Theory HOL-Complex_Analysis.Complex_Singularities

theory Complex_Singularities
  imports Conformal_Mappings
begin

subsection ‹Non-essential singular points›

definitiontag important› is_pole ::
  "('a::topological_space  'b::real_normed_vector)  'a  bool" where
  "is_pole f a =  (LIM x (at a). f x :> at_infinity)"

lemma is_pole_cong:
  assumes "eventually (λx. f x = g x) (at a)" "a=b"
  shows "is_pole f a  is_pole g b"
  unfolding is_pole_def using assms by (intro filterlim_cong,auto)

lemma is_pole_transform:
  assumes "is_pole f a" "eventually (λx. f x = g x) (at a)" "a=b"
  shows "is_pole g b"
  using is_pole_cong assms by auto

lemma is_pole_shift_iff:
  fixes f :: "('a::real_normed_vector  'b::real_normed_vector)"
  shows "is_pole (f  (+) d) a = is_pole f (a + d)"
  by (metis add_diff_cancel_right' filterlim_shift_iff is_pole_def)

lemma is_pole_tendsto:
  fixes f::"('a::topological_space  'b::real_normed_div_algebra)"
  shows "is_pole f x  ((inverse o f)  0) (at x)"
  unfolding is_pole_def
  by (auto simp add:filterlim_inverse_at_iff[symmetric] comp_def filterlim_at)

lemma is_pole_shift_0:
  fixes f :: "('a::real_normed_vector  'b::real_normed_vector)"
  shows "is_pole f z  is_pole (λx. f (z + x)) 0"
  unfolding is_pole_def by (subst at_to_0) (auto simp: filterlim_filtermap add_ac)

lemma is_pole_shift_0':
  fixes f :: "('a::real_normed_vector  'b::real_normed_vector)"
  shows "NO_MATCH 0 z  is_pole f z  is_pole (λx. f (z + x)) 0"
  by (metis is_pole_shift_0)

lemma is_pole_compose_iff:
  assumes "filtermap g (at x) = (at y)"
  shows   "is_pole (f  g) x  is_pole f y"
  unfolding is_pole_def filterlim_def filtermap_compose assms ..

lemma is_pole_inverse_holomorphic:
  assumes "open s"
    and f_holo:"f holomorphic_on (s-{z})"
    and pole:"is_pole f z"
    and non_z:"xs-{z}. f x0"
  shows "(λx. if x=z then 0 else inverse (f x)) holomorphic_on s"
proof -
  define g where "g  λx. if x=z then 0 else inverse (f x)"
  have "isCont g z" unfolding isCont_def  using is_pole_tendsto[OF pole]
    by (simp add: g_def cong: LIM_cong)
  moreover have "continuous_on (s-{z}) f" using f_holo holomorphic_on_imp_continuous_on by auto
  hence "continuous_on (s-{z}) (inverse o f)" unfolding comp_def
    by (auto elim!:continuous_on_inverse simp add:non_z)
  hence "continuous_on (s-{z}) g" unfolding g_def
    using continuous_on_eq by fastforce
  ultimately have "continuous_on s g" using open_delete[OF open s] open s
    by (auto simp add:continuous_on_eq_continuous_at)
  moreover have "(inverse o f) holomorphic_on (s-{z})"
    unfolding comp_def using f_holo
    by (auto elim!:holomorphic_on_inverse simp add:non_z)
  hence "g holomorphic_on (s-{z})"
    using g_def holomorphic_transform by force
  ultimately show ?thesis unfolding g_def using open s
    by (auto elim!: no_isolated_singularity)
qed

lemma not_is_pole_holomorphic:
  assumes "open A" "x  A" "f holomorphic_on A"
  shows   "¬is_pole f x"
proof -
  have "continuous_on A f" 
    by (intro holomorphic_on_imp_continuous_on) fact
  with assms have "isCont f x" 
    by (simp add: continuous_on_eq_continuous_at)
  hence "f x f x" 
    by (simp add: isCont_def)
  thus "¬is_pole f x" 
    unfolding is_pole_def
    using not_tendsto_and_filterlim_at_infinity[of "at x" f "f x"] by auto
qed

lemma is_pole_inverse_power: "n > 0  is_pole (λz::complex. 1 / (z - a) ^ n) a"
  unfolding is_pole_def inverse_eq_divide [symmetric]
  by (intro filterlim_compose[OF filterlim_inverse_at_infinity] tendsto_intros)
     (auto simp: filterlim_at eventually_at intro!: exI[of _ 1] tendsto_eq_intros)

lemma is_pole_cmult_iff [simp]:
  assumes "c  0"
  shows "is_pole (λz. c * f z :: 'a :: real_normed_field) z  is_pole f z"
proof
  assume "is_pole (λz. c * f z) z"
  with c0 have "is_pole (λz. inverse c * (c * f z)) z" 
    unfolding is_pole_def
    by (force intro: tendsto_mult_filterlim_at_infinity)
  thus "is_pole f z"
    using c0 by (simp add: field_simps)
next
  assume "is_pole f z"
  with c0 show "is_pole (λz. c * f z) z"  
    by (auto intro!: tendsto_mult_filterlim_at_infinity simp: is_pole_def)
qed

lemma is_pole_uminus_iff [simp]: "is_pole (λz. -f z :: 'a :: real_normed_field) z  is_pole f z"
  using is_pole_cmult_iff[of "-1" f] by (simp del: is_pole_cmult_iff)

lemma is_pole_inverse: "is_pole (λz::complex. 1 / (z - a)) a"
  using is_pole_inverse_power[of 1 a] by simp

lemma is_pole_divide:
  fixes f :: "'a :: t2_space  'b :: real_normed_field"
  assumes "isCont f z" "filterlim g (at 0) (at z)" "f z  0"
  shows   "is_pole (λz. f z / g z) z"
proof -
  have "filterlim (λz. f z * inverse (g z)) at_infinity (at z)"
    using assms filterlim_compose filterlim_inverse_at_infinity isCont_def
      tendsto_mult_filterlim_at_infinity by blast
  thus ?thesis by (simp add: field_split_simps is_pole_def)
qed

lemma is_pole_basic:
  assumes "f holomorphic_on A" "open A" "z  A" "f z  0" "n > 0"
  shows   "is_pole (λw. f w / (w - z) ^ n) z"
proof (rule is_pole_divide)
  have "continuous_on A f" by (rule holomorphic_on_imp_continuous_on) fact
  with assms show "isCont f z" by (auto simp: continuous_on_eq_continuous_at)
  have "filterlim (λw. (w - z) ^ n) (nhds 0) (at z)"
    using assms by (auto intro!: tendsto_eq_intros)
  thus "filterlim (λw. (w - z) ^ n) (at 0) (at z)"
    by (intro filterlim_atI tendsto_eq_intros)
       (insert assms, auto simp: eventually_at_filter)
qed fact+

lemma is_pole_basic':
  assumes "f holomorphic_on A" "open A" "0  A" "f 0  0" "n > 0"
  shows   "is_pole (λw. f w / w ^ n) 0"
  using is_pole_basic[of f A 0] assms by simp

lemma is_pole_compose: 
  assumes "is_pole f w" "g z w" "eventually (λz. g z  w) (at z)"
  shows   "is_pole (λx. f (g x)) z"
  using assms(1) unfolding is_pole_def
  by (rule filterlim_compose) (use assms in auto simp: filterlim_at)

lemma is_pole_plus_const_iff:
  "is_pole f z  is_pole (λx. f x + c) z"
proof 
  assume "is_pole f z"
  then have "filterlim f at_infinity (at z)" unfolding is_pole_def .
  moreover have "((λ_. c)  c) (at z)" by auto
  ultimately have " LIM x (at z). f x + c :> at_infinity"
    using tendsto_add_filterlim_at_infinity'[of f "at z"] by auto
  then show "is_pole (λx. f x + c) z" unfolding is_pole_def .
next
  assume "is_pole (λx. f x + c) z"
  then have "filterlim (λx. f x + c) at_infinity (at z)" 
    unfolding is_pole_def .
  moreover have "((λ_. -c)  -c) (at z)" by auto
  ultimately have " LIM x (at z). f x :> at_infinity"
    using tendsto_add_filterlim_at_infinity'[of "(λx. f x + c)"
        "at z" "(λ_. - c)" "-c"] 
    by auto
  then show "is_pole f z" unfolding is_pole_def .
qed

lemma is_pole_minus_const_iff:
  "is_pole (λx. f x - c) z  is_pole f z"
  using is_pole_plus_const_iff [of f z "-c"] by simp

lemma is_pole_alt:
  "is_pole f x  = (B>0. U. open U  xU  (yU. yx  norm (f y)B))"
  unfolding is_pole_def
  unfolding filterlim_at_infinity[of 0,simplified] eventually_at_topological
  by auto

lemma is_pole_mult_analytic_nonzero1:
  assumes "is_pole g x" "f analytic_on {x}" "f x  0"
  shows   "is_pole (λx. f x * g x) x"
  unfolding is_pole_def
proof (rule tendsto_mult_filterlim_at_infinity)
  show "f x f x"
    using assms by (simp add: analytic_at_imp_isCont isContD)
qed (use assms in auto simp: is_pole_def)

lemma is_pole_mult_analytic_nonzero2:
  assumes "is_pole f x" "g analytic_on {x}" "g x  0"
  shows   "is_pole (λx. f x * g x) x"
proof -
  have g: "g analytic_on {x}"
    using assms by auto
  show ?thesis
    using is_pole_mult_analytic_nonzero1 [OF is_pole f x g] g x  0
    by (simp add: mult.commute)
qed

lemma is_pole_mult_analytic_nonzero1_iff:
  assumes "f analytic_on {x}" "f x  0"
  shows   "is_pole (λx. f x * g x) x  is_pole g x"
proof
  assume "is_pole g x"
  thus "is_pole (λx. f x * g x) x"
    by (intro is_pole_mult_analytic_nonzero1 assms)
next
  assume "is_pole (λx. f x * g x) x"
  hence "is_pole (λx. inverse (f x) * (f x * g x)) x"
    by (rule is_pole_mult_analytic_nonzero1)
       (use assms in auto intro!: analytic_intros)
  also have "?this  is_pole g x"
  proof (rule is_pole_cong)
    have "eventually (λx. f x  0) (at x)"
      using assms by (simp add: analytic_at_neq_imp_eventually_neq)
    thus "eventually (λx. inverse (f x) * (f x * g x) = g x) (at x)"
      by eventually_elim auto
  qed auto
  finally show "is_pole g x" .
qed

lemma is_pole_mult_analytic_nonzero2_iff:
  assumes "g analytic_on {x}" "g x  0"
  shows   "is_pole (λx. f x * g x) x  is_pole f x"
  by (subst mult.commute, rule is_pole_mult_analytic_nonzero1_iff) (fact assms)+

lemma frequently_const_imp_not_is_pole:
  fixes z :: "'a::first_countable_topology"
  assumes "frequently (λw. f w = c) (at z)"
  shows   "¬ is_pole f z"
proof
  assume "is_pole f z"
  from assms have "z islimpt {w. f w = c}"
    by (simp add: islimpt_conv_frequently_at)
  then obtain g where g: "n. g n  {w. f w = c} - {z}" "g  z"
    unfolding islimpt_sequential by blast
  then have "(f  g)  c"
    by (simp add: tendsto_eventually)
  moreover have *: "filterlim g (at z) sequentially"
    using g by (auto simp: filterlim_at)
  have "filterlim (f  g) at_infinity sequentially"
    unfolding o_def by (rule filterlim_compose [OF _ *])
                       (use is_pole f z in simp add: is_pole_def)
  ultimately show False
    using not_tendsto_and_filterlim_at_infinity trivial_limit_sequentially by blast
qed
  
 text ‹The proposition
              termx. ((f::complexcomplex)  x) (at z)  is_pole f z
can be interpreted as the complex function termf has a non-essential singularity at termz
(i.e. the singularity is either removable or a pole).›
definition not_essential::"[complex  complex, complex]  bool" where
  "not_essential f z = (x. fzx  is_pole f z)"

definition isolated_singularity_at::"[complex  complex, complex]  bool" where
  "isolated_singularity_at f z = (r>0. f analytic_on ball z r-{z})"

lemma not_essential_cong:
  assumes "eventually (λx. f x = g x) (at z)" "z = z'"
  shows   "not_essential f z  not_essential g z'"
  unfolding not_essential_def using assms filterlim_cong is_pole_cong by fastforce

lemma not_essential_compose_iff:
  assumes "filtermap g (at z) = at z'"
  shows   "not_essential (f  g) z = not_essential f z'"
  unfolding not_essential_def filterlim_def filtermap_compose assms is_pole_compose_iff[OF assms]
  by blast

lemma isolated_singularity_at_cong:
  assumes "eventually (λx. f x = g x) (at z)" "z = z'"
  shows   "isolated_singularity_at f z  isolated_singularity_at g z'"
proof -
  have "isolated_singularity_at g z"
    if "isolated_singularity_at f z" "eventually (λx. f x = g x) (at z)" for f g
  proof -
    from that(1) obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
      by (auto simp: isolated_singularity_at_def)
    from that(2) obtain r' where r': "r' > 0" "xball z r'-{z}. f x = g x"
      unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_commute)

    have "f holomorphic_on ball z r - {z}"
      using r(2) by (subst (asm) analytic_on_open) auto
    hence "f holomorphic_on ball z (min r r') - {z}"
      by (rule holomorphic_on_subset) auto
    also have "?this  g holomorphic_on ball z (min r r') - {z}"
      using r' by (intro holomorphic_cong) auto
    also have "  g analytic_on ball z (min r r') - {z}"
      by (subst analytic_on_open) auto
    finally show ?thesis
      unfolding isolated_singularity_at_def
      by (intro exI[of _ "min r r'"]) (use r > 0 r' > 0 in auto)
  qed
  from this[of f g] this[of g f] assms show ?thesis
    by (auto simp: eq_commute)
qed
  
lemma removable_singularity:
  assumes "f holomorphic_on A - {x}" "open A"
  assumes "f x c"
  shows   "(λy. if y = x then c else f y) holomorphic_on A" (is "?g holomorphic_on _")
proof -
  have "continuous_on A ?g"
    unfolding continuous_on_def
  proof
    fix y assume y: "y  A"
    show "(?g  ?g y) (at y within A)"
    proof (cases "y = x")
      case False
      have "continuous_on (A - {x}) f"
        using assms(1) by (meson holomorphic_on_imp_continuous_on)
      hence "(f  ?g y) (at y within A - {x})"
        using False y by (auto simp: continuous_on_def)
      also have "?this  (?g  ?g y) (at y within A - {x})"
        by (intro filterlim_cong refl) (auto simp: eventually_at_filter)
      also have "at y within A - {x} = at y within A"
        using y assms False
        by (intro at_within_nhd[of _ "A - {x}"]) auto
      finally show ?thesis .
    next
      case [simp]: True
      have "f x c"
        by fact
      also have "?this  (?g  ?g y) (at y)"
        by (intro filterlim_cong) (auto simp: eventually_at_filter)
      finally show ?thesis
        by (meson Lim_at_imp_Lim_at_within)
    qed
  qed
  moreover {
    have "?g holomorphic_on A - {x}"
      using assms(1) holomorphic_transform by fastforce
  }
  ultimately show ?thesis
    by (rule no_isolated_singularity) (use assms in auto)
qed

lemma removable_singularity':
  assumes "isolated_singularity_at f z"
  assumes "f z c"
  shows   "(λy. if y = z then c else f y) analytic_on {z}"
proof -
  from assms obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
    by (auto simp: isolated_singularity_at_def)
  have "(λy. if y = z then c else f y) holomorphic_on ball z r"
  proof (rule removable_singularity)
    show "f holomorphic_on ball z r - {z}"
      using r(2) by (subst (asm) analytic_on_open) auto
  qed (use assms in auto)
  thus ?thesis
    using r(1) unfolding analytic_at by (intro exI[of _ "ball z r"]) auto
qed

named_theorems singularity_intros "introduction rules for singularities"

lemma holomorphic_factor_unique:
  fixes f::"complex  complex" and z::complex and r::real and m n::int
  assumes "r>0" "g z0" "h z0"
    and asm:"wball z r-{z}. f w = g w * (w-z) powi n  g w0  f w =  h w * (w - z) powi m  h w0"
    and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
  shows "n=m"
proof -
  have [simp]:"at z within ball z r  bot" using r>0
      by (auto simp add:at_within_ball_bot_iff)
  have False when "n>m"
  proof -
    have "(h  0) (at z within ball z r)"
    proof (rule Lim_transform_within[OF _ r>0, where f="λw. (w - z) powi (n - m) * g w"])
      have "wball z r-{z}. h w = (w-z)powi(n-m) * g w"
        using n>m asm r>0 by (simp add: field_simps power_int_diff) force
      then show "x'  ball z r; 0 < dist x' z;dist x' z < r
             (x' - z) powi (n - m) * g x' = h x'" for x' by auto
    next
      define F where "F  at z within ball z r"
      define f' where "f'  λx. (x - z) powi (n-m)"
      have "f' z=0" using n>m unfolding f'_def by auto
      moreover have "continuous F f'" unfolding f'_def F_def continuous_def
        using n>m
          by (auto simp add: Lim_ident_at  intro!:tendsto_powr_complex_0 tendsto_eq_intros)
      ultimately have "(f'  0) F" unfolding F_def
        by (simp add: continuous_within)
      moreover have "(g  g z) F"
        unfolding F_def
        using r>0 centre_in_ball continuous_on_def g_holo holomorphic_on_imp_continuous_on by blast
      ultimately show " ((λw. f' w * g w)  0) F" using tendsto_mult by fastforce
    qed
    moreover have "(h  h z) (at z within ball z r)"
      using holomorphic_on_imp_continuous_on[OF h_holo]
      by (auto simp add:continuous_on_def r>0)
    ultimately have "h z=0" by (auto intro!: tendsto_unique)
    thus False using h z0 by auto
  qed
  moreover have False when "m>n"
  proof -
    have "(g  0) (at z within ball z r)"
    proof (rule Lim_transform_within[OF _ r>0, where f="λw. (w - z) powi (m - n) * h w"])
      have "wball z r -{z}. g w = (w-z) powi (m-n) * h w" using m>n asm
        by (simp add:field_simps power_int_diff) force
      then show "x'  ball z r; 0 < dist x' z;dist x' z < r
             (x' - z) powi (m - n) * h x' = g x'" for x' by auto
    next
      define F where "F  at z within ball z r"
      define f' where "f' λx. (x - z) powi (m-n)"
      have "f' z=0" using m>n unfolding f'_def by auto
      moreover have "continuous F f'" unfolding f'_def F_def continuous_def
        using m>n
        by (auto simp: Lim_ident_at intro!:tendsto_powr_complex_0 tendsto_eq_intros)
      ultimately have "(f'  0) F" unfolding F_def
        by (simp add: continuous_within)
      moreover have "(h  h z) F"
        using holomorphic_on_imp_continuous_on[OF h_holo,unfolded continuous_on_def] r>0
        unfolding F_def by auto
      ultimately show " ((λw. f' w * h w)  0) F" using tendsto_mult by fastforce
    qed
    moreover have "(g  g z) (at z within ball z r)"
      using holomorphic_on_imp_continuous_on[OF g_holo]
      by (auto simp add:continuous_on_def r>0)
    ultimately have "g z=0" by (auto intro!: tendsto_unique)
    thus False using g z0 by auto
  qed
  ultimately show "n=m" by fastforce
qed

lemma holomorphic_factor_puncture:
  assumes f_iso:"isolated_singularity_at f z"
      and "not_essential f z" ― ‹termf has either a removable singularity or a pole at termz
      and non_zero:"Fw in (at z). f w0" ― ‹termf will not be constantly zero in a neighbour of termz
  shows "∃!n::int. g r. 0 < r  g holomorphic_on cball z r  g z0
           (wcball z r-{z}. f w = g w * (w-z) powi n  g w0)"
proof -
  define P where "P = (λf n g r. 0 < r  g holomorphic_on cball z r  g z0
           (wcball z r - {z}. f w = g w * (w-z) powi n   g w0))"
  have imp_unique:"∃!n::int. g r. P f n g r" when "n g r. P f n g r"
  proof (rule ex_ex1I[OF that])
    fix n1 n2 :: int
    assume g1_asm:"g1 r1. P f n1 g1 r1" and g2_asm:"g2 r2. P f n2 g2 r2"
    define fac where "fac  λn g r. wcball z r-{z}. f w = g w * (w - z) powi n  g w  0"
    obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z0"
        and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
    obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z0"
        and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
    define r where "r  min r1 r2"
    have "r>0" using r1>0 r2>0 unfolding r_def by auto
    moreover have "wball z r-{z}. f w = g1 w * (w-z) powi n1  g1 w0
         f w = g2 w * (w - z) powi n2   g2 w0"
      using fac n1 g1 r1 fac n2 g2 r2   unfolding fac_def r_def
      by fastforce
    ultimately show "n1=n2" 
      using g1_holo g2_holo g1 z0 g2 z0
      apply (elim holomorphic_factor_unique)
      by (auto simp add:r_def)
  qed

  have P_exist:" n g r. P h n g r" when
      "z'. (h  z') (at z)" "isolated_singularity_at h z"  "Fw in (at z). h w0"
    for h
  proof -
    from that(2) obtain r where "r>0" and r: "h analytic_on ball z r - {z}"
      unfolding isolated_singularity_at_def by auto
    obtain z' where "(h  z') (at z)" using z'. (h  z') (at z) by auto
    define h' where "h'=(λx. if x=z then z' else h x)"
    have "h' holomorphic_on ball z r"
    proof (rule no_isolated_singularity'[of "{z}"])
      show "w. w  {z}  (h'  h' w) (at w within ball z r)"
        by (simp add: LIM_cong Lim_at_imp_Lim_at_within h z z' h'_def)
      show "h' holomorphic_on ball z r - {z}"
        using r analytic_imp_holomorphic h'_def holomorphic_transform by fastforce
    qed auto
    have ?thesis when "z'=0"
    proof -
      have "h' z=0" using that unfolding h'_def by auto
      moreover have "¬ h' constant_on ball z r"
        using Fw in (at z). h w0 unfolding constant_on_def frequently_def eventually_at h'_def
        by (metis 0 < r centre_in_ball dist_commute mem_ball that)
      moreover note h' holomorphic_on ball z r
      ultimately obtain g r1 n where "0 < n" "0 < r1" "ball z r1  ball z r" and
          g:"g holomorphic_on ball z r1"
          "w. w  ball z r1  h' w = (w - z) ^ n * g w"
          "w. w  ball z r1  g w  0"
        using holomorphic_factor_zero_nonconstant[of _ "ball z r" z thesis,simplified,
                OF h' holomorphic_on ball z r r>0 h' z=0 ¬ h' constant_on ball z r]
        by (auto simp add:dist_commute)
      define rr where "rr=r1/2"
      have "P h' n g rr"
        unfolding P_def rr_def
        using n>0 r1>0 g by (auto simp add:powr_nat)
      then have "P h n g rr"
        unfolding h'_def P_def by auto
      then show ?thesis unfolding P_def by blast
    qed
    moreover have ?thesis when "z'0"
    proof -
      have "h' z0" using that unfolding h'_def by auto
      obtain r1 where "r1>0" "cball z r1  ball z r" "xcball z r1. h' x0"
      proof -
        have "isCont h' z" "h' z0"
          by (auto simp add: Lim_cong_within h z z' z'0 continuous_at h'_def)
        then obtain r2 where r2:"r2>0" "xball z r2. h' x0"
          using continuous_at_avoid[of z h' 0 ] unfolding ball_def by auto
        define r1 where "r1=min r2 r / 2"
        have "0 < r1" "cball z r1  ball z r"
          using r2>0 r>0 unfolding r1_def by auto
        moreover have "xcball z r1. h' x  0"
          using r2 unfolding r1_def by simp
        ultimately show ?thesis using that by auto
      qed
      then have "P h' 0 h' r1" using h' holomorphic_on ball z r unfolding P_def by auto
      then have "P h 0 h' r1" unfolding P_def h'_def by auto
      then show ?thesis unfolding P_def by blast
    qed
    ultimately show ?thesis by auto
  qed

  have ?thesis when "x. (f  x) (at z)"
    apply (rule_tac imp_unique[unfolded P_def])
    using P_exist[OF that(1) f_iso non_zero] unfolding P_def .
  moreover have ?thesis when "is_pole f z"
  proof (rule imp_unique[unfolded P_def])
    obtain e where [simp]:"e>0" and e_holo:"f holomorphic_on ball z e - {z}" and e_nz: "xball z e-{z}. f x0"
    proof -
      have "F z in at z. f z  0"
        using is_pole f z filterlim_at_infinity_imp_eventually_ne unfolding is_pole_def
        by auto
      then obtain e1 where e1:"e1>0" "xball z e1-{z}. f x0"
        using that eventually_at[of "λx. f x0" z UNIV,simplified] by (auto simp add:dist_commute)
      obtain e2 where e2:"e2>0" "f holomorphic_on ball z e2 - {z}"
        using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by auto
      show ?thesis
        using e1 e2 by (force intro: that[of "min e1 e2"])
    qed

    define h where "h  λx. inverse (f x)"
    have "n g r. P h n g r"
    proof -
      have "(λx. inverse (f x)) analytic_on ball z e - {z}"
        by (metis e_holo e_nz open_ball analytic_on_open holomorphic_on_inverse open_delete)
      moreover have "h z 0"
        using Lim_transform_within_open assms(2) h_def is_pole_tendsto that by fastforce
      moreover have "Fw in (at z). h w0"
        using non_zero by (simp add: h_def)
      ultimately show ?thesis
        using P_exist[of h] e > 0
        unfolding isolated_singularity_at_def h_def
        by blast
    qed
    then obtain n g r
      where "0 < r" and
            g_holo:"g holomorphic_on cball z r" and "g z0" and
            g_fac:"(wcball z r-{z}. h w = g w * (w - z) powi n   g w  0)"
      unfolding P_def by auto
    have "P f (-n) (inverse o g) r"
    proof -
      have "f w = inverse (g w) * (w - z) powi (- n)" when "wcball z r - {z}" for w
        by (metis g_fac h_def inverse_inverse_eq inverse_mult_distrib power_int_minus that)
      then show ?thesis
        unfolding P_def comp_def
        using r>0 g_holo g_fac g z0 by (auto intro:holomorphic_intros)
    qed
    then show "x g r. 0 < r  g holomorphic_on cball z r  g z  0
                   (wcball z r - {z}. f w = g w * (w - z) powi x   g w  0)"
      unfolding P_def by blast
  qed
  ultimately show ?thesis using not_essential f z unfolding not_essential_def  by presburger
qed

lemma not_essential_transform:
  assumes "not_essential g z"
  assumes "F w in (at z). g w = f w"
  shows "not_essential f z"
  using assms unfolding not_essential_def
  by (simp add: filterlim_cong is_pole_cong)

lemma isolated_singularity_at_transform:
  assumes "isolated_singularity_at g z"
  assumes "F w in (at z). g w = f w"
  shows "isolated_singularity_at f z"
  using assms isolated_singularity_at_cong by blast

lemma not_essential_powr[singularity_intros]:
  assumes "LIM w (at z). f w :> (at x)"
  shows "not_essential (λw. (f w) powi n) z"
proof -
  define fp where "fp=(λw. (f w) powi n)"
  have ?thesis when "n>0"
  proof -
    have "(λw.  (f w) ^ (nat n)) z x ^ nat n"
      using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
    then have "fp z x ^ nat n" unfolding fp_def
      by (smt (verit) LIM_cong power_int_def that)
    then show ?thesis unfolding not_essential_def fp_def by auto
  qed
  moreover have ?thesis when "n=0"
  proof -
    have "F x in at z. fp x = 1"
      using that filterlim_at_within_not_equal[OF assms] by (auto simp: fp_def)
    then have "fp z 1"
      by (simp add: tendsto_eventually)
    then show ?thesis unfolding fp_def not_essential_def by auto
  qed
  moreover have ?thesis when "n<0"
  proof (cases "x=0")
    case True
    have "(λx. f x ^ nat (- n)) z 0"
      using assms True that unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
    moreover have "F x in at z. f x ^ nat (- n)  0"
      by (smt (verit) True assms eventually_at_topological filterlim_at power_eq_0_iff)
    ultimately have "LIM w (at z). inverse ((f w) ^ (nat (-n))) :> at_infinity"
      by (metis filterlim_atI filterlim_compose filterlim_inverse_at_infinity)
    then have "LIM w (at z). fp w :> at_infinity"
    proof (elim filterlim_mono_eventually)
      show "F x in at z. inverse (f x ^ nat (- n)) = fp x"
        using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
        by (smt (verit) eventuallyI power_int_def power_inverse that)
    qed auto
    then show ?thesis unfolding fp_def not_essential_def is_pole_def by auto
  next
    case False
    let ?xx= "inverse (x ^ (nat (-n)))"
    have "(λw. inverse ((f w) ^ (nat (-n)))) z?xx"
      using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
    then have "fp z ?xx"
      by (smt (verit, best) LIM_cong fp_def power_int_def power_inverse that)
    then show ?thesis unfolding fp_def not_essential_def by auto
  qed
  ultimately show ?thesis by linarith
qed

lemma isolated_singularity_at_powr[singularity_intros]:
  assumes "isolated_singularity_at f z" "F w in (at z). f w0"
  shows "isolated_singularity_at (λw. (f w) powi n) z"
proof -
  obtain r1 where "r1>0" "f analytic_on ball z r1 - {z}"
    using assms(1) unfolding isolated_singularity_at_def by auto
  then have r1:"f holomorphic_on ball z r1 - {z}"
    using analytic_on_open[of "ball z r1-{z}" f] by blast
  obtain r2 where "r2>0" and r2:"w. w  z  dist w z < r2  f w  0"
    using assms(2) unfolding eventually_at by auto
  define r3 where "r3=min r1 r2"
  have "(λw. (f w) powi n) holomorphic_on ball z r3 - {z}"
    by (intro holomorphic_on_power_int) (use r1 r2 in auto simp: dist_commute r3_def)
  moreover have "r3>0" unfolding r3_def using 0 < r1 0 < r2 by linarith
  ultimately show ?thesis
    by (meson open_ball analytic_on_open isolated_singularity_at_def open_delete)
qed

lemma non_zero_neighbour:
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z"
      and f_nconst:"Fw in (at z). f w0"
    shows "F w in (at z). f w0"
proof -
  obtain fn fp fr where [simp]:"fp z  0" and "fr > 0"
          and fr: "fp holomorphic_on cball z fr"
                  "wcball z fr - {z}. f w = fp w * (w - z) powi fn  fp w  0"
    using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
  have "f w  0" when " w  z" "dist w z < fr" for w
  proof -
    have "f w = fp w * (w - z) powi fn" "fp w  0"
      using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
    moreover have "(w - z) powi fn 0"
      unfolding powr_eq_0_iff using wz by auto
    ultimately show ?thesis by auto
  qed
  then show ?thesis using fr>0 unfolding eventually_at by auto
qed

lemma non_zero_neighbour_pole:
  assumes "is_pole f z"
  shows "F w in (at z). f w0"
  using assms filterlim_at_infinity_imp_eventually_ne[of f "at z" 0]
  unfolding is_pole_def by auto

lemma non_zero_neighbour_alt:
  assumes holo: "f holomorphic_on S"
      and "open S" "connected S" "z  S"  "β  S" "f β  0"
    shows "F w in (at z). f w0  wS"
proof (cases "f z = 0")
  case True
  from isolated_zeros[OF holo open S connected S z  S True β  S f β  0]
  obtain r where "0 < r" "ball z r  S" "w  ball z r - {z}.f w  0" by metis
  then show ?thesis
    by (smt (verit) open_ball centre_in_ball eventually_at_topological insertE insert_Diff subsetD)
next
  case False
  obtain r1 where r1:"r1>0" "y. dist z y < r1  f y  0"
    using continuous_at_avoid[of z f, OF _ False] assms(2,4) continuous_on_eq_continuous_at
      holo holomorphic_on_imp_continuous_on by blast
  obtain r2 where r2:"r2>0" "ball z r2  S"
    using assms openE by blast
  show ?thesis unfolding eventually_at
    by (metis (no_types) dist_commute dual_order.strict_trans linorder_less_linear mem_ball r1 r2 subsetD)
qed

lemma not_essential_times[singularity_intros]:
  assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
  assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
  shows "not_essential (λw. f w * g w) z"
proof -
  define fg where "fg = (λw. f w * g w)"
  have ?thesis when "¬ ((Fw in (at z). f w0)  (Fw in (at z). g w0))"
  proof -
    have "Fw in (at z). fg w=0"
      using fg_def frequently_elim1 not_eventually that by fastforce
    from tendsto_cong[OF this] have "fg z0" by auto
    then show ?thesis unfolding not_essential_def fg_def by auto
  qed
  moreover have ?thesis when f_nconst:"Fw in (at z). f w0" and g_nconst:"Fw in (at z). g w0"
  proof -
    obtain fn fp fr where [simp]:"fp z  0" and "fr > 0"
          and fr: "fp holomorphic_on cball z fr"
                  "wcball z fr - {z}. f w = fp w * (w - z) powi fn  fp w  0"
      using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
    obtain gn gp gr where [simp]:"gp z  0" and "gr > 0"
          and gr: "gp holomorphic_on cball z gr"
                  "wcball z gr - {z}. g w = gp w * (w - z) powi gn  gp w  0"
      using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto

    define r1 where "r1=(min fr gr)"
    have "r1>0" unfolding r1_def using  fr>0 gr>0 by auto
    have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w0"
      when "wball z r1 - {z}" for w
    proof -
      have "f w = fp w * (w - z) powi fn" "fp w0"
        using fr(2)[rule_format,of w] that unfolding r1_def by auto
      moreover have "g w = gp w * (w - z) powi gn" "gp w  0"
        using gr(2)[rule_format, of w] that unfolding r1_def by auto
      ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w0"
        using that unfolding fg_def by (auto simp add:power_int_add)
    qed

    have [intro]: "fp zfp z" "gp zgp z"
        using fr(1) fr>0 gr(1) gr>0
        by (meson open_ball ball_subset_cball centre_in_ball
            continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
            holomorphic_on_subset)+
    have ?thesis when "fn+gn>0"
    proof -
      have "(λw. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) z0"
        using that by (auto intro!:tendsto_eq_intros)
      then have "fg z 0"
         apply (elim Lim_transform_within[OF _ r1>0])
        by (smt (verit, best) Diff_iff dist_commute fg_times mem_ball power_int_def singletonD that zero_less_dist_iff)
      then show ?thesis unfolding not_essential_def fg_def by auto
    qed
    moreover have ?thesis when "fn+gn=0"
    proof -
      have "(λw. fp w * gp w) zfp z*gp z"
        using that by (auto intro!:tendsto_eq_intros)
      then have "fg z fp z*gp z"
        apply (elim Lim_transform_within[OF _ r1>0])
        apply (subst fg_times)
        by (auto simp add:dist_commute that)
      then show ?thesis unfolding not_essential_def fg_def by auto
    qed
    moreover have ?thesis when "fn+gn<0"
    proof -
      have "LIM x at z. (x - z) ^ nat (- (fn + gn)) :> at 0"
        using eventually_at_topological that
        by (force intro!: tendsto_eq_intros filterlim_atI)
      moreover have "c. (λc. fp c * gp c) z c  0  c"
        using fp z fp z gp z gp z tendsto_mult by fastforce
      ultimately have "LIM w (at z). fp w * gp w / (w-z)^nat (-(fn+gn)) :> at_infinity"
        using filterlim_divide_at_infinity by blast
      then have "is_pole fg z" unfolding is_pole_def
        apply (elim filterlim_transform_within[OF _ _ r1>0])
        using that
        by (simp_all add: dist_commute fg_times of_int_of_nat divide_simps power_int_def del: minus_add_distrib)
      then show ?thesis unfolding not_essential_def fg_def by auto
    qed
    ultimately show ?thesis unfolding not_essential_def fg_def by fastforce
  qed
  ultimately show ?thesis by auto
qed

lemma not_essential_inverse[singularity_intros]:
  assumes f_ness:"not_essential f z"
  assumes f_iso:"isolated_singularity_at f z"
  shows "not_essential (λw. inverse (f w)) z"
proof -
  define vf where "vf = (λw. inverse (f w))"
  have ?thesis when "¬(Fw in (at z). f w0)"
  proof -
    have "Fw in (at z). f w=0"
      using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
    then have "vf z0" 
      unfolding vf_def by (simp add: tendsto_eventually)
    then show ?thesis unfolding not_essential_def vf_def by auto
  qed
  moreover have ?thesis when "is_pole f z"
  proof -
    have "vf z0"
      using that filterlim_at filterlim_inverse_at_iff unfolding is_pole_def vf_def by blast
    then show ?thesis unfolding not_essential_def vf_def by auto
  qed
  moreover have ?thesis when "x. fzx " and f_nconst:"Fw in (at z). f w0"
  proof -
    from that obtain fz where fz:"fzfz" by auto
    have ?thesis when "fz=0"

    proof -
      have "(λw. inverse (vf w)) z0"
        using fz that unfolding vf_def by auto
      moreover have "F w in at z. inverse (vf w)  0"
        using non_zero_neighbour[OF f_iso f_ness f_nconst]
        unfolding vf_def by auto
      ultimately show ?thesis unfolding not_essential_def vf_def
         using filterlim_atI filterlim_inverse_at_iff is_pole_def by blast
    qed
    moreover have ?thesis when "fz0"
      using fz not_essential_def tendsto_inverse that by blast
    ultimately show ?thesis by auto
  qed
  ultimately show ?thesis using f_ness unfolding not_essential_def by auto
qed

lemma isolated_singularity_at_inverse[singularity_intros]:
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z"
  shows "isolated_singularity_at (λw. inverse (f w)) z"
proof -
  define vf where "vf = (λw. inverse (f w))"
  have ?thesis when "¬(Fw in (at z). f w0)"
  proof -
    have "Fw in (at z). f w=0"
      using that[unfolded frequently_def, simplified] by (auto elim: eventually_rev_mp)
    then have "Fw in (at z). vf w=0"
      unfolding vf_def by auto
    then obtain d1 where "d1>0" and d1:"x. x  z  dist x z < d1  vf x = 0"
      unfolding eventually_at by auto
    then have "vf holomorphic_on ball z d1-{z}"
      apply (rule_tac holomorphic_transform[of "λ_. 0"])
      by (auto simp add:dist_commute)
    then have "vf analytic_on ball z d1 - {z}"
      by (simp add: analytic_on_open open_delete)
    then show ?thesis using d1>0 unfolding isolated_singularity_at_def vf_def by auto
  qed
  moreover have ?thesis when f_nconst:"Fw in (at z). f w0"
  proof -
    have "F w in at z. f w  0" using non_zero_neighbour[OF f_iso f_ness f_nconst] .
    then obtain d1 where d1:"d1>0" "x. x  z  dist x z < d1  f x  0"
      unfolding eventually_at by auto
    obtain d2 where "d2>0" and d2:"f analytic_on ball z d2 - {z}"
      using f_iso unfolding isolated_singularity_at_def by auto
    define d3 where "d3=min d1 d2"
    have "d3>0" unfolding d3_def using d1>0 d2>0 by auto
    moreover
    have "f analytic_on ball z d3 - {z}"
      by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball)
    then have "vf analytic_on ball z d3 - {z}"
      unfolding vf_def
      by (intro analytic_on_inverse; simp add: d1(2) d3_def dist_commute)
    ultimately show ?thesis unfolding isolated_singularity_at_def vf_def by auto
  qed
  ultimately show ?thesis by auto
qed

lemma not_essential_divide[singularity_intros]:
  assumes f_ness:"not_essential f z" and g_ness:"not_essential g z"
  assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
  shows "not_essential (λw. f w / g w) z"
proof -
  have "not_essential (λw. f w * inverse (g w)) z"
    by (simp add: f_iso f_ness g_iso g_ness isolated_singularity_at_inverse not_essential_inverse not_essential_times)
  then show ?thesis by (simp add:field_simps)
qed

lemma
  assumes f_iso:"isolated_singularity_at f z"
      and g_iso:"isolated_singularity_at g z"
    shows isolated_singularity_at_times[singularity_intros]:
              "isolated_singularity_at (λw. f w * g w) z"
      and isolated_singularity_at_add[singularity_intros]:
              "isolated_singularity_at (λw. f w + g w) z"
proof -
  obtain d1 d2 where "d1>0" "d2>0"
      and d1:"f analytic_on ball z d1 - {z}" and d2:"g analytic_on ball z d2 - {z}"
    using f_iso g_iso unfolding isolated_singularity_at_def by auto
  define d3 where "d3=min d1 d2"
  have "d3>0" unfolding d3_def using d1>0 d2>0 by auto

  have fan: "f analytic_on ball z d3 - {z}"
    by (smt (verit, best) Diff_iff analytic_on_analytic_at d1 d3_def mem_ball)
  have gan: "g analytic_on ball z d3 - {z}"
    by (smt (verit, best) Diff_iff analytic_on_analytic_at d2 d3_def mem_ball)
  have "(λw. f w * g w) analytic_on ball z d3 - {z}"
    using analytic_on_mult fan gan by blast
  then show "isolated_singularity_at (λw. f w * g w) z"
    using d3>0 unfolding isolated_singularity_at_def by auto
  have "(λw. f w + g w) analytic_on ball z d3 - {z}"
    using analytic_on_add fan gan by blast
  then show "isolated_singularity_at (λw. f w + g w) z"
    using d3>0 unfolding isolated_singularity_at_def by auto
qed

lemma isolated_singularity_at_uminus[singularity_intros]:
  assumes f_iso:"isolated_singularity_at f z"
  shows "isolated_singularity_at (λw. - f w) z"
  using assms unfolding isolated_singularity_at_def using analytic_on_neg by blast

lemma isolated_singularity_at_id[singularity_intros]:
     "isolated_singularity_at (λw. w) z"
  unfolding isolated_singularity_at_def by (simp add: gt_ex)

lemma isolated_singularity_at_minus[singularity_intros]:
  assumes "isolated_singularity_at f z" and "isolated_singularity_at g z"
  shows "isolated_singularity_at (λw. f w - g w) z"
  unfolding diff_conv_add_uminus
  using assms isolated_singularity_at_add isolated_singularity_at_uminus by blast

lemma isolated_singularity_at_divide[singularity_intros]:
  assumes "isolated_singularity_at f z"
      and "isolated_singularity_at g z"
      and "not_essential g z"
    shows "isolated_singularity_at (λw. f w / g w) z"
  unfolding divide_inverse
  by (simp add: assms isolated_singularity_at_inverse isolated_singularity_at_times)

lemma isolated_singularity_at_const[singularity_intros]:
    "isolated_singularity_at (λw. c) z"
  unfolding isolated_singularity_at_def by (simp add: gt_ex)

lemma isolated_singularity_at_holomorphic:
  assumes "f holomorphic_on s-{z}" "open s" "zs"
  shows "isolated_singularity_at f z"
  using assms unfolding isolated_singularity_at_def
  by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)

lemma isolated_singularity_at_altdef:
  "isolated_singularity_at f z  eventually (λz. f analytic_on {z}) (at z)"
proof
  assume "isolated_singularity_at f z"
  then obtain r where r: "r > 0" "f analytic_on ball z r - {z}"
    unfolding isolated_singularity_at_def by blast
  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r(1) by (intro eventually_at_in_open) auto
  thus "eventually (λz. f analytic_on {z}) (at z)"
    by eventually_elim (use r analytic_on_subset in auto)
next
  assume "eventually (λz. f analytic_on {z}) (at z)"
  then obtain A where A: "open A" "z  A" "w. w  A - {z}  f analytic_on {w}"
    unfolding eventually_at_topological by blast
  then show "isolated_singularity_at f z"
    by (meson analytic_imp_holomorphic analytic_on_analytic_at isolated_singularity_at_holomorphic)
qed

lemma isolated_singularity_at_shift:
  assumes "isolated_singularity_at (λx. f (x + w)) z"
  shows   "isolated_singularity_at f (z + w)"
proof -
  from assms obtain r where r: "r > 0" and ana: "(λx. f (x + w)) analytic_on ball z r - {z}"
    unfolding isolated_singularity_at_def by blast
  have "((λx. f (x + w))  (λx. x - w)) analytic_on (ball (z + w) r - {z + w})"
    by (rule analytic_on_compose_gen[OF _ ana])
       (auto simp: dist_norm algebra_simps intro!: analytic_intros)
  hence "f analytic_on (ball (z + w) r - {z + w})"
    by (simp add: o_def)
  thus ?thesis using r
    unfolding isolated_singularity_at_def by blast
qed

lemma isolated_singularity_at_shift_iff:
  "isolated_singularity_at f (z + w)  isolated_singularity_at (λx. f (x + w)) z"
  using isolated_singularity_at_shift[of f w z]
        isolated_singularity_at_shift[of "λx. f (x + w)" "-w" "w + z"]
  by (auto simp: algebra_simps)

lemma isolated_singularity_at_shift_0:
  "NO_MATCH 0 z  isolated_singularity_at f z  isolated_singularity_at (λx. f (z + x)) 0"
  using isolated_singularity_at_shift_iff[of f 0 z] by (simp add: add_ac)

lemma not_essential_shift:
  assumes "not_essential (λx. f (x + w)) z"
  shows   "not_essential f (z + w)"
proof -
  from assms consider c where "(λx. f (x + w)) z c" | "is_pole (λx. f (x + w)) z"
    unfolding not_essential_def by blast
  thus ?thesis
  proof cases
    case (1 c)
    hence "f z + w c"
      by (smt (verit, ccfv_SIG) LIM_cong add.assoc filterlim_at_to_0)
    thus ?thesis
      by (auto simp: not_essential_def)
  next
    case 2
    hence "is_pole f (z + w)"
      by (subst is_pole_shift_iff [symmetric]) (auto simp: o_def add_ac)
    thus ?thesis
      by (auto simp: not_essential_def)
  qed
qed

lemma not_essential_shift_iff: "not_essential f (z + w)  not_essential (λx. f (x + w)) z"
  using not_essential_shift[of f w z]
        not_essential_shift[of "λx. f (x + w)" "-w" "w + z"]
  by (auto simp: algebra_simps)

lemma not_essential_shift_0:
  "NO_MATCH 0 z  not_essential f z  not_essential (λx. f (z + x)) 0"
  using not_essential_shift_iff[of f 0 z] by (simp add: add_ac)

lemma not_essential_holomorphic:
  assumes "f holomorphic_on A" "x  A" "open A"
  shows   "not_essential f x"
  by (metis assms at_within_open continuous_on holomorphic_on_imp_continuous_on not_essential_def)

lemma not_essential_analytic:
  assumes "f analytic_on {z}"
  shows   "not_essential f z"
  using analytic_at assms not_essential_holomorphic by blast

lemma not_essential_id [singularity_intros]: "not_essential (λw. w) z"
  by (simp add: not_essential_analytic)

lemma is_pole_imp_not_essential [intro]: "is_pole f z  not_essential f z"
  by (auto simp: not_essential_def)

lemma tendsto_imp_not_essential [intro]: "f z c  not_essential f z"
  by (auto simp: not_essential_def)

lemma eventually_not_pole:
  assumes "isolated_singularity_at f z"
  shows   "eventually (λw. ¬is_pole f w) (at z)"
proof -
  from assms obtain r where "r > 0" and r: "f analytic_on ball z r - {z}"
    by (auto simp: isolated_singularity_at_def)
  then have "eventually (λw. w  ball z r - {z}) (at z)"
    by (intro eventually_at_in_open) auto
  thus "eventually (λw. ¬is_pole f w) (at z)"
    by (metis (no_types, lifting) analytic_at analytic_on_analytic_at eventually_mono not_is_pole_holomorphic r)
qed

lemma not_islimpt_poles:
  assumes "isolated_singularity_at f z"
  shows   "¬z islimpt {w. is_pole f w}"
  using eventually_not_pole [OF assms]
  by (auto simp: islimpt_conv_frequently_at frequently_def)

lemma analytic_at_imp_no_pole: "f analytic_on {z}  ¬is_pole f z"
  using analytic_at not_is_pole_holomorphic by blast

lemma not_essential_const [singularity_intros]: "not_essential (λ_. c) z"
  by blast

lemma not_essential_uminus [singularity_intros]:
  assumes f_ness: "not_essential f z"
  assumes f_iso:"isolated_singularity_at f z"
  shows "not_essential (λw. -f w) z"
proof -
  have "not_essential (λw. -1 * f w) z"
    by (intro assms singularity_intros)
  thus ?thesis by simp
qed

lemma isolated_singularity_at_analytic:
  assumes "f analytic_on {z}"
  shows   "isolated_singularity_at f z"
  by (meson Diff_subset analytic_at assms holomorphic_on_subset isolated_singularity_at_holomorphic)

subsection ‹The order of non-essential singularities (i.e. removable singularities or poles)›

definitiontag important› zorder :: "(complex  complex)  complex  int" where
  "zorder f z = (THE n. (h r. r>0  h holomorphic_on cball z r  h z0
                    (wcball z r - {z}. f w =  h w * (w-z) powi n
                    h w 0)))"

definitiontag important› zor_poly
    ::"[complex  complex, complex]  complex  complex" where
  "zor_poly f z = (SOME h. r. r > 0  h holomorphic_on cball z r  h z  0
                    (wcball z r - {z}. f w =  h w * (w - z) powi (zorder f z)
                    h w 0))"

lemma zorder_exist:
  fixes f::"complex  complex" and z::complex
  defines "n  zorder f z" and "g  zor_poly f z"
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z"
      and f_nconst:"Fw in (at z). f w0"
  shows "g z0  (r. r>0  g holomorphic_on cball z r
     (wcball z r - {z}. f w  = g w * (w-z) powi n   g w 0))"
proof -
  define P where "P = (λn g r. 0 < r  g holomorphic_on cball z r  g z0
           (wcball z r - {z}. f w = g w * (w-z) powi n  g w0))"
  have "∃!k. g r. P k g r"
    using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
  then have "g r. P n g r"
    unfolding n_def P_def zorder_def
    by (drule_tac theI',argo)
  then have "r. P n g r"
    unfolding P_def zor_poly_def g_def n_def
    by (drule_tac someI_ex,argo)
  then obtain r1 where "P n g r1" by auto
  then show ?thesis unfolding P_def by auto
qed

lemma zorder_shift:
  shows  "zorder f z = zorder (λu. f (u + z)) 0"
  unfolding zorder_def
  apply (rule arg_cong [of concl: The])
  apply (auto simp: fun_eq_iff Ball_def dist_norm)
  subgoal for x h r
    apply (rule_tac x="h o (+)z" in exI)
    apply (rule_tac x="r" in exI)
    apply (intro conjI holomorphic_on_compose holomorphic_intros)
       apply (simp_all flip: cball_translation)
    apply (simp add: add.commute)
    done
  subgoal for x h r
    apply (rule_tac x="h o (λu. u-z)" in exI)
    apply (rule_tac x="r" in exI)
    apply (intro conjI holomorphic_on_compose holomorphic_intros)
       apply (simp_all add: flip: cball_translation_subtract)
    by (metis diff_add_cancel eq_iff_diff_eq_0 norm_minus_commute)
  done

lemma zorder_shift': "NO_MATCH 0 z  zorder f z = zorder (λu. f (u + z)) 0"
  by (rule zorder_shift)

lemma
  fixes f::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z"
      and f_nconst:"Fw in (at z). f w0"
    shows zorder_inverse: "zorder (λw. inverse (f w)) z = - zorder f z"
      and zor_poly_inverse: "Fw in (at z). zor_poly (λw. inverse (f w)) z w
                                                = inverse (zor_poly f z w)"
proof -
  define vf where "vf = (λw. inverse (f w))"
  define fn vfn where
    "fn = zorder f z"  and "vfn = zorder vf z"
  define fp vfp where
    "fp = zor_poly f z" and "vfp = zor_poly vf z"

  obtain fr where [simp]:"fp z  0" and "fr > 0"
          and fr: "fp holomorphic_on cball z fr"
                  "wcball z fr - {z}. f w = fp w * (w - z) powi fn  fp w  0"
    using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
    by auto
  have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powi (-fn)"
        and fr_nz: "inverse (fp w)  0"
    when "wball z fr - {z}" for w
  proof -
    have "f w = fp w * (w - z) powi fn" "fp w  0"
      using fr(2) that by auto
    then show "vf w = (inverse (fp w)) * (w - z) powi (-fn)" "inverse (fp w)0"
      by (simp_all add: power_int_minus vf_def)
  qed
  obtain vfr where [simp]:"vfp z  0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
      "(wcball z vfr - {z}. vf w = vfp w * (w - z) powi vfn  vfp w  0)"
  proof -
    have "isolated_singularity_at vf z"
      using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
    moreover have "not_essential vf z"
      using not_essential_inverse[OF f_ness f_iso] unfolding vf_def .
    moreover have "F w in at z. vf w  0"
      using f_nconst unfolding vf_def by (auto elim:frequently_elim1)
    ultimately show ?thesis using zorder_exist[of vf z, folded vfn_def vfp_def] that by auto
  qed


  define r1 where "r1 = min fr vfr"
  have "r1>0" using fr>0 vfr>0 unfolding r1_def by simp
  show "vfn = - fn"
  proof (rule holomorphic_factor_unique)
    have §: "w. fp w = 0; dist z w < fr  False"
      using fr_nz by force
    then show "wball z r1 - {z}.
               vf w = vfp w * (w - z) powi vfn 
               vfp w  0  vf w = inverse (fp w) * (w - z) powi (- fn) 
               inverse (fp w)  0"
      using fr_inverse r1_def vfr(2)
      by (smt (verit) Diff_iff inverse_nonzero_iff_nonzero mem_ball mem_cball)
    show "vfp holomorphic_on ball z r1"
      using r1_def vfr(1) by auto
    show "(λw. inverse (fp w)) holomorphic_on ball z r1"
      by (metis § ball_subset_cball fr(1) holomorphic_on_inverse holomorphic_on_subset mem_ball min.cobounded2 min.commute r1_def subset_ball)
  qed (use r1>0 in auto)
  have "vfp w = inverse (fp w)" when "wball z r1-{z}" for w
  proof -
    have "w  ball z fr - {z}" "w  cball z vfr - {z}"  "wz" using that unfolding r1_def by auto
    from fr_inverse[OF this(1)] fr_nz[OF this(1)] vfr(2)[rule_format,OF this(2)] vfn = - fn wz
    show ?thesis by auto
  qed
  then show "Fw in (at z). vfp w = inverse (fp w)"
    unfolding eventually_at using r1>0
    by (metis DiffI dist_commute mem_ball singletonD)
qed

lemma zor_poly_shift:
  assumes iso1: "isolated_singularity_at f z"
    and ness1: "not_essential f z"
    and nzero1: "F w in at z. f w  0"
  shows "F w in nhds z. zor_poly f z w = zor_poly (λu. f (z + u)) 0 (w-z)"
proof -
  obtain r1 where "r1>0" "zor_poly f z z  0" and
      holo1:"zor_poly f z holomorphic_on cball z r1" and
      rball1:"wcball z r1 - {z}.
           f w = zor_poly f z w * (w - z) powi (zorder f z) 
           zor_poly f z w  0"
    using zorder_exist[OF iso1 ness1 nzero1] by blast

  define ff where "ff=(λu. f (z + u))"
  have "isolated_singularity_at ff 0"
    unfolding ff_def
    using iso1 isolated_singularity_at_shift_iff[of f 0 z]
    by (simp add:algebra_simps)
  moreover have "not_essential ff 0"
    unfolding ff_def
    using ness1 not_essential_shift_iff[of f 0 z]
    by (simp add:algebra_simps)
  moreover have "F w in at 0. ff w  0"
    unfolding ff_def using nzero1
    by (smt (verit, ccfv_SIG) add.commute eventually_at_to_0
        eventually_mono not_frequently)
  ultimately obtain r2 where "r2>0" "zor_poly ff 0 0  0" and
      holo2:"zor_poly ff 0 holomorphic_on cball 0 r2" and
      rball2:"wcball 0 r2 - {0}.
           ff w = zor_poly ff 0 w * w powi (zorder ff 0) 
           zor_poly ff 0 w  0"
    using zorder_exist[of ff 0] by auto

  define r where "r=min r1 r2"
  have "r>0" using r1>0 r2>0 unfolding r_def by auto

  have "zor_poly f z w = zor_poly ff 0 (w - z)"
    if "wball z r - {z}" for w
  proof -
    define n where "n  zorder f z"

    have "f w = zor_poly f z w * (w - z) powi n"
      using n_def r_def rball1 that by auto
    moreover have "f w = zor_poly ff 0 (w - z) * (w - z) powi n"
    proof -
      have "w-zcball 0 r2 - {0}"
        using r_def that by (auto simp:dist_complex_def)
      from rball2[rule_format, OF this]
      have "ff (w - z) = zor_poly ff 0 (w - z) * (w - z)
                            powi (zorder ff 0)"
        by simp
      moreover have "of_int (zorder ff 0) = n"
        unfolding n_def ff_def by (simp add:zorder_shift' add.commute)
      ultimately show ?thesis unfolding ff_def by auto
    qed
    ultimately have "zor_poly f z w * (w - z) powi n
                = zor_poly ff 0 (w - z) * (w - z) powi n"
      by auto
    moreover have "(w - z) powi n 0"
      using that by auto
    ultimately show ?thesis
      using mult_cancel_right by blast
  qed
  then have "F w in at z. zor_poly f z w
                  = zor_poly ff 0 (w - z)"
    unfolding eventually_at
    by (metis DiffI 0 < r dist_commute mem_ball singletonD)
  moreover have "isCont (zor_poly f z) z"
    using holo1[THEN holomorphic_on_imp_continuous_on]
    by (simp add: 0 < r1 continuous_on_interior)
  moreover 
  have "isCont (zor_poly ff 0) 0"
    using 0 < r2 centre_in_ball continuous_on_interior holo2 holomorphic_on_imp_continuous_on interior_cball by blast  
  then have "isCont (λw. zor_poly ff 0 (w - z)) z"
      unfolding isCont_iff by simp
  ultimately show "F w in nhds z. zor_poly f z w = zor_poly ff 0 (w - z)"
    by (elim at_within_isCont_imp_nhds;auto)
qed

lemma
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
      and f_ness:"not_essential f z" and g_ness:"not_essential g z"
      and fg_nconst: "Fw in (at z). f w * g w 0"
  shows zorder_times:"zorder (λw. f w * g w) z = zorder f z + zorder g z" and
        zor_poly_times:"Fw in (at z). zor_poly (λw. f w * g w) z w
                                                  = zor_poly f z w *zor_poly g z w"
proof -
  define fg where "fg = (λw. f w * g w)"
  define fn gn fgn where
    "fn = zorder f z" and "gn = zorder g z" and "fgn = zorder fg z"
  define fp gp fgp where
    "fp = zor_poly f z" and "gp = zor_poly g z" and "fgp = zor_poly fg z"
  have f_nconst:"Fw in (at z). f w  0" and g_nconst:"Fw in (at z).g w 0"
    using fg_nconst by (auto elim!:frequently_elim1)
  obtain fr where [simp]:"fp z  0" and "fr > 0"
          and fr: "fp holomorphic_on cball z fr"
                  "wcball z fr - {z}. f w = fp w * (w - z) powi fn  fp w  0"
    using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
  obtain gr where [simp]:"gp z  0" and "gr > 0"
          and gr: "gp holomorphic_on cball z gr"
                  "wcball z gr - {z}. g w = gp w * (w - z) powi gn  gp w  0"
    using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
  define r1 where "r1=min fr gr"
  have "r1>0" unfolding r1_def using fr>0 gr>0 by auto
  have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w0"
    when "wball z r1 - {z}" for w
  proof -
    have "f w = fp w * (w - z) powi fn" "fp w  0"
      using fr(2)[rule_format,of w] that unfolding r1_def by auto
    moreover have "g w = gp w * (w - z) powi gn" "gp w  0"
      using gr(2) that unfolding r1_def by auto
    ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w0"
      using that unfolding fg_def by (auto simp add:power_int_add)
  qed

  obtain fgr where [simp]:"fgp z  0" and "fgr > 0"
          and fgr: "fgp holomorphic_on cball z fgr"
                  "wcball z fgr - {z}. fg w = fgp w * (w - z) powi fgn  fgp w  0"
  proof -
    have "isolated_singularity_at fg z"
      unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
    moreover have "not_essential fg z"
      by (simp add: f_iso f_ness fg_def g_iso g_ness not_essential_times)
    moreover have "F w in at z. fg w  0"
      using fg_def fg_nconst by blast
    ultimately show ?thesis 
      using that zorder_exist[of fg z] fgn_def fgp_def by fastforce
  qed
  define r2 where "r2 = min fgr r1"
  have "r2>0" using r1>0 fgr>0 unfolding r2_def by simp
  show "fgn = fn + gn "
    apply (rule holomorphic_factor_unique[of r2 fgp z "λw. fp w * gp w" fg])
    subgoal using r2>0 by simp
    subgoal by simp
    subgoal by simp
    subgoal
    proof (rule ballI)
      fix w assume "w  ball z r2 - {z}"
      then have "w  ball z r1 - {z}" "w  cball z fgr - {z}"  unfolding r2_def by auto
      then show "fg w = fgp w * (w - z) powi fgn  fgp w  0
               fg w = fp w * gp w * (w - z) powi (fn + gn)  fp w * gp w  0"
        using fg_times fgp_nz fgr(2) by blast
    qed
    subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
    subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
    done

  have "fgp w = fp w *gp w" when "wball z r2-{z}" for w
  proof -
    have "w  ball z r1 - {z}" "w  cball z fgr - {z}" "wz" using that  unfolding r2_def by auto
    from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)] fgn = fn + gn wz
    show ?thesis by auto
  qed
  then show "Fw in (at z). fgp w = fp w * gp w"
    using r2>0 unfolding eventually_at by (auto simp add:dist_commute)
qed

lemma
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
      and f_ness:"not_essential f z" and g_ness:"not_essential g z"
      and fg_nconst: "Fw in (at z). f w * g w 0"
  shows zorder_divide:"zorder (λw. f w / g w) z = zorder f z - zorder g z" and
        zor_poly_divide:"Fw in (at z). zor_poly (λw. f w / g w) z w
                                                  = zor_poly f z w  / zor_poly g z w"
proof -
  have f_nconst:"Fw in (at z). f w  0" and g_nconst:"Fw in (at z).g w 0"
    using fg_nconst by (auto elim!:frequently_elim1)
  define vg where "vg=(λw. inverse (g w))"
  have 1: "isolated_singularity_at vg z"
    by (simp add: g_iso g_ness isolated_singularity_at_inverse vg_def)
  moreover have 2: "not_essential vg z"
    by (simp add: g_iso g_ness not_essential_inverse vg_def)
  moreover have 3: "F w in at z. f w * vg w  0"
    using fg_nconst vg_def by auto
  ultimately  have "zorder (λw. f w * vg w) z = zorder f z + zorder vg z"
    using zorder_times[OF f_iso _ f_ness] by blast
  then show "zorder (λw. f w / g w) z = zorder f z - zorder g z"
    using zorder_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
    by (auto simp add:field_simps)

  have "F w in at z. zor_poly (λw. f w * vg w) z w = zor_poly f z w * zor_poly vg z w"
    using zor_poly_times[OF f_iso _ f_ness,of vg] 1 2 3 by blast
  then show "Fw in (at z). zor_poly (λw. f w / g w) z w = zor_poly f z w  / zor_poly g z w"
    using zor_poly_inverse[OF g_iso g_ness g_nconst,folded vg_def] unfolding vg_def
    by eventually_elim (auto simp add:field_simps)
qed

lemma zorder_exist_zero:
  fixes f::"complex  complex" and z::complex
  defines "nzorder f z" and "gzor_poly f z"
  assumes  holo: "f holomorphic_on s" and
          "open s" "connected s" "zs"
      and non_const: "ws. f w  0"
  shows "(if f z=0 then n > 0 else n=0)  (r. r>0  cball z r  s  g holomorphic_on cball z r
     (wcball z r. f w  = g w * (w-z) ^ nat n   g w 0))"
proof -
  obtain r where "g z  0" and r: "r>0" "cball z r  s" "g holomorphic_on cball z r"
            "(wcball z r - {z}. f w = g w * (w - z) powi n  g w  0)"
  proof -
    have "g z  0  (r>0. g holomorphic_on cball z r
             (wcball z r - {z}. f w = g w * (w - z) powi n  g w  0))"
    proof (rule zorder_exist[of f z,folded g_def n_def])
      show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
        using holo assms(4,6)
        by (meson Diff_subset open_ball analytic_on_holomorphic holomorphic_on_subset openE)
      show "not_essential f z" unfolding not_essential_def
        using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
        by fastforce
      have "F w in at z. f w  0  ws"
      proof -
        obtain w where "ws" "f w0" using non_const by auto
        then show ?thesis
          by (rule non_zero_neighbour_alt[OF holo open s connected s zs])
      qed
      then show "F w in at z. f w  0"
        by (auto elim: eventually_frequentlyE)
    qed
    then obtain r1 where "g z  0" "r1>0" and r1:"g holomorphic_on cball z r1"
            "(wcball z r1 - {z}. f w = g w * (w - z) powi n  g w  0)"
      by auto
    obtain r2 where r2: "r2>0" "cball z r2  s"
      using assms(4,6) open_contains_cball_eq by blast
    define r3 where "r3  min r1 r2"
    have "r3>0" "cball z r3  s" using r1>0 r2 unfolding r3_def by auto
    moreover have "g holomorphic_on cball z r3"
      using r1(1) unfolding r3_def by auto
    moreover have "(wcball z r3 - {z}. f w = g w * (w - z) powi n  g w  0)"
      using r1(2) unfolding r3_def by auto
    ultimately show ?thesis using that[of r3] g z0 by auto
  qed

  have fz_lim: "f z  f z"
    by (metis assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
  have gz_lim: "g zg z"
    using r
    by (meson Elementary_Metric_Spaces.open_ball analytic_at analytic_at_imp_isCont 
        ball_subset_cball centre_in_ball holomorphic_on_subset isContD)
  have if_0:"if f z=0 then n > 0 else n=0"
  proof -
    have "(λw. g w * (w - z) powi n) z f z"
      using fz_lim Lim_transform_within_open[where s="ball z r"] r by fastforce
    then have "(λw. (g w * (w - z) powi n) / g w) z f z/g z"
      using gz_lim g z  0 tendsto_divide by blast
    then have powi_tendsto:"(λw. (w - z) powi n) z f z/g z"
      using Lim_transform_within_open[where s="ball z r"] r by fastforce

    have ?thesis when "n0" "f z=0"
    proof -
      have "(λw. (w - z) ^ nat n) z f z/g z"
        using Lim_transform_within[OF powi_tendsto, where d=r]
        by (meson power_int_def r(1) that(1))
      then have *:"(λw. (w - z) ^ nat n) z 0" using f z=0 by simp
      moreover have False when "n=0"
      proof -
        have "(λw. (w - z) ^ nat n) z 1"
          using n=0 by auto
        then show False using * using LIM_unique zero_neq_one by blast
      qed
      ultimately show ?thesis using that by fastforce
    qed
    moreover have ?thesis when "n0" "f z0"
    proof -
      have False when "n>0"
      proof -
        have "(λw. (w - z) ^ nat n) z f z/g z"
          using Lim_transform_within[OF powi_tendsto, where d=r]
          by (meson 0  n power_int_def r(1))
        moreover have "(λw. (w - z) ^ nat n) z 0"
          using n>0 by (auto intro!:tendsto_eq_intros)
        ultimately show False using f z0 g z0 using LIM_unique divide_eq_0_iff by blast
      qed
      then show ?thesis using that by force
    qed
    moreover have False when "n<0"
    proof -
      have "(λw. inverse ((w - z) ^ nat (- n))) z f z/g z"
        by (smt (verit) LIM_cong power_int_def power_inverse powi_tendsto that)
      moreover
      have "(λw.((w - z) ^ nat (- n))) z 0"
        using that by (auto intro!:tendsto_eq_intros)
      ultimately
      have "(λx. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) z 0" 
        using tendsto_mult by fastforce
      then have "(λx. 1::complex) z 0"
        using Lim_transform_within_open by fastforce
      then show False using LIM_const_eq by fastforce
    qed
    ultimately show ?thesis by fastforce
  qed
  moreover have "f w  = g w * (w-z) ^ nat n   g w 0" when "wcball z r" for w
  proof (cases "w=z")
    case True
    then have "f zf w"
      using fz_lim by blast
    then have "(λw. g w * (w-z) ^ nat n) zf w"
    proof (elim Lim_transform_within[OF _ r>0])
      fix x assume "0 < dist x z" "dist x z < r"
      then have "x  cball z r - {z}" "xz"
        unfolding cball_def by (auto simp add: dist_commute)
      then have "f x = g x * (x - z) powi n"
        using r(4)[rule_format,of x] by simp
      also have "... = g x * (x - z) ^ nat n"
        by (smt (verit, best) if_0 int_nat_eq power_int_of_nat)
      finally show "f x = g x * (x - z) ^ nat n" .
    qed
    moreover have "(λw. g w * (w-z) ^ nat n) z g w * (w-z) ^ nat n"
      using True by (auto intro!:tendsto_eq_intros gz_lim)
    ultimately have "f w = g w * (w-z) ^ nat n" using LIM_unique by blast
    then show ?thesis using g z0 True by auto
  next
    case False
    then have "f w = g w * (w - z) powi n  g w  0"
      using r(4) that by auto
    then show ?thesis
      by (smt (verit, best) False if_0 int_nat_eq power_int_of_nat)
  qed
  ultimately show ?thesis using r by auto
qed

lemma zorder_exist_pole:
  fixes f::"complex  complex" and z::complex
  defines "nzorder f z" and "gzor_poly f z"
  assumes  holo: "f holomorphic_on S-{z}" and "open S" "zS" and "is_pole f z"
  shows "n < 0  g z0  (r. r>0  cball z r  S  g holomorphic_on cball z r
     (wcball z r - {z}. f w  = g w / (w-z) ^ nat (- n)  g w 0))"
proof -
  obtain r where "g z  0" and r: "r>0" "cball z r  S" "g holomorphic_on cball z r"
            "(wcball z r - {z}. f w = g w * (w - z) powi n  g w  0)"
  proof -
    have "g z  0  (r>0. g holomorphic_on cball z r
             (wcball z r - {z}. f w = g w * (w - z) powi n  g w  0))"
    proof (rule zorder_exist[of f z,folded g_def n_def])
      show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
        using holo assms(4,5)
        by (metis analytic_on_holomorphic centre_in_ball insert_Diff openE open_delete subset_insert_iff)
      show "not_essential f z" unfolding not_essential_def
        using assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on
        by fastforce
      from non_zero_neighbour_pole[OF is_pole f z] show "F w in at z. f w  0"
        by (auto elim: eventually_frequentlyE)
    qed
    then obtain r1 where "g z  0" "r1>0" and r1:"g holomorphic_on cball z r1"
            "(wcball z r1 - {z}. f w = g w * (w - z) powi n  g w  0)"
      by auto
    obtain r2 where r2: "r2>0" "cball z r2  S"
      using assms(4,5) open_contains_cball_eq by metis
    define r3 where "r3=min r1 r2"
    have "r3>0" "cball z r3  S" using r1>0 r2 unfolding r3_def by auto
    moreover have "g holomorphic_on cball z r3"
      using r1(1) unfolding r3_def by auto
    moreover have "(wcball z r3 - {z}. f w = g w * (w - z) powi n  g w  0)"
      using r1(2) unfolding r3_def by auto
    ultimately show ?thesis using that[of r3] g z0 by auto
  qed

  have "n<0"
  proof (rule ccontr)
    assume " ¬ n < 0"
    define c where "c=(if n=0 then g z else 0)"
    have [simp]:"g z g z"
      by (metis open_ball at_within_open ball_subset_cball centre_in_ball
            continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
    have "F x in at z. f x = g x * (x - z) ^ nat n"
      unfolding eventually_at_topological
      apply (rule_tac exI[where x="ball z r"])
      by (simp add: ¬ n < 0 linorder_not_le power_int_def r(1) r(4))
    moreover have "(λx. g x * (x - z) ^ nat n) z c"
    proof (cases "n=0")
      case True
      then show ?thesis unfolding c_def by simp
    next
      case False
      then have "(λx. (x - z) ^ nat n) z 0" using ¬ n < 0
        by (auto intro!:tendsto_eq_intros)
      from tendsto_mult[OF _ this,of g "g z",simplified]
      show ?thesis unfolding c_def using False by simp
    qed
    ultimately have "f zc" using tendsto_cong by fast
    then show False using is_pole f z at_neq_bot not_tendsto_and_filterlim_at_infinity
      unfolding is_pole_def by blast
  qed
  moreover have "wcball z r - {z}. f w  = g w / (w-z) ^ nat (- n)  g w 0"
    using r(4) n<0
    by (smt (verit) inverse_eq_divide mult.right_neutral power_int_def power_inverse times_divide_eq_right)
  ultimately show ?thesis using r(1-3) g z0 by auto
qed

lemma zorder_eqI:
  assumes "open S" "z  S" "g holomorphic_on S" "g z  0"
  assumes fg_eq:"w. w  S;wz  f w = g w * (w - z) powi n"
  shows   "zorder f z = n"
proof -
  have "continuous_on S g" by (rule holomorphic_on_imp_continuous_on) fact
  moreover have "open (-{0::complex})" by auto
  ultimately have "open ((g -` (-{0}))  S)"
    unfolding continuous_on_open_vimage[OF open S] by blast
  moreover from assms have "z  (g -` (-{0}))  S" by auto
  ultimately obtain r where r: "r > 0" "cball z r   S  (g -` (-{0}))"
    unfolding open_contains_cball by blast

  let ?gg= "(λw. g w * (w - z) powi n)"
  define P where "P = (λn g r. 0 < r  g holomorphic_on cball z r  g z0
           (wcball z r - {z}. f w = g w * (w-z) powi n  g w0))"
  have "P n g r"
    unfolding P_def using r assms(3,4,5) by auto
  then have "g r. P n g r" by auto
  moreover have unique: "∃!n. g r. P n g r" unfolding P_def
  proof (rule holomorphic_factor_puncture)
    have "ball z r-{z}  S" using r using ball_subset_cball by blast
    then have "?gg holomorphic_on ball z r-{z}"
      using g holomorphic_on S r by (auto intro!: holomorphic_intros)
    then have "f holomorphic_on ball z r - {z}"
      by (smt (verit, best) DiffD2 ball z r-{z}  S fg_eq holomorphic_cong singleton_iff subset_iff)
    then show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
      using analytic_on_open open_delete r(1) by blast
  next
    have "not_essential ?gg z"
    proof (intro singularity_intros)
      show "not_essential g z"
        by (meson continuous_on S g assms continuous_on_eq_continuous_at
            isCont_def not_essential_def)
      show " F w in at z. w - z  0" by (simp add: eventually_at_filter)
      then show "LIM w at z. w - z :> at 0"
        unfolding filterlim_at by (auto intro:tendsto_eq_intros)
      show "isolated_singularity_at g z"
        by (meson Diff_subset open_ball analytic_on_holomorphic
            assms holomorphic_on_subset isolated_singularity_at_def openE)
    qed
    moreover
    have "F w in at z. g w * (w - z) powi n = f w"
      unfolding eventually_at_topological using assms fg_eq by force
    ultimately show "not_essential f z"
      using not_essential_transform by blast
    show "F w in at z. f w  0" unfolding frequently_at
    proof (intro strip)
      fix d::real assume "0 < d"
      define z' where "z'  z+min d r / 2"
      have "z'  z" " dist z' z < d "
        unfolding z'_def using d>0 r>0 by (auto simp add:dist_norm)
      moreover have "f z'  0"
      proof (subst fg_eq[OF _ z'z])
        have "z'  cball z r"
          unfolding z'_def using r>0 d>0 by (auto simp add:dist_norm)
        then show " z'  S" using r(2) by blast
        show "g z' * (z' - z) powi n  0"
          using P_def P n g r z'  cball z r z'  z by auto
      qed
      ultimately show "xUNIV. x  z  dist x z < d  f x  0" by auto
    qed
  qed
  ultimately have "(THE n. g r. P n g r) = n"
    by (rule_tac the1_equality)
  then show ?thesis unfolding zorder_def P_def by blast
qed

lemma simple_zeroI:
  assumes "open S" "z  S" "g holomorphic_on S" "g z  0"
  assumes "w. w  S  f w = g w * (w - z)"
  shows   "zorder f z = 1"
  using assms zorder_eqI by force

lemma higher_deriv_power:
  shows   "(deriv ^^ j) (λw. (w - z) ^ n) w =
             pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
proof (induction j arbitrary: w)
  case 0
  thus ?case by auto
next
  case (Suc j w)
  have "(deriv ^^ Suc j) (λw. (w - z) ^ n) w = deriv ((deriv ^^ j) (λw. (w - z) ^ n)) w"
    by simp
  also have "(deriv ^^ j) (λw. (w - z) ^ n) =
               (λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j))"
    using Suc by (intro Suc.IH ext)
  also {
    have "( has_field_derivative of_nat (n - j) *
               pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - Suc j)) (at w)"
      using Suc.prems by (auto intro!: derivative_eq_intros)
    also have "of_nat (n - j) * pochhammer (of_nat (Suc n - j)) j =
                 pochhammer (of_nat (Suc n - Suc j)) (Suc j)"
      by (cases "Suc j  n", subst pochhammer_rec)
         (insert Suc.prems, simp_all add: algebra_simps Suc_diff_le pochhammer_0_left)
    finally have "deriv (λw. pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)) w =
                     * (w - z) ^ (n - Suc j)"
      by (rule DERIV_imp_deriv)
  }
  finally show ?case .
qed

lemma zorder_zero_eqI:
  assumes  f_holo:"f holomorphic_on S" and "open S" "z  S"
  assumes zero: "i. i < nat n  (deriv ^^ i) f z = 0"
  assumes nz: "(deriv ^^ nat n) f z  0" and "n0"
  shows   "zorder f z = n"
proof -
  obtain r where [simp]:"r>0" and "ball z r  S"
    using open S zS openE by blast
  have nz':"wball z r. f w  0"
  proof (rule ccontr)
    assume "¬ (wball z r. f w  0)"
    then have "eventually (λu. f u = 0) (nhds z)"
      using open_ball 0 < r centre_in_ball eventually_nhds by blast
    then have "(deriv ^^ nat n) f z = (deriv ^^ nat n) (λ_. 0) z"
      by (intro higher_deriv_cong_ev) auto
    also have "(deriv ^^ nat n) (λ_. 0) z = 0"
      by (induction n) simp_all
    finally show False using nz by contradiction
  qed

  define zn g where "zn = zorder f z" and "g = zor_poly f z"
  obtain e where e_if: "if f z = 0 then 0 < zn else zn = 0" and
            [simp]: "e>0" and "cball z e  ball z r" and
            g_holo: "g holomorphic_on cball z e" and
            e_fac: "(wcball z e. f w = g w * (w - z) ^ nat zn  g w  0)"
  proof -
    have "f holomorphic_on ball z r"
      using f_holo ball z r  S by auto
    from that zorder_exist_zero[of f "ball z r" z,simplified,OF this nz',folded zn_def g_def]
    show thesis by blast
  qed
  then obtain "zn  0" "g z  0"
    by (metis centre_in_cball less_le_not_le order_refl)

  define A where "A  (λi. of_nat (i choose (nat zn)) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
  have deriv_A:"(deriv ^^ i) f z = (if zn  int i then A i else 0)" for i
  proof -
    have "eventually (λw. w  ball z e) (nhds z)"
      using cball z e  ball z r e>0 by (intro eventually_nhds_in_open) auto
    hence "eventually (λw. f w = (w - z) ^ (nat zn) * g w) (nhds z)"
      using e_fac eventually_mono by fastforce
    hence "(deriv ^^ i) f z = (deriv ^^ i) (λw. (w - z) ^ nat zn * g w) z"
      by (intro higher_deriv_cong_ev) auto
    also have " = (j=0..i. of_nat (i choose j) *
                       (deriv ^^ j) (λw. (w - z) ^ nat zn) z * (deriv ^^ (i - j)) g z)"
      using g_holo e>0
      by (intro higher_deriv_mult[of _ "ball z e"]) (auto intro!: holomorphic_intros)
    also have " = (j=0..i. if j = nat zn then
                    of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z else 0)"
    proof (intro sum.cong refl, goal_cases)
      case (1 j)
      have "(deriv ^^ j) (λw. (w - z) ^ nat zn) z =
              pochhammer (of_nat (Suc (nat zn) - j)) j * 0 ^ (nat zn - j)"
        by (subst higher_deriv_power) auto
      also have " = (if j = nat zn then fact j else 0)"
        by (auto simp: not_less pochhammer_0_left pochhammer_fact)
      also have "of_nat (i choose j) *  * (deriv ^^ (i - j)) g z =
                   (if j = nat zn then of_nat (i choose (nat zn)) * fact (nat zn)
                        * (deriv ^^ (i - nat zn)) g z else 0)"
        by simp
      finally show ?case .
    qed
    also have " = (if i  zn then A i else 0)"
      by (auto simp: A_def)
    finally show "(deriv ^^ i) f z = " .
  qed

  have False when "n<zn"
  proof -
    have "(deriv ^^ nat n) f z = 0"
      using deriv_A[of "nat n"] that n0 by auto
    with nz show False by auto
  qed
  moreover have "nzn"
  proof -
    have "g z  0" using e_fac[rule_format,of z] e>0 by simp
    then have "(deriv ^^ nat zn) f z  0"
      using deriv_A[of "nat zn"] by(auto simp add:A_def)
    then have "nat zn  nat n" using zero[of "nat zn"] by linarith
    moreover have "zn0" using e_if by (auto split:if_splits)
    ultimately show ?thesis using nat_le_eq_zle by blast
  qed
  ultimately show ?thesis unfolding zn_def by fastforce
qed

lemma
  assumes "eventually (λz. f z = g z) (at z)" "z = z'"
  shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
proof -
  define P where "P = (λff n h r. 0 < r  h holomorphic_on cball z r  h z0
                     (wcball z r - {z}. ff w = h w * (w-z) powi n  h w0))"
  have "(r. P f n h r) = (r. P g n h r)" for n h
  proof -
    have *: "r. P g n h r" if "r. P f n h r" and "eventually (λx. f x = g x) (at z)" for f g
    proof -
      from that(1) obtain r1 where r1_P:"P f n h r1" by auto
      from that(2) obtain r2 where "r2>0" and r2_dist:"x. x  z  dist x z  r2  f x = g x"
        unfolding eventually_at_le by auto
      define r where "r=min r1 r2"
      have "r>0" "h z0" using r1_P r2>0 unfolding r_def P_def by auto
      moreover have "h holomorphic_on cball z r"
        using r1_P unfolding P_def r_def by auto
      moreover have "g w = h w * (w - z) powi n  h w  0" when "wcball z r - {z}" for w
      proof -
        have "f w = h w * (w - z) powi n  h w  0"
          using r1_P that unfolding P_def r_def by auto
        moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
          by (simp add: dist_commute)
        ultimately show ?thesis by simp
      qed
      ultimately show ?thesis unfolding P_def by auto
    qed
    from assms have eq': "eventually (λz. g z = f z) (at z)"
      by (simp add: eq_commute)
    show ?thesis
      by (rule iffI[OF *[OF _ assms(1)] *[OF _ eq']])
  qed
  then show "zorder f z = zorder g z'" "zor_poly f z = zor_poly g z'"
      using z=z' unfolding P_def zorder_def zor_poly_def by auto
qed

lemma zorder_times_analytic':
  assumes "isolated_singularity_at f z" "not_essential f z"
  assumes "g analytic_on {z}" "frequently (λz. f z * g z  0) (at z)"
  shows   "zorder (λx. f x * g x) z = zorder f z + zorder g z"
  using assms isolated_singularity_at_analytic not_essential_analytic zorder_times by blast

lemma zorder_cmult:
  assumes "c  0"
  shows   "zorder (λz. c * f z) z = zorder f z"
proof -
  define P where
    "P = (λf n h r. 0 < r  h holomorphic_on cball z r 
              h z  0  (wcball z r - {z}. f w = h w * (w - z) powi n  h w  0))"
  have *: "P (λx. c * f x) n (λx. c * h x) r" if "P f n h r" "c  0" for f n h r c
    using that unfolding P_def by (auto intro!: holomorphic_intros)
  have "(h r. P (λx. c * f x) n h r)  (h r. P f n h r)" for n
    using *[of f n _ _ c] *[of "λx. c * f x" n _ _ "inverse c"] c  0
    by (fastforce simp: field_simps)
  hence "(THE n. h r. P (λx. c * f x) n h r) = (THE n. h r. P f n h r)"
    by simp
  thus ?thesis
    by (simp add: zorder_def P_def)
qed

lemma zorder_uminus [simp]: "zorder (λz. -f z) z = zorder f z"
  using zorder_cmult[of "-1" f] by simp

lemma zorder_nonzero_div_power:
  assumes sz: "open S" "z  S" "f holomorphic_on S" "f z  0" and "n > 0"
  shows  "zorder (λw. f w / (w - z) ^ n) z = - n"
  by (intro zorder_eqI [OF sz]) (simp add: inverse_eq_divide power_int_minus)

lemma zor_poly_eq:
  assumes "isolated_singularity_at f z" "not_essential f z" "F w in at z. f w  0"
  shows "eventually (λw. zor_poly f z w = f w * (w - z) powi - zorder f z) (at z)"
proof -
  obtain r where r:"r>0"
       "(wcball z r - {z}. f w = zor_poly f z w * (w - z) powi (zorder f z))"
    using zorder_exist[OF assms] by blast
  then have *: "wball z r - {z}. zor_poly f z w = f w * (w - z) powi - zorder f z"
    by (auto simp: field_simps power_int_minus)
  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r eventually_at_ball'[of r z UNIV] by auto
  thus ?thesis by eventually_elim (insert *, auto)
qed

lemma zor_poly_zero_eq:
  assumes "f holomorphic_on S" "open S" "connected S" "z  S" "wS. f w  0"
  shows "eventually (λw. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)) (at z)"
proof -
  obtain r where r:"r>0"
       "(wcball z r - {z}. f w = zor_poly f z w * (w - z) ^ nat (zorder f z))"
    using zorder_exist_zero[OF assms] by auto
  then have *: "wball z r - {z}. zor_poly f z w = f w / (w - z) ^ nat (zorder f z)"
    by (auto simp: field_simps powr_minus)
  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r eventually_at_ball'[of r z UNIV] by auto
  thus ?thesis by eventually_elim (insert *, auto)
qed

lemma zor_poly_pole_eq:
  assumes f_iso:"isolated_singularity_at f z" "is_pole f z"
  shows "eventually (λw. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)) (at z)"
proof -
  obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
    using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
  obtain r where r:"r>0"
       "(wcball z r - {z}. f w = zor_poly f z w / (w - z) ^ nat (- zorder f z))"
    using zorder_exist_pole[OF f_holo,simplified,OF is_pole f z] by auto
  then have *: "wball z r - {z}. zor_poly f z w = f w * (w - z) ^ nat (- zorder f z)"
    by (auto simp: field_simps)
  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r eventually_at_ball'[of r z UNIV] by auto
  thus ?thesis by eventually_elim (insert *, auto)
qed

lemma zor_poly_eqI:
  fixes f :: "complex  complex" and z0 :: complex
  defines "n  zorder f z0"
  assumes "isolated_singularity_at f z0" "not_essential f z0" "F w in at z0. f w  0"
  assumes lim: "((λx. f (g x) * (g x - z0) powi - n)  c) F"
  assumes g: "filterlim g (at z0) F" and "F  bot"
  shows   "zor_poly f z0 z0 = c"
proof -
  from zorder_exist[OF assms(2-4)] obtain r where
    r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
       "w. w  cball z0 r - {z0}  f w = zor_poly f z0 w * (w - z0) powi n"
    unfolding n_def by blast
  from r(1) have "eventually (λw. w  ball z0 r  w  z0) (at z0)"
    using eventually_at_ball'[of r z0 UNIV] by auto
  hence "eventually (λw. zor_poly f z0 w = f w * (w - z0) powi - n) (at z0)"
    by eventually_elim (insert r, auto simp: field_simps power_int_minus)
  moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
    using r by (intro holomorphic_on_imp_continuous_on) auto
  with r(1,2) have "isCont (zor_poly f z0) z0"
    by (auto simp: continuous_on_eq_continuous_at)
  hence "(zor_poly f z0  zor_poly f z0 z0) (at z0)"
    unfolding isCont_def .
  ultimately have "((λw. f w * (w - z0) powi - n)  zor_poly f z0 z0) (at z0)"
    by (blast intro: Lim_transform_eventually)
  hence "((λx. f (g x) * (g x - z0) powi - n)  zor_poly f z0 z0) F"
    by (rule filterlim_compose[OF _ g])
  from tendsto_unique[OF F  bot this lim] show ?thesis .
qed

lemma zor_poly_zero_eqI:
  fixes f :: "complex  complex" and z0 :: complex
  defines "n  zorder f z0"
  assumes "f holomorphic_on A" "open A" "connected A" "z0  A" "zA. f z  0"
  assumes lim: "((λx. f (g x) / (g x - z0) ^ nat n)  c) F"
  assumes g: "filterlim g (at z0) F" and "F  bot"
  shows   "zor_poly f z0 z0 = c"
proof -
  from zorder_exist_zero[OF assms(2-6)] obtain r where
    r: "r > 0" "cball z0 r  A" "zor_poly f z0 holomorphic_on cball z0 r"
       "w. w  cball z0 r  f w = zor_poly f z0 w * (w - z0) ^ nat n"
    unfolding n_def by blast
  from r(1) have "eventually (λw. w  ball z0 r  w  z0) (at z0)"
    using eventually_at_ball'[of r z0 UNIV] by auto
  hence "eventually (λw. zor_poly f z0 w = f w / (w - z0) ^ nat n) (at z0)"
    by eventually_elim (insert r, auto simp: field_simps)
  moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
    using r by (intro holomorphic_on_imp_continuous_on) auto
  with r(1,2) have "isCont (zor_poly f z0) z0"
    by (auto simp: continuous_on_eq_continuous_at)
  hence "(zor_poly f z0  zor_poly f z0 z0) (at z0)"
    unfolding isCont_def .
  ultimately have "((λw. f w / (w - z0) ^ nat n)  zor_poly f z0 z0) (at z0)"
    by (blast intro: Lim_transform_eventually)
  hence "((λx. f (g x) / (g x - z0) ^ nat n)  zor_poly f z0 z0) F"
    by (rule filterlim_compose[OF _ g])
  from tendsto_unique[OF F  bot this lim] show ?thesis .
qed

lemma zor_poly_pole_eqI:
  fixes f :: "complex  complex" and z0 :: complex
  defines "n  zorder f z0"
  assumes f_iso:"isolated_singularity_at f z0" and "is_pole f z0"
  assumes lim: "((λx. f (g x) * (g x - z0) ^ nat (-n))  c) F"
  assumes g: "filterlim g (at z0) F" and "F  bot"
  shows   "zor_poly f z0 z0 = c"
proof -
  obtain r where r: "r > 0"  "zor_poly f z0 holomorphic_on cball z0 r"
  proof -
    have "F w in at z0. f w  0"
      using non_zero_neighbour_pole[OF is_pole f z0] by (auto elim:eventually_frequentlyE)
    moreover have "not_essential f z0" unfolding not_essential_def using is_pole f z0 by simp
    ultimately show ?thesis using that zorder_exist[OF f_iso,folded n_def] by auto
  qed
  from r(1) have "eventually (λw. w  ball z0 r  w  z0) (at z0)"
    using eventually_at_ball'[of r z0 UNIV] by auto
  have "eventually (λw. zor_poly f z0 w = f w * (w - z0) ^ nat (-n)) (at z0)"
    using zor_poly_pole_eq[OF f_iso is_pole f z0] unfolding n_def .
  moreover have "continuous_on (ball z0 r) (zor_poly f z0)"
    using r by (intro holomorphic_on_imp_continuous_on) auto
  with r(1,2) have "isCont (zor_poly f z0) z0"
    by (auto simp: continuous_on_eq_continuous_at)
  hence "(zor_poly f z0  zor_poly f z0 z0) (at z0)"
    unfolding isCont_def .
  ultimately have "((λw. f w * (w - z0) ^ nat (-n))  zor_poly f z0 z0) (at z0)"
    by (blast intro: Lim_transform_eventually)
  hence "((λx. f (g x) * (g x - z0) ^ nat (-n))  zor_poly f z0 z0) F"
    by (rule filterlim_compose[OF _ g])
  from tendsto_unique[OF F  bot this lim] show ?thesis .
qed

lemma
  assumes "is_pole f (x :: complex)" "open A" "x  A"
  assumes "y. y  A - {x}  (f has_field_derivative f' y) (at y)"
  shows   is_pole_deriv': "is_pole f' x"
    and   zorder_deriv':  "zorder f' x = zorder f x - 1"
proof -
  have holo: "f holomorphic_on A - {x}"
    using assms by (subst holomorphic_on_open) auto
  obtain r where r: "r > 0" "ball x r  A"
    using assms(2,3) openE by blast
  moreover have "open (ball x r - {x})"
    by auto
  ultimately have "isolated_singularity_at f x"
    by (auto simp: isolated_singularity_at_def analytic_on_open
             intro!: exI[of _ r] holomorphic_on_subset[OF holo])
  hence ev: "F w in at x. zor_poly f x w = f w * (w - x) ^ nat (- zorder f x)"
    using is_pole f x zor_poly_pole_eq by blast

  define P where "P = zor_poly f x"
  define n where "n = nat (-zorder f x)"

  obtain r where r: "r > 0" "cball x r  A" "P holomorphic_on cball x r" "zorder f x < 0" "P x  0"
    "wcball x r - {x}. f w = P w / (w - x) ^ n  P w  0"
    unfolding P_def n_def using zorder_exist_pole[OF holo assms(2,3,1)] by blast
  have n: "n > 0"
    using r(4) by (auto simp: n_def)

  have [derivative_intros]: "(P has_field_derivative deriv P w) (at w)"
    if "w  ball x r" for w
    using that by (intro holomorphic_derivI[OF holomorphic_on_subset[OF r(3), of "ball x r"]]) auto

  define D where "D = (λw. (deriv P w * (w - x) - of_nat n * P w) / (w - x) ^ (n + 1))"
  define n' where "n' = n - 1"
  have n': "n = Suc n'"
    using n by (simp add: n'_def)

  have "eventually (λw. w  ball x r) (nhds x)"
    using r > 0 by (intro eventually_nhds_in_open) auto
  hence ev'': "eventually (λw. w  ball x r - {x}) (at x)"
    by (auto simp: eventually_at_filter elim: eventually_mono)

  {
    fix w assume w: "w  ball x r - {x}"
    have ev': "eventually (λw. w  ball x r - {x}) (nhds w)"
      using w by (intro eventually_nhds_in_open) auto

    have "((λw. P w / (w - x) ^ n) has_field_derivative D w) (at w)"
      apply (rule derivative_eq_intros refl | use w in force)+
      using w
      apply (simp add: divide_simps D_def)
      apply (simp add: n' algebra_simps)
      done
    also have "?this  (f has_field_derivative D w) (at w)"
      using r by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto
    finally have "(f has_field_derivative D w) (at w)" .
    moreover have "(f has_field_derivative f' w) (at w)"
      using w r by (intro assms) auto
    ultimately have "D w = f' w"
      using DERIV_unique by blast
  } note D_eq = this

  have "is_pole D x"
    unfolding D_def using n r > 0 P x  0
    by (intro is_pole_basic[where A = "ball x r"] holomorphic_intros holomorphic_on_subset[OF r(3)]) auto
  also have "?this  is_pole f' x"
    by (intro is_pole_cong eventually_mono[OF ev''] D_eq) auto
  finally show "is_pole f' x" .

  have "zorder f' x = -int (Suc n)"
  proof (rule zorder_eqI)
    show "open (ball x r)" "x  ball x r"
      using r > 0 by auto
    show "f' w = (deriv P w * (w - x) - of_nat n * P w) * (w - x) powi (- int (Suc n))"
      if "w  ball x r" "w  x" for w
      using that D_eq[of w] n by (auto simp: D_def power_int_diff power_int_minus powr_nat' divide_simps)
  qed (use r n in auto intro!: holomorphic_intros)
  thus "zorder f' x = zorder f x - 1"
    using n by (simp add: n_def)
qed

lemma
  assumes "is_pole f (x :: complex)" "isolated_singularity_at f x"
  shows   is_pole_deriv: "is_pole (deriv f) x"
    and   zorder_deriv:  "zorder (deriv f) x = zorder f x - 1"
proof -
  from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
    by (auto simp: isolated_singularity_at_def)
  hence holo: "f holomorphic_on ball x r - {x}"
    by (subst (asm) analytic_on_open) auto
  have *: "x  ball x r" "open (ball x r)" "open (ball x r - {x})"
    using r > 0 by auto
  show "is_pole (deriv f) x" "zorder (deriv f) x = zorder f x - 1"
    by (meson "*" assms(1) holo holomorphic_derivI is_pole_deriv' zorder_deriv')+
qed

lemma removable_singularity_deriv':
  assumes "f x c" "x  A" "open (A :: complex set)"
  assumes "y. y  A - {x}  (f has_field_derivative f' y) (at y)"
  shows   "c. f' x c"
proof -
  have holo: "f holomorphic_on A - {x}"
    using assms by (subst holomorphic_on_open) auto

  define g where "g = (λy. if y = x then c else f y)"
  have deriv_g_eq: "deriv g y = f' y" if "y  A - {x}" for y
  proof -
    have ev: "eventually (λy. y  A - {x}) (nhds y)"
      using that assms by (intro eventually_nhds_in_open) auto
    have "(f has_field_derivative f' y) (at y)"
      using assms that by auto
    also have "?this  (g has_field_derivative f' y) (at y)"
      by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev]) (auto simp: g_def)
    finally show ?thesis
      by (intro DERIV_imp_deriv assms)
  qed

  have "g holomorphic_on A"
    unfolding g_def using assms assms(1) holo by (intro removable_singularity) auto
  hence "deriv g holomorphic_on A"
    by (intro holomorphic_deriv assms)
  hence "continuous_on A (deriv g)"
    by (meson holomorphic_on_imp_continuous_on)
  hence "(deriv g  deriv g x) (at x within A)"
    using assms by (auto simp: continuous_on_def)
  also have "?this  (f'  deriv g x) (at x within A)"
    by (intro filterlim_cong refl) (auto simp: eventually_at_filter deriv_g_eq)
  finally have "f' x deriv g x"
    using open A x  A by (meson tendsto_within_open)
  thus ?thesis
    by blast
qed

lemma removable_singularity_deriv:
  assumes "f x c" "isolated_singularity_at f x"
  shows   "c. deriv f x c"
proof -
  from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
    by (auto simp: isolated_singularity_at_def)
  hence holo: "f holomorphic_on ball x r - {x}"
    using analytic_imp_holomorphic by blast
  show ?thesis
    using assms(1)
  proof (rule removable_singularity_deriv')
    show "x  ball x r" "open (ball x r)"
      using r > 0 by auto
  qed (auto intro!: holomorphic_derivI[OF holo])
qed

lemma not_essential_deriv':
  assumes "not_essential f x" "x  A" "open A"
  assumes "y. y  A - {x}  (f has_field_derivative f' y) (at y)"
  shows   "not_essential f' x"
proof -
  have holo: "f holomorphic_on A - {x}"
    using assms by (subst holomorphic_on_open) auto
  from assms consider "is_pole f x" | c where "f x c"
    by (auto simp: not_essential_def)
  thus ?thesis
  proof cases
    case 1
    hence "is_pole f' x"
      using is_pole_deriv' assms by blast
    thus ?thesis by (auto simp: not_essential_def)
  next
    case (2 c)
    from 2 have "c. f' x c"
      by (rule removable_singularity_deriv'[OF _ assms(2-4)])
    thus ?thesis
      by (auto simp: not_essential_def)
  qed
qed

lemma not_essential_deriv[singularity_intros]:
  assumes "not_essential f x" "isolated_singularity_at f x"
  shows   "not_essential (deriv f) x"
proof -
  from assms(2) obtain r where r: "r > 0" "f analytic_on ball x r - {x}"
    by (auto simp: isolated_singularity_at_def)
  hence holo: "f holomorphic_on ball x r - {x}"
    by (subst (asm) analytic_on_open) auto
  show ?thesis
    using assms(1)
  proof (rule not_essential_deriv')
    show "x  ball x r" "open (ball x r)"
      using r > 0 by auto
  qed (auto intro!: holomorphic_derivI[OF holo])
qed

lemma not_essential_frequently_0_imp_tendsto_0:
  fixes f :: "complex  complex"
  assumes sing: "isolated_singularity_at f z" "not_essential f z"
  assumes freq: "frequently (λz. f z = 0) (at z)"
  shows   "f z 0"
proof -
  from freq obtain g :: "nat  complex" where g: "filterlim g (at z) at_top" "n. f (g n) = 0"
    using frequently_atE by blast
  have "eventually (λx. f (g x) = 0) sequentially"
    using g by auto
  hence fg: "(λx. f (g x))  0"
    by (simp add: tendsto_eventually)

  from assms(2) consider c where "f z c" | "is_pole f z"
    unfolding not_essential_def by blast
  thus ?thesis
  proof cases
    case (1 c)
    have "(λx. f (g x))  c"
      by (rule filterlim_compose[OF 1 g(1)])
    with fg have "c = 0"
      using LIMSEQ_unique by blast
    with 1 show ?thesis by simp
  next
    case 2
    have "filterlim (λx. f (g x)) at_infinity sequentially"
      by (rule filterlim_compose[OF _ g(1)]) (use 2 in auto simp: is_pole_def)
    with fg have False
      by (meson not_tendsto_and_filterlim_at_infinity sequentially_bot)
    thus ?thesis ..
  qed
qed

lemma not_essential_frequently_0_imp_eventually_0:
  fixes f :: "complex  complex"
  assumes sing: "isolated_singularity_at f z" "not_essential f z"
  assumes freq: "frequently (λz. f z = 0) (at z)"
  shows   "eventually (λz. f z = 0) (at z)"
proof -
  from sing obtain r where r: "r > 0" and "f analytic_on ball z r - {z}"
    by (auto simp: isolated_singularity_at_def)
  hence holo: "f holomorphic_on ball z r - {z}"
    by (subst (asm) analytic_on_open) auto
  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r by (intro eventually_at_in_open) auto
  from freq and this have "frequently (λw. f w = 0  w  ball z r - {z}) (at z)"
    using frequently_eventually_frequently by blast
  hence "frequently (λw. w  {wball z r - {z}. f w = 0}) (at z)"
    by (simp add: conj_commute)
  hence limpt: "z islimpt {wball z r - {z}. f w = 0}"
    using islimpt_conv_frequently_at by blast

  define g where "g = (λw. if w = z then 0 else f w)"
  have "f z 0"
    by (intro not_essential_frequently_0_imp_tendsto_0 assms)
  hence g_holo: "g holomorphic_on ball z r"
    unfolding g_def by (intro removable_singularity holo) auto

  have g_eq_0: "g w = 0" if "w  ball z r" for w
  proof (rule analytic_continuation[where f = g])
    show "open (ball z r)" "connected (ball z r)"
      using r by auto
    show "z islimpt {wball z r - {z}. f w = 0}"
      by fact
    show "g w = 0" if "w  {w  ball z r - {z}. f w = 0}" for w
      using that by (auto simp: g_def)
  qed (use r that g_holo in auto)

  have "eventually (λw. w  ball z r - {z}) (at z)"
    using r by (intro eventually_at_in_open) auto
  thus "eventually (λw. f w = 0) (at z)"
    by (metis freq non_zero_neighbour not_eventually not_frequently sing)
qed

lemma pole_imp_not_constant:
  fixes f :: "'a :: {perfect_space}  _"
  assumes "is_pole f x" "open A" "x  A" "A  insert x B"
  shows   "¬f constant_on B"
proof
  assume *: "f constant_on B"
  then obtain c where c: "xB. f x = c"
    by (auto simp: constant_on_def)
  have "eventually (λy. y  A - {x}) (at x)"
    using assms by (intro eventually_at_in_open) auto
  hence "eventually (λy. f y = c) (at x)"
    by eventually_elim (use c assms in auto)
  hence **: "f x c"
    by (simp add: tendsto_eventually)
  show False
    using not_tendsto_and_filterlim_at_infinity[OF _ ** assms(1)[unfolded is_pole_def]] by simp
qed


lemma neg_zorder_imp_is_pole:
  assumes iso:"isolated_singularity_at f z" and f_ness:"not_essential f z"
      and "zorder f z < 0" and fre_nz:"F w in at z. f w  0 "
    shows "is_pole f z"
proof -
  define P where "P = zor_poly f z"
  define n where "n = zorder f z"
  have "n<0" unfolding n_def by (simp add: assms(3))
  define nn where "nn = nat (-n)"

  obtain r where "P z  0" "r>0" and r_holo:"P holomorphic_on cball z r" and
       w_Pn:"(wcball z r - {z}. f w = P w * (w - z) powi n  P w  0)"
    using zorder_exist[OF iso f_ness fre_nz,folded P_def n_def] by auto

  have "is_pole (λw. P w * (w - z) powi n) z"
    unfolding is_pole_def
  proof (rule tendsto_mult_filterlim_at_infinity)
    show "P z P z"
      by (meson open_ball 0 < r ball_subset_cball centre_in_ball
          continuous_on_eq_continuous_at continuous_on_subset
          holomorphic_on_imp_continuous_on isContD r_holo)
    show "P z0" by (simp add: P z  0)

    have "LIM x at z. inverse ((x - z) ^ nat (-n)) :> at_infinity"
      apply (subst filterlim_inverse_at_iff[symmetric])
      using n<0
      by (auto intro!:tendsto_eq_intros filterlim_atI
              simp add:eventually_at_filter)
    then show "LIM x at z. (x - z) powi n :> at_infinity"
    proof (elim filterlim_mono_eventually)
      have "inverse ((x - z) ^ nat (-n)) = (x - z) powi n"
        if "xz" for x
        by (metis n < 0 linorder_not_le power_int_def power_inverse)
      then show "F x in at z. inverse ((x - z) ^ nat (-n))
                  = (x - z) powi n"
        by (simp add: eventually_at_filter)
    qed auto
  qed
  moreover have "F w in at z. f w =  P w * (w - z) powi n"
    unfolding eventually_at_le
    using w_Pn r>0 by (force simp add: dist_commute)
  ultimately show ?thesis using is_pole_cong by fast
qed

lemma is_pole_divide_zorder:
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z" and g_iso:"isolated_singularity_at g z"
      and f_ness:"not_essential f z" and g_ness:"not_essential g z"
      and fg_nconst: "Fw in (at z). f w * g w 0"
      and z_less:"zorder f z < zorder g z"
    shows "is_pole (λz. f z / g z) z"
proof -
  define fn gn fg where "fn=zorder f z" and "gn=zorder g z"
                        and "fg=(λw. f w / g w)"

  have "isolated_singularity_at fg z"
    unfolding fg_def using f_iso g_iso g_ness
    by (auto intro:singularity_intros)
  moreover have "not_essential fg z"
    unfolding fg_def using f_iso g_iso g_ness f_ness
    by (auto intro:singularity_intros)
  moreover have "zorder fg z < 0"
  proof -
    have "zorder fg z = fn - gn"
      using zorder_divide[OF f_iso g_iso f_ness g_ness
            fg_nconst,folded fn_def gn_def fg_def] .
    then show ?thesis
      using z_less by (simp add: fn_def gn_def)
  qed
  moreover have "F w in at z. fg w  0"
    using fg_nconst unfolding fg_def by force
  ultimately show "is_pole fg z"
    using neg_zorder_imp_is_pole by auto
qed

lemma isolated_pole_imp_nzero_times:
  assumes f_iso:"isolated_singularity_at f z"
    and "is_pole f z"
  shows "Fw in (at z). deriv f w * f w  0"
proof (rule ccontr)
  assume "¬ (F w in at z.  deriv f w  * f w  0)"
  then have "F x in at z. deriv f x * f x = 0"
    unfolding not_frequently by simp
  moreover have "F w in at z. f w  0"
    using non_zero_neighbour_pole[OF is_pole f z] .
  moreover have "F w in at z. deriv f w  0"
    using is_pole_deriv[OF is_pole f z f_iso,THEN non_zero_neighbour_pole]
    .
  ultimately have "F w in at z. False"
    by eventually_elim auto
  then show False by auto
qed

lemma isolated_pole_imp_neg_zorder:
  assumes "isolated_singularity_at f z" and "is_pole f z"
  shows "zorder f z < 0"
  using analytic_imp_holomorphic assms centre_in_ball isolated_singularity_at_def zorder_exist_pole by blast


lemma isolated_singularity_at_deriv[singularity_intros]:
  assumes "isolated_singularity_at f x"
  shows "isolated_singularity_at (deriv f) x"
  by (meson analytic_deriv assms isolated_singularity_at_def)

lemma zorder_deriv_minus_1:
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z"
      and f_nconst:"F w in at z. f w  0"
      and f_ord:"zorder f z 0"
    shows "zorder (deriv f) z = zorder f z - 1"
proof -
  define P where "P = zor_poly f z"
  define n where "n = zorder f z"
  have "n0" unfolding n_def using f_ord by auto

  obtain r where "P z  0" "r>0" and P_holo:"P holomorphic_on cball z r"
          and "(wcball z r - {z}. f w
                            = P w * (w - z) powi n  P w  0)"
    using zorder_exist[OF f_iso f_ness f_nconst,folded P_def n_def] by auto
  from this(4)
  have f_eq:"(wcball z r - {z}. f w
                            = P w * (w - z) powi n  P w  0)"
    using complex_powr_of_int f_ord n_def by presburger

  define D where "D = (λw. (deriv P w * (w - z) + of_int n * P w)
                          * (w - z) powi (n - 1))"

  have deriv_f_eq:"deriv f w = D w" if "w  ball z r - {z}" for w
  proof -
    have ev': "eventually (λw. w  ball z r - {z}) (nhds w)"
      using that by (intro eventually_nhds_in_open) auto

    define wz where "wz = w - z"

    have "wz 0" unfolding wz_def using that by auto
    moreover have "(P has_field_derivative deriv P w) (at w)"
      by (meson DiffD1 Elementary_Metric_Spaces.open_ball P_holo
          ball_subset_cball holomorphic_derivI holomorphic_on_subset that)
    ultimately have "((λw. P w * (w - z) powi n) has_field_derivative D w) (at w)"
      unfolding D_def using that
      apply (auto intro!: derivative_eq_intros)
      apply (fold wz_def)
      by (auto simp:algebra_simps simp flip:power_int_add_1')
    also have "?this  (f has_field_derivative D w) (at w)"
      using f_eq
      by (intro has_field_derivative_cong_ev refl eventually_mono[OF ev']) auto
    ultimately have "(f has_field_derivative D w) (at w)" by simp
    moreover have "(f has_field_derivative deriv f w) (at w)"
      by (metis DERIV_imp_deriv calculation)
    ultimately show ?thesis using DERIV_imp_deriv by blast
  qed

  show "zorder (deriv f) z = n - 1"
  proof (rule zorder_eqI)
    show "open (ball z r)" "z  ball z r"
      using r > 0 by auto
    define g where "g=(λw. (deriv P w * (w - z) + of_int n * P w))"
    show "g holomorphic_on ball z r"
      unfolding g_def using P_holo
      by (auto intro!:holomorphic_intros)
    show "g z  0"
      unfolding g_def using P z  0 n0 by auto
    show "deriv f w =
         (deriv P w * (w - z) + of_int n * P w) * (w - z) powi (n - 1)"
      if "w  ball z r" "w  z" for w
      using D_def deriv_f_eq that by blast
  qed
qed


lemma deriv_divide_is_pole: ―‹Generalises @{thm zorder_deriv}
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z"
      and f_ness:"not_essential f z" 
      and fg_nconst: "Fw in (at z). deriv f w *  f w  0"
      and f_ord:"zorder f z 0"
    shows "is_pole (λz. deriv f z / f z) z"
proof (rule neg_zorder_imp_is_pole)
  define ff where "ff=(λw. deriv f w / f w)"
  show "isolated_singularity_at ff z" 
    using f_iso f_ness unfolding ff_def
    by (auto intro:singularity_intros)
  show "not_essential ff z" 
    unfolding ff_def using f_ness f_iso
    by (auto intro:singularity_intros)

  have "zorder ff z =  zorder (deriv f) z - zorder f z"
    unfolding ff_def using f_iso f_ness fg_nconst
    using isolated_singularity_at_deriv not_essential_deriv zorder_divide by blast
  moreover have "zorder (deriv f) z = zorder f z - 1"
    using f_iso f_ness f_ord fg_nconst frequently_elim1 zorder_deriv_minus_1 by fastforce
  ultimately show "zorder ff z < 0" by auto
    
  show "F w in at z. ff w  0" 
    unfolding ff_def using fg_nconst by auto
qed

lemma is_pole_deriv_divide_is_pole:
  fixes f g::"complex  complex" and z::complex
  assumes f_iso:"isolated_singularity_at f z"
      and "is_pole f z" 
    shows "is_pole (λz. deriv f z / f z) z"
proof (rule deriv_divide_is_pole[OF f_iso])
  show "not_essential f z" 
    using is_pole f z unfolding not_essential_def by auto
  show "F w in at z. deriv f w * f w  0"
    using assms f_iso isolated_pole_imp_nzero_times by blast
  show "zorder f z  0"
    using isolated_pole_imp_neg_zorder assms by fastforce
qed

subsection ‹Isolated zeroes›

definition isolated_zero :: "(complex  complex)  complex  bool" where
  "isolated_zero f z  f z = 0  eventually (λz. f z  0) (at z)"

lemma isolated_zero_altdef: "isolated_zero f z  f z = 0  ¬z islimpt {z. f z = 0}"
  unfolding isolated_zero_def eventually_at_filter eventually_nhds islimpt_def by blast

lemma isolated_zero_mult1:
  assumes "isolated_zero f x" "isolated_zero g x"
  shows   "isolated_zero (λx. f x * g x) x"
proof -
  have "F x in at x. f x  0" "F x in at x. g x  0"
    using assms unfolding isolated_zero_def by auto
  hence "eventually (λx. f x * g x  0) (at x)"
    by eventually_elim auto
  with assms show ?thesis
    by (auto simp: isolated_zero_def)
qed

lemma isolated_zero_mult2:
  assumes "isolated_zero f x" "g x  0" "g analytic_on {x}"
  shows   "isolated_zero (λx. f x * g x) x"
proof -
  have "eventually (λx. f x  0) (at x)"
    using assms unfolding isolated_zero_def by auto
  moreover have "eventually (λx. g x  0) (at x)"
    using analytic_at_neq_imp_eventually_neq[of g x 0] assms by auto
  ultimately have "eventually (λx. f x * g x  0) (at x)"
    by eventually_elim auto
  thus ?thesis
    using assms(1) by (auto simp: isolated_zero_def)
qed

lemma isolated_zero_mult3:
  assumes "isolated_zero f x" "g x  0" "g analytic_on {x}"
  shows   "isolated_zero (λx. g x * f x) x"
  using isolated_zero_mult2[OF assms] by (simp add: mult_ac)
  
lemma isolated_zero_prod:
  assumes "x. x  I  isolated_zero (f x) z" "I  {}" "finite I"
  shows   "isolated_zero (λy. xI. f x y) z"
  using assms(3,2,1) by (induction rule: finite_ne_induct) (auto intro: isolated_zero_mult1)

lemma non_isolated_zero':
  assumes "isolated_singularity_at f z" "not_essential f z" "f z = 0" "¬isolated_zero f z"
  shows   "eventually (λz. f z = 0) (at z)"
  by (metis assms isolated_zero_def non_zero_neighbour not_eventually)

lemma non_isolated_zero:
  assumes "¬isolated_zero f z" "f analytic_on {z}" "f z = 0"
  shows   "eventually (λz. f z = 0) (nhds z)"
proof -
  have "eventually (λz. f z = 0) (at z)"
    by (rule non_isolated_zero')
       (use assms in auto intro: not_essential_analytic isolated_singularity_at_analytic)
  with f z = 0 show ?thesis
    unfolding eventually_at_filter by (auto elim!: eventually_mono)
qed

lemma not_essential_compose:
  assumes "not_essential f (g z)" "g analytic_on {z}"
  shows   "not_essential (λx. f (g x)) z"
proof (cases "isolated_zero (λw. g w - g z) z")
  case False
  hence "eventually (λw. g w - g z = 0) (nhds z)"
    by (rule non_isolated_zero) (use assms in auto intro!: analytic_intros)
  hence "not_essential (λx. f (g x)) z  not_essential (λ_. f (g z)) z"
    by (intro not_essential_cong refl)
       (auto elim!: eventually_mono simp: eventually_at_filter)
  thus ?thesis
    by (simp add: not_essential_const)
next
  case True
  hence ev: "eventually (λw. g w  g z) (at z)"
    by (auto simp: isolated_zero_def)
  from assms consider c where "f g z c" | "is_pole f (g z)"
    by (auto simp: not_essential_def)  
  have "isCont g z"
    by (rule analytic_at_imp_isCont) fact
  hence lim: "g z g z"
    using isContD by blast

  from assms(1) consider c where "f g z c" | "is_pole f (g z)"
    unfolding not_essential_def by blast
  thus ?thesis
  proof cases
    fix c assume "f g z c"
    hence "(λx. f (g x)) z c"
      by (rule filterlim_compose) (use lim ev in auto simp: filterlim_at)
    thus ?thesis
      by (auto simp: not_essential_def)
  next
    assume "is_pole f (g z)"
    hence "is_pole (λx. f (g x)) z"
      by (rule is_pole_compose) fact+
    thus ?thesis
      by (auto simp: not_essential_def)
  qed
qed
  
subsection ‹Isolated points›

definition isolated_points_of :: "complex set  complex set" where
  "isolated_points_of A = {zA. eventually (λw. w  A) (at z)}"

lemma isolated_points_of_altdef: "isolated_points_of A = {zA. ¬z islimpt A}"
  unfolding isolated_points_of_def islimpt_def eventually_at_filter eventually_nhds by blast

lemma isolated_points_of_empty [simp]: "isolated_points_of {} = {}"
  and isolated_points_of_UNIV [simp]:  "isolated_points_of UNIV = {}"
  by (auto simp: isolated_points_of_def)

lemma isolated_points_of_open_is_empty [simp]: "open A  isolated_points_of A = {}"
  unfolding isolated_points_of_altdef 
  by (simp add: interior_limit_point interior_open)

lemma isolated_points_of_subset: "isolated_points_of A  A"
  by (auto simp: isolated_points_of_def)

lemma isolated_points_of_discrete:
  assumes "discrete A"
  shows   "isolated_points_of A = A"
  using assms by (auto simp: isolated_points_of_def discrete_altdef)

lemmas uniform_discreteI1 = uniformI1
lemmas uniform_discreteI2 = uniformI2

lemma isolated_singularity_at_compose:
  assumes "isolated_singularity_at f (g z)" "g analytic_on {z}"
  shows   "isolated_singularity_at (λx. f (g x)) z"
proof (cases "isolated_zero (λw. g w - g z) z")
  case False
  hence "eventually (λw. g w - g z = 0) (nhds z)"
    by (rule non_isolated_zero) (use assms in auto intro!: analytic_intros)
  hence "isolated_singularity_at (λx. f (g x)) z  isolated_singularity_at (λ_. f (g z)) z"
    by (intro isolated_singularity_at_cong refl)
       (auto elim!: eventually_mono simp: eventually_at_filter)
  thus ?thesis
    by (simp add: isolated_singularity_at_const)
next
  case True
  from assms(1) obtain r where r: "r > 0" "f analytic_on ball (g z) r - {g z}"
    by (auto simp: isolated_singularity_at_def)
  hence holo_f: "f holomorphic_on ball (g z) r - {g z}"
    by (subst (asm) analytic_on_open) auto
  from assms(2) obtain r' where r': "r' > 0" "g holomorphic_on ball z r'"
    by (auto simp: analytic_on_def)

  have "continuous_on (ball z r') g"
    using holomorphic_on_imp_continuous_on r' by blast
  hence "isCont g z"
    using r' by (subst (asm) continuous_on_eq_continuous_at) auto
  hence "g z g z"
    using isContD by blast
  hence "eventually (λw. g w  ball (g z) r) (at z)"
    using r > 0 unfolding tendsto_def by force
  moreover have "eventually (λw. g w  g z) (at z)" using True
    by (auto simp: isolated_zero_def elim!: eventually_mono)
  ultimately have "eventually (λw. g w  ball (g z) r - {g z}) (at z)"
    by eventually_elim auto
  then obtain r'' where r'': "r'' > 0" "wball z r''-{z}. g w  ball (g z) r - {g z}"
    unfolding eventually_at_filter eventually_nhds_metric ball_def
    by (auto simp: dist_commute)
  have "f  g holomorphic_on ball z (min r' r'') - {z}"
  proof (rule holomorphic_on_compose_gen)
    show "g holomorphic_on ball z (min r' r'') - {z}"
      by (rule holomorphic_on_subset[OF r'(2)]) auto
    show "f holomorphic_on ball (g z) r - {g z}"
      by fact
    show "g ` (ball z (min r' r'') - {z})  ball (g z) r - {g z}"
      using r'' by force
  qed
  hence "f  g analytic_on ball z (min r' r'') - {z}"
    by (subst analytic_on_open) auto
  thus ?thesis using r' > 0 r'' > 0
    by (auto simp: isolated_singularity_at_def o_def intro!: exI[of _ "min r' r''"])
qed

lemma is_pole_power_int_0:
  assumes "f analytic_on {x}" "isolated_zero f x" "n < 0"
  shows   "is_pole (λx. f x powi n) x"
proof -
  have "f x f x"
    using assms(1) by (simp add: analytic_at_imp_isCont isContD)
  with assms show ?thesis
    unfolding is_pole_def
    by (intro filterlim_power_int_neg_at_infinity) (auto simp: isolated_zero_def)
qed

lemma isolated_zero_imp_not_constant_on:
  assumes "isolated_zero f x" "x  A" "open A"
  shows   "¬f constant_on A"
proof
  assume "f constant_on A"
  then obtain c where c: "x. x  A  f x = c"
    by (auto simp: constant_on_def)
  from assms and c[of x] have [simp]: "c = 0"
    by (auto simp: isolated_zero_def)
  have "eventually (λx. f x  0) (at x)"
    using assms by (auto simp: isolated_zero_def)
  moreover have "eventually (λx. x  A) (at x)"
    using assms by (intro eventually_at_in_open') auto
  ultimately have "eventually (λx. False) (at x)"
    by eventually_elim (use c in auto)
  thus False
    by simp
qed

end