--- a/src/HOL/Complex_Analysis/Complex_Singularities.thy Sun Feb 19 21:21:19 2023 +0000
+++ b/src/HOL/Complex_Analysis/Complex_Singularities.thy Mon Feb 20 15:19:53 2023 +0000
@@ -344,7 +344,7 @@
lemma holomorphic_factor_unique:
fixes f::"complex \<Rightarrow> complex" and z::complex and r::real and m n::int
assumes "r>0" "g z\<noteq>0" "h z\<noteq>0"
- and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powr m \<and> h w\<noteq>0"
+ and asm:"\<forall>w\<in>ball z r-{z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0 \<and> f w = h w * (w - z) powi m \<and> h w\<noteq>0"
and g_holo:"g holomorphic_on ball z r" and h_holo:"h holomorphic_on ball z r"
shows "n=m"
proof -
@@ -353,14 +353,14 @@
have False when "n>m"
proof -
have "(h \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (n - m) * g w"])
- have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powr(n-m) * g w"
- using \<open>n>m\<close> asm \<open>r>0\<close> by (simp add: field_simps powr_diff) force
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powi (n - m) * g w"])
+ have "\<forall>w\<in>ball z r-{z}. h w = (w-z)powi(n-m) * g w"
+ using \<open>n>m\<close> asm \<open>r>0\<close> by (simp add: field_simps power_int_diff) force
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (n - m) * g x' = h x'" for x' by auto
+ \<Longrightarrow> (x' - z) powi (n - m) * g x' = h x'" for x' by auto
next
define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv> \<lambda>x. (x - z) powr (n-m)"
+ define f' where "f' \<equiv> \<lambda>x. (x - z) powi (n-m)"
have "f' z=0" using \<open>n>m\<close> unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
using \<open>n>m\<close>
@@ -381,14 +381,14 @@
moreover have False when "m>n"
proof -
have "(g \<longlongrightarrow> 0) (at z within ball z r)"
- proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powr (m - n) * h w"])
- have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powr (m-n) * h w" using \<open>m>n\<close> asm
- by (simp add:field_simps powr_diff) force
+ proof (rule Lim_transform_within[OF _ \<open>r>0\<close>, where f="\<lambda>w. (w - z) powi (m - n) * h w"])
+ have "\<forall>w\<in>ball z r -{z}. g w = (w-z) powi (m-n) * h w" using \<open>m>n\<close> asm
+ by (simp add:field_simps power_int_diff) force
then show "\<lbrakk>x' \<in> ball z r; 0 < dist x' z;dist x' z < r\<rbrakk>
- \<Longrightarrow> (x' - z) powr (m - n) * h x' = g x'" for x' by auto
+ \<Longrightarrow> (x' - z) powi (m - n) * h x' = g x'" for x' by auto
next
define F where "F \<equiv> at z within ball z r"
- define f' where "f' \<equiv>\<lambda>x. (x - z) powr (m-n)"
+ define f' where "f' \<equiv>\<lambda>x. (x - z) powi (m-n)"
have "f' z=0" using \<open>m>n\<close> unfolding f'_def by auto
moreover have "continuous F f'" unfolding f'_def F_def continuous_def
using \<open>m>n\<close>
@@ -414,23 +414,23 @@
and "not_essential f z" \<comment> \<open>\<^term>\<open>f\<close> has either a removable singularity or a pole at \<^term>\<open>z\<close>\<close>
and non_zero:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0" \<comment> \<open>\<^term>\<open>f\<close> will not be constantly zero in a neighbour of \<^term>\<open>z\<close>\<close>
shows "\<exists>!n::int. \<exists>g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powr n \<and> g w\<noteq>0)"
+ \<and> (\<forall>w\<in>cball z r-{z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0)"
proof -
define P where "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))"
have imp_unique:"\<exists>!n::int. \<exists>g r. P f n g r" when "\<exists>n g r. P f n g r"
proof (rule ex_ex1I[OF that])
fix n1 n2 :: int
assume g1_asm:"\<exists>g1 r1. P f n1 g1 r1" and g2_asm:"\<exists>g2 r2. P f n2 g2 r2"
- define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powr (of_int n) \<and> g w \<noteq> 0"
+ define fac where "fac \<equiv> \<lambda>n g r. \<forall>w\<in>cball z r-{z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0"
obtain g1 r1 where "0 < r1" and g1_holo: "g1 holomorphic_on cball z r1" and "g1 z\<noteq>0"
and "fac n1 g1 r1" using g1_asm unfolding P_def fac_def by auto
obtain g2 r2 where "0 < r2" and g2_holo: "g2 holomorphic_on cball z r2" and "g2 z\<noteq>0"
and "fac n2 g2 r2" using g2_asm unfolding P_def fac_def by auto
define r where "r \<equiv> min r1 r2"
have "r>0" using \<open>r1>0\<close> \<open>r2>0\<close> unfolding r_def by auto
- moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powr n1 \<and> g1 w\<noteq>0
- \<and> f w = g2 w * (w - z) powr n2 \<and> g2 w\<noteq>0"
+ moreover have "\<forall>w\<in>ball z r-{z}. f w = g1 w * (w-z) powi n1 \<and> g1 w\<noteq>0
+ \<and> f w = g2 w * (w - z) powi n2 \<and> g2 w\<noteq>0"
using \<open>fac n1 g1 r1\<close> \<open>fac n2 g2 r2\<close> unfolding fac_def r_def
by fastforce
ultimately show "n1=n2" using g1_holo g2_holo \<open>g1 z\<noteq>0\<close> \<open>g2 z\<noteq>0\<close>
@@ -533,19 +533,18 @@
then obtain n g r
where "0 < r" and
g_holo:"g holomorphic_on cball z r" and "g z\<noteq>0" and
- g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ g_fac:"(\<forall>w\<in>cball z r-{z}. h w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
unfolding P_def by auto
have "P f (-n) (inverse o g) r"
proof -
- have "f w = inverse (g w) * (w - z) powr of_int (- n)" when "w\<in>cball z r - {z}" for w
- by (metis g_fac h_def inverse_inverse_eq inverse_mult_distrib of_int_minus
- powr_minus that)
+ have "f w = inverse (g w) * (w - z) powi (- n)" when "w\<in>cball 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 \<open>r>0\<close> g_holo g_fac \<open>g z\<noteq>0\<close> by (auto intro:holomorphic_intros)
qed
then show "\<exists>x g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int x \<and> g w \<noteq> 0)"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi x \<and> g w \<noteq> 0)"
unfolding P_def by blast
qed
ultimately show ?thesis using \<open>not_essential f z\<close> unfolding not_essential_def by presburger
@@ -583,15 +582,15 @@
lemma not_essential_powr[singularity_intros]:
assumes "LIM w (at z). f w :> (at x)"
- shows "not_essential (\<lambda>w. (f w) powr (of_int n)) z"
+ shows "not_essential (\<lambda>w. (f w) powi n) z"
proof -
- define fp where "fp=(\<lambda>w. (f w) powr (of_int n))"
+ define fp where "fp=(\<lambda>w. (f w) powi n)"
have ?thesis when "n>0"
proof -
have "(\<lambda>w. (f w) ^ (nat n)) \<midarrow>z\<rightarrow> x ^ nat n"
using that assms unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp \<midarrow>z\<rightarrow> x ^ nat n" unfolding fp_def
- by (smt (verit, best) LIM_equal of_int_of_nat power_eq_0_iff powr_nat that zero_less_nat_eq)
+ 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"
@@ -615,7 +614,7 @@
proof (elim filterlim_mono_eventually)
show "\<forall>\<^sub>F x in at z. inverse (f x ^ nat (- n)) = fp x"
using filterlim_at_within_not_equal[OF assms,of 0] unfolding fp_def
- by (smt (verit, ccfv_threshold) eventually_mono powr_of_int that)
+ 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
@@ -624,8 +623,7 @@
have "(\<lambda>w. inverse ((f w) ^ (nat (-n)))) \<midarrow>z\<rightarrow>?xx"
using assms False unfolding filterlim_at by (auto intro!:tendsto_eq_intros)
then have "fp \<midarrow>z\<rightarrow> ?xx"
- by (smt (verit) LIM_cong fp_def inverse_nonzero_iff_nonzero power_eq_0_iff powr_eq_0_iff
- powr_of_int that zero_less_nat_eq)
+ 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
@@ -633,7 +631,7 @@
lemma isolated_singularity_at_powr[singularity_intros]:
assumes "isolated_singularity_at f z" "\<forall>\<^sub>F w in (at z). f w\<noteq>0"
- shows "isolated_singularity_at (\<lambda>w. (f w) powr (of_int n)) z"
+ shows "isolated_singularity_at (\<lambda>w. (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
@@ -642,8 +640,8 @@
obtain r2 where "r2>0" and r2:"\<forall>w. w \<noteq> z \<and> dist w z < r2 \<longrightarrow> f w \<noteq> 0"
using assms(2) unfolding eventually_at by auto
define r3 where "r3=min r1 r2"
- have "(\<lambda>w. (f w) powr of_int n) holomorphic_on ball z r3 - {z}"
- by (intro holomorphic_on_powr_of_int) (use r1 r2 in \<open>auto simp: dist_commute r3_def\<close>)
+ have "(\<lambda>w. (f w) powi n) holomorphic_on ball z r3 - {z}"
+ by (intro holomorphic_on_power_int) (use r1 r2 in \<open>auto simp: dist_commute r3_def\<close>)
moreover have "r3>0" unfolding r3_def using \<open>0 < r1\<close> \<open>0 < r2\<close> by linarith
ultimately show ?thesis
by (meson open_ball analytic_on_open isolated_singularity_at_def open_delete)
@@ -657,13 +655,13 @@
proof -
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
have "f w \<noteq> 0" when " w \<noteq> z" "dist w z < fr" for w
proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w \<noteq> 0"
+ have "f w = fp w * (w - z) powi fn" "fp w \<noteq> 0"
using fr(2)[rule_format, of w] using that by (auto simp add:dist_commute)
- moreover have "(w - z) powr of_int fn \<noteq>0"
+ moreover have "(w - z) powi fn \<noteq>0"
unfolding powr_eq_0_iff using \<open>w\<noteq>z\<close> by auto
ultimately show ?thesis by auto
qed
@@ -715,24 +713,24 @@
proof -
obtain fn fp fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0"
using holomorphic_factor_puncture[OF f_iso f_ness f_nconst,THEN ex1_implies_ex] by auto
obtain gn gp gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powi gn \<and> gp w \<noteq> 0"
using holomorphic_factor_puncture[OF g_iso g_ness g_nconst,THEN ex1_implies_ex] by auto
define r1 where "r1=(min fr gr)"
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w\<noteq>0"
when "w\<in>ball z r1 - {z}" for w
proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ moreover have "g w = gp w * (w - z) powi gn" "gp w \<noteq> 0"
using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
+ ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w\<noteq>0"
+ using that unfolding fg_def by (auto simp add:power_int_add)
qed
have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
@@ -745,9 +743,8 @@
have "(\<lambda>w. (fp w * gp w) * (w - z) ^ (nat (fn+gn))) \<midarrow>z\<rightarrow>0"
using that by (auto intro!:tendsto_eq_intros)
then have "fg \<midarrow>z\<rightarrow> 0"
- using Lim_transform_within[OF _ \<open>r1>0\<close>]
- by (smt (verit, ccfv_SIG) DiffI dist_commute dist_nz fg_times mem_ball powr_of_int right_minus_eq
- singletonD that)
+ apply (elim Lim_transform_within[OF _ \<open>r1>0\<close>])
+ 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"
@@ -771,7 +768,8 @@
using filterlim_divide_at_infinity by blast
then have "is_pole fg z" unfolding is_pole_def
apply (elim filterlim_transform_within[OF _ _ \<open>r1>0\<close>])
- by (simp_all add: dist_commute fg_times of_int_of_nat powr_minus_divide that)
+ 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
@@ -1091,13 +1089,13 @@
definition\<^marker>\<open>tag important\<close> zorder :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> int" where
"zorder f z = (THE n. (\<exists>h r. r>0 \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powr (of_int n)
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w-z) powi n
\<and> h w \<noteq>0)))"
definition\<^marker>\<open>tag important\<close> zor_poly
::"[complex \<Rightarrow> complex, complex] \<Rightarrow> complex \<Rightarrow> complex" where
"zor_poly f z = (SOME h. \<exists>r. r > 0 \<and> h holomorphic_on cball z r \<and> h z \<noteq> 0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr (zorder f z)
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powi (zorder f z)
\<and> h w \<noteq>0))"
lemma zorder_exist:
@@ -1107,10 +1105,10 @@
and f_ness:"not_essential f z"
and f_nconst:"\<exists>\<^sub>Fw in (at z). f w\<noteq>0"
shows "g z\<noteq>0 \<and> (\<exists>r. r>0 \<and> g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr n \<and> g w \<noteq>0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w \<noteq>0))"
proof -
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))"
have "\<exists>!n. \<exists>g r. P n g r"
using holomorphic_factor_puncture[OF assms(3-)] unfolding P_def by auto
then have "\<exists>g r. P n g r"
@@ -1163,20 +1161,20 @@
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fn_def fp_def]
by auto
- have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))"
+ have fr_inverse: "vf w = (inverse (fp w)) * (w - z) powi (-fn)"
and fr_nz: "inverse (fp w) \<noteq> 0"
when "w\<in>ball z fr - {z}" for w
proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0"
using fr(2)[rule_format,of w] that by auto
- then show "vf w = (inverse (fp w)) * (w - z) powr (of_int (-fn))" "inverse (fp w)\<noteq>0"
- unfolding vf_def by (auto simp add:powr_minus)
+ then show "vf w = (inverse (fp w)) * (w - z) powi (-fn)" "inverse (fp w)\<noteq>0"
+ by (simp_all add: power_int_minus vf_def)
qed
obtain vfr where [simp]:"vfp z \<noteq> 0" and "vfr>0" and vfr:"vfp holomorphic_on cball z vfr"
- "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powr of_int vfn \<and> vfp w \<noteq> 0)"
+ "(\<forall>w\<in>cball z vfr - {z}. vf w = vfp w * (w - z) powi vfn \<and> vfp w \<noteq> 0)"
proof -
have "isolated_singularity_at vf z"
using isolated_singularity_at_inverse[OF f_iso f_ness] unfolding vf_def .
@@ -1195,8 +1193,8 @@
have \<section>: "\<And>w. \<lbrakk>fp w = 0; dist z w < fr\<rbrakk> \<Longrightarrow> False"
using fr_nz by force
then show "\<forall>w\<in>ball z r1 - {z}.
- vf w = vfp w * (w - z) powr complex_of_int vfn \<and>
- vfp w \<noteq> 0 \<and> vf w = inverse (fp w) * (w - z) powr complex_of_int (- fn) \<and>
+ vf w = vfp w * (w - z) powi vfn \<and>
+ vfp w \<noteq> 0 \<and> vf w = inverse (fp w) * (w - z) powi (- fn) \<and>
inverse (fp w) \<noteq> 0"
using fr_inverse r1_def vfr(2)
by (smt (verit) Diff_iff inverse_nonzero_iff_nonzero mem_ball mem_cball)
@@ -1225,7 +1223,7 @@
obtain r1 where "r1>0" "zor_poly f z z \<noteq> 0" and
holo1:"zor_poly f z holomorphic_on cball z r1" and
rball1:"\<forall>w\<in>cball z r1 - {z}.
- f w = zor_poly f z w * (w - z) powr of_int (zorder f z) \<and>
+ f w = zor_poly f z w * (w - z) powi (zorder f z) \<and>
zor_poly f z w \<noteq> 0"
using zorder_exist[OF iso1 ness1 nzero1] by blast
@@ -1245,7 +1243,7 @@
ultimately obtain r2 where "r2>0" "zor_poly ff 0 0 \<noteq> 0" and
holo2:"zor_poly ff 0 holomorphic_on cball 0 r2" and
rball2:"\<forall>w\<in>cball 0 r2 - {0}.
- ff w = zor_poly ff 0 w * w powr of_int (zorder ff 0) \<and>
+ ff w = zor_poly ff 0 w * w powi (zorder ff 0) \<and>
zor_poly ff 0 w \<noteq> 0"
using zorder_exist[of ff 0] by auto
@@ -1255,9 +1253,9 @@
have "zor_poly f z w = zor_poly ff 0 (w - z)"
if "w\<in>ball z r - {z}" for w
proof -
- define n::complex where "n= of_int (zorder f z)"
+ define n where "n \<equiv> zorder f z"
- have "f w = zor_poly f z w * (w - z) powr n"
+ have "f w = zor_poly f z w * (w - z) powi n"
proof -
have "w\<in>cball z r1 - {z}"
using r_def that by auto
@@ -1265,26 +1263,25 @@
show ?thesis unfolding n_def by auto
qed
- moreover have "f w = zor_poly ff 0 (w - z) * (w - z) powr n"
+ moreover have "f w = zor_poly ff 0 (w - z) * (w - z) powi n"
proof -
have "w-z\<in>cball 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)
- powr of_int (zorder ff 0)"
+ 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) powr n
- = zor_poly ff 0 (w - z) * (w - z) powr n"
+ 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) powr n \<noteq>0"
+ moreover have "(w - z) powi n \<noteq>0"
using that by auto
ultimately show ?thesis
- apply (subst (asm) mult_cancel_right)
- by (simp add:ff_def)
+ using mult_cancel_right by blast
qed
then have "\<forall>\<^sub>F w in at z. zor_poly f z w
= zor_poly ff 0 (w - z)"
@@ -1327,31 +1324,32 @@
using fg_nconst by (auto elim!:frequently_elim1)
obtain fr where [simp]:"fp z \<noteq> 0" and "fr > 0"
and fr: "fp holomorphic_on cball z fr"
- "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
+ "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powi fn \<and> fp w \<noteq> 0"
using zorder_exist[OF f_iso f_ness f_nconst,folded fp_def fn_def] by auto
obtain gr where [simp]:"gp z \<noteq> 0" and "gr > 0"
and gr: "gp holomorphic_on cball z gr"
- "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
+ "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powi gn \<and> gp w \<noteq> 0"
using zorder_exist[OF g_iso g_ness g_nconst,folded gn_def gp_def] by auto
define r1 where "r1=min fr gr"
have "r1>0" unfolding r1_def using \<open>fr>0\<close> \<open>gr>0\<close> by auto
- have fg_times:"fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" and fgp_nz:"fp w*gp w\<noteq>0"
+ have fg_times:"fg w = (fp w * gp w) * (w - z) powi (fn+gn)" and fgp_nz:"fp w*gp w\<noteq>0"
when "w\<in>ball z r1 - {z}" for w
proof -
- have "f w = fp w * (w - z) powr of_int fn" "fp w\<noteq>0"
+ have "f w = fp w * (w - z) powi fn" "fp w\<noteq>0"
using fr(2)[rule_format,of w] that unfolding r1_def by auto
- moreover have "g w = gp w * (w - z) powr of_int gn" "gp w \<noteq> 0"
+ moreover have "g w = gp w * (w - z) powi gn" "gp w \<noteq> 0"
using gr(2)[rule_format, of w] that unfolding r1_def by auto
- ultimately show "fg w = (fp w * gp w) * (w - z) powr (of_int (fn+gn))" "fp w*gp w\<noteq>0"
- unfolding fg_def by (auto simp add:powr_add)
+ ultimately show "fg w = (fp w * gp w) * (w - z) powi (fn+gn)" "fp w*gp w\<noteq>0"
+ using that
+ unfolding fg_def by (auto simp add:power_int_add)
qed
obtain fgr where [simp]:"fgp z \<noteq> 0" and "fgr > 0"
and fgr: "fgp holomorphic_on cball z fgr"
- "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0"
+ "\<forall>w\<in>cball z fgr - {z}. fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0"
proof -
have "fgp z \<noteq> 0 \<and> (\<exists>r>0. fgp holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0))"
apply (rule zorder_exist[of fg z, folded fgn_def fgp_def])
subgoal unfolding fg_def using isolated_singularity_at_times[OF f_iso g_iso] .
subgoal unfolding fg_def using not_essential_times[OF f_ness g_ness f_iso g_iso] .
@@ -1371,8 +1369,8 @@
fix w assume "w \<in> ball z r2 - {z}"
then have "w \<in> ball z r1 - {z}" "w \<in> cball z fgr - {z}" unfolding r2_def by auto
from fg_times[OF this(1)] fgp_nz[OF this(1)] fgr(2)[rule_format,OF this(2)]
- show "fg w = fgp w * (w - z) powr of_int fgn \<and> fgp w \<noteq> 0
- \<and> fg w = fp w * gp w * (w - z) powr of_int (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
+ show "fg w = fgp w * (w - z) powi fgn \<and> fgp w \<noteq> 0
+ \<and> fg w = fp w * gp w * (w - z) powi (fn + gn) \<and> fp w * gp w \<noteq> 0" by auto
qed
subgoal using fgr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
subgoal using fr(1) gr(1) unfolding r2_def r1_def by (auto intro!:holomorphic_intros)
@@ -1431,10 +1429,10 @@
\<and> (\<forall>w\<in>cball z r. f w = g w * (w-z) ^ nat n \<and> g w \<noteq>0))"
proof -
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
proof -
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,6)
@@ -1453,7 +1451,7 @@
by auto
qed
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
using assms(4,6) open_contains_cball_eq by blast
@@ -1461,7 +1459,7 @@
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
qed
@@ -1469,22 +1467,23 @@
have fz_lim: "f\<midarrow> z \<rightarrow> f z"
by (metis assms(4,6) at_within_open continuous_on holo holomorphic_on_imp_continuous_on)
have gz_lim: "g \<midarrow>z\<rightarrow>g z"
- by (metis r open_ball at_within_open ball_subset_cball centre_in_ball
- continuous_on_def continuous_on_subset holomorphic_on_imp_continuous_on)
+ 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 "(\<lambda>w. g w * (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z"
+ have "(\<lambda>w. g w * (w - z) powi n) \<midarrow>z\<rightarrow> f z"
using fz_lim Lim_transform_within_open[where s="ball z r"] r by fastforce
- then have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
+ then have "(\<lambda>w. (g w * (w - z) powi n) / g w) \<midarrow>z\<rightarrow> f z/g z"
using gz_lim \<open>g z \<noteq> 0\<close> tendsto_divide by blast
- then have powr_tendsto:"(\<lambda>w. (w - z) powr of_int n) \<midarrow>z\<rightarrow> f z/g z"
+ then have powi_tendsto:"(\<lambda>w. (w - z) powi n) \<midarrow>z\<rightarrow> f z/g z"
using Lim_transform_within_open[where s="ball z r"] r by fastforce
have ?thesis when "n\<ge>0" "f z=0"
proof -
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto Lim_transform_within[where d=r]
- by (fastforce simp: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ using Lim_transform_within[OF powi_tendsto, where d=r]
+ by (meson power_int_def r(1) that(1))
then have *:"(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0" using \<open>f z=0\<close> by simp
moreover have False when "n=0"
proof -
@@ -1499,8 +1498,8 @@
have False when "n>0"
proof -
have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> f z/g z"
- using powr_tendsto Lim_transform_within[where d=r]
- by (fastforce simp add: powr_of_int \<open>n\<ge>0\<close> \<open>r>0\<close>)
+ using Lim_transform_within[OF powi_tendsto, where d=r]
+ by (meson \<open>0 \<le> n\<close> power_int_def r(1))
moreover have "(\<lambda>w. (w - z) ^ nat n) \<midarrow>z\<rightarrow> 0"
using \<open>n>0\<close> by (auto intro!:tendsto_eq_intros)
ultimately show False using \<open>f z\<noteq>0\<close> \<open>g z\<noteq>0\<close> using LIM_unique divide_eq_0_iff by blast
@@ -1510,11 +1509,13 @@
moreover have False when "n<0"
proof -
have "(\<lambda>w. inverse ((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> f z/g z"
- "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
- apply (smt (verit, ccfv_SIG) LIM_cong eq_iff_diff_eq_0 powr_of_int powr_tendsto that)
+ by (smt (verit) LIM_cong power_int_def power_inverse powi_tendsto that)
+ moreover
+ have "(\<lambda>w.((w - z) ^ nat (- n))) \<midarrow>z\<rightarrow> 0"
using that by (auto intro!:tendsto_eq_intros)
- from tendsto_mult[OF this,simplified]
- have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0" .
+ ultimately
+ have "(\<lambda>x. inverse ((x - z) ^ nat (- n)) * (x - z) ^ nat (- n)) \<midarrow>z\<rightarrow> 0"
+ using tendsto_mult by fastforce
then have "(\<lambda>x. 1::complex) \<midarrow>z\<rightarrow> 0"
by (elim Lim_transform_within_open[where s=UNIV],auto)
then show False using LIM_const_eq by fastforce
@@ -1531,10 +1532,10 @@
fix x assume "0 < dist x z" "dist x z < r"
then have "x \<in> cball z r - {z}" "x\<noteq>z"
unfolding cball_def by (auto simp add: dist_commute)
- then have "f x = g x * (x - z) powr of_int n"
+ 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) \<open>x \<noteq> z\<close> if_0 powr_of_int right_minus_eq)
+ 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 "(\<lambda>w. g w * (w-z) ^ nat n) \<midarrow>z\<rightarrow> g w * (w-z) ^ nat n"
@@ -1543,9 +1544,10 @@
then show ?thesis using \<open>g z\<noteq>0\<close> True by auto
next
case False
- then have "f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
+ then have "f w = g w * (w - z) powi n \<and> g w \<noteq> 0"
using r(4) that by auto
- then show ?thesis using False if_0 powr_of_int by (auto split:if_splits)
+ 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
@@ -1560,10 +1562,10 @@
\<and> (\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0))"
proof -
obtain r where "g z \<noteq> 0" and r: "r>0" "cball z r \<subseteq> s" "g holomorphic_on cball z r"
- "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ "(\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
proof -
have "g z \<noteq> 0 \<and> (\<exists>r>0. g holomorphic_on cball z r
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0))"
proof (rule zorder_exist[of f z,folded g_def n_def])
show "isolated_singularity_at f z" unfolding isolated_singularity_at_def
using holo assms(4,5)
@@ -1576,7 +1578,7 @@
by auto
qed
then obtain r1 where "g z \<noteq> 0" "r1>0" and r1:"g holomorphic_on cball z r1"
- "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ "(\<forall>w\<in>cball z r1 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
by auto
obtain r2 where r2: "r2>0" "cball z r2 \<subseteq> s"
using assms(4,5) open_contains_cball_eq by metis
@@ -1584,7 +1586,7 @@
have "r3>0" "cball z r3 \<subseteq> s" using \<open>r1>0\<close> r2 unfolding r3_def by auto
moreover have "g holomorphic_on cball z r3"
using r1(1) unfolding r3_def by auto
- moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0)"
+ moreover have "(\<forall>w\<in>cball z r3 - {z}. f w = g w * (w - z) powi n \<and> g w \<noteq> 0)"
using r1(2) unfolding r3_def by auto
ultimately show ?thesis using that[of r3] \<open>g z\<noteq>0\<close> by auto
qed
@@ -1599,7 +1601,7 @@
have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
unfolding eventually_at_topological
apply (rule_tac exI[where x="ball z r"])
- using r powr_of_int \<open>\<not> n < 0\<close> by auto
+ by (simp add: \<open>\<not> n < 0\<close> linorder_not_le power_int_def r(1) r(4))
moreover have "(\<lambda>x. g x * (x - z) ^ nat n) \<midarrow>z\<rightarrow> c"
proof (cases "n=0")
case True
@@ -1616,14 +1618,14 @@
unfolding is_pole_def by blast
qed
moreover have "\<forall>w\<in>cball z r - {z}. f w = g w / (w-z) ^ nat (- n) \<and> g w \<noteq>0"
- using r(4) \<open>n<0\<close> powr_of_int
- by (metis Diff_iff divide_inverse eq_iff_diff_eq_0 insert_iff linorder_not_le)
+ using r(4) \<open>n<0\<close>
+ 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) \<open>g z\<noteq>0\<close> by auto
qed
lemma zorder_eqI:
assumes "open s" "z \<in> s" "g holomorphic_on s" "g z \<noteq> 0"
- assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> f w = g w * (w - z) powr n"
+ assumes fg_eq:"\<And>w. \<lbrakk>w \<in> s;w\<noteq>z\<rbrakk> \<Longrightarrow> 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
@@ -1634,9 +1636,9 @@
ultimately obtain r where r: "r > 0" "cball z r \<subseteq> s \<inter> (g -` (-{0}))"
unfolding open_contains_cball by blast
- let ?gg= "(\<lambda>w. g w * (w - z) powr n)"
+ let ?gg= "(\<lambda>w. g w * (w - z) powi n)"
define P where "P = (\<lambda>n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powr (of_int n) \<and> g w\<noteq>0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w-z) powi n \<and> g w\<noteq>0))"
have "P n g r"
unfolding P_def using r assms(3,4,5) by auto
then have "\<exists>g r. P n g r" by auto
@@ -1663,7 +1665,7 @@
assms holomorphic_on_subset isolated_singularity_at_def openE)
qed
moreover
- have "\<forall>\<^sub>F w in at z. g w * (w - z) powr n = f w"
+ have "\<forall>\<^sub>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
@@ -1678,7 +1680,7 @@
have "z' \<in> cball z r"
unfolding z'_def using \<open>r>0\<close> \<open>d>0\<close> by (auto simp add:dist_norm)
then show " z' \<in> s" using r(2) by blast
- show "g z' * (z' - z) powr of_int n \<noteq> 0"
+ show "g z' * (z' - z) powi n \<noteq> 0"
using P_def \<open>P n g r\<close> \<open>z' \<in> cball z r\<close> \<open>z' \<noteq> z\<close> by auto
qed
ultimately show "\<exists>x\<in>UNIV. x \<noteq> z \<and> dist x z < d \<and> f x \<noteq> 0" by auto
@@ -1813,7 +1815,7 @@
shows zorder_cong:"zorder f z = zorder g z'" and zor_poly_cong:"zor_poly f z = zor_poly g z'"
proof -
define P where "P = (\<lambda>ff n h r. 0 < r \<and> h holomorphic_on cball z r \<and> h z\<noteq>0
- \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powr (of_int n) \<and> h w\<noteq>0))"
+ \<and> (\<forall>w\<in>cball z r - {z}. ff w = h w * (w-z) powi n \<and> h w\<noteq>0))"
have "(\<exists>r. P f n h r) = (\<exists>r. P g n h r)" for n h
proof -
have *: "\<exists>r. P g n h r" if "\<exists>r. P f n h r" and "eventually (\<lambda>x. f x = g x) (at z)" for f g
@@ -1825,9 +1827,9 @@
have "r>0" "h z\<noteq>0" using r1_P \<open>r2>0\<close> unfolding r_def P_def by auto
moreover have "h holomorphic_on cball z r"
using r1_P unfolding P_def r_def by auto
- moreover have "g w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
+ moreover have "g w = h w * (w - z) powi n \<and> h w \<noteq> 0" when "w\<in>cball z r - {z}" for w
proof -
- have "f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0"
+ have "f w = h w * (w - z) powi n \<and> h w \<noteq> 0"
using r1_P that unfolding P_def r_def by auto
moreover have "f w=g w" using r2_dist[rule_format,of w] that unfolding r_def
by (simp add: dist_commute)
@@ -1859,7 +1861,7 @@
proof -
define P where
"P = (\<lambda>f n h r. 0 < r \<and> h holomorphic_on cball z r \<and>
- h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powr of_int n \<and> h w \<noteq> 0))"
+ h z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = h w * (w - z) powi n \<and> h w \<noteq> 0))"
have *: "P (\<lambda>x. c * f x) n (\<lambda>x. c * h x) r" if "P f n h r" "c \<noteq> 0" for f n h r c
using that unfolding P_def by (auto intro!: holomorphic_intros)
have "(\<exists>h r. P (\<lambda>x. c * f x) n h r) \<longleftrightarrow> (\<exists>h r. P f n h r)" for n
@@ -1874,17 +1876,17 @@
lemma zorder_nonzero_div_power:
assumes sz: "open s" "z \<in> s" "f holomorphic_on s" "f z \<noteq> 0" and "n > 0"
shows "zorder (\<lambda>w. f w / (w - z) ^ n) z = - n"
- using zorder_eqI [OF sz] by (simp add: powr_minus_divide)
+ 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" "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
- shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powr - zorder f z) (at z)"
+ shows "eventually (\<lambda>w. zor_poly f z w = f w * (w - z) powi - zorder f z) (at z)"
proof -
obtain r where r:"r>0"
- "(\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z))"
+ "(\<forall>w\<in>cball 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 *: "\<forall>w\<in>ball z r - {z}. zor_poly f z w = f w * (w - z) powr - zorder f z"
- by (auto simp: field_simps powr_minus)
+ then have *: "\<forall>w\<in>ball 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 (\<lambda>w. w \<in> ball z r - {z}) (at z)"
using r eventually_at_ball'[of r z UNIV] by auto
thus ?thesis by eventually_elim (insert *, auto)
@@ -1924,27 +1926,27 @@
fixes f :: "complex \<Rightarrow> complex" and z0 :: complex
defines "n \<equiv> zorder f z0"
assumes "isolated_singularity_at f z0" "not_essential f z0" "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
- assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> c) F"
+ assumes lim: "((\<lambda>x. f (g x) * (g x - z0) powi - n) \<longlongrightarrow> c) F"
assumes g: "filterlim g (at z0) F" and "F \<noteq> bot"
shows "zor_poly f z0 z0 = c"
proof -
from zorder_exist[OF assms(2-4)] obtain r where
r: "r > 0" "zor_poly f z0 holomorphic_on cball z0 r"
- "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powr n"
+ "\<And>w. w \<in> cball z0 r - {z0} \<Longrightarrow> f w = zor_poly f z0 w * (w - z0) powi n"
unfolding n_def by blast
from r(1) have "eventually (\<lambda>w. w \<in> ball z0 r \<and> w \<noteq> z0) (at z0)"
using eventually_at_ball'[of r z0 UNIV] by auto
- hence "eventually (\<lambda>w. zor_poly f z0 w = f w * (w - z0) powr - n) (at z0)"
- by eventually_elim (insert r, auto simp: field_simps powr_minus)
+ hence "eventually (\<lambda>w. 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 \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
unfolding isCont_def .
- ultimately have "((\<lambda>w. f w * (w - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
+ ultimately have "((\<lambda>w. f w * (w - z0) powi - n) \<longlongrightarrow> zor_poly f z0 z0) (at z0)"
by (blast intro: Lim_transform_eventually)
- hence "((\<lambda>x. f (g x) * (g x - z0) powr - n) \<longlongrightarrow> zor_poly f z0 z0) F"
+ hence "((\<lambda>x. f (g x) * (g x - z0) powi - n) \<longlongrightarrow> zor_poly f z0 z0) F"
by (rule filterlim_compose[OF _ g])
from tendsto_unique[OF \<open>F \<noteq> bot\<close> this lim] show ?thesis .
qed
@@ -2082,9 +2084,9 @@
proof (rule zorder_eqI)
show "open (ball x r)" "x \<in> ball x r"
using \<open>r > 0\<close> by auto
- show "f' w = (deriv P w * (w - x) - of_nat n * P w) * (w - x) powr of_int (- int (Suc n))"
+ show "f' w = (deriv P w * (w - x) - of_nat n * P w) * (w - x) powi (- int (Suc n))"
if "w \<in> ball x r" "w \<noteq> x" for w
- using that D_eq[of w] n by (auto simp: D_def powr_diff powr_minus powr_nat' divide_simps)
+ 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 \<open>auto intro!: holomorphic_intros\<close>)
thus "zorder f' x = zorder f x - 1"
using n by (simp add: n_def)
@@ -2306,10 +2308,10 @@
define nn where "nn = nat (-n)"
obtain r where "P z \<noteq> 0" "r>0" and r_holo:"P holomorphic_on cball z r" and
- w_Pn:"(\<forall>w\<in>cball z r - {z}. f w = P w * (w - z) powr of_int n \<and> P w \<noteq> 0)"
+ w_Pn:"(\<forall>w\<in>cball z r - {z}. f w = P w * (w - z) powi n \<and> P w \<noteq> 0)"
using zorder_exist[OF iso f_ness fre_nz,folded P_def n_def] by auto
- have "is_pole (\<lambda>w. P w * (w - z) powr of_int n) z"
+ have "is_pole (\<lambda>w. P w * (w - z) powi n) z"
unfolding is_pole_def
proof (rule tendsto_mult_filterlim_at_infinity)
show "P \<midarrow>z\<rightarrow> P z"
@@ -2323,18 +2325,17 @@
using \<open>n<0\<close>
by (auto intro!:tendsto_eq_intros filterlim_atI
simp add:eventually_at_filter)
- then show "LIM x at z. (x - z) powr of_int n :> at_infinity"
+ 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) powr of_int n"
+ have "inverse ((x - z) ^ nat (-n)) = (x - z) powi n"
if "x\<noteq>z" for x
- apply (subst powr_of_int)
- using \<open>n<0\<close> using that by auto
+ by (metis \<open>n < 0\<close> linorder_not_le power_int_def power_inverse)
then show "\<forall>\<^sub>F x in at z. inverse ((x - z) ^ nat (-n))
- = (x - z) powr of_int n"
+ = (x - z) powi n"
by (simp add: eventually_at_filter)
qed auto
qed
- moreover have "\<forall>\<^sub>F w in at z. f w = P w * (w - z) powr of_int n"
+ moreover have "\<forall>\<^sub>F w in at z. f w = P w * (w - z) powi n"
unfolding eventually_at_le
apply (rule exI[where x=r])
using w_Pn \<open>r>0\<close> by (simp add: dist_commute)
@@ -2429,7 +2430,7 @@
obtain r where "P z \<noteq> 0" "r>0" and P_holo:"P holomorphic_on cball z r"
and "(\<forall>w\<in>cball z r - {z}. f w
- = P w * (w - z) powr of_int n \<and> P w \<noteq> 0)"
+ = P w * (w - z) powi n \<and> P w \<noteq> 0)"
using zorder_exist[OF f_iso f_ness f_nconst,folded P_def n_def] by auto
from this(4)
have f_eq:"(\<forall>w\<in>cball z r - {z}. f w
@@ -2475,10 +2476,9 @@
show "g z \<noteq> 0"
unfolding g_def using \<open>P z \<noteq> 0\<close> \<open>n\<noteq>0\<close> by auto
show "deriv f w =
- (deriv P w * (w - z) + of_int n * P w) * (w - z) powr of_int (n - 1)"
+ (deriv P w * (w - z) + of_int n * P w) * (w - z) powi (n - 1)"
if "w \<in> ball z r" "w \<noteq> z" for w
- apply (subst complex_powr_of_int)
- using deriv_f_eq that unfolding D_def by auto
+ using D_def deriv_f_eq that by blast
qed
qed