# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1744135560 -3600
# Node ID a1de627d417ae66a0d05fa64e0e65df7de49441c
# Parent  5a0d1075911cdf330a0a47af37cac09c16c27df3
More of Manuel's material

diff -r 5a0d1075911c -r a1de627d417a src/HOL/Analysis/Complex_Transcendental.thy
--- a/src/HOL/Analysis/Complex_Transcendental.thy	Mon Apr 07 12:36:56 2025 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy	Tue Apr 08 19:06:00 2025 +0100
@@ -2919,6 +2919,16 @@
   qed (use z in auto)
 qed
 
+lemma has_field_derivative_csqrt' [derivative_intros]:
+  assumes "(f has_field_derivative f') (at x within A)" "f x \<notin> \<real>\<^sub>\<le>\<^sub>0"
+  shows   "((\<lambda>x. csqrt (f x)) has_field_derivative (f' / (2 * csqrt (f x)))) (at x within A)"
+proof -
+  have "((csqrt \<circ> f) has_field_derivative (inverse (2 * csqrt (f x)) * f')) (at x within A)"
+    using has_field_derivative_csqrt assms(1) by (rule DERIV_chain) fact
+  thus ?thesis
+    by (simp add: o_def field_simps)
+qed
+
 lemma field_differentiable_at_csqrt:
     "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> csqrt field_differentiable at z"
   using field_differentiable_def has_field_derivative_csqrt by blast
diff -r 5a0d1075911c -r a1de627d417a src/HOL/Analysis/Gamma_Function.thy
--- a/src/HOL/Analysis/Gamma_Function.thy	Mon Apr 07 12:36:56 2025 +0200
+++ b/src/HOL/Analysis/Gamma_Function.thy	Tue Apr 08 19:06:00 2025 +0100
@@ -35,66 +35,6 @@
   finally show ?thesis .
 qed
 
-lemma plus_one_in_nonpos_Ints_imp: "z + 1 \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> z \<in> \<int>\<^sub>\<le>\<^sub>0"
-  using nonpos_Ints_diff_Nats[of "z+1" "1"] by simp_all
-
-lemma of_int_in_nonpos_Ints_iff:
-  "(of_int n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n \<le> 0"
-  by (auto simp: nonpos_Ints_def)
-
-lemma one_plus_of_int_in_nonpos_Ints_iff:
-  "(1 + of_int n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n \<le> -1"
-proof -
-  have "1 + of_int n = (of_int (n + 1) :: 'a)" by simp
-  also have "\<dots> \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n + 1 \<le> 0" by (subst of_int_in_nonpos_Ints_iff) simp_all
-  also have "\<dots> \<longleftrightarrow> n \<le> -1" by presburger
-  finally show ?thesis .
-qed
-
-lemma one_minus_of_nat_in_nonpos_Ints_iff:
-  "(1 - of_nat n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n > 0"
-proof -
-  have "(1 - of_nat n :: 'a) = of_int (1 - int n)" by simp
-  also have "\<dots> \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n > 0" by (subst of_int_in_nonpos_Ints_iff) presburger
-  finally show ?thesis .
-qed
-
-lemma fraction_not_in_ints:
-  assumes "\<not>(n dvd m)" "n \<noteq> 0"
-  shows   "of_int m / of_int n \<notin> (\<int> :: 'a :: {division_ring,ring_char_0} set)"
-proof
-  assume "of_int m / (of_int n :: 'a) \<in> \<int>"
-  then obtain k where "of_int m / of_int n = (of_int k :: 'a)" by (elim Ints_cases)
-  with assms have "of_int m = (of_int (k * n) :: 'a)" by (auto simp add: field_split_simps)
-  hence "m = k * n" by (subst (asm) of_int_eq_iff)
-  hence "n dvd m" by simp
-  with assms(1) show False by contradiction
-qed
-
-lemma fraction_not_in_nats:
-  assumes "\<not>n dvd m" "n \<noteq> 0"
-  shows   "of_int m / of_int n \<notin> (\<nat> :: 'a :: {division_ring,ring_char_0} set)"
-proof
-  assume "of_int m / of_int n \<in> (\<nat> :: 'a set)"
-  also note Nats_subset_Ints
-  finally have "of_int m / of_int n \<in> (\<int> :: 'a set)" .
-  moreover have "of_int m / of_int n \<notin> (\<int> :: 'a set)"
-    using assms by (intro fraction_not_in_ints)
-  ultimately show False by contradiction
-qed
-
-lemma not_in_Ints_imp_not_in_nonpos_Ints: "z \<notin> \<int> \<Longrightarrow> z \<notin> \<int>\<^sub>\<le>\<^sub>0"
-  by (auto simp: Ints_def nonpos_Ints_def)
-
-lemma double_in_nonpos_Ints_imp:
-  assumes "2 * (z :: 'a :: field_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0"
-  shows   "z \<in> \<int>\<^sub>\<le>\<^sub>0 \<or> z + 1/2 \<in> \<int>\<^sub>\<le>\<^sub>0"
-proof-
-  from assms obtain k where k: "2 * z = - of_nat k" by (elim nonpos_Ints_cases')
-  thus ?thesis by (cases "even k") (auto elim!: evenE oddE simp: field_simps)
-qed
-
-
 lemma sin_series: "(\<lambda>n. ((-1)^n / fact (2*n+1)) *\<^sub>R z^(2*n+1)) sums sin z"
 proof -
   from sin_converges[of z] have "(\<lambda>n. sin_coeff n *\<^sub>R z^n) sums sin z" .
diff -r 5a0d1075911c -r a1de627d417a src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy
--- a/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy	Mon Apr 07 12:36:56 2025 +0200
+++ b/src/HOL/Complex_Analysis/Cauchy_Integral_Formula.thy	Tue Apr 08 19:06:00 2025 +0100
@@ -412,12 +412,88 @@
   finally show ?case by simp
 qed simp_all
 
+lemma higher_deriv_cmult':
+  assumes "f analytic_on {x}"
+  shows   "(deriv ^^ j) (\<lambda>x. c * f x) x = c * (deriv ^^ j) f x"
+  using assms higher_deriv_cmult[of f _ x j c] assms
+  using analytic_at_two by blast
+
+lemma deriv_cmult':
+  assumes "f analytic_on {x}"
+  shows   "deriv (\<lambda>x. c * f x) x = c * deriv f x"
+  using higher_deriv_cmult'[OF assms, of 1 c] by simp
+
+lemma analytic_derivI:
+  assumes "f analytic_on {z}"
+  shows   "(f has_field_derivative (deriv f z)) (at z within A)"
+  using assms holomorphic_derivI[of f _ z] analytic_at by blast
+
+lemma deriv_compose_analytic:
+  fixes f g :: "complex \<Rightarrow> complex"
+  assumes "f analytic_on {g z}" "g analytic_on {z}"
+  shows "deriv (\<lambda>x. f (g x)) z = deriv f (g z) * deriv g z"
+proof -
+  have "((f \<circ> g) has_field_derivative (deriv f (g z) * deriv g z)) (at z)"
+    by (intro DERIV_chain analytic_derivI assms)
+  thus ?thesis
+    by (auto intro!: DERIV_imp_deriv simp add: o_def)
+qed
+
 lemma valid_path_compose_holomorphic:
   assumes "valid_path g" "f holomorphic_on S" and "open S" "path_image g \<subseteq> S"
   shows "valid_path (f \<circ> g)"
   by (meson assms holomorphic_deriv holomorphic_on_imp_continuous_on holomorphic_on_imp_differentiable_at
       holomorphic_on_subset subsetD valid_path_compose)
 
+lemma valid_path_compose_analytic:
+  assumes "valid_path g" and holo:"f analytic_on S" and "path_image g \<subseteq> S"
+  shows "valid_path (f \<circ> g)"
+proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
+  fix x assume "x \<in> path_image g"
+  then show "f field_differentiable at x"
+    using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
+next
+  show "continuous_on (path_image g) (deriv f)"
+    by (intro holomorphic_on_imp_continuous_on analytic_imp_holomorphic analytic_intros
+              analytic_on_subset[OF holo] assms)
+qed
+
+lemma analytic_on_deriv [analytic_intros]:
+  assumes "f analytic_on g ` A"
+  assumes "g analytic_on A"
+  shows   "(\<lambda>x. deriv f (g x)) analytic_on A"
+proof -
+  have "(deriv f \<circ> g) analytic_on A"
+    by (rule analytic_on_compose_gen[OF assms(2) analytic_deriv[OF assms(1)]]) auto
+  thus ?thesis
+    by (simp add: o_def)
+qed
+    
+lemma contour_integral_comp_analyticW:
+  assumes "f analytic_on s" "valid_path \<gamma>" "path_image \<gamma> \<subseteq> s"
+  shows "contour_integral (f \<circ> \<gamma>) g = contour_integral \<gamma> (\<lambda>w. deriv f w * g (f w))"
+proof -
+  obtain spikes where "finite spikes" and \<gamma>_diff: "\<gamma> C1_differentiable_on {0..1} - spikes"
+    using \<open>valid_path \<gamma>\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto  
+  show "contour_integral (f \<circ> \<gamma>) g 
+      = contour_integral \<gamma> (\<lambda>w. deriv f w * g (f w))"
+    unfolding contour_integral_integral
+  proof (rule integral_spike[rule_format,OF negligible_finite[OF \<open>finite spikes\<close>]])
+    fix t::real assume t:"t \<in> {0..1} - spikes"
+    then have "\<gamma> differentiable at t" 
+      using \<gamma>_diff unfolding C1_differentiable_on_eq by auto
+    moreover have "f field_differentiable at (\<gamma> t)" 
+    proof -
+      have "\<gamma> t \<in> s" using t assms unfolding path_image_def by auto 
+      thus ?thesis 
+        using \<open>f analytic_on s\<close>  analytic_on_imp_differentiable_at by blast
+    qed
+    ultimately show "deriv f (\<gamma> t) * g (f (\<gamma> t)) * vector_derivative \<gamma> (at t) =
+         g ((f \<circ> \<gamma>) t) * vector_derivative (f \<circ> \<gamma>) (at t)"
+      by (subst vector_derivative_chain_at_general) (simp_all add:field_simps)
+  qed
+qed
+
 subsection\<open>Morera's theorem\<close>
 
 lemma Morera_local_triangle_ball:
diff -r 5a0d1075911c -r a1de627d417a src/HOL/Complex_Analysis/Meromorphic.thy
--- a/src/HOL/Complex_Analysis/Meromorphic.thy	Mon Apr 07 12:36:56 2025 +0200
+++ b/src/HOL/Complex_Analysis/Meromorphic.thy	Tue Apr 08 19:06:00 2025 +0100
@@ -618,6 +618,11 @@
     by (auto simp: isCont_def)
 qed
   
+lemma analytic_on_imp_nicely_meromorphic_on:
+  "f analytic_on A \<Longrightarrow> f nicely_meromorphic_on A"
+  by (meson analytic_at_imp_isCont analytic_on_analytic_at
+            analytic_on_imp_meromorphic_on isContD nicely_meromorphic_on_def)
+
 lemma remove_sings_meromorphic [meromorphic_intros]:
   assumes "f meromorphic_on A"
   shows   "remove_sings f meromorphic_on A"
diff -r 5a0d1075911c -r a1de627d417a src/HOL/Library/Nonpos_Ints.thy
--- a/src/HOL/Library/Nonpos_Ints.thy	Mon Apr 07 12:36:56 2025 +0200
+++ b/src/HOL/Library/Nonpos_Ints.thy	Tue Apr 08 19:06:00 2025 +0100
@@ -305,4 +305,101 @@
 lemma ii_not_nonpos_Reals [iff]: "\<i> \<notin> \<real>\<^sub>\<le>\<^sub>0"
   by (simp add: complex_nonpos_Reals_iff)
 
+lemma plus_one_in_nonpos_Ints_imp: "z + 1 \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> z \<in> \<int>\<^sub>\<le>\<^sub>0"
+  using nonpos_Ints_diff_Nats[of "z+1" "1"] by simp_all
+
+lemma of_int_in_nonpos_Ints_iff:
+  "(of_int n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n \<le> 0"
+  by (auto simp: nonpos_Ints_def)
+
+lemma one_plus_of_int_in_nonpos_Ints_iff:
+  "(1 + of_int n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n \<le> -1"
+proof -
+  have "1 + of_int n = (of_int (n + 1) :: 'a)" by simp
+  also have "\<dots> \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n + 1 \<le> 0" by (subst of_int_in_nonpos_Ints_iff) simp_all
+  also have "\<dots> \<longleftrightarrow> n \<le> -1" by presburger
+  finally show ?thesis .
+qed
+
+lemma one_minus_of_nat_in_nonpos_Ints_iff:
+  "(1 - of_nat n :: 'a :: ring_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n > 0"
+proof -
+  have "(1 - of_nat n :: 'a) = of_int (1 - int n)" by simp
+  also have "\<dots> \<in> \<int>\<^sub>\<le>\<^sub>0 \<longleftrightarrow> n > 0" by (subst of_int_in_nonpos_Ints_iff) presburger
+  finally show ?thesis .
+qed
+
+lemma fraction_not_in_ints:
+  assumes "\<not>(n dvd m)" "n \<noteq> 0"
+  shows   "of_int m / of_int n \<notin> (\<int> :: 'a :: {division_ring,ring_char_0} set)"
+proof
+  assume "of_int m / (of_int n :: 'a) \<in> \<int>"
+  then obtain k where "of_int m / of_int n = (of_int k :: 'a)" by (elim Ints_cases)
+  with assms have "of_int m = (of_int (k * n) :: 'a)" by (auto simp add: field_split_simps)
+  hence "m = k * n" by (subst (asm) of_int_eq_iff)
+  hence "n dvd m" by simp
+  with assms(1) show False by contradiction
+qed
+
+lemma fraction_not_in_nats:
+  assumes "\<not>n dvd m" "n \<noteq> 0"
+  shows   "of_int m / of_int n \<notin> (\<nat> :: 'a :: {division_ring,ring_char_0} set)"
+proof
+  assume "of_int m / of_int n \<in> (\<nat> :: 'a set)"
+  also note Nats_subset_Ints
+  finally have "of_int m / of_int n \<in> (\<int> :: 'a set)" .
+  moreover have "of_int m / of_int n \<notin> (\<int> :: 'a set)"
+    using assms by (intro fraction_not_in_ints)
+  ultimately show False by contradiction
+qed
+
+lemma not_in_Ints_imp_not_in_nonpos_Ints: "z \<notin> \<int> \<Longrightarrow> z \<notin> \<int>\<^sub>\<le>\<^sub>0"
+  by (auto simp: Ints_def nonpos_Ints_def)
+
+lemma double_in_nonpos_Ints_imp:
+  assumes "2 * (z :: 'a :: field_char_0) \<in> \<int>\<^sub>\<le>\<^sub>0"
+  shows   "z \<in> \<int>\<^sub>\<le>\<^sub>0 \<or> z + 1/2 \<in> \<int>\<^sub>\<le>\<^sub>0"
+proof-
+  from assms obtain k where k: "2 * z = - of_nat k" by (elim nonpos_Ints_cases')
+  thus ?thesis by (cases "even k") (auto elim!: evenE oddE simp: field_simps)
+qed
+
+lemma fraction_numeral_Ints_iff [simp]:
+  "numeral a / numeral b \<in> (\<int> :: 'a :: {division_ring, ring_char_0} set)
+   \<longleftrightarrow> (numeral b :: int) dvd numeral a"  (is "?L=?R")
+proof
+  show "?L \<Longrightarrow> ?R"
+    by (metis fraction_not_in_ints of_int_numeral zero_neq_numeral)
+  assume ?R
+  then obtain k::int where "numeral a = numeral b * (of_int k :: 'a)"
+    unfolding dvd_def by (metis of_int_mult of_int_numeral)
+  then show ?L
+    by (metis Ints_of_int divide_eq_eq mult.commute of_int_mult of_int_numeral)
+qed
+
+lemma fraction_numeral_Ints_iff1 [simp]:
+  "1 / numeral b \<in> (\<int> :: 'a :: {division_ring, ring_char_0} set)
+   \<longleftrightarrow> b = Num.One"  (is "?L=?R")
+  using fraction_numeral_Ints_iff [of Num.One, where 'a='a] by simp
+
+lemma fraction_numeral_Nats_iff [simp]:
+  "numeral a / numeral b \<in> (\<nat> :: 'a :: {division_ring, ring_char_0} set)
+   \<longleftrightarrow> (numeral b :: int) dvd numeral a"  (is "?L=?R")
+proof
+  show "?L \<Longrightarrow> ?R"
+    using Nats_subset_Ints fraction_numeral_Ints_iff by blast
+  assume ?R
+  then obtain k::nat where "numeral a = numeral b * (of_nat k :: 'a)"
+    unfolding dvd_def
+    by (metis dvd_def int_dvd_int_iff of_nat_mult of_nat_numeral)
+  then show ?L
+    by (metis mult_of_nat_commute nonzero_divide_eq_eq of_nat_in_Nats
+        zero_neq_numeral)
+qed
+
+lemma fraction_numeral_Nats_iff1 [simp]:
+  "1 / numeral b \<in> (\<nat> :: 'a :: {division_ring, ring_char_0} set)
+   \<longleftrightarrow> b = Num.One"  (is "?L=?R")
+  using fraction_numeral_Nats_iff [of Num.One, where 'a='a] by simp
+
 end