Un meromorphic_on_def)
+ next
+ show "pts \<union> {z \<in> A. isolated_zero f z} \<subseteq> A"
+ using mero by (auto simp: meromorphic_on_def)
+ next
+ have "(\<lambda>z. inverse (f z)) analytic_on (\<Union>w\<in>A - (pts \<union> {z \<in> A. isolated_zero f z}) . {w})"
+ unfolding analytic_on_UN
+ proof (intro ballI)
+ fix w assume w: "w \<in> A - (pts \<union> {z \<in> A. isolated_zero f z})"
+ show "(\<lambda>z. inverse (f z)) analytic_on {w}"
+ proof (cases "f w = 0")
+ case False
+ thus ?thesis using w
+ by (intro analytic_intros analytic_on_subset[OF ana]) auto
+ next
+ case True
+ have "eventually (\<lambda>w. f w = 0) (nhds w)"
+ using True w by (intro non_isolated_zero analytic_on_subset[OF ana]) auto
+ hence "(\<lambda>z. inverse (f z)) analytic_on {w} \<longleftrightarrow> (\<lambda>_. 0) analytic_on {w}"
+ using w by (intro analytic_at_cong refl) auto
+ thus ?thesis
+ by simp
+ qed
+ qed
+ also have "\<dots> = A - (pts \<union> {z \<in> A. isolated_zero f z})"
+ by blast
+ finally have "(\<lambda>z. inverse (f z)) analytic_on \<dots>" .
+ moreover have "open (A - (pts \<union> {z \<in> A. isolated_zero f z}))"
+ using closedin_isolated_zeros[OF mero] open' \<open>open A\<close>
+ by (metis (no_types, lifting) Diff_Diff_Int Diff_Un closedin_closed open_Diff open_Int)
+ ultimately show "(\<lambda>z. inverse (f z)) holomorphic_on A - (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (subst (asm) analytic_on_open) auto
+ qed (use assms in \<open>auto simp: meromorphic_on_def islimpt_Un
+ intro!: holomorphic_intros singularity_intros\<close>)
+qed
+
+lemma meromorphic_on_inverse'' [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "{z\<in>A. f z = 0} \<subseteq> pts"
+ shows "(\<lambda>z. inverse (f z)) meromorphic_on A pts"
+proof -
+ have "(\<lambda>z. inverse (f z)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (intro meromorphic_on_inverse assms)
+ also have "(pts \<union> {z \<in> A. isolated_zero f z}) = pts"
+ using assms(2) by (auto simp: isolated_zero_def)
+ finally show ?thesis .
+qed
+
+lemma meromorphic_on_divide [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" and "g meromorphic_on A pts"
+ shows "(\<lambda>z. f z / g z) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
+proof -
+ have mero1: "(\<lambda>z. inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
+ by (intro meromorphic_intros assms)
+ have sparse: "\<forall>x\<in>A. \<not> x islimpt pts \<union> {z\<in>A. isolated_zero g z}" and "pts \<subseteq> A"
+ using mero1 by (auto simp: meromorphic_on_def)
+ have mero2: "f meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
+ by (rule meromorphic_on_superset_pts[OF assms(1)]) (use sparse \<open>pts \<subseteq> A\<close> in auto)
+ have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero g z})"
+ by (intro meromorphic_on_mult mero1 mero2)
+ thus ?thesis
+ by (simp add: field_simps)
+qed
+
+lemma meromorphic_on_divide' [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "g meromorphic_on A pts" "{z\<in>A. g z = 0} \<subseteq> pts"
+ shows "(\<lambda>z. f z / g z) meromorphic_on A pts"
+proof -
+ have "(\<lambda>z. f z * inverse (g z)) meromorphic_on A pts"
+ by (intro meromorphic_intros assms)
+ thus ?thesis
+ by (simp add: field_simps)
+qed
+
+lemma meromorphic_on_cmult_left [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>x. c * f x) meromorphic_on A pts"
+ using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
+
+lemma meromorphic_on_cmult_right [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>x. f x * c) meromorphic_on A pts"
+ using assms by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
+
+lemma meromorphic_on_scaleR [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>x. c *\<^sub>R f x) meromorphic_on A pts"
+ using assms unfolding scaleR_conv_of_real
+ by (intro meromorphic_intros) (auto simp: meromorphic_on_def)
+
+lemma meromorphic_on_sum [meromorphic_intros]:
+ assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts"
+ assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
+ shows "(\<lambda>x. \<Sum>y\<in>I. f y x) meromorphic_on A pts"
+proof -
+ have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
+ using assms(2)
+ proof
+ assume "I \<noteq> {}"
+ then obtain x where "x \<in> I"
+ by blast
+ from assms(1)[OF this] show ?thesis
+ by (auto simp: meromorphic_on_def)
+ qed auto
+ show ?thesis
+ using assms(1)
+ by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>)
+qed
+
+lemma meromorphic_on_prod [meromorphic_intros]:
+ assumes "\<And>y. y \<in> I \<Longrightarrow> f y meromorphic_on A pts"
+ assumes "I \<noteq> {} \<or> open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
+ shows "(\<lambda>x. \<Prod>y\<in>I. f y x) meromorphic_on A pts"
+proof -
+ have *: "open A \<and> pts \<subseteq> A \<and> (\<forall>x\<in>A. \<not>x islimpt pts)"
+ using assms(2)
+ proof
+ assume "I \<noteq> {}"
+ then obtain x where "x \<in> I"
+ by blast
+ from assms(1)[OF this] show ?thesis
+ by (auto simp: meromorphic_on_def)
+ qed auto
+ show ?thesis
+ using assms(1)
+ by (induction I rule: infinite_finite_induct) (use * in \<open>auto intro!: meromorphic_intros\<close>)
+qed
+
+lemma meromorphic_on_power [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>x. f x ^ n) meromorphic_on A pts"
+proof -
+ have "(\<lambda>x. \<Prod>i\<in>{..<n}. f x) meromorphic_on A pts"
+ by (intro meromorphic_intros assms(1)) (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+ thus ?thesis
+ by simp
+qed
+
+lemma meromorphic_on_power_int [meromorphic_intros]:
+ assumes "f meromorphic_on A pts"
+ shows "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+proof -
+ have inv: "(\<lambda>x. inverse (f x)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (intro meromorphic_intros assms)
+ have *: "f meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (intro meromorphic_on_superset_pts [OF assms(1)])
+ (use inv in \<open>auto simp: meromorphic_on_def\<close>)
+ show ?thesis
+ proof (cases "n \<ge> 0")
+ case True
+ have "(\<lambda>x. f x ^ nat n) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (intro meromorphic_intros *)
+ thus ?thesis
+ using True by (simp add: power_int_def)
+ next
+ case False
+ have "(\<lambda>x. inverse (f x) ^ nat (-n)) meromorphic_on A (pts \<union> {z \<in> A. isolated_zero f z})"
+ by (intro meromorphic_intros assms)
+ thus ?thesis
+ using False by (simp add: power_int_def)
+ qed
+qed
+
+lemma meromorphic_on_power_int' [meromorphic_intros]:
+ assumes "f meromorphic_on A pts" "n \<ge> 0 \<or> (\<forall>z\<in>A. isolated_zero f z \<longrightarrow> z \<in> pts)"
+ shows "(\<lambda>z. f z powi n) meromorphic_on A pts"
+proof (cases "n \<ge> 0")
+ case True
+ have "(\<lambda>z. f z ^ nat n) meromorphic_on A pts"
+ by (intro meromorphic_intros assms)
+ thus ?thesis
+ using True by (simp add: power_int_def)
+next
+ case False
+ have "(\<lambda>z. f z powi n) meromorphic_on A (pts \<union> {z\<in>A. isolated_zero f z})"
+ by (rule meromorphic_on_power_int) fact
+ also from assms(2) False have "pts \<union> {z\<in>A. isolated_zero f z} = pts"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma has_laurent_expansion_on_imp_meromorphic_on:
+ assumes "open A"
+ assumes laurent: "\<And>z. z \<in> A \<Longrightarrow> \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F"
+ shows "f meromorphic_on A {z\<in>A. \<not>f analytic_on {z}}"
+ unfolding meromorphic_on_def
+proof (intro conjI ballI)
+ fix z assume "z \<in> {z\<in>A. \<not>f analytic_on {z}}"
+ then obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ using laurent[of z] by blast
+ from F show "not_essential f z" "isolated_singularity_at f z"
+ using has_laurent_expansion_not_essential has_laurent_expansion_isolated by blast+
+next
+ fix z assume z: "z \<in> A"
+ obtain F where F: "(\<lambda>w. f (z + w)) has_laurent_expansion F"
+ using laurent[of z] \<open>z \<in> A\<close> by blast
+ from F have "isolated_singularity_at f z"
+ using has_laurent_expansion_isolated z by blast
+ then obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
+ unfolding isolated_singularity_at_def by blast
+ have "f analytic_on {w}" if "w \<in> ball z r - {z}" for w
+ by (rule analytic_on_subset[OF r(2)]) (use that in auto)
+ hence "eventually (\<lambda>w. f analytic_on {w}) (at z)"
+ using eventually_at_in_open[of "ball z r" z] \<open>r > 0\<close> by (auto elim!: eventually_mono)
+ hence "\<not>z islimpt {w. \<not>f analytic_on {w}}"
+ by (auto simp: islimpt_conv_frequently_at frequently_def)
+ thus "\<not>z islimpt {w\<in>A. \<not>f analytic_on {w}}"
+ using islimpt_subset[of z "{w\<in>A. \<not>f analytic_on {w}}" "{w. \<not>f analytic_on {w}}"] by blast
+next
+ have "f analytic_on A - {w\<in>A. \<not>f analytic_on {w}}"
+ by (subst analytic_on_analytic_at) auto
+ thus "f holomorphic_on A - {w\<in>A. \<not>f analytic_on {w}}"
+ by (meson analytic_imp_holomorphic)
+qed (use assms in auto)
+
+lemma meromorphic_on_imp_has_laurent_expansion:
+ assumes "f meromorphic_on A pts" "z \<in> A"
+ shows "(\<lambda>w. f (z + w)) has_laurent_expansion laurent_expansion f z"
+proof (cases "z \<in> pts")
+ case True
+ thus ?thesis
+ using assms by (intro not_essential_has_laurent_expansion) (auto simp: meromorphic_on_def)
+next
+ case False
+ have "f holomorphic_on (A - pts)"
+ using assms by (auto simp: meromorphic_on_def)
+ moreover have "z \<in> A - pts" "open (A - pts)"
+ using assms(2) False by (auto intro!: meromorphic_imp_open_diff[OF assms(1)])
+ ultimately have "f analytic_on {z}"
+ unfolding analytic_at by blast
+ thus ?thesis
+ using isolated_singularity_at_analytic not_essential_analytic
+ not_essential_has_laurent_expansion by blast
+qed
+
+lemma
+ assumes "isolated_singularity_at f z" "f \<midarrow>z\<rightarrow> c"
+ shows eventually_remove_sings_eq_nhds':
+ "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)"
+ and remove_sings_analytic_at_singularity: "remove_sings f analytic_on {z}"
+proof -
+ have "eventually (\<lambda>w. w \<noteq> z) (at z)"
+ by (auto simp: eventually_at_filter)
+ hence "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (at z)"
+ using eventually_remove_sings_eq_at[OF assms(1)]
+ by eventually_elim auto
+ moreover have "remove_sings f z = c"
+ using assms by auto
+ ultimately show ev: "eventually (\<lambda>w. remove_sings f w = (if w = z then c else f w)) (nhds z)"
+ by (simp add: eventually_at_filter)
+
+ have "(\<lambda>w. if w = z then c else f w) analytic_on {z}"
+ by (intro removable_singularity' assms)
+ also have "?this \<longleftrightarrow> remove_sings f analytic_on {z}"
+ using ev by (intro analytic_at_cong) (auto simp: eq_commute)
+ finally show \<dots> .
+qed
+
+lemma remove_sings_meromorphic_on:
+ assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts - pts' \<Longrightarrow> \<not>is_pole f z" "pts' \<subseteq> pts"
+ shows "remove_sings f meromorphic_on A pts'"
+ unfolding meromorphic_on_def
+proof safe
+ have "remove_sings f analytic_on {z}" if "z \<in> A - pts'" for z
+ proof (cases "z \<in> pts")
+ case False
+ hence *: "f analytic_on {z}"
+ using assms meromorphic_imp_open_diff[OF assms(1)] that
+ by (force simp: meromorphic_on_def analytic_at)
+ have "remove_sings f analytic_on {z} \<longleftrightarrow> f analytic_on {z}"
+ by (intro analytic_at_cong eventually_remove_sings_eq_nhds * refl)
+ thus ?thesis using * by simp
+ next
+ case True
+ have isol: "isolated_singularity_at f z"
+ using True using assms by (auto simp: meromorphic_on_def)
+ from assms(1) have "not_essential f z"
+ using True by (auto simp: meromorphic_on_def)
+ with assms(2) True that obtain c where "f \<midarrow>z\<rightarrow> c"
+ by (auto simp: not_essential_def)
+ thus "remove_sings f analytic_on {z}"
+ by (intro remove_sings_analytic_at_singularity isol)
+ qed
+ hence "remove_sings f analytic_on A - pts'"
+ by (subst analytic_on_analytic_at) auto
+ thus "remove_sings f holomorphic_on A - pts'"
+ using meromorphic_imp_open_diff'[OF assms(1,3)] by (subst (asm) analytic_on_open)
+qed (use assms islimpt_subset[OF _ assms(3)] in \<open>auto simp: meromorphic_on_def\<close>)
+
+lemma remove_sings_holomorphic_on:
+ assumes "f meromorphic_on A pts" "\<And>z. z \<in> pts \<Longrightarrow> \<not>is_pole f z"
+ shows "remove_sings f holomorphic_on A"
+ using remove_sings_meromorphic_on[OF assms(1), of "{}"] assms(2)
+ by (auto simp: meromorphic_on_no_singularities)
+
+lemma meromorphic_on_Ex_iff:
+ "(\<exists>pts. f meromorphic_on A pts) \<longleftrightarrow>
+ open A \<and> (\<forall>z\<in>A. \<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F)"
+proof safe
+ fix pts assume *: "f meromorphic_on A pts"
+ from * show "open A"
+ by (auto simp: meromorphic_on_def)
+ show "\<exists>F. (\<lambda>w. f (z + w)) has_laurent_expansion F" if "z \<in> A" for z
+ using that *
+ by (intro exI[of _ "laurent_expansion f z"] meromorphic_on_imp_has_laurent_expansion)
+qed (blast intro!: has_laurent_expansion_on_imp_meromorphic_on)
+
+lemma is_pole_inverse_holomorphic_pts:
+ fixes pts::"complex set" and f::"complex \<Rightarrow> complex"
+ defines "g \<equiv> \<lambda>x. (if x\<in>pts then 0 else inverse (f x))"
+ assumes mer: "f meromorphic_on D pts"
+ and non_z: "\<And>z. z \<in> D - pts \<Longrightarrow> f z \<noteq> 0"
+ and all_poles:"\<forall>x. is_pole f x \<longleftrightarrow> x\<in>pts"
+ shows "g holomorphic_on D"
+proof -
+ have "open D" and f_holo: "f holomorphic_on (D-pts)"
+ using mer by (auto simp: meromorphic_on_def)
+ have "\<exists>r. r>0 \<and> f analytic_on ball z r - {z}
+ \<and> (\<forall>x \<in> ball z r - {z}. f x\<noteq>0)" if "z\<in>pts" for z
+ proof -
+ have "isolated_singularity_at f z" "is_pole f z"
+ using mer meromorphic_on_def that all_poles by blast+
+ then obtain r1 where "r1>0" and fan: "f analytic_on ball z r1 - {z}"
+ by (meson isolated_singularity_at_def)
+ obtain r2 where "r2>0" "\<forall>x \<in> ball z r2 - {z}. f x\<noteq>0"
+ using non_zero_neighbour_pole[OF \<open>is_pole f z\<close>]
+ unfolding eventually_at by (metis Diff_iff UNIV_I dist_commute insertI1 mem_ball)
+ define r where "r = min r1 r2"
+ have "r>0" by (simp add: \<open>0 < r2\<close> \<open>r1>0\<close> r_def)
+ moreover have "f analytic_on ball z r - {z}"
+ using r_def by (force intro: analytic_on_subset [OF fan])
+ moreover have "\<forall>x \<in> ball z r - {z}. f x\<noteq>0"
+ by (simp add: \<open>\<forall>x\<in>ball z r2 - {z}. f x \<noteq> 0\<close> r_def)
+ ultimately show ?thesis by auto
+ qed
+ then obtain get_r where r_pos:"get_r z>0"
+ and r_ana:"f analytic_on ball z (get_r z) - {z}"
+ and r_nz:"\<forall>x \<in> ball z (get_r z) - {z}. f x\<noteq>0"
+ if "z\<in>pts" for z
+ by metis
+ define p_balls where "p_balls \<equiv> \<Union>z\<in>pts. ball z (get_r z)"
+ have g_ball:"g holomorphic_on ball z (get_r z)" if "z\<in>pts" for z
+ proof -
+ have "(\<lambda>x. if x = z then 0 else inverse (f x)) holomorphic_on ball z (get_r z)"
+ proof (rule is_pole_inverse_holomorphic)
+ show "f holomorphic_on ball z (get_r z) - {z}"
+ using analytic_imp_holomorphic r_ana that by blast
+ show "is_pole f z"
+ using mer meromorphic_on_def that all_poles by force
+ show "\<forall>x\<in>ball z (get_r z) - {z}. f x \<noteq> 0"
+ using r_nz that by metis
+ qed auto
+ then show ?thesis unfolding g_def
+ by (smt (verit, ccfv_SIG) Diff_iff Elementary_Metric_Spaces.open_ball
+ all_poles analytic_imp_holomorphic empty_iff
+ holomorphic_transform insert_iff not_is_pole_holomorphic
+ open_delete r_ana that)
+ qed
+ then have "g holomorphic_on p_balls"
+ proof -
+ have "g analytic_on p_balls"
+ unfolding p_balls_def analytic_on_UN
+ using g_ball by (simp add: analytic_on_open)
+ moreover have "open p_balls" using p_balls_def by blast
+ ultimately show ?thesis
+ by (simp add: analytic_imp_holomorphic)
+ qed
+ moreover have "g holomorphic_on D-pts"
+ proof -
+ have "(\<lambda>z. inverse (f z)) holomorphic_on D - pts"
+ using f_holo holomorphic_on_inverse non_z by blast
+ then show ?thesis
+ by (metis DiffD2 g_def holomorphic_transform)
+ qed
+ moreover have "open p_balls"
+ using p_balls_def by blast
+ ultimately have "g holomorphic_on (p_balls \<union> (D-pts))"
+ by (simp add: holomorphic_on_Un meromorphic_imp_open_diff[OF mer])
+ moreover have "D \<subseteq> p_balls \<union> (D-pts)"
+ unfolding p_balls_def using \<open>\<And>z. z \<in> pts \<Longrightarrow> 0 < get_r z\<close> by force
+ ultimately show "g holomorphic_on D" by (meson holomorphic_on_subset)
+qed
+
+lemma meromorphic_imp_analytic_on:
+ assumes "f meromorphic_on D pts"
+ shows "f analytic_on (D - pts)"
+ by (metis assms analytic_on_open meromorphic_imp_open_diff meromorphic_on_def)
+
+lemma meromorphic_imp_constant_on:
+ assumes merf: "f meromorphic_on D pts"
+ and "f constant_on (D - pts)"
+ and "\<forall>x\<in>pts. is_pole f x"
+ shows "f constant_on D"
+proof -
+ obtain c where c:"\<And>z. z \<in> D-pts \<Longrightarrow> f z = c"
+ by (meson assms constant_on_def)
+
+ have "f z = c" if "z \<in> D" for z
+ proof (cases "is_pole f z")
+ case True
+ then obtain r0 where "r0 > 0" and r0: "f analytic_on ball z r0 - {z}" and pol: "is_pole f z"
+ using merf unfolding meromorphic_on_def isolated_singularity_at_def
+ by (metis \<open>z \<in> D\<close> insert_Diff insert_Diff_if insert_iff merf
+ meromorphic_imp_open_diff not_is_pole_holomorphic)
+ have "open D"
+ using merf meromorphic_on_def by auto
+ then obtain r where "r > 0" "ball z r \<subseteq> D" "r \<le> r0"
+ by (smt (verit, best) \<open>0 < r0\<close> \<open>z \<in> D\<close> openE order_subst2 subset_ball)
+ have r: "f analytic_on ball z r - {z}"
+ by (meson Diff_mono \<open>r \<le> r0\<close> analytic_on_subset order_refl r0 subset_ball)
+ have "ball z r - {z} \<subseteq> -pts"
+ using merf r unfolding meromorphic_on_def
+ by (meson ComplI Elementary_Metric_Spaces.open_ball
+ analytic_imp_holomorphic assms(3) not_is_pole_holomorphic open_delete subsetI)
+ with \<open>ball z r \<subseteq> D\<close> have "ball z r - {z} \<subseteq> D-pts"
+ by fastforce
+ with c have c': "\<And>u. u \<in> ball z r - {z} \<Longrightarrow> f u = c"
+ by blast
+ have False if "\<forall>\<^sub>F x in at z. cmod c + 1 \<le> cmod (f x)"
+ proof -
+ have "\<forall>\<^sub>F x in at z within ball z r - {z}. cmod c + 1 \<le> cmod (f x)"
+ by (smt (verit, best) Diff_UNIV Diff_eq_empty_iff eventually_at_topological insert_subset that)
+ with \<open>r > 0\<close> show ?thesis
+ apply (simp add: c' eventually_at_filter topological_space_class.eventually_nhds open_dist)
+ by (metis dist_commute min_less_iff_conj perfect_choose_dist)
+ qed
+ with pol show ?thesis
+ by (auto simp: is_pole_def filterlim_at_infinity_conv_norm_at_top filterlim_at_top)
+ next
+ case False
+ then show ?thesis by (meson DiffI assms(3) c that)
+ qed
+ then show ?thesis
+ by (simp add: constant_on_def)
+qed
+
+
+lemma meromorphic_isolated:
+ assumes merf: "f meromorphic_on D pts" and "p\<in>pts"
+ obtains r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}"
+proof -
+ have "\<forall>z\<in>D. \<exists>e>0. finite (pts \<inter> ball z e)"
+ using merf unfolding meromorphic_on_def islimpt_eq_infinite_ball
+ by auto
+ then obtain r0 where r0:"r0>0" "finite (pts \<inter> ball p r0)"
+ by (metis assms(2) in_mono merf meromorphic_on_def)
+ moreover define pts' where "pts' = pts \<inter> ball p r0 - {p}"
+ ultimately have "finite pts'"
+ by simp
+
+ define r1 where "r1=(if pts'={} then r0 else
+ min (Min {dist p' p |p'. p'\<in>pts'}/2) r0)"
+ have "r1>0 \<and> pts \<inter> ball p r1 - {p} = {}"
+ proof (cases "pts'={}")
+ case True
+ then show ?thesis
+ using pts'_def r0(1) r1_def by presburger
+ next
+ case False
+ define S where "S={dist p' p |p'. p'\<in>pts'}"
+
+ have nempty:"S \<noteq> {}"
+ using False S_def by blast
+ have finite:"finite S"
+ using \<open>finite pts'\<close> S_def by simp
+
+ have "r1>0"
+ proof -
+ have "r1=min (Min S/2) r0"
+ using False unfolding S_def r1_def by auto
+ moreover have "Min S\<in>S"
+ using \<open>S\<noteq>{}\<close> \<open>finite S\<close> Min_in by auto
+ then have "Min S>0" unfolding S_def
+ using pts'_def by force
+ ultimately show ?thesis using \<open>r0>0\<close> by auto
+ qed
+ moreover have "pts \<inter> ball p r1 - {p} = {}"
+ proof (rule ccontr)
+ assume "pts \<inter> ball p r1 - {p} \<noteq> {}"
+ then obtain p' where "p'\<in>pts \<inter> ball p r1 - {p}" by blast
+ moreover have "r1\<le>r0" using r1_def by auto
+ ultimately have "p'\<in>pts'" unfolding pts'_def
+ by auto
+ then have "dist p' p\<ge>Min S"
+ using S_def eq_Min_iff local.finite by blast
+ moreover have "dist p' p < Min S"
+ using \<open>p'\<in>pts \<inter> ball p r1 - {p}\<close> False unfolding r1_def
+ apply (fold S_def)
+ by (smt (verit, ccfv_threshold) DiffD1 Int_iff dist_commute
+ dist_triangle_half_l mem_ball)
+ ultimately show False by auto
+ qed
+ ultimately show ?thesis by auto
+ qed
+ then have "r1>0" and r1_pts:"pts \<inter> ball p r1 - {p} = {}" by auto
+
+ obtain r2 where "r2>0" "ball p r2 \<subseteq> D"
+ by (metis assms(2) merf meromorphic_on_def openE subset_eq)
+ define r where "r=min r1 r2"
+ have "r > 0" unfolding r_def
+ by (simp add: \<open>0 < r1\<close> \<open>0 < r2\<close>)
+ moreover have "ball p r \<subseteq> D"
+ using \<open>ball p r2 \<subseteq> D\<close> r_def by auto
+ moreover have "ball p r \<inter> pts = {p}"
+ using assms(2) \<open>r>0\<close> r1_pts
+ unfolding r_def by auto
+ ultimately show ?thesis using that by auto
+qed
+
+lemma meromorphic_pts_closure:
+ assumes merf: "f meromorphic_on D pts"
+ shows "pts \<subseteq> closure (D - pts)"
+proof -
+ have "p islimpt (D - pts)" if "p\<in>pts" for p
+ proof -
+ obtain r where "r>0" "ball p r \<subseteq> D" "ball p r \<inter> pts = {p}"
+ using meromorphic_isolated[OF merf \<open>p\<in>pts\<close>] by auto
+ from \<open>r>0\<close>
+ have "p islimpt ball p r - {p}"
+ by (meson open_ball ball_subset_cball in_mono islimpt_ball
+ islimpt_punctured le_less open_contains_ball_eq)
+ moreover have " ball p r - {p} \<subseteq> D - pts"
+ using \<open>ball p r \<inter> pts = {p}\<close> \<open>ball p r \<subseteq> D\<close> by fastforce
+ ultimately show ?thesis
+ using islimpt_subset by auto
+ qed
+ then show ?thesis by (simp add: islimpt_in_closure subset_eq)
+qed
+
+lemma nconst_imp_nzero_neighbour:
+ assumes merf: "f meromorphic_on D pts"
+ and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
+ and "z\<in>D" and "connected D"
+ shows "(\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts)"
+proof -
+ obtain \<beta> where \<beta>:"\<beta> \<in> D - pts" "f \<beta>\<noteq>0"
+ using f_nconst by auto
+
+ have ?thesis if "z\<notin>pts"
+ proof -
+ have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts"
+ apply (rule non_zero_neighbour_alt[of f "D-pts" z \<beta>])
+ subgoal using merf meromorphic_on_def by blast
+ subgoal using merf meromorphic_imp_open_diff by auto
+ subgoal using assms(4) merf meromorphic_imp_connected_diff by blast
+ subgoal by (simp add: assms(3) that)
+ using \<beta> by auto
+ then show ?thesis by (auto elim:eventually_mono)
+ qed
+ moreover have ?thesis if "z\<in>pts" "\<not> f \<midarrow>z\<rightarrow> 0"
+ proof -
+ have "\<forall>\<^sub>F w in at z. w \<in> D - pts"
+ using merf[unfolded meromorphic_on_def islimpt_iff_eventually] \<open>z\<in>D\<close>
+ using eventually_at_in_open' eventually_elim2 by fastforce
+ moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
+ proof (cases "is_pole f z")
+ case True
+ then show ?thesis using non_zero_neighbour_pole by auto
+ next
+ case False
+ moreover have "not_essential f z"
+ using merf meromorphic_on_def that(1) by fastforce
+ ultimately obtain c where "c\<noteq>0" "f \<midarrow>z\<rightarrow> c"
+ by (metis \<open>\<not> f \<midarrow>z\<rightarrow> 0\<close> not_essential_def)
+ then show ?thesis
+ using tendsto_imp_eventually_ne by auto
+ qed
+ ultimately show ?thesis by eventually_elim auto
+ qed
+ moreover have ?thesis if "z\<in>pts" "f \<midarrow>z\<rightarrow> 0"
+ proof -
+ define ff where "ff=(\<lambda>x. if x=z then 0 else f x)"
+ define A where "A=D - (pts - {z})"
+
+ have "f holomorphic_on A - {z}"
+ by (metis A_def Diff_insert analytic_imp_holomorphic
+ insert_Diff merf meromorphic_imp_analytic_on that(1))
+ moreover have "open A"
+ using A_def merf meromorphic_imp_open_diff' by force
+ ultimately have "ff holomorphic_on A"
+ using \<open>f \<midarrow>z\<rightarrow> 0\<close> unfolding ff_def
+ by (rule removable_singularity)
+ moreover have "connected A"
+ proof -
+ have "connected (D - pts)"
+ using assms(4) merf meromorphic_imp_connected_diff by auto
+ moreover have "D - pts \<subseteq> A"
+ unfolding A_def by auto
+ moreover have "A \<subseteq> closure (D - pts)" unfolding A_def
+ by (smt (verit, ccfv_SIG) Diff_empty Diff_insert
+ closure_subset insert_Diff_single insert_absorb
+ insert_subset merf meromorphic_pts_closure that(1))
+ ultimately show ?thesis using connected_intermediate_closure
+ by auto
+ qed
+ moreover have "z \<in> A" using A_def assms(3) by blast
+ moreover have "ff z = 0" unfolding ff_def by auto
+ moreover have "\<beta> \<in> A " using A_def \<beta>(1) by blast
+ moreover have "ff \<beta> \<noteq> 0" using \<beta>(1) \<beta>(2) ff_def that(1) by auto
+ ultimately obtain r where "0 < r"
+ "ball z r \<subseteq> A" "\<And>x. x \<in> ball z r - {z} \<Longrightarrow> ff x \<noteq> 0"
+ using \<open>open A\<close> isolated_zeros[of ff A z \<beta>] by auto
+ then show ?thesis unfolding eventually_at ff_def
+ by (intro exI[of _ r]) (auto simp: A_def dist_commute ball_def)
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma nconst_imp_nzero_neighbour':
+ assumes merf: "f meromorphic_on D pts"
+ and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
+ and "z\<in>D" and "connected D"
+ shows "\<forall>\<^sub>F w in at z. f w \<noteq> 0"
+ using nconst_imp_nzero_neighbour[OF assms]
+ by (auto elim:eventually_mono)
+
+lemma meromorphic_compact_finite_zeros:
+ assumes merf:"f meromorphic_on D pts"
+ and "compact S" "S \<subseteq> D" "connected D"
+ and f_nconst:"\<not>(\<forall>w\<in>D-pts. f w=0)"
+ shows "finite ({x\<in>S. f x=0})"
+proof -
+ have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts})"
+ proof (rule ccontr)
+ assume "infinite {x \<in> S. f x = 0 \<and> x \<notin> pts}"
+ then obtain z where "z\<in>S" and z_lim:"z islimpt {x \<in> S. f x = 0
+ \<and> x \<notin> pts}"
+ using \<open>compact S\<close> unfolding compact_eq_Bolzano_Weierstrass
+ by auto
+
+ from z_lim
+ have "\<exists>\<^sub>F x in at z. f x = 0 \<and> x \<in> S \<and> x \<notin> pts"
+ unfolding islimpt_iff_eventually not_eventually by simp
+ moreover have "\<forall>\<^sub>F w in at z. f w \<noteq> 0 \<and> w \<in> D - pts"
+ using nconst_imp_nzero_neighbour[OF merf f_nconst _ \<open>connected D\<close>]
+ \<open>z\<in>S\<close> \<open>S \<subseteq> D\<close>
+ by auto
+ ultimately have "\<exists>\<^sub>F x in at z. False"
+ by (simp add: eventually_mono frequently_def)
+ then show False by auto
+ qed
+ moreover have "finite (S \<inter> pts)"
+ using meromorphic_compact_finite_pts[OF merf \<open>compact S\<close> \<open>S \<subseteq> D\<close>] .
+ ultimately have "finite ({x\<in>S. f x=0 \<and> x \<notin> pts} \<union> (S \<inter> pts))"
+ unfolding finite_Un by auto
+ then show ?thesis by (elim rev_finite_subset) auto
+qed
+
+lemma meromorphic_onI [intro?]:
+ assumes "open A" "pts \<subseteq> A"
+ assumes "f holomorphic_on A - pts" "\<And>z. z \<in> A \<Longrightarrow> \<not>z islimpt pts"
+ assumes "\<And>z. z \<in> pts \<Longrightarrow> isolated_singularity_at f z"
+ assumes "\<And>z. z \<in> pts \<Longrightarrow> not_essential f z"
+ shows "f meromorphic_on A pts"
+ using assms unfolding meromorphic_on_def by blast
+
+lemma Polygamma_plus_of_nat:
+ assumes "\<forall>k<m. z \<noteq> -of_nat k"
+ shows "Polygamma n (z + of_nat m) =
+ Polygamma n z + (-1) ^ n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n)"
+ using assms
+proof (induction m)
+ case (Suc m)
+ have "Polygamma n (z + of_nat (Suc m)) = Polygamma n (z + of_nat m + 1)"
+ by (simp add: add_ac)
+ also have "\<dots> = Polygamma n (z + of_nat m) + (-1) ^ n * fact n * (1 / ((z + of_nat m) ^ Suc n))"
+ using Suc.prems by (subst Polygamma_plus1) (auto simp: add_eq_0_iff2)
+ also have "Polygamma n (z + of_nat m) =
+ Polygamma n z + (-1) ^ n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n) * fact n"
+ using Suc.prems by (subst Suc.IH) auto
+ finally show ?case
+ by (simp add: algebra_simps)
+qed auto
+
+lemma tendsto_Gamma [tendsto_intros]:
+ assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "((\<lambda>z. Gamma (f z)) \<longlongrightarrow> Gamma c) F"
+ by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
+
+lemma tendsto_Polygamma [tendsto_intros]:
+ fixes f :: "_ \<Rightarrow> 'a :: {real_normed_field,euclidean_space}"
+ assumes "(f \<longlongrightarrow> c) F" "c \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "((\<lambda>z. Polygamma n (f z)) \<longlongrightarrow> Polygamma n c) F"
+ by (intro isCont_tendsto_compose[OF _ assms(1)] continuous_intros assms)
+
+lemma analytic_on_Gamma' [analytic_intros]:
+ assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "(\<lambda>z. Gamma (f z)) analytic_on A"
+ using analytic_on_compose_gen[OF assms(1) analytic_Gamma[of "f ` A"]] assms(2)
+ by (auto simp: o_def)
+
+lemma analytic_on_Polygamma' [analytic_intros]:
+ assumes "f analytic_on A" "\<forall>x\<in>A. f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
+ shows "(\<lambda>z. Polygamma n (f z)) analytic_on A"
+ using analytic_on_compose_gen[OF assms(1) analytic_on_Polygamma[of "f ` A" n]] assms(2)
+ by (auto simp: o_def)
+
+lemma
+ shows is_pole_Polygamma: "is_pole (Polygamma n) (-of_nat m :: complex)"
+ and zorder_Polygamma: "zorder (Polygamma n) (-of_nat m) = -int (Suc n)"
+ and residue_Polygamma: "residue (Polygamma n) (-of_nat m) = (if n = 0 then -1 else 0)"
+proof -
+ define g1 :: "complex \<Rightarrow> complex" where
+ "g1 = (\<lambda>z. Polygamma n (z + of_nat (Suc m)) +
+ (-1) ^ Suc n * fact n * (\<Sum>k<m. 1 / (z + of_nat k) ^ Suc n))"
+ define g :: "complex \<Rightarrow> complex" where
+ "g = (\<lambda>z. g1 z + (-1) ^ Suc n * fact n / (z + of_nat m) ^ Suc n)"
+ define F where "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_const ((-1) ^ Suc n * fact n) / fls_X ^ Suc n"
+ have F_altdef: "F = fps_to_fls (fps_expansion g1 (-of_nat m)) + fls_shift (n+1) (fls_const ((-1) ^ Suc n * fact n))"
+ by (simp add: F_def del: power_Suc)
+
+ have "\<not>(-of_nat m) islimpt (\<int>\<^sub>\<le>\<^sub>0 :: complex set)"
+ by (intro discrete_imp_not_islimpt[where e = 1])
+ (auto elim!: nonpos_Ints_cases simp: dist_of_int)
+ hence "eventually (\<lambda>z::complex. z \<notin> \<int>\<^sub>\<le>\<^sub>0) (at (-of_nat m))"
+ by (auto simp: islimpt_conv_frequently_at frequently_def)
+ hence ev: "eventually (\<lambda>z. Polygamma n z = g z) (at (-of_nat m))"
+ proof eventually_elim
+ case (elim z)
+ hence *: "\<forall>k<Suc m. z \<noteq> - of_nat k"
+ by auto
+ thus ?case
+ using Polygamma_plus_of_nat[of "Suc m" z n, OF *]
+ by (auto simp: g_def g1_def algebra_simps)
+ qed
+
+ have "(\<lambda>w. g (-of_nat m + w)) has_laurent_expansion F"
+ unfolding g_def F_def
+ by (intro laurent_expansion_intros has_laurent_expansion_fps analytic_at_imp_has_fps_expansion)
+ (auto simp: g1_def intro!: laurent_expansion_intros analytic_intros)
+ also have "?this \<longleftrightarrow> (\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F"
+ using ev by (intro has_laurent_expansion_cong refl)
+ (simp_all add: eq_commute at_to_0' eventually_filtermap)
+ finally have *: "(\<lambda>w. Polygamma n (-of_nat m + w)) has_laurent_expansion F" .
+
+ have subdegree: "fls_subdegree F = -int (Suc n)" unfolding F_def
+ by (subst fls_subdegree_add_eq2) (simp_all add: fls_subdegree_fls_to_fps fls_divide_subdegree)
+ have [simp]: "F \<noteq> 0"
+ using subdegree by auto
+
+ show "is_pole (Polygamma n) (-of_nat m :: complex)"
+ using * by (rule has_laurent_expansion_imp_is_pole) (auto simp: subdegree)
+ show "zorder (Polygamma n) (-of_nat m :: complex) = -int (Suc n)"
+ by (subst has_laurent_expansion_zorder[OF *]) (auto simp: subdegree)
+ show "residue (Polygamma n) (-of_nat m :: complex) = (if n = 0 then -1 else 0)"
+ by (subst has_laurent_expansion_residue[OF *]) (auto simp: F_altdef)
+qed
+
+lemma Gamma_meromorphic_on [meromorphic_intros]: "Gamma meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0"
+proof
+ show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex
+ by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
+next
+ fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0"
+ then obtain n where n: "z = -of_nat n"
+ by (elim nonpos_Ints_cases')
+ show "not_essential Gamma z"
+ by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Gamma)
+ have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))"
+ by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
+ have "Gamma holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}"
+ by (intro holomorphic_intros) auto
+ thus "isolated_singularity_at Gamma z"
+ by (rule isolated_singularity_at_holomorphic) (use z * in auto)
+qed (auto intro!: holomorphic_intros)
+
+lemma Polygamma_meromorphic_on [meromorphic_intros]: "Polygamma n meromorphic_on UNIV \<int>\<^sub>\<le>\<^sub>0"
+proof
+ show "\<not>z islimpt \<int>\<^sub>\<le>\<^sub>0" for z :: complex
+ by (intro discrete_imp_not_islimpt[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
+next
+ fix z :: complex assume z: "z \<in> \<int>\<^sub>\<le>\<^sub>0"
+ then obtain m where n: "z = -of_nat m"
+ by (elim nonpos_Ints_cases')
+ show "not_essential (Polygamma n) z"
+ by (auto simp: n intro!: is_pole_imp_not_essential is_pole_Polygamma)
+ have *: "open (-(\<int>\<^sub>\<le>\<^sub>0 - {z}))"
+ by (intro open_Compl discrete_imp_closed[of 1]) (auto elim!: nonpos_Ints_cases simp: dist_of_int)
+ have "Polygamma n holomorphic_on -(\<int>\<^sub>\<le>\<^sub>0 - {z}) - {z}"
+ by (intro holomorphic_intros) auto
+ thus "isolated_singularity_at (Polygamma n) z"
+ by (rule isolated_singularity_at_holomorphic) (use z * in auto)
+qed (auto intro!: holomorphic_intros)
+
+
+theorem argument_principle':
+ fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
+ \<comment> \<open>\<^term>\<open>pz\<close> is the set of non-essential singularities and zeros\<close>
+ defines "pz \<equiv> {w\<in>s. f w = 0 \<or> w \<in> poles}"
+ assumes "open s" and
+ "connected s" and
+ f_holo:"f holomorphic_on s-poles" and
+ h_holo:"h holomorphic_on s" and
+ "valid_path g" and
+ loop:"pathfinish g = pathstart g" and
+ path_img:"path_image g \<subseteq> s - pz" and
+ homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
+ finite:"finite pz" and
+ poles:"\<forall>p\<in>s\<inter>poles. not_essential f p"
+ shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
+ (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
+proof -
+ define ff where "ff = remove_sings f"
+
+ have finite':"finite (s \<inter> poles)"
+ using finite unfolding pz_def by (auto elim:rev_finite_subset)
+
+ have isolated:"isolated_singularity_at f z" if "z\<in>s" for z
+ proof (rule isolated_singularity_at_holomorphic)
+ show "f holomorphic_on (s-(poles-{z})) - {z}"
+ by (metis Diff_empty Diff_insert Diff_insert0 Diff_subset
+ f_holo holomorphic_on_subset insert_Diff)
+ show "open (s - (poles - {z}))"
+ by (metis Diff_Diff_Int Int_Diff assms(2) finite' finite_Diff
+ finite_imp_closed inf.idem open_Diff)
+ show "z \<in> s - (poles - {z})"
+ using assms(4) that by auto
+ qed
+
+ have not_ess:"not_essential f w" if "w\<in>s" for w
+ by (metis Diff_Diff_Int Diff_iff Int_Diff Int_absorb assms(2)
+ f_holo finite' finite_imp_closed not_essential_holomorphic
+ open_Diff poles that)
+
+ have nzero:"\<forall>\<^sub>F x in at w. f x \<noteq> 0" if "w\<in>s" for w
+ proof (rule ccontr)
+ assume "\<not> (\<forall>\<^sub>F x in at w. f x \<noteq> 0)"
+ then have "\<exists>\<^sub>F x in at w. f x = 0"
+ unfolding not_eventually by simp
+ moreover have "\<forall>\<^sub>F x in at w. x\<in>s"
+ by (simp add: assms(2) eventually_at_in_open' that)
+ ultimately have "\<exists>\<^sub>F x in at w. x\<in>{w\<in>s. f w = 0}"
+ apply (elim frequently_rev_mp)
+ by (auto elim:eventually_mono)
+ from frequently_at_imp_islimpt[OF this]
+ have "w islimpt {w \<in> s. f w = 0}" .
+ then have "infinite({w \<in> s. f w = 0} \<inter> ball w 1)"
+ unfolding islimpt_eq_infinite_ball by auto
+ then have "infinite({w \<in> s. f w = 0})"
+ by auto
+ then have "infinite pz" unfolding pz_def
+ by (smt (verit) Collect_mono_iff rev_finite_subset)
+ then show False using finite by auto
+ qed
+
+ obtain pts' where pts':"pts' \<subseteq> s \<inter> poles"
+ "finite pts'" "ff holomorphic_on s - pts'" "\<forall>x\<in>pts'. is_pole ff x"
+ apply (elim get_all_poles_from_remove_sings
+ [of f,folded ff_def,rotated -1])
+ subgoal using f_holo by fastforce
+ using \<open>open s\<close> poles finite' by auto
+
+ have pts'_sub_pz:"{w \<in> s. ff w = 0 \<or> w \<in> pts'} \<subseteq> pz"
+ proof -
+ have "w\<in>poles" if "w\<in>s" "w\<in>pts'" for w
+ by (meson in_mono le_infE pts'(1) that(2))
+ moreover have "f w=0" if" w\<in>s" "w\<notin>poles" "ff w=0" for w
+ proof -
+ have "\<not> is_pole f w"
+ by (metis DiffI Diff_Diff_Int Diff_subset assms(2) f_holo
+ finite' finite_imp_closed inf.absorb_iff2
+ not_is_pole_holomorphic open_Diff that(1) that(2))
+ then have "f \<midarrow>w\<rightarrow> 0"
+ using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w=0\<close>
+ unfolding ff_def by auto
+ moreover have "f analytic_on {w}"
+ using that(1,2) finite' f_holo assms(2)
+ by (metis Diff_Diff_Int Diff_empty Diff_iff Diff_subset
+ double_diff finite_imp_closed
+ holomorphic_on_imp_analytic_at open_Diff)
+ ultimately show ?thesis
+ using ff_def remove_sings_at_analytic that(3) by presburger
+ qed
+ ultimately show ?thesis unfolding pz_def by auto
+ qed
+
+
+ have "contour_integral g (\<lambda>x. deriv f x * h x / f x)
+ = contour_integral g (\<lambda>x. deriv ff x * h x / ff x)"
+ proof (rule contour_integral_eq)
+ fix x assume "x \<in> path_image g"
+ have "f analytic_on {x}"
+ proof (rule holomorphic_on_imp_analytic_at[of _ "s-poles"])
+ from finite'
+ show "open (s - poles)"
+ using \<open>open s\<close>
+ by (metis Diff_Compl Diff_Diff_Int Diff_eq finite_imp_closed
+ open_Diff)
+ show "x \<in> s - poles"
+ using path_img \<open>x \<in> path_image g\<close> unfolding pz_def by auto
+ qed (use f_holo in simp)
+ then show "deriv f x * h x / f x = deriv ff x * h x / ff x"
+ unfolding ff_def by auto
+ qed
+ also have "... = complex_of_real (2 * pi) * \<i> *
+ (\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}.
+ winding_number g p * h p * of_int (zorder ff p))"
+ proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close>, of ff pts' h g])
+ show "path_image g \<subseteq> s - {w \<in> s. ff w = 0 \<or> w \<in> pts'}"
+ using path_img pts'_sub_pz by auto
+ show "finite {w \<in> s. ff w = 0 \<or> w \<in> pts'}"
+ using pts'_sub_pz finite
+ using rev_finite_subset by blast
+ qed (use pts' assms in auto)
+ also have "... = 2 * pi * \<i> *
+ (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
+ proof -
+ have "(\<Sum>p\<in>{w \<in> s. ff w = 0 \<or> w \<in> pts'}.
+ winding_number g p * h p * of_int (zorder ff p)) =
+ (\<Sum>p\<in>pz. winding_number g p * h p * of_int (zorder f p))"
+ proof (rule sum.mono_neutral_cong_left)
+ have "zorder f w = 0"
+ if "w\<in>s" " f w = 0 \<or> w \<in> poles" "ff w \<noteq> 0" " w \<notin> pts'"
+ for w
+ proof -
+ define F where "F=laurent_expansion f w"
+ have has_l:"(\<lambda>x. f (w + x)) has_laurent_expansion F"
+ unfolding F_def
+ apply (rule not_essential_has_laurent_expansion)
+ using isolated not_ess \<open>w\<in>s\<close> by auto
+ from has_laurent_expansion_eventually_nonzero_iff[OF this]
+ have "F \<noteq>0"
+ using nzero \<open>w\<in>s\<close> by auto
+ from tendsto_0_subdegree_iff[OF has_l this]
+ have "f \<midarrow>w\<rightarrow> 0 = (0 < fls_subdegree F)" .
+ moreover have "\<not> (is_pole f w \<or> f \<midarrow>w\<rightarrow> 0)"
+ using remove_sings_eq_0_iff[OF not_ess[OF \<open>w\<in>s\<close>]] \<open>ff w \<noteq> 0\<close>
+ unfolding ff_def by auto
+ moreover have "is_pole f w = (fls_subdegree F < 0)"
+ using is_pole_fls_subdegree_iff[OF has_l] .
+ ultimately have "fls_subdegree F = 0" by auto
+ then show ?thesis
+ using has_laurent_expansion_zorder[OF has_l \<open>F\<noteq>0\<close>] by auto
+ qed
+ then show "\<forall>i\<in>pz - {w \<in> s. ff w = 0 \<or> w \<in> pts'}.
+ winding_number g i * h i * of_int (zorder f i) = 0"
+ unfolding pz_def by auto
+ show "\<And>x. x \<in> {w \<in> s. ff w = 0 \<or> w \<in> pts'} \<Longrightarrow>
+ winding_number g x * h x * of_int (zorder ff x) =
+ winding_number g x * h x * of_int (zorder f x)"
+ using isolated zorder_remove_sings[of f,folded ff_def] by auto
+ qed (use pts'_sub_pz finite in auto)
+ then show ?thesis by auto
+ qed
+ finally show ?thesis .
+qed
+
+lemma meromorphic_on_imp_isolated_singularity:
+ assumes "f meromorphic_on D pts" "z \<in> D"
+ shows "isolated_singularity_at f z"
+ by (meson DiffI assms(1) assms(2) holomorphic_on_imp_analytic_at isolated_singularity_at_analytic
+ meromorphic_imp_open_diff meromorphic_on_def)
+
+lemma meromorphic_imp_not_is_pole:
+ assumes "f meromorphic_on D pts" "z \<in> D - pts"
+ shows "\<not>is_pole f z"
+proof -
+ from assms have "f analytic_on {z}"
+ using meromorphic_on_imp_analytic_at by blast
+ thus ?thesis
+ using analytic_at not_is_pole_holomorphic by blast
+qed
+
+lemma meromorphic_all_poles_iff_empty [simp]: "f meromorphic_on pts pts \<longleftrightarrow> pts = {}"
+ by (auto simp: meromorphic_on_def holomorphic_on_def open_imp_islimpt)
+
+lemma meromorphic_imp_nonsingular_point_exists:
+ assumes "f meromorphic_on A pts" "A \<noteq> {}"
+ obtains x where "x \<in> A - pts"
+proof -
+ have "A \<noteq> pts"
+ using assms by auto
+ moreover have "pts \<subseteq> A"
+ using assms by (auto simp: meromorphic_on_def)
+ ultimately show ?thesis
+ using that by blast
+qed
+
+lemma meromorphic_frequently_const_imp_const:
+ assumes "f meromorphic_on A pts" "connected A"
+ assumes "frequently (\<lambda>w. f w = c) (at z)"
+ assumes "z \<in> A - pts"
+ assumes "w \<in> A - pts"
+ shows "f w = c"
+proof -
+ have "f w - c = 0"
+ proof (rule analytic_continuation[where f = "\<lambda>z. f z - c"])
+ show "(\<lambda>z. f z - c) holomorphic_on (A - pts)"
+ by (intro holomorphic_intros meromorphic_imp_holomorphic[OF assms(1)])
+ show [intro]: "open (A - pts)"
+ using assms meromorphic_imp_open_diff by blast
+ show "connected (A - pts)"
+ using assms meromorphic_imp_connected_diff by blast
+ show "{z\<in>A-pts. f z = c} \<subseteq> A - pts"
+ by blast
+ have "eventually (\<lambda>z. z \<in> A - pts) (at z)"
+ using assms by (intro eventually_at_in_open') auto
+ hence "frequently (\<lambda>z. f z = c \<and> z \<in> A - pts) (at z)"
+ by (intro frequently_eventually_frequently assms)
+ thus "z islimpt {z\<in>A-pts. f z = c}"
+ by (simp add: islimpt_conv_frequently_at conj_commute)
+ qed (use assms in auto)
+ thus ?thesis
+ by simp
+qed
+
+lemma meromorphic_imp_eventually_neq:
+ assumes "f meromorphic_on A pts" "connected A" "\<not>f constant_on A - pts"
+ assumes "z \<in> A - pts"
+ shows "eventually (\<lambda>z. f z \<noteq> c) (at z)"
+proof (rule ccontr)
+ assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)"
+ hence *: "frequently (\<lambda>z. f z = c) (at z)"
+ by (auto simp: frequently_def)
+ have "\<forall>w\<in>A-pts. f w = c"
+ using meromorphic_frequently_const_imp_const [OF assms(1,2) * assms(4)] by blast
+ hence "f constant_on A - pts"
+ by (auto simp: constant_on_def)
+ thus False
+ using assms(3) by contradiction
+qed
+
+lemma meromorphic_frequently_const_imp_const':
+ assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w"
+ assumes "frequently (\<lambda>w. f w = c) (at z)"
+ assumes "z \<in> A"
+ assumes "w \<in> A"
+ shows "f w = c"
+proof -
+ have "\<not>is_pole f z"
+ using frequently_const_imp_not_is_pole[OF assms(4)] .
+ with assms have z: "z \<in> A - pts"
+ by auto
+ have *: "f w = c" if "w \<in> A - pts" for w
+ using that meromorphic_frequently_const_imp_const [OF assms(1,2,4) z] by auto
+ have "\<not>is_pole f u" if "u \<in> A" for u
+ proof -
+ have "is_pole f u \<longleftrightarrow> is_pole (\<lambda>_. c) u"
+ proof (rule is_pole_cong)
+ have "eventually (\<lambda>w. w \<in> A - (pts - {u}) - {u}) (at u)"
+ by (intro eventually_at_in_open meromorphic_imp_open_diff' [OF assms(1)]) (use that in auto)
+ thus "eventually (\<lambda>w. f w = c) (at u)"
+ by eventually_elim (use * in auto)
+ qed auto
+ thus ?thesis
+ by auto
+ qed
+ moreover have "pts \<subseteq> A"
+ using assms(1) by (simp add: meromorphic_on_def)
+ ultimately have "pts = {}"
+ using assms(3) by auto
+ with * and \<open>w \<in> A\<close> show ?thesis
+ by blast
+qed
+
+lemma meromorphic_imp_eventually_neq':
+ assumes "f meromorphic_on A pts" "connected A" "\<forall>w\<in>pts. is_pole f w" "\<not>f constant_on A"
+ assumes "z \<in> A"
+ shows "eventually (\<lambda>z. f z \<noteq> c) (at z)"
+proof (rule ccontr)
+ assume "\<not>eventually (\<lambda>z. f z \<noteq> c) (at z)"
+ hence *: "frequently (\<lambda>z. f z = c) (at z)"
+ by (auto simp: frequently_def)
+ have "\<forall>w\<in>A. f w = c"
+ using meromorphic_frequently_const_imp_const' [OF assms(1,2,3) * assms(5)] by blast
+ hence "f constant_on A"
+ by (auto simp: constant_on_def)
+ thus False
+ using assms(4) by contradiction
+qed
+
+lemma zorder_eq_0_iff_meromorphic:
+ assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
+ assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
+ shows "zorder f z = 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z \<noteq> 0"
+proof (cases "z \<in> pts")
+ case True
+ from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F"
+ by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *)
+ from F and assms(4) have [simp]: "F \<noteq> 0"
+ using has_laurent_expansion_eventually_nonzero_iff by blast
+ show ?thesis using True assms(2)
+ using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F]
+ by auto
+next
+ case False
+ have ana: "f analytic_on {z}"
+ using meromorphic_on_imp_analytic_at False assms by blast
+ hence "\<not>is_pole f z"
+ using analytic_at not_is_pole_holomorphic by blast
+ moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)"
+ using assms(4) by (intro eventually_frequently) auto
+ ultimately show ?thesis using zorder_eq_0_iff[OF ana] False
+ by auto
+qed
+
+lemma zorder_pos_iff_meromorphic:
+ assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
+ assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
+ shows "zorder f z > 0 \<longleftrightarrow> \<not>is_pole f z \<and> f z = 0"
+proof (cases "z \<in> pts")
+ case True
+ from assms obtain F where F: "(\<lambda>x. f (z + x)) has_laurent_expansion F"
+ by (metis True meromorphic_on_def not_essential_has_laurent_expansion) (* TODO: better lemmas *)
+ from F and assms(4) have [simp]: "F \<noteq> 0"
+ using has_laurent_expansion_eventually_nonzero_iff by blast
+ show ?thesis using True assms(2)
+ using is_pole_fls_subdegree_iff [OF F] has_laurent_expansion_zorder [OF F]
+ by auto
+next
+ case False
+ have ana: "f analytic_on {z}"
+ using meromorphic_on_imp_analytic_at False assms by blast
+ hence "\<not>is_pole f z"
+ using analytic_at not_is_pole_holomorphic by blast
+ moreover have "frequently (\<lambda>w. f w \<noteq> 0) (at z)"
+ using assms(4) by (intro eventually_frequently) auto
+ ultimately show ?thesis using zorder_pos_iff'[OF ana] False
+ by auto
+qed
+
+lemma zorder_neg_iff_meromorphic:
+ assumes "f meromorphic_on A pts" "\<forall>z\<in>pts. is_pole f z" "z \<in> A"
+ assumes "eventually (\<lambda>x. f x \<noteq> 0) (at z)"
+ shows "zorder f z < 0 \<longleftrightarrow> is_pole f z"
+proof -
+ have "frequently (\<lambda>x. f x \<noteq> 0) (at z)"
+ using assms by (intro eventually_frequently) auto
+ moreover from assms have "isolated_singularity_at f z" "not_essential f z"
+ using meromorphic_on_imp_isolated_singularity meromorphic_on_imp_not_essential by blast+
+ ultimately show ?thesis
+ using isolated_pole_imp_neg_zorder neg_zorder_imp_is_pole by blast
+qed
+
+lemma meromorphic_on_imp_discrete:
+ assumes mero:"f meromorphic_on S pts" and "connected S"
+ and nconst:"\<not> (\<forall>w\<in>S - pts. f w = c)"
+ shows "discrete {x\<in>S. f x=c}"
+proof -
+ define g where "g=(\<lambda>x. f x - c)"
+ have "\<forall>\<^sub>F w in at z. g w \<noteq> 0" if "z \<in> S" for z
+ proof (rule nconst_imp_nzero_neighbour'[of g S pts z])
+ show "g meromorphic_on S pts" using mero unfolding g_def
+ by (auto intro:meromorphic_intros)
+ show "\<not> (\<forall>w\<in>S - pts. g w = 0)" using nconst unfolding g_def by auto
+ qed fact+
+ then show ?thesis
+ unfolding discrete_altdef g_def
+ using eventually_mono by fastforce
+qed
+
+lemma meromorphic_isolated_in:
+ assumes merf: "f meromorphic_on D pts" "p\<in>pts"
+ shows "p isolated_in pts"
+ by (meson assms isolated_in_islimpt_iff meromorphic_on_def subsetD)
+
+lemma remove_sings_constant_on:
+ assumes merf: "f meromorphic_on D pts" and "connected D"
+ and const:"f constant_on (D - pts)"
+ shows "(remove_sings f) constant_on D"
+proof -
+ have remove_sings_const: "remove_sings f constant_on D - pts"
+ using const
+ by (metis constant_onE merf meromorphic_on_imp_analytic_at remove_sings_at_analytic)
+
+ have ?thesis if "D = {}"
+ using that unfolding constant_on_def by auto
+ moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} = {}"
+ proof -
+ obtain \<xi> where "\<xi> \<in> (D - pts)" "\<xi> islimpt (D - pts)"
+ proof -
+ have "open (D - pts)"
+ using meromorphic_imp_open_diff[OF merf] .
+ moreover have "(D - pts) \<noteq> {}" using \<open>D\<noteq>{}\<close>
+ by (metis Diff_empty closure_empty merf
+ meromorphic_pts_closure subset_empty)
+ ultimately show ?thesis using open_imp_islimpt that by auto
+ qed
+ moreover have "remove_sings f holomorphic_on D"
+ using remove_sings_holomorphic_on[OF merf] that by auto
+ moreover note remove_sings_const
+ moreover have "open D"
+ using assms(1) meromorphic_on_def by blast
+ ultimately show ?thesis
+ using Conformal_Mappings.analytic_continuation'
+ [of "remove_sings f" D "D-pts" \<xi>] \<open>connected D\<close>
+ by auto
+ qed
+ moreover have ?thesis if "D\<noteq>{}" "{x\<in>pts. is_pole f x} \<noteq> {}"
+ proof -
+ define PP where "PP={x\<in>D. is_pole f x}"
+ have "remove_sings f meromorphic_on D PP"
+ using merf unfolding PP_def
+ apply (elim remove_sings_meromorphic_on)
+ subgoal using assms(1) meromorphic_on_def by force
+ subgoal using meromorphic_pole_subset merf by auto
+ done
+ moreover have "remove_sings f constant_on D - PP"
+ proof -
+ obtain \<xi> where "\<xi> \<in> f ` (D - pts)"
+ by (metis Diff_empty Diff_eq_empty_iff \<open>D \<noteq> {}\<close> assms(1)
+ closure_empty ex_in_conv imageI meromorphic_pts_closure)
+ have \<xi>:"\<forall>x\<in>D - pts. f x = \<xi>"
+ by (metis \<open>\<xi> \<in> f ` (D - pts)\<close> assms(3) constant_on_def image_iff)
+
+ have "remove_sings f x = \<xi>" if "x\<in>D - PP" for x
+ proof (cases "x\<in>pts")
+ case True
+ then have"x isolated_in pts"
+ using meromorphic_isolated_in[OF merf] by auto
+ then obtain T0 where T0:"open T0" "T0 \<inter> pts = {x}"
+ unfolding isolated_in_def by auto
+ obtain T1 where T1:"open T1" "x\<in>T1" "T1 \<subseteq> D"
+ using merf unfolding meromorphic_on_def
+ using True by blast
+ define T2 where "T2 = T1 \<inter> T0"
+ have "open T2" "x\<in>T2" "T2 - {x} \<subseteq> D - pts"
+ using T0 T1 unfolding T2_def by auto
+ then have "\<forall>w\<in>T2. w\<noteq>x \<longrightarrow> f w =\<xi>"
+ using \<xi> by auto
+ then have "\<forall>\<^sub>F x in at x. f x = \<xi>"
+ unfolding eventually_at_topological
+ using \<open>open T2\<close> \<open>x\<in>T2\<close> by auto
+ then have "f \<midarrow>x\<rightarrow> \<xi>"
+ using tendsto_eventually by auto
+ then show ?thesis by blast
+ next
+ case False
+ then show ?thesis
+ using \<open>\<forall>x\<in>D - pts. f x = \<xi>\<close> assms(1)
+ meromorphic_on_imp_analytic_at that by auto
+ qed
+
+ then show ?thesis unfolding constant_on_def by auto
+ qed
+
+ moreover have "is_pole (remove_sings f) x" if "x\<in>PP" for x
+ proof -
+ have "isolated_singularity_at f x"
+ by (metis (mono_tags, lifting) DiffI PP_def assms(1)
+ isolated_singularity_at_analytic mem_Collect_eq
+ meromorphic_on_def meromorphic_on_imp_analytic_at that)
+ then show ?thesis using that unfolding PP_def by simp
+ qed
+ ultimately show ?thesis
+ using meromorphic_imp_constant_on
+ [of "remove_sings f" D PP]
+ by auto
+ qed
+ ultimately show ?thesis by auto
+qed
+
+lemma meromorphic_eq_meromorphic_extend:
+ assumes "f meromorphic_on A pts1" "g meromorphic_on A pts1" "\<not>z islimpt pts2"
+ assumes "\<And>z. z \<in> A - pts2 \<Longrightarrow> f z = g z" "pts1 \<subseteq> pts2" "z \<in> A - pts1"
+ shows "f z = g z"
+proof -
+ have "g analytic_on {z}"
+ using assms by (intro meromorphic_on_imp_analytic_at[OF assms(2)]) auto
+ hence "g \<midarrow>z\<rightarrow> g z"
+ using analytic_at_imp_isCont isContD by blast
+ also have "?this \<longleftrightarrow> f \<midarrow>z\<rightarrow> g z"
+ proof (intro filterlim_cong)
+ have "eventually (\<lambda>w. w \<notin> pts2) (at z)"
+ using assms by (auto simp: islimpt_conv_frequently_at frequently_def)
+ moreover have "eventually (\<lambda>w. w \<in> A) (at z)"
+ using assms by (intro eventually_at_in_open') (auto simp: meromorphic_on_def)
+ ultimately show "\<forall>\<^sub>F x in at z. g x = f x"
+ by eventually_elim (use assms in auto)
+ qed auto
+ finally have "f \<midarrow>z\<rightarrow> g z" .
+ moreover have "f analytic_on {z}"
+ using assms by (intro meromorphic_on_imp_analytic_at[OF assms(1)]) auto
+ hence "f \<midarrow>z\<rightarrow> f z"
+ using analytic_at_imp_isCont isContD by blast
+ ultimately show ?thesis
+ using tendsto_unique by force
+qed
+
+lemma meromorphic_constant_on_extend:
+ assumes "f constant_on A - pts1" "f meromorphic_on A pts1" "f meromorphic_on A pts2" "pts2 \<subseteq> pts1"
+ shows "f constant_on A - pts2"
+proof -
+ from assms(1) obtain c where c: "\<And>z. z \<in> A - pts1 \<Longrightarrow> f z = c"
+ unfolding constant_on_def by auto
+ have "f z = c" if "z \<in> A - pts2" for z
+ using assms(3)
+ proof (rule meromorphic_eq_meromorphic_extend[where z = z])
+ show "(\<lambda>a. c) meromorphic_on A pts2"
+ by (intro meromorphic_on_const) (use assms in \<open>auto simp: meromorphic_on_def\<close>)
+ show "\<not>z islimpt pts1"
+ using that assms by (auto simp: meromorphic_on_def)
+ qed (use assms c that in auto)
+ thus ?thesis
+ by (auto simp: constant_on_def)
+qed
+
+lemma meromorphic_remove_sings_constant_on_imp_constant_on:
+ assumes "f meromorphic_on A pts"
+ assumes "remove_sings f constant_on A"
+ shows "f constant_on A - pts"
+proof -
+ from assms(2) obtain c where c: "\<And>z. z \<in> A \<Longrightarrow> remove_sings f z = c"
+ by (auto simp: constant_on_def)
+ have "f z = c" if "z \<in> A - pts" for z
+ using meromorphic_on_imp_analytic_at[OF assms(1) that] c[of z] that
+ by auto
+ thus ?thesis
+ by (auto simp: constant_on_def)
+qed
+
+
+
+
+definition singularities_on :: "complex set \<Rightarrow> (complex \<Rightarrow> complex) \<Rightarrow> complex set" where
+ "singularities_on A f =
+ {z\<in>A. isolated_singularity_at f z \<and> not_essential f z \<and> \<not>f analytic_on {z}}"
+
+lemma singularities_on_subset: "singularities_on A f \<subseteq> A"
+ by (auto simp: singularities_on_def)
+
+lemma pole_in_singularities_on:
+ assumes "f meromorphic_on A pts" "z \<in> A" "is_pole f z"
+ shows "z \<in> singularities_on A f"
+ unfolding singularities_on_def not_essential_def using assms
+ using analytic_at_imp_no_pole meromorphic_on_imp_isolated_singularity by force
+
+
+lemma meromorphic_on_subset_pts:
+ assumes "f meromorphic_on A pts" "pts' \<subseteq> pts" "f analytic_on pts - pts'"
+ shows "f meromorphic_on A pts'"
+proof
+ show "open A" "pts' \<subseteq> A"
+ using assms by (auto simp: meromorphic_on_def)
+ show "isolated_singularity_at f z" "not_essential f z" if "z \<in> pts'" for z
+ using assms that by (auto simp: meromorphic_on_def)
+ show "\<not>z islimpt pts'" if "z \<in> A" for z
+ using assms that islimpt_subset unfolding meromorphic_on_def by blast
+ have "f analytic_on A - pts"
+ using assms(1) meromorphic_imp_analytic by blast
+ with assms have "f analytic_on (A - pts) \<union> (pts - pts')"
+ by (subst analytic_on_Un) auto
+ also have "(A - pts) \<union> (pts - pts') = A - pts'"
+ using assms by (auto simp: meromorphic_on_def)
+ finally show "f holomorphic_on A - pts'"
+ using analytic_imp_holomorphic by blast
+qed
+
+lemma meromorphic_on_imp_superset_singularities_on:
+ assumes "f meromorphic_on A pts"
+ shows "singularities_on A f \<subseteq> pts"
+proof
+ fix z assume "z \<in> singularities_on A f"
+ hence "z \<in> A" "\<not>f analytic_on {z}"
+ by (auto simp: singularities_on_def)
+ with assms show "z \<in> pts"
+ by (meson DiffI meromorphic_on_imp_analytic_at)
+qed
+
+lemma meromorphic_on_singularities_on:
+ assumes "f meromorphic_on A pts"
+ shows "f meromorphic_on A (singularities_on A f)"
+ using assms meromorphic_on_imp_superset_singularities_on[OF assms]
+proof (rule meromorphic_on_subset_pts)
+ have "f analytic_on {z}" if "z \<in> pts - singularities_on A f" for z
+ using that assms by (auto simp: singularities_on_def meromorphic_on_def)
+ thus "f analytic_on pts - singularities_on A f"
+ using analytic_on_analytic_at by blast
+qed
+
+theorem Residue_theorem_inside:
+ assumes f: "f meromorphic_on s pts"
+ "simply_connected s"
+ assumes g: "valid_path g"
+ "pathfinish g = pathstart g"
+ "path_image g \<subseteq> s - pts"
+ defines "pts1 \<equiv> pts \<inter> inside (path_image g)"
+ shows "finite pts1"
+ and "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
+proof -
+ note [dest] = valid_path_imp_path
+ have cl_g [intro]: "closed (path_image g)"
+ using g by (auto intro!: closed_path_image)
+ have "open s"
+ using f(1) by (auto simp: meromorphic_on_def)
+ define pts2 where "pts2 = pts - pts1"
+
+ define A where "A = path_image g \<union> inside (path_image g)"
+ have "closed A"
+ unfolding A_def using g by (intro closed_path_image_Un_inside) auto
+ moreover have "bounded A"
+ unfolding A_def using g by (auto intro!: bounded_path_image bounded_inside)
+ ultimately have 1: "compact A"
+ using compact_eq_bounded_closed by blast
+ have 2: "open (s - pts2)"
+ using f by (auto intro!: meromorphic_imp_open_diff' [OF f(1)] simp: pts2_def)
+ have 3: "A \<subseteq> s - pts2"
+ unfolding A_def pts2_def pts1_def
+ using f(2) g(3) 2 subset_simply_connected_imp_inside_subset[of s "path_image g"] \<open>open s\<close>
+ by auto
+
+ obtain \<epsilon> where \<epsilon>: "\<epsilon> > 0" "(\<Union>x\<in>A. ball x \<epsilon>) \<subseteq> s - pts2"
+ using compact_subset_open_imp_ball_epsilon_subset[OF 1 2 3] by blast
+ define B where "B = (\<Union>x\<in>A. ball x \<epsilon>)"
+
+ have "finite (A \<inter> pts)"
+ using 1 3 by (intro meromorphic_compact_finite_pts[OF f(1)]) auto
+ also have "A \<inter> pts = pts1"
+ unfolding pts1_def using g by (auto simp: A_def)
+ finally show fin: "finite pts1" .
+
+ show "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
+ proof (rule Residue_theorem)
+ show "open B"
+ by (auto simp: B_def)
+ next
+ have "connected A"
+ unfolding A_def using g
+ by (intro connected_with_inside closed_path_image connected_path_image) auto
+ hence "connected (A \<union> B)"
+ unfolding B_def using g \<open>\<epsilon> > 0\<close> f(2)
+ by (intro connected_Un_UN connected_path_image valid_path_imp_path)
+ (auto simp: simply_connected_imp_connected)
+ also have "A \<union> B = B"
+ using \<epsilon>(1) by (auto simp: B_def)
+ finally show "connected B" .
+ next
+ have "f holomorphic_on (s - pts)"
+ by (intro meromorphic_imp_holomorphic f)
+ moreover have "B - pts1 \<subseteq> s - pts"
+ using \<epsilon> unfolding B_def by (auto simp: pts1_def pts2_def)
+ ultimately show "f holomorphic_on (B - pts1)"
+ by (rule holomorphic_on_subset)
+ next
+ have "path_image g \<subseteq> A - pts1"
+ using g unfolding pts1_def by (auto simp: A_def)
+ also have "\<dots> \<subseteq> B - pts1"
+ unfolding B_def using \<epsilon>(1) by auto
+ finally show "path_image g \<subseteq> B - pts1" .
+ next
+ show "\<forall>z. z \<notin> B \<longrightarrow> winding_number g z = 0"
+ proof safe
+ fix z assume z: "z \<notin> B"
+ hence "z \<notin> A"
+ using \<epsilon>(1) by (auto simp: B_def)
+ hence "z \<in> outside (path_image g)"
+ unfolding A_def by (simp add: union_with_inside)
+ thus "winding_number g z = 0"
+ using g by (intro winding_number_zero_in_outside) auto
+ qed
+ qed (use g fin in auto)
+qed
+
+theorem Residue_theorem':
+ assumes f: "f meromorphic_on s pts"
+ "simply_connected s"
+ assumes g: "valid_path g"
+ "pathfinish g = pathstart g"
+ "path_image g \<subseteq> s - pts"
+ assumes pts': "finite pts'"
+ "pts' \<subseteq> s"
+ "\<And>z. z \<in> pts - pts' \<Longrightarrow> winding_number g z = 0"
+ shows "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts'. winding_number g p * residue f p)"
+proof -
+ note [dest] = valid_path_imp_path
+ define pts1 where "pts1 = pts \<inter> inside (path_image g)"
+
+ have "contour_integral g f = 2 * pi * \<i> * (\<Sum>p\<in>pts1. winding_number g p * residue f p)"
+ unfolding pts1_def by (intro Residue_theorem_inside[OF f g])
+ also have "(\<Sum>p\<in>pts1. winding_number g p * residue f p) =
+ (\<Sum>p\<in>pts'. winding_number g p * residue f p)"
+ proof (intro sum.mono_neutral_cong refl)
+ show "finite pts1"
+ unfolding pts1_def by (intro Residue_theorem_inside[OF f g])
+ show "finite pts'"
+ by fact
+ next
+ fix z assume z: "z \<in> pts' - pts1"
+ show "winding_number g z * residue f z = 0"
+ proof (cases "z \<in> pts")
+ case True
+ with z have "z \<notin> path_image g \<union> inside (path_image g)"
+ using g(3) by (auto simp: pts1_def)
+ hence "z \<in> outside (path_image g)"
+ by (simp add: union_with_inside)
+ hence "winding_number g z = 0"
+ using g by (intro winding_number_zero_in_outside) auto
+ thus ?thesis
+ by simp
+ next
+ case False
+ with z pts' have "z \<in> s - pts"
+ by auto
+ with f(1) have "f analytic_on {z}"
+ by (intro meromorphic_on_imp_analytic_at)
+ hence "residue f z = 0"
+ using analytic_at residue_holo by blast
+ thus ?thesis
+ by simp
+ qed
+ next
+ fix z assume z: "z \<in> pts1 - pts'"
+ hence "winding_number g z = 0"
+ using pts' by (auto simp: pts1_def)
+ thus "winding_number g z * residue f z = 0"
+ by simp
+ qed
+ finally show ?thesis .
+qed
+
+end
\ No newline at end of file