src/HOL/Complex_Analysis/Residue_Theorem.thy

Reorganised HOL-Complex_Analysis

section \<open>The Residue Theorem, the Argument Principle and Rouch\'{e}'s Theorem\<close>
theory Residue_Theorem
  imports Complex_Residues
begin

subsection \<open>Cauchy's residue theorem\<close>

lemma get_integrable_path:
  assumes "open s" "connected (s-pts)" "finite pts" "f holomorphic_on (s-pts) " "a\<in>s-pts" "b\<in>s-pts"
  obtains g where "valid_path g" "pathstart g = a" "pathfinish g = b"
    "path_image g \<subseteq> s-pts" "f contour_integrable_on g" using assms
proof (induct arbitrary:s thesis a rule:finite_induct[OF \<open>finite pts\<close>])
  case 1
  obtain g where "valid_path g" "path_image g \<subseteq> s" "pathstart g = a" "pathfinish g = b"
    using connected_open_polynomial_connected[OF \<open>open s\<close>,of a b ] \<open>connected (s - {})\<close>
      valid_path_polynomial_function "1.prems"(6) "1.prems"(7) by auto
  moreover have "f contour_integrable_on g"
    using contour_integrable_holomorphic_simple[OF _ \<open>open s\<close> \<open>valid_path g\<close> \<open>path_image g \<subseteq> s\<close>,of f]
      \<open>f holomorphic_on s - {}\<close>
    by auto
  ultimately show ?case using "1"(1)[of g] by auto
next
  case idt:(2 p pts)
  obtain e where "e>0" and e:"\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
    using finite_ball_avoid[OF \<open>open s\<close> \<open>finite (insert p pts)\<close>, of a]
      \<open>a \<in> s - insert p pts\<close>
    by auto
  define a' where "a' \<equiv> a+e/2"
  have "a'\<in>s-{p} -pts"  using e[rule_format,of "a+e/2"] \<open>e>0\<close>
    by (auto simp add:dist_complex_def a'_def)
  then obtain g' where g'[simp]:"valid_path g'" "pathstart g' = a'" "pathfinish g' = b"
    "path_image g' \<subseteq> s - {p} - pts" "f contour_integrable_on g'"
    using idt.hyps(3)[of a' "s-{p}"] idt.prems idt.hyps(1)
    by (metis Diff_insert2 open_delete)
  define g where "g \<equiv> linepath a a' +++ g'"
  have "valid_path g" unfolding g_def by (auto intro: valid_path_join)
  moreover have "pathstart g = a" and  "pathfinish g = b" unfolding g_def by auto
  moreover have "path_image g \<subseteq> s - insert p pts" unfolding g_def
    proof (rule subset_path_image_join)
      have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
        by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
      then show "path_image (linepath a a') \<subseteq> s - insert p pts" using e idt(9)
        by auto
    next
      show "path_image g' \<subseteq> s - insert p pts" using g'(4) by blast
    qed
  moreover have "f contour_integrable_on g"
    proof -
      have "closed_segment a a' \<subseteq> ball a e" using \<open>e>0\<close>
        by (auto dest!:segment_bound1 simp:a'_def dist_complex_def norm_minus_commute)
      then have "continuous_on (closed_segment a a') f"
        using e idt.prems(6) holomorphic_on_imp_continuous_on[OF idt.prems(5)]
        apply (elim continuous_on_subset)
        by auto
      then have "f contour_integrable_on linepath a a'"
        using contour_integrable_continuous_linepath by auto
      then show ?thesis unfolding g_def
        apply (rule contour_integrable_joinI)
        by (auto simp add: \<open>e>0\<close>)
    qed
  ultimately show ?case using idt.prems(1)[of g] by auto
qed

lemma Cauchy_theorem_aux:
  assumes "open s" "connected (s-pts)" "finite pts" "pts \<subseteq> s" "f holomorphic_on s-pts"
          "valid_path g" "pathfinish g = pathstart g" "path_image g \<subseteq> s-pts"
          "\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
          "\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
  shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
    using assms
proof (induct arbitrary:s g rule:finite_induct[OF \<open>finite pts\<close>])
  case 1
  then show ?case by (simp add: Cauchy_theorem_global contour_integral_unique)
next
  case (2 p pts)
  note fin[simp] = \<open>finite (insert p pts)\<close>
    and connected = \<open>connected (s - insert p pts)\<close>
    and valid[simp] = \<open>valid_path g\<close>
    and g_loop[simp] = \<open>pathfinish g = pathstart g\<close>
    and holo[simp]= \<open>f holomorphic_on s - insert p pts\<close>
    and path_img = \<open>path_image g \<subseteq> s - insert p pts\<close>
    and winding = \<open>\<forall>z. z \<notin> s \<longrightarrow> winding_number g z = 0\<close>
    and h = \<open>\<forall>pa\<in>s. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s \<and> (w \<noteq> pa \<longrightarrow> w \<notin> insert p pts))\<close>
  have "h p>0" and "p\<in>s"
    and h_p: "\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> insert p pts)"
    using h \<open>insert p pts \<subseteq> s\<close> by auto
  obtain pg where pg[simp]: "valid_path pg" "pathstart pg = pathstart g" "pathfinish pg=p+h p"
      "path_image pg \<subseteq> s-insert p pts" "f contour_integrable_on pg"
    proof -
      have "p + h p\<in>cball p (h p)" using h[rule_format,of p]
        by (simp add: \<open>p \<in> s\<close> dist_norm)
      then have "p + h p \<in> s - insert p pts" using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close>
        by fastforce
      moreover have "pathstart g \<in> s - insert p pts " using path_img by auto
      ultimately show ?thesis
        using get_integrable_path[OF \<open>open s\<close> connected fin holo,of "pathstart g" "p+h p"] that
        by blast
    qed
  obtain n::int where "n=winding_number g p"
    using integer_winding_number[OF _ g_loop,of p] valid path_img
    by (metis DiffD2 Ints_cases insertI1 subset_eq valid_path_imp_path)
  define p_circ where "p_circ \<equiv> circlepath p (h p)"
  define p_circ_pt where "p_circ_pt \<equiv> linepath (p+h p) (p+h p)"
  define n_circ where "n_circ \<equiv> \<lambda>n. ((+++) p_circ ^^ n) p_circ_pt"
  define cp where "cp \<equiv> if n\<ge>0 then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
  have n_circ:"valid_path (n_circ k)"
      "winding_number (n_circ k) p = k"
      "pathstart (n_circ k) = p + h p" "pathfinish (n_circ k) = p + h p"
      "path_image (n_circ k) =  (if k=0 then {p + h p} else sphere p (h p))"
      "p \<notin> path_image (n_circ k)"
      "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number (n_circ k) p'=0 \<and> p'\<notin>path_image (n_circ k)"
      "f contour_integrable_on (n_circ k)"
      "contour_integral (n_circ k) f = k *  contour_integral p_circ f"
      for k
    proof (induct k)
      case 0
      show "valid_path (n_circ 0)"
        and "path_image (n_circ 0) =  (if 0=0 then {p + h p} else sphere p (h p))"
        and "winding_number (n_circ 0) p = of_nat 0"
        and "pathstart (n_circ 0) = p + h p"
        and "pathfinish (n_circ 0) = p + h p"
        and "p \<notin> path_image (n_circ 0)"
        unfolding n_circ_def p_circ_pt_def using \<open>h p > 0\<close>
        by (auto simp add: dist_norm)
      show "winding_number (n_circ 0) p'=0 \<and> p'\<notin>path_image (n_circ 0)" when "p'\<notin>s- pts" for p'
        unfolding n_circ_def p_circ_pt_def
        apply (auto intro!:winding_number_trivial)
        by (metis Diff_iff pathfinish_in_path_image pg(3) pg(4) subsetCE subset_insertI that)+
      show "f contour_integrable_on (n_circ 0)"
        unfolding n_circ_def p_circ_pt_def
        by (auto intro!:contour_integrable_continuous_linepath simp add:continuous_on_sing)
      show "contour_integral (n_circ 0) f = of_nat 0  *  contour_integral p_circ f"
        unfolding n_circ_def p_circ_pt_def by auto
    next
      case (Suc k)
      have n_Suc:"n_circ (Suc k) = p_circ +++ n_circ k" unfolding n_circ_def by auto
      have pcirc:"p \<notin> path_image p_circ" "valid_path p_circ" "pathfinish p_circ = pathstart (n_circ k)"
        using Suc(3) unfolding p_circ_def using \<open>h p > 0\<close> by (auto simp add: p_circ_def)
      have pcirc_image:"path_image p_circ \<subseteq> s - insert p pts"
        proof -
          have "path_image p_circ \<subseteq> cball p (h p)" using \<open>0 < h p\<close> p_circ_def by auto
          then show ?thesis using h_p pcirc(1) by auto
        qed
      have pcirc_integrable:"f contour_integrable_on p_circ"
        by (auto simp add:p_circ_def intro!: pcirc_image[unfolded p_circ_def]
          contour_integrable_continuous_circlepath holomorphic_on_imp_continuous_on
          holomorphic_on_subset[OF holo])
      show "valid_path (n_circ (Suc k))"
        using valid_path_join[OF pcirc(2) Suc(1) pcirc(3)] unfolding n_circ_def by auto
      show "path_image (n_circ (Suc k))
          = (if Suc k = 0 then {p + complex_of_real (h p)} else sphere p (h p))"
        proof -
          have "path_image p_circ = sphere p (h p)"
            unfolding p_circ_def using \<open>0 < h p\<close> by auto
          then show ?thesis unfolding n_Suc  using Suc.hyps(5)  \<open>h p>0\<close>
            by (auto simp add:  path_image_join[OF pcirc(3)]  dist_norm)
        qed
      then show "p \<notin> path_image (n_circ (Suc k))" using \<open>h p>0\<close> by auto
      show "winding_number (n_circ (Suc k)) p = of_nat (Suc k)"
        proof -
          have "winding_number p_circ p = 1"
            by (simp add: \<open>h p > 0\<close> p_circ_def winding_number_circlepath_centre)
          moreover have "p \<notin> path_image (n_circ k)" using Suc(5) \<open>h p>0\<close> by auto
          then have "winding_number (p_circ +++ n_circ k) p
              = winding_number p_circ p + winding_number (n_circ k) p"
            using  valid_path_imp_path Suc.hyps(1) Suc.hyps(2) pcirc
            apply (intro winding_number_join)
            by auto
          ultimately show ?thesis using Suc(2) unfolding n_circ_def
            by auto
        qed
      show "pathstart (n_circ (Suc k)) = p + h p"
        by (simp add: n_circ_def p_circ_def)
      show "pathfinish (n_circ (Suc k)) = p + h p"
        using Suc(4) unfolding n_circ_def by auto
      show "winding_number (n_circ (Suc k)) p'=0 \<and>  p'\<notin>path_image (n_circ (Suc k))" when "p'\<notin>s-pts" for p'
        proof -
          have " p' \<notin> path_image p_circ" using \<open>p \<in> s\<close> h p_circ_def that using pcirc_image by blast
          moreover have "p' \<notin> path_image (n_circ k)"
            using Suc.hyps(7) that by blast
          moreover have "winding_number p_circ p' = 0"
            proof -
              have "path_image p_circ \<subseteq> cball p (h p)"
                using h unfolding p_circ_def using \<open>p \<in> s\<close> by fastforce
              moreover have "p'\<notin>cball p (h p)" using \<open>p \<in> s\<close> h that "2.hyps"(2) by fastforce
              ultimately show ?thesis unfolding p_circ_def
                apply (intro winding_number_zero_outside)
                by auto
            qed
          ultimately show ?thesis
            unfolding n_Suc
            apply (subst winding_number_join)
            by (auto simp: valid_path_imp_path pcirc Suc that not_in_path_image_join Suc.hyps(7)[OF that])
        qed
      show "f contour_integrable_on (n_circ (Suc k))"
        unfolding n_Suc
        by (rule contour_integrable_joinI[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)])
      show "contour_integral (n_circ (Suc k)) f = (Suc k) *  contour_integral p_circ f"
        unfolding n_Suc
        by (auto simp add:contour_integral_join[OF pcirc_integrable Suc(8) pcirc(2) Suc(1)]
          Suc(9) algebra_simps)
    qed
  have cp[simp]:"pathstart cp = p + h p"  "pathfinish cp = p + h p"
         "valid_path cp" "path_image cp \<subseteq> s - insert p pts"
         "winding_number cp p = - n"
         "\<And>p'. p'\<notin>s - pts \<Longrightarrow> winding_number cp p'=0 \<and> p' \<notin> path_image cp"
         "f contour_integrable_on cp"
         "contour_integral cp f = - n * contour_integral p_circ f"
    proof -
      show "pathstart cp = p + h p" and "pathfinish cp = p + h p" and "valid_path cp"
        using n_circ unfolding cp_def by auto
    next
      have "sphere p (h p) \<subseteq>  s - insert p pts"
        using h[rule_format,of p] \<open>insert p pts \<subseteq> s\<close> by force
      moreover  have "p + complex_of_real (h p) \<in> s - insert p pts"
        using pg(3) pg(4) by (metis pathfinish_in_path_image subsetCE)
      ultimately show "path_image cp \<subseteq>  s - insert p pts" unfolding cp_def
        using n_circ(5)  by auto
    next
      show "winding_number cp p = - n"
        unfolding cp_def using winding_number_reversepath n_circ \<open>h p>0\<close>
        by (auto simp: valid_path_imp_path)
    next
      show "winding_number cp p'=0 \<and> p' \<notin> path_image cp" when "p'\<notin>s - pts" for p'
        unfolding cp_def
        apply (auto)
        apply (subst winding_number_reversepath)
        by (auto simp add: valid_path_imp_path n_circ(7)[OF that] n_circ(1))
    next
      show "f contour_integrable_on cp" unfolding cp_def
        using contour_integrable_reversepath_eq n_circ(1,8) by auto
    next
      show "contour_integral cp f = - n * contour_integral p_circ f"
        unfolding cp_def using contour_integral_reversepath[OF n_circ(1)] n_circ(9)
        by auto
    qed
  define g' where "g' \<equiv> g +++ pg +++ cp +++ (reversepath pg)"
  have "contour_integral g' f = (\<Sum>p\<in>pts. winding_number g' p * contour_integral (circlepath p (h p)) f)"
    proof (rule "2.hyps"(3)[of "s-{p}" "g'",OF _ _ \<open>finite pts\<close> ])
      show "connected (s - {p} - pts)" using connected by (metis Diff_insert2)
      show "open (s - {p})" using \<open>open s\<close> by auto
      show " pts \<subseteq> s - {p}" using \<open>insert p pts \<subseteq> s\<close> \<open> p \<notin> pts\<close>  by blast
      show "f holomorphic_on s - {p} - pts" using holo \<open>p \<notin> pts\<close> by (metis Diff_insert2)
      show "valid_path g'"
        unfolding g'_def cp_def using n_circ valid pg g_loop
        by (auto intro!:valid_path_join )
      show "pathfinish g' = pathstart g'"
        unfolding g'_def cp_def using pg(2) by simp
      show "path_image g' \<subseteq> s - {p} - pts"
        proof -
          define s' where "s' \<equiv> s - {p} - pts"
          have s':"s' = s-insert p pts " unfolding s'_def by auto
          then show ?thesis using path_img pg(4) cp(4)
            unfolding g'_def
            apply (fold s'_def s')
            apply (intro subset_path_image_join)
            by auto
        qed
      note path_join_imp[simp]
      show "\<forall>z. z \<notin> s - {p} \<longrightarrow> winding_number g' z = 0"
        proof clarify
          fix z assume z:"z\<notin>s - {p}"
          have "winding_number (g +++ pg +++ cp +++ reversepath pg) z = winding_number g z
              + winding_number (pg +++ cp +++ (reversepath pg)) z"
            proof (rule winding_number_join)
              show "path g" using \<open>valid_path g\<close> by (simp add: valid_path_imp_path)
              show "z \<notin> path_image g" using z path_img by auto
              show "path (pg +++ cp +++ reversepath pg)" using pg(3) cp
                by (simp add: valid_path_imp_path)
            next
              have "path_image (pg +++ cp +++ reversepath pg) \<subseteq> s - insert p pts"
                using pg(4) cp(4) by (auto simp:subset_path_image_join)
              then show "z \<notin> path_image (pg +++ cp +++ reversepath pg)" using z by auto
            next
              show "pathfinish g = pathstart (pg +++ cp +++ reversepath pg)" using g_loop by auto
            qed
          also have "... = winding_number g z + (winding_number pg z
              + winding_number (cp +++ (reversepath pg)) z)"
            proof (subst add_left_cancel,rule winding_number_join)
              show "path pg" and "path (cp +++ reversepath pg)"
               and "pathfinish pg = pathstart (cp +++ reversepath pg)"
                by (auto simp add: valid_path_imp_path)
              show "z \<notin> path_image pg" using pg(4) z by blast
              show "z \<notin> path_image (cp +++ reversepath pg)" using z
                by (metis Diff_iff \<open>z \<notin> path_image pg\<close> contra_subsetD cp(4) insertI1
                  not_in_path_image_join path_image_reversepath singletonD)
            qed
          also have "... = winding_number g z + (winding_number pg z
              + (winding_number cp z + winding_number (reversepath pg) z))"
            apply (auto intro!:winding_number_join simp: valid_path_imp_path)
            apply (metis Diff_iff contra_subsetD cp(4) insertI1 singletonD z)
            by (metis Diff_insert2 Diff_subset contra_subsetD pg(4) z)
          also have "... = winding_number g z + winding_number cp z"
            apply (subst winding_number_reversepath)
            apply (auto simp: valid_path_imp_path)
            by (metis Diff_iff contra_subsetD insertI1 pg(4) singletonD z)
          finally have "winding_number g' z = winding_number g z + winding_number cp z"
            unfolding g'_def .
          moreover have "winding_number g z + winding_number cp z = 0"
            using winding z \<open>n=winding_number g p\<close> by auto
          ultimately show "winding_number g' z = 0" unfolding g'_def by auto
        qed
      show "\<forall>pa\<in>s - {p}. 0 < h pa \<and> (\<forall>w\<in>cball pa (h pa). w \<in> s - {p} \<and> (w \<noteq> pa \<longrightarrow> w \<notin> pts))"
        using h by fastforce
    qed
  moreover have "contour_integral g' f = contour_integral g f
      - winding_number g p * contour_integral p_circ f"
    proof -
      have "contour_integral g' f =  contour_integral g f
        + contour_integral (pg +++ cp +++ reversepath pg) f"
        unfolding g'_def
        apply (subst contour_integral_join)
        by (auto simp add:open_Diff[OF \<open>open s\<close>,OF finite_imp_closed[OF fin]]
          intro!: contour_integrable_holomorphic_simple[OF holo _ _ path_img]
          contour_integrable_reversepath)
      also have "... = contour_integral g f + contour_integral pg f
          + contour_integral (cp +++ reversepath pg) f"
        apply (subst contour_integral_join)
        by (auto simp add:contour_integrable_reversepath)
      also have "... = contour_integral g f + contour_integral pg f
          + contour_integral cp f + contour_integral (reversepath pg) f"
        apply (subst contour_integral_join)
        by (auto simp add:contour_integrable_reversepath)
      also have "... = contour_integral g f + contour_integral cp f"
        using contour_integral_reversepath
        by (auto simp add:contour_integrable_reversepath)
      also have "... = contour_integral g f - winding_number g p * contour_integral p_circ f"
        using \<open>n=winding_number g p\<close> by auto
      finally show ?thesis .
    qed
  moreover have "winding_number g' p' = winding_number g p'" when "p'\<in>pts" for p'
    proof -
      have [simp]: "p' \<notin> path_image g" "p' \<notin> path_image pg" "p'\<notin>path_image cp"
        using "2.prems"(8) that
        apply blast
        apply (metis Diff_iff Diff_insert2 contra_subsetD pg(4) that)
        by (meson DiffD2 cp(4) rev_subsetD subset_insertI that)
      have "winding_number g' p' = winding_number g p'
          + winding_number (pg +++ cp +++ reversepath pg) p'" unfolding g'_def
        apply (subst winding_number_join)
        apply (simp_all add: valid_path_imp_path)
        apply (intro not_in_path_image_join)
        by auto
      also have "... = winding_number g p' + winding_number pg p'
          + winding_number (cp +++ reversepath pg) p'"
        apply (subst winding_number_join)
        apply (simp_all add: valid_path_imp_path)
        apply (intro not_in_path_image_join)
        by auto
      also have "... = winding_number g p' + winding_number pg p'+ winding_number cp p'
          + winding_number (reversepath pg) p'"
        apply (subst winding_number_join)
        by (simp_all add: valid_path_imp_path)
      also have "... = winding_number g p' + winding_number cp p'"
        apply (subst winding_number_reversepath)
        by (simp_all add: valid_path_imp_path)
      also have "... = winding_number g p'" using that by auto
      finally show ?thesis .
    qed
  ultimately show ?case unfolding p_circ_def
    apply (subst (asm) sum.cong[OF refl,
        of pts _ "\<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"])
    by (auto simp add:sum.insert[OF \<open>finite pts\<close> \<open>p\<notin>pts\<close>] algebra_simps)
qed

lemma Cauchy_theorem_singularities:
  assumes "open s" "connected s" "finite pts" and
          holo:"f holomorphic_on s-pts" and
          "valid_path g" and
          loop:"pathfinish g = pathstart g" and
          "path_image g \<subseteq> s-pts" and
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0" and
          avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
  shows "contour_integral g f = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
    (is "?L=?R")
proof -
  define circ where "circ \<equiv> \<lambda>p. winding_number g p * contour_integral (circlepath p (h p)) f"
  define pts1 where "pts1 \<equiv> pts \<inter> s"
  define pts2 where "pts2 \<equiv> pts - pts1"
  have "pts=pts1 \<union> pts2" "pts1 \<inter> pts2 = {}" "pts2 \<inter> s={}" "pts1\<subseteq>s"
    unfolding pts1_def pts2_def by auto
  have "contour_integral g f =  (\<Sum>p\<in>pts1. circ p)" unfolding circ_def
    proof (rule Cauchy_theorem_aux[OF \<open>open s\<close> _ _ \<open>pts1\<subseteq>s\<close> _ \<open>valid_path g\<close> loop _ homo])
      have "finite pts1" unfolding pts1_def using \<open>finite pts\<close> by auto
      then show "connected (s - pts1)"
        using \<open>open s\<close> \<open>connected s\<close> connected_open_delete_finite[of s] by auto
    next
      show "finite pts1" using \<open>pts = pts1 \<union> pts2\<close> assms(3) by auto
      show "f holomorphic_on s - pts1" by (metis Diff_Int2 Int_absorb holo pts1_def)
      show "path_image g \<subseteq> s - pts1" using assms(7) pts1_def by auto
      show "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts1))"
        by (simp add: avoid pts1_def)
    qed
  moreover have "sum circ pts2=0"
    proof -
      have "winding_number g p=0" when "p\<in>pts2" for p
        using  \<open>pts2 \<inter> s={}\<close> that homo[rule_format,of p] by auto
      thus ?thesis unfolding circ_def
        apply (intro sum.neutral)
        by auto
    qed
  moreover have "?R=sum circ pts1 + sum circ pts2"
    unfolding circ_def
    using sum.union_disjoint[OF _ _ \<open>pts1 \<inter> pts2 = {}\<close>] \<open>finite pts\<close> \<open>pts=pts1 \<union> pts2\<close>
    by blast
  ultimately show ?thesis
    apply (fold circ_def)
    by auto
qed

theorem Residue_theorem:
  fixes s pts::"complex set" and f::"complex \<Rightarrow> complex"
    and g::"real \<Rightarrow> complex"
  assumes "open s" "connected s" "finite pts" and
          holo:"f holomorphic_on s-pts" and
          "valid_path g" and
          loop:"pathfinish g = pathstart g" and
          "path_image g \<subseteq> s-pts" and
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z  = 0"
  shows "contour_integral g f = 2 * pi * \<i> *(\<Sum>p\<in>pts. winding_number g p * residue f p)"
proof -
  define c where "c \<equiv>  2 * pi * \<i>"
  obtain h where avoid:"\<forall>p\<in>s. h p>0 \<and> (\<forall>w\<in>cball p (h p). w\<in>s \<and> (w\<noteq>p \<longrightarrow> w \<notin> pts))"
    using finite_cball_avoid[OF \<open>open s\<close> \<open>finite pts\<close>] by metis
  have "contour_integral g f
      = (\<Sum>p\<in>pts. winding_number g p * contour_integral (circlepath p (h p)) f)"
    using Cauchy_theorem_singularities[OF assms avoid] .
  also have "... = (\<Sum>p\<in>pts.  c * winding_number g p * residue f p)"
    proof (intro sum.cong)
      show "pts = pts" by simp
    next
      fix x assume "x \<in> pts"
      show "winding_number g x * contour_integral (circlepath x (h x)) f
          = c * winding_number g x * residue f x"
        proof (cases "x\<in>s")
          case False
          then have "winding_number g x=0" using homo by auto
          thus ?thesis by auto
        next
          case True
          have "contour_integral (circlepath x (h x)) f = c* residue f x"
            using \<open>x\<in>pts\<close> \<open>finite pts\<close> avoid[rule_format,OF True]
            apply (intro base_residue[of "s-(pts-{x})",THEN contour_integral_unique,folded c_def])
            by (auto intro:holomorphic_on_subset[OF holo] open_Diff[OF \<open>open s\<close> finite_imp_closed])
          then show ?thesis by auto
        qed
    qed
  also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
    by (simp add: sum_distrib_left algebra_simps)
  finally show ?thesis unfolding c_def .
qed

subsection \<open>The argument principle\<close>

theorem argument_principle:
  fixes f::"complex \<Rightarrow> complex" and poles s:: "complex set"
  defines "pz \<equiv> {w. f w = 0 \<or> w \<in> poles}" \<comment> \<open>\<^term>\<open>pz\<close> is the set of poles and zeros\<close>
  assumes "open s" and
          "connected s" and
          f_holo:"f holomorphic_on s-poles" and
          h_holo:"h holomorphic_on s" and
          "valid_path g" and
          loop:"pathfinish g = pathstart g" and
          path_img:"path_image g \<subseteq> s - pz" and
          homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number g z = 0" and
          finite:"finite pz" and
          poles:"\<forall>p\<in>poles. is_pole f p"
  shows "contour_integral g (\<lambda>x. deriv f x * h x / f x) = 2 * pi * \<i> *
          (\<Sum>p\<in>pz. winding_number g p * h p * zorder f p)"
    (is "?L=?R")
proof -
  define c where "c \<equiv> 2 * complex_of_real pi * \<i> "
  define ff where "ff \<equiv> (\<lambda>x. deriv f x * h x / f x)"
  define cont where "cont \<equiv> \<lambda>ff p e. (ff has_contour_integral c * zorder f p * h p ) (circlepath p e)"
  define avoid where "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"

  have "\<exists>e>0. avoid p e \<and> (p\<in>pz \<longrightarrow> cont ff p e)" when "p\<in>s" for p
  proof -
    obtain e1 where "e1>0" and e1_avoid:"avoid p e1"
      using finite_cball_avoid[OF \<open>open s\<close> finite] \<open>p\<in>s\<close> unfolding avoid_def by auto
    have "\<exists>e2>0. cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2" when "p\<in>pz"
    proof -
      define po where "po \<equiv> zorder f p"
      define pp where "pp \<equiv> zor_poly f p"
      define f' where "f' \<equiv> \<lambda>w. pp w * (w - p) powr po"
      define ff' where "ff' \<equiv> (\<lambda>x. deriv f' x * h x / f' x)"
      obtain r where "pp p\<noteq>0" "r>0" and
          "r<e1" and
          pp_holo:"pp holomorphic_on cball p r" and
          pp_po:"(\<forall>w\<in>cball p r-{p}. f w = pp w * (w - p) powr po \<and> pp w \<noteq> 0)"
      proof -
        have "isolated_singularity_at f p"
        proof -
          have "f holomorphic_on ball p e1 - {p}"
            apply (intro holomorphic_on_subset[OF f_holo])
            using e1_avoid \<open>p\<in>pz\<close> unfolding avoid_def pz_def by force
          then show ?thesis unfolding isolated_singularity_at_def
            using \<open>e1>0\<close> analytic_on_open open_delete by blast
        qed
        moreover have "not_essential f p"
        proof (cases "is_pole f p")
          case True
          then show ?thesis unfolding not_essential_def by auto
        next
          case False
          then have "p\<in>s-poles" using \<open>p\<in>s\<close> poles unfolding pz_def by auto
          moreover have "open (s-poles)"
            using \<open>open s\<close>
            apply (elim open_Diff)
            apply (rule finite_imp_closed)
            using finite unfolding pz_def by simp
          ultimately have "isCont f p"
            using holomorphic_on_imp_continuous_on[OF f_holo] continuous_on_eq_continuous_at
            by auto
          then show ?thesis unfolding isCont_def not_essential_def by auto
        qed
        moreover have "\<exists>\<^sub>F w in at p. f w \<noteq> 0 "
        proof (rule ccontr)
          assume "\<not> (\<exists>\<^sub>F w in at p. f w \<noteq> 0)"
          then have "\<forall>\<^sub>F w in at p. f w= 0" unfolding frequently_def by auto
          then obtain rr where "rr>0" "\<forall>w\<in>ball p rr - {p}. f w =0"
            unfolding eventually_at by (auto simp add:dist_commute)
          then have "ball p rr - {p} \<subseteq> {w\<in>ball p rr-{p}. f w=0}" by blast
          moreover have "infinite (ball p rr - {p})" using \<open>rr>0\<close> using finite_imp_not_open by fastforce
          ultimately have "infinite {w\<in>ball p rr-{p}. f w=0}" using infinite_super by blast
          then have "infinite pz"
            unfolding pz_def infinite_super by auto
          then show False using \<open>finite pz\<close> by auto
        qed
        ultimately obtain r where "pp p \<noteq> 0" and r:"r>0" "pp holomorphic_on cball p r"
                  "(\<forall>w\<in>cball p r - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
          using zorder_exist[of f p,folded po_def pp_def] by auto
        define r1 where "r1=min r e1 / 2"
        have "r1<e1" unfolding r1_def using \<open>e1>0\<close> \<open>r>0\<close> by auto
        moreover have "r1>0" "pp holomorphic_on cball p r1"
                  "(\<forall>w\<in>cball p r1 - {p}. f w = pp w * (w - p) powr of_int po \<and> pp w \<noteq> 0)"
          unfolding r1_def using \<open>e1>0\<close> r by auto
        ultimately show ?thesis using that \<open>pp p\<noteq>0\<close> by auto
      qed

      define e2 where "e2 \<equiv> r/2"
      have "e2>0" using \<open>r>0\<close> unfolding e2_def by auto
      define anal where "anal \<equiv> \<lambda>w. deriv pp w * h w / pp w"
      define prin where "prin \<equiv> \<lambda>w. po * h w / (w - p)"
      have "((\<lambda>w.  prin w + anal w) has_contour_integral c * po * h p) (circlepath p e2)"
      proof (rule has_contour_integral_add[of _ _ _ _ 0,simplified])
        have "ball p r \<subseteq> s"
          using \<open>r<e1\<close> avoid_def ball_subset_cball e1_avoid by (simp add: subset_eq)
        then have "cball p e2 \<subseteq> s"
          using \<open>r>0\<close> unfolding e2_def by auto
        then have "(\<lambda>w. po * h w) holomorphic_on cball p e2"
          using h_holo by (auto intro!: holomorphic_intros)
        then show "(prin has_contour_integral c * po * h p ) (circlepath p e2)"
          using Cauchy_integral_circlepath_simple[folded c_def, of "\<lambda>w. po * h w"] \<open>e2>0\<close>
          unfolding prin_def by (auto simp add: mult.assoc)
        have "anal holomorphic_on ball p r" unfolding anal_def
          using pp_holo h_holo pp_po \<open>ball p r \<subseteq> s\<close> \<open>pp p\<noteq>0\<close>
          by (auto intro!: holomorphic_intros)
        then show "(anal has_contour_integral 0) (circlepath p e2)"
          using e2_def \<open>r>0\<close>
          by (auto elim!: Cauchy_theorem_disc_simple)
      qed
      then have "cont ff' p e2" unfolding cont_def po_def
      proof (elim has_contour_integral_eq)
        fix w assume "w \<in> path_image (circlepath p e2)"
        then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
        define wp where "wp \<equiv> w-p"
        have "wp\<noteq>0" and "pp w \<noteq>0"
          unfolding wp_def using \<open>w\<noteq>p\<close> \<open>w\<in>ball p r\<close> pp_po by auto
        moreover have der_f':"deriv f' w = po * pp w * (w-p) powr (po - 1) + deriv pp w * (w-p) powr po"
        proof (rule DERIV_imp_deriv)
          have "(pp has_field_derivative (deriv pp w)) (at w)"
            using DERIV_deriv_iff_has_field_derivative pp_holo \<open>w\<noteq>p\<close>
            by (meson open_ball \<open>w \<in> ball p r\<close> ball_subset_cball holomorphic_derivI holomorphic_on_subset)
          then show " (f' has_field_derivative of_int po * pp w * (w - p) powr of_int (po - 1)
                  + deriv pp w * (w - p) powr of_int po) (at w)"
            unfolding f'_def using \<open>w\<noteq>p\<close>
            by (auto intro!: derivative_eq_intros DERIV_cong[OF has_field_derivative_powr_of_int])
        qed
        ultimately show "prin w + anal w = ff' w"
          unfolding ff'_def prin_def anal_def
          apply simp
          apply (unfold f'_def)
          apply (fold wp_def)
          apply (auto simp add:field_simps)
          by (metis (no_types, lifting) diff_add_cancel mult.commute powr_add powr_to_1)
      qed
      then have "cont ff p e2" unfolding cont_def
      proof (elim has_contour_integral_eq)
        fix w assume "w \<in> path_image (circlepath p e2)"
        then have "w\<in>ball p r" and "w\<noteq>p" unfolding e2_def using \<open>r>0\<close> by auto
        have "deriv f' w =  deriv f w"
        proof (rule complex_derivative_transform_within_open[where s="ball p r - {p}"])
          show "f' holomorphic_on ball p r - {p}" unfolding f'_def using pp_holo
            by (auto intro!: holomorphic_intros)
        next
          have "ball p e1 - {p} \<subseteq> s - poles"
            using ball_subset_cball e1_avoid[unfolded avoid_def] unfolding pz_def
            by auto
          then have "ball p r - {p} \<subseteq> s - poles"
            apply (elim dual_order.trans)
            using \<open>r<e1\<close> by auto
          then show "f holomorphic_on ball p r - {p}" using f_holo
            by auto
        next
          show "open (ball p r - {p})" by auto
          show "w \<in> ball p r - {p}" using \<open>w\<in>ball p r\<close> \<open>w\<noteq>p\<close> by auto
        next
          fix x assume "x \<in> ball p r - {p}"
          then show "f' x = f x"
            using pp_po unfolding f'_def by auto
        qed
        moreover have " f' w  =  f w "
          using \<open>w \<in> ball p r\<close> ball_subset_cball subset_iff pp_po \<open>w\<noteq>p\<close>
          unfolding f'_def by auto
        ultimately show "ff' w = ff w"
          unfolding ff'_def ff_def by simp
      qed
      moreover have "cball p e2 \<subseteq> ball p e1"
        using \<open>0 < r\<close> \<open>r<e1\<close> e2_def by auto
      ultimately show ?thesis using \<open>e2>0\<close> by auto
    qed
    then obtain e2 where e2:"p\<in>pz \<longrightarrow> e2>0 \<and> cball p e2 \<subseteq> ball p e1 \<and> cont ff p e2"
      by auto
    define e4 where "e4 \<equiv> if p\<in>pz then e2 else  e1"
    have "e4>0" using e2 \<open>e1>0\<close> unfolding e4_def by auto
    moreover have "avoid p e4" using e2 \<open>e1>0\<close> e1_avoid unfolding e4_def avoid_def by auto
    moreover have "p\<in>pz \<longrightarrow> cont ff p e4"
      by (auto simp add: e2 e4_def)
    ultimately show ?thesis by auto
  qed
  then obtain get_e where get_e:"\<forall>p\<in>s. get_e p>0 \<and> avoid p (get_e p)
      \<and> (p\<in>pz \<longrightarrow> cont ff p (get_e p))"
    by metis
  define ci where "ci \<equiv> \<lambda>p. contour_integral (circlepath p (get_e p)) ff"
  define w where "w \<equiv> \<lambda>p. winding_number g p"
  have "contour_integral g ff = (\<Sum>p\<in>pz. w p * ci p)" unfolding ci_def w_def
  proof (rule Cauchy_theorem_singularities[OF \<open>open s\<close> \<open>connected s\<close> finite _ \<open>valid_path g\<close> loop
        path_img homo])
    have "open (s - pz)" using open_Diff[OF _ finite_imp_closed[OF finite]] \<open>open s\<close> by auto
    then show "ff holomorphic_on s - pz" unfolding ff_def using f_holo h_holo
      by (auto intro!: holomorphic_intros simp add:pz_def)
  next
    show "\<forall>p\<in>s. 0 < get_e p \<and> (\<forall>w\<in>cball p (get_e p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz))"
      using get_e using avoid_def by blast
  qed
  also have "... = (\<Sum>p\<in>pz. c * w p * h p * zorder f p)"
  proof (rule sum.cong[of pz pz,simplified])
    fix p assume "p \<in> pz"
    show "w p * ci p = c * w p * h p * (zorder f p)"
    proof (cases "p\<in>s")
      assume "p \<in> s"
      have "ci p = c * h p * (zorder f p)" unfolding ci_def
        apply (rule contour_integral_unique)
        using get_e \<open>p\<in>s\<close> \<open>p\<in>pz\<close> unfolding cont_def by (metis mult.assoc mult.commute)
      thus ?thesis by auto
    next
      assume "p\<notin>s"
      then have "w p=0" using homo unfolding w_def by auto
      then show ?thesis by auto
    qed
  qed
  also have "... = c*(\<Sum>p\<in>pz. w p * h p * zorder f p)"
    unfolding sum_distrib_left by (simp add:algebra_simps)
  finally have "contour_integral g ff = c * (\<Sum>p\<in>pz. w p * h p * of_int (zorder f p))" .
  then show ?thesis unfolding ff_def c_def w_def by simp
qed

subsection \<open>Rouche's theorem \<close>

theorem Rouche_theorem:
  fixes f g::"complex \<Rightarrow> complex" and s:: "complex set"
  defines "fg\<equiv>(\<lambda>p. f p + g p)"
  defines "zeros_fg\<equiv>{p. fg p = 0}" and "zeros_f\<equiv>{p. f p = 0}"
  assumes
    "open s" and "connected s" and
    "finite zeros_fg" and
    "finite zeros_f" and
    f_holo:"f holomorphic_on s" and
    g_holo:"g holomorphic_on s" and
    "valid_path \<gamma>" and
    loop:"pathfinish \<gamma> = pathstart \<gamma>" and
    path_img:"path_image \<gamma> \<subseteq> s " and
    path_less:"\<forall>z\<in>path_image \<gamma>. cmod(f z) > cmod(g z)" and
    homo:"\<forall>z. (z \<notin> s) \<longrightarrow> winding_number \<gamma> z = 0"
  shows "(\<Sum>p\<in>zeros_fg. winding_number \<gamma> p * zorder fg p)
          = (\<Sum>p\<in>zeros_f. winding_number \<gamma> p * zorder f p)"
proof -
  have path_fg:"path_image \<gamma> \<subseteq> s - zeros_fg"
  proof -
    have False when "z\<in>path_image \<gamma>" and "f z + g z=0" for z
    proof -
      have "cmod (f z) > cmod (g z)" using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
      moreover have "f z = - g z"  using \<open>f z + g z =0\<close> by (simp add: eq_neg_iff_add_eq_0)
      then have "cmod (f z) = cmod (g z)" by auto
      ultimately show False by auto
    qed
    then show ?thesis unfolding zeros_fg_def fg_def using path_img by auto
  qed
  have path_f:"path_image \<gamma> \<subseteq> s - zeros_f"
  proof -
    have False when "z\<in>path_image \<gamma>" and "f z =0" for z
    proof -
      have "cmod (g z) < cmod (f z) " using \<open>z\<in>path_image \<gamma>\<close> path_less by auto
      then have "cmod (g z) < 0" using \<open>f z=0\<close> by auto
      then show False by auto
    qed
    then show ?thesis unfolding zeros_f_def using path_img by auto
  qed
  define w where "w \<equiv> \<lambda>p. winding_number \<gamma> p"
  define c where "c \<equiv> 2 * complex_of_real pi * \<i>"
  define h where "h \<equiv> \<lambda>p. g p / f p + 1"
  obtain spikes
    where "finite spikes" and spikes: "\<forall>x\<in>{0..1} - spikes. \<gamma> differentiable at x"
    using \<open>valid_path \<gamma>\<close>
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  have h_contour:"((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
  proof -
    have outside_img:"0 \<in> outside (path_image (h o \<gamma>))"
    proof -
      have "h p \<in> ball 1 1" when "p\<in>path_image \<gamma>" for p
      proof -
        have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
          apply (cases "cmod (f p) = 0")
          by (auto simp add: norm_divide)
        then show ?thesis unfolding h_def by (auto simp add:dist_complex_def)
      qed
      then have "path_image (h o \<gamma>) \<subseteq> ball 1 1"
        by (simp add: image_subset_iff path_image_compose)
      moreover have " (0::complex) \<notin> ball 1 1" by (simp add: dist_norm)
      ultimately show "?thesis"
        using  convex_in_outside[of "ball 1 1" 0] outside_mono by blast
    qed
    have valid_h:"valid_path (h \<circ> \<gamma>)"
    proof (rule valid_path_compose_holomorphic[OF \<open>valid_path \<gamma>\<close> _ _ path_f])
      show "h holomorphic_on s - zeros_f"
        unfolding h_def using f_holo g_holo
        by (auto intro!: holomorphic_intros simp add:zeros_f_def)
    next
      show "open (s - zeros_f)" using \<open>finite zeros_f\<close> \<open>open s\<close> finite_imp_closed
        by auto
    qed
    have "((\<lambda>z. 1/z) has_contour_integral 0) (h \<circ> \<gamma>)"
    proof -
      have "0 \<notin> path_image (h \<circ> \<gamma>)" using outside_img by (simp add: outside_def)
      then have "((\<lambda>z. 1/z) has_contour_integral c * winding_number (h \<circ> \<gamma>) 0) (h \<circ> \<gamma>)"
        using has_contour_integral_winding_number[of "h o \<gamma>" 0,simplified] valid_h
        unfolding c_def by auto
      moreover have "winding_number (h o \<gamma>) 0 = 0"
      proof -
        have "0 \<in> outside (path_image (h \<circ> \<gamma>))" using outside_img .
        moreover have "path (h o \<gamma>)"
          using valid_h  by (simp add: valid_path_imp_path)
        moreover have "pathfinish (h o \<gamma>) = pathstart (h o \<gamma>)"
          by (simp add: loop pathfinish_compose pathstart_compose)
        ultimately show ?thesis using winding_number_zero_in_outside by auto
      qed
      ultimately show ?thesis by auto
    qed
    moreover have "vector_derivative (h \<circ> \<gamma>) (at x) = vector_derivative \<gamma> (at x) * deriv h (\<gamma> x)"
      when "x\<in>{0..1} - spikes" for x
    proof (rule vector_derivative_chain_at_general)
      show "\<gamma> differentiable at x" using that \<open>valid_path \<gamma>\<close> spikes by auto
    next
      define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
      define t where "t \<equiv> \<gamma> x"
      have "f t\<noteq>0" unfolding zeros_f_def t_def
        by (metis DiffD1 image_eqI norm_not_less_zero norm_zero path_defs(4) path_less that)
      moreover have "t\<in>s"
        using contra_subsetD path_image_def path_fg t_def that by fastforce
      ultimately have "(h has_field_derivative der t) (at t)"
        unfolding h_def der_def using g_holo f_holo \<open>open s\<close>
        by (auto intro!: holomorphic_derivI derivative_eq_intros)
      then show "h field_differentiable at (\<gamma> x)"
        unfolding t_def field_differentiable_def by blast
    qed
    then have " ((/) 1 has_contour_integral 0) (h \<circ> \<gamma>)
                  = ((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
      unfolding has_contour_integral
      apply (intro has_integral_spike_eq[OF negligible_finite, OF \<open>finite spikes\<close>])
      by auto
    ultimately show ?thesis by auto
  qed
  then have "contour_integral \<gamma> (\<lambda>x. deriv h x / h x) = 0"
    using  contour_integral_unique by simp
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = contour_integral \<gamma> (\<lambda>x. deriv f x / f x)
      + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
  proof -
    have "(\<lambda>p. deriv f p / f p) contour_integrable_on \<gamma>"
    proof (rule contour_integrable_holomorphic_simple[OF _ _ \<open>valid_path \<gamma>\<close> path_f])
      show "open (s - zeros_f)" using finite_imp_closed[OF \<open>finite zeros_f\<close>] \<open>open s\<close>
        by auto
      then show "(\<lambda>p. deriv f p / f p) holomorphic_on s - zeros_f"
        using f_holo
        by (auto intro!: holomorphic_intros simp add:zeros_f_def)
    qed
    moreover have "(\<lambda>p. deriv h p / h p) contour_integrable_on \<gamma>"
      using h_contour
      by (simp add: has_contour_integral_integrable)
    ultimately have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x + deriv h x / h x) =
                        contour_integral \<gamma> (\<lambda>p. deriv f p / f p) + contour_integral \<gamma> (\<lambda>p. deriv h p / h p)"
      using contour_integral_add[of "(\<lambda>p. deriv f p / f p)" \<gamma> "(\<lambda>p. deriv h p / h p)" ]
      by auto
    moreover have "deriv fg p / fg p =  deriv f p / f p + deriv h p / h p"
                      when "p\<in> path_image \<gamma>" for p
    proof -
      have "fg p\<noteq>0" and "f p\<noteq>0" using path_f path_fg that unfolding zeros_f_def zeros_fg_def
        by auto
      have "h p\<noteq>0"
      proof (rule ccontr)
        assume "\<not> h p \<noteq> 0"
        then have "g p / f p= -1" unfolding h_def by (simp add: add_eq_0_iff2)
        then have "cmod (g p/f p) = 1" by auto
        moreover have "cmod (g p/f p) <1" using path_less[rule_format,OF that]
          apply (cases "cmod (f p) = 0")
          by (auto simp add: norm_divide)
        ultimately show False by auto
      qed
      have der_fg:"deriv fg p =  deriv f p + deriv g p" unfolding fg_def
        using f_holo g_holo holomorphic_on_imp_differentiable_at[OF _  \<open>open s\<close>] path_img that
        by auto
      have der_h:"deriv h p = (deriv g p * f p - g p * deriv f p)/(f p * f p)"
      proof -
        define der where "der \<equiv> \<lambda>p. (deriv g p * f p - g p * deriv f p)/(f p * f p)"
        have "p\<in>s" using path_img that by auto
        then have "(h has_field_derivative der p) (at p)"
          unfolding h_def der_def using g_holo f_holo \<open>open s\<close> \<open>f p\<noteq>0\<close>
          by (auto intro!: derivative_eq_intros holomorphic_derivI)
        then show ?thesis unfolding der_def using DERIV_imp_deriv by auto
      qed
      show ?thesis
        apply (simp only:der_fg der_h)
        apply (auto simp add:field_simps \<open>h p\<noteq>0\<close> \<open>f p\<noteq>0\<close> \<open>fg p\<noteq>0\<close>)
        by (auto simp add:field_simps h_def \<open>f p\<noteq>0\<close> fg_def)
    qed
    then have "contour_integral \<gamma> (\<lambda>p. deriv fg p / fg p)
                  = contour_integral \<gamma> (\<lambda>p. deriv f p / f p + deriv h p / h p)"
      by (elim contour_integral_eq)
    ultimately show ?thesis by auto
  qed
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv fg x / fg x) = c * (\<Sum>p\<in>zeros_fg. w p * zorder fg p)"
    unfolding c_def zeros_fg_def w_def
  proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
        , of _ "{}" "\<lambda>_. 1",simplified])
    show "fg holomorphic_on s" unfolding fg_def using f_holo g_holo holomorphic_on_add by auto
    show "path_image \<gamma> \<subseteq> s - {p. fg p = 0}" using path_fg unfolding zeros_fg_def .
    show " finite {p. fg p = 0}" using \<open>finite zeros_fg\<close> unfolding zeros_fg_def .
  qed
  moreover have "contour_integral \<gamma> (\<lambda>x. deriv f x / f x) = c * (\<Sum>p\<in>zeros_f. w p * zorder f p)"
    unfolding c_def zeros_f_def w_def
  proof (rule argument_principle[OF \<open>open s\<close> \<open>connected s\<close> _ _ \<open>valid_path \<gamma>\<close> loop _ homo
        , of _ "{}" "\<lambda>_. 1",simplified])
    show "f holomorphic_on s" using f_holo g_holo holomorphic_on_add by auto
    show "path_image \<gamma> \<subseteq> s - {p. f p = 0}" using path_f unfolding zeros_f_def .
    show " finite {p. f p = 0}" using \<open>finite zeros_f\<close> unfolding zeros_f_def .
  qed
  ultimately have " c* (\<Sum>p\<in>zeros_fg. w p * (zorder fg p)) = c* (\<Sum>p\<in>zeros_f. w p * (zorder f p))"
    by auto
  then show ?thesis unfolding c_def using w_def by auto
qed

end