# HG changeset patch
# User Manuel Eberl <eberlm@in.tum.de>
# Date 1513069274 -3600
# Node ID 88d1c9d86f4853bfb2cabe6da3553ea720572da6
# Parent  a77d54ef718b905696f291dd36a7e902903bae17
Moved analysis material from AFP

diff -r a77d54ef718b -r 88d1c9d86f48 src/HOL/Analysis/Complex_Analysis_Basics.thy
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy	Mon Dec 11 17:28:32 2017 +0100
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy	Tue Dec 12 10:01:14 2017 +0100
@@ -379,6 +379,28 @@
   "f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
   by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)
 
+lemma holomorphic_on_Un [holomorphic_intros]:
+  assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
+  shows   "f holomorphic_on (A \<union> B)"
+  using assms by (auto simp: holomorphic_on_def  at_within_open[of _ A] 
+                             at_within_open[of _ B]  at_within_open[of _ "A \<union> B"] open_Un)
+
+lemma holomorphic_on_If_Un [holomorphic_intros]:
+  assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
+  assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
+  shows   "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
+proof (intro holomorphic_on_Un)
+  note \<open>f holomorphic_on A\<close>
+  also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
+    by (intro holomorphic_cong) auto
+  finally show \<dots> .
+next
+  note \<open>g holomorphic_on B\<close>
+  also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
+    using assms by (intro holomorphic_cong) auto
+  finally show \<dots> .
+qed (insert assms, auto)
+
 lemma DERIV_deriv_iff_field_differentiable:
   "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
   unfolding field_differentiable_def by (metis DERIV_imp_deriv)
diff -r a77d54ef718b -r 88d1c9d86f48 src/HOL/Analysis/Infinite_Set_Sum.thy
--- a/src/HOL/Analysis/Infinite_Set_Sum.thy	Mon Dec 11 17:28:32 2017 +0100
+++ b/src/HOL/Analysis/Infinite_Set_Sum.thy	Tue Dec 12 10:01:14 2017 +0100
@@ -256,6 +256,27 @@
     show "f abs_summable_on insert x A" by simp
 qed
 
+lemma abs_summable_sum: 
+  assumes "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B"
+  shows   "(\<lambda>y. \<Sum>x\<in>A. f x y) abs_summable_on B"
+  using assms unfolding abs_summable_on_def by (intro Bochner_Integration.integrable_sum)
+
+lemma abs_summable_Re: "f abs_summable_on A \<Longrightarrow> (\<lambda>x. Re (f x)) abs_summable_on A"
+  by (simp add: abs_summable_on_def)
+
+lemma abs_summable_Im: "f abs_summable_on A \<Longrightarrow> (\<lambda>x. Im (f x)) abs_summable_on A"
+  by (simp add: abs_summable_on_def)
+
+lemma abs_summable_on_finite_diff:
+  assumes "f abs_summable_on A" "A \<subseteq> B" "finite (B - A)"
+  shows   "f abs_summable_on B"
+proof -
+  have "f abs_summable_on (A \<union> (B - A))"
+    by (intro abs_summable_on_union assms abs_summable_on_finite)
+  also from assms have "A \<union> (B - A) = B" by blast
+  finally show ?thesis .
+qed
+
 lemma abs_summable_on_reindex_bij_betw:
   assumes "bij_betw g A B"
   shows   "(\<lambda>x. f (g x)) abs_summable_on A \<longleftrightarrow> f abs_summable_on B"
@@ -407,6 +428,27 @@
 lemma infsetsum_all_0: "(\<And>x. x \<in> A \<Longrightarrow> f x = 0) \<Longrightarrow> infsetsum f A = 0"
   by simp
 
+lemma infsetsum_nonneg: "(\<And>x. x \<in> A \<Longrightarrow> f x \<ge> (0::real)) \<Longrightarrow> infsetsum f A \<ge> 0"
+  unfolding infsetsum_def by (rule Bochner_Integration.integral_nonneg) auto
+
+lemma sum_infsetsum:
+  assumes "\<And>x. x \<in> A \<Longrightarrow> f x abs_summable_on B"
+  shows   "(\<Sum>x\<in>A. \<Sum>\<^sub>ay\<in>B. f x y) = (\<Sum>\<^sub>ay\<in>B. \<Sum>x\<in>A. f x y)"
+  using assms by (simp add: infsetsum_def abs_summable_on_def Bochner_Integration.integral_sum)
+
+lemma Re_infsetsum: "f abs_summable_on A \<Longrightarrow> Re (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Re (f x))"
+  by (simp add: infsetsum_def abs_summable_on_def)
+
+lemma Im_infsetsum: "f abs_summable_on A \<Longrightarrow> Im (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Im (f x))"
+  by (simp add: infsetsum_def abs_summable_on_def)
+
+lemma infsetsum_of_real: 
+  shows "infsetsum (\<lambda>x. of_real (f x) 
+           :: 'a :: {real_normed_algebra_1,banach,second_countable_topology,real_inner}) A = 
+             of_real (infsetsum f A)"
+  unfolding infsetsum_def
+  by (rule integral_bounded_linear'[OF bounded_linear_of_real bounded_linear_inner_left[of 1]]) auto
+
 lemma infsetsum_finite [simp]: "finite A \<Longrightarrow> infsetsum f A = (\<Sum>x\<in>A. f x)"
   by (simp add: infsetsum_def lebesgue_integral_count_space_finite)
 
diff -r a77d54ef718b -r 88d1c9d86f48 src/HOL/Computational_Algebra/Euclidean_Algorithm.thy
--- a/src/HOL/Computational_Algebra/Euclidean_Algorithm.thy	Mon Dec 11 17:28:32 2017 +0100
+++ b/src/HOL/Computational_Algebra/Euclidean_Algorithm.thy	Tue Dec 12 10:01:14 2017 +0100
@@ -602,6 +602,9 @@
     by (simp add: fun_eq_iff Euclidean_Algorithm.Gcd_def semiring_Gcd_class.Gcd_Lcm)
 qed
 
+lemma prime_factorization_Suc_0 [simp]: "prime_factorization (Suc 0) = {#}"
+  unfolding One_nat_def [symmetric] using prime_factorization_1 .
+
 instance int :: normalization_euclidean_semiring ..
 
 instance int :: euclidean_ring_gcd
diff -r a77d54ef718b -r 88d1c9d86f48 src/HOL/Series.thy
--- a/src/HOL/Series.thy	Mon Dec 11 17:28:32 2017 +0100
+++ b/src/HOL/Series.thy	Tue Dec 12 10:01:14 2017 +0100
@@ -1223,4 +1223,32 @@
   ultimately show ?thesis by simp
 qed
 
+lemma summable_bounded_partials:
+  fixes f :: "nat \<Rightarrow> 'a :: {real_normed_vector,complete_space}"
+  assumes bound: "eventually (\<lambda>x0. \<forall>a\<ge>x0. \<forall>b>a. norm (sum f {a<..b}) \<le> g a) sequentially"
+  assumes g: "g \<longlonglongrightarrow> 0"
+  shows   "summable f" unfolding summable_iff_convergent'
+proof (intro Cauchy_convergent CauchyI', goal_cases)
+  case (1 \<epsilon>)
+  with g have "eventually (\<lambda>x. \<bar>g x\<bar> < \<epsilon>) sequentially"
+    by (auto simp: tendsto_iff)
+  from eventually_conj[OF this bound] obtain x0 where x0:
+    "\<And>x. x \<ge> x0 \<Longrightarrow> \<bar>g x\<bar> < \<epsilon>" "\<And>a b. x0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> norm (sum f {a<..b}) \<le> g a" 
+    unfolding eventually_at_top_linorder by auto
+
+  show ?case
+  proof (intro exI[of _ x0] allI impI)
+    fix m n assume mn: "x0 \<le> m" "m < n"
+    have "dist (sum f {..m}) (sum f {..n}) = norm (sum f {..n} - sum f {..m})"
+      by (simp add: dist_norm norm_minus_commute)
+    also have "sum f {..n} - sum f {..m} = sum f ({..n} - {..m})"
+      using mn by (intro Groups_Big.sum_diff [symmetric]) auto
+    also have "{..n} - {..m} = {m<..n}" using mn by auto
+    also have "norm (sum f {m<..n}) \<le> g m" using mn by (intro x0) auto
+    also have "\<dots> \<le> \<bar>g m\<bar>" by simp
+    also have "\<dots> < \<epsilon>" using mn by (intro x0) auto
+    finally show "dist (sum f {..m}) (sum f {..n}) < \<epsilon>" .
+  qed
+qed
+
 end