--- a/src/HOL/Analysis/Infinite_Products.thy Fri Apr 27 12:43:05 2018 +0100
+++ b/src/HOL/Analysis/Infinite_Products.thy Wed May 02 12:47:56 2018 +0100
@@ -1,6 +1,5 @@
-(*
- File: HOL/Analysis/Infinite_Product.thy
- Author: Manuel Eberl
+(*File: HOL/Analysis/Infinite_Product.thy
+ Author: Manuel Eberl & LC Paulson
Basic results about convergence and absolute convergence of infinite products
and their connection to summability.
@@ -9,7 +8,7 @@
theory Infinite_Products
imports Complex_Main
begin
-
+
lemma sum_le_prod:
fixes f :: "'a \<Rightarrow> 'b :: linordered_semidom"
assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<ge> 0"
@@ -51,8 +50,27 @@
by (rule tendsto_eq_intros refl | simp)+
qed auto
+definition gen_has_prod :: "[nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}, nat, 'a] \<Rightarrow> bool"
+ where "gen_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
+
+text\<open>The nonzero and zero cases, as in \emph{Complex Analysis} by Joseph Bak and Donald J.Newman, page 241\<close>
+definition has_prod :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a \<Rightarrow> bool" (infixr "has'_prod" 80)
+ where "f has_prod p \<equiv> gen_has_prod f 0 p \<or> (\<exists>i q. p = 0 \<and> f i = 0 \<and> gen_has_prod f (Suc i) q)"
+
definition convergent_prod :: "(nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}) \<Rightarrow> bool" where
- "convergent_prod f \<longleftrightarrow> (\<exists>M L. (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
+ "convergent_prod f \<equiv> \<exists>M p. gen_has_prod f M p"
+
+definition prodinf :: "(nat \<Rightarrow> 'a::{t2_space, comm_semiring_1}) \<Rightarrow> 'a"
+ (binder "\<Prod>" 10)
+ where "prodinf f = (THE p. f has_prod p)"
+
+lemmas prod_defs = gen_has_prod_def has_prod_def convergent_prod_def prodinf_def
+
+lemma has_prod_subst[trans]: "f = g \<Longrightarrow> g has_prod z \<Longrightarrow> f has_prod z"
+ by simp
+
+lemma has_prod_cong: "(\<And>n. f n = g n) \<Longrightarrow> f has_prod c \<longleftrightarrow> g has_prod c"
+ by presburger
lemma convergent_prod_altdef:
fixes f :: "nat \<Rightarrow> 'a :: {t2_space,comm_semiring_1}"
@@ -60,7 +78,7 @@
proof
assume "convergent_prod f"
then obtain M L where *: "(\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L" "L \<noteq> 0"
- by (auto simp: convergent_prod_def)
+ by (auto simp: prod_defs)
have "f i \<noteq> 0" if "i \<ge> M" for i
proof
assume "f i = 0"
@@ -79,7 +97,7 @@
qed
with * show "(\<exists>M L. (\<forall>n\<ge>M. f n \<noteq> 0) \<and> (\<lambda>n. \<Prod>i\<le>n. f (i+M)) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
by blast
-qed (auto simp: convergent_prod_def)
+qed (auto simp: prod_defs)
definition abs_convergent_prod :: "(nat \<Rightarrow> _) \<Rightarrow> bool" where
"abs_convergent_prod f \<longleftrightarrow> convergent_prod (\<lambda>i. 1 + norm (f i - 1))"
@@ -101,12 +119,12 @@
qed
qed (use L in simp_all)
hence "L \<noteq> 0" by auto
- with L show ?thesis unfolding abs_convergent_prod_def convergent_prod_def
+ with L show ?thesis unfolding abs_convergent_prod_def prod_defs
by (intro exI[of _ "0::nat"] exI[of _ L]) auto
qed
lemma
- fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,idom}"
+ fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
assumes "convergent_prod f"
shows convergent_prod_imp_convergent: "convergent (\<lambda>n. \<Prod>i\<le>n. f i)"
and convergent_prod_to_zero_iff: "(\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> 0 \<longleftrightarrow> (\<exists>i. f i = 0)"
@@ -141,8 +159,30 @@
qed
qed
+lemma convergent_prod_iff_nz_lim:
+ fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
+ assumes "\<And>i. f i \<noteq> 0"
+ shows "convergent_prod f \<longleftrightarrow> (\<exists>L. (\<lambda>n. \<Prod>i\<le>n. f i) \<longlonglongrightarrow> L \<and> L \<noteq> 0)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?lhs then show ?rhs
+ using assms convergentD convergent_prod_imp_convergent convergent_prod_to_zero_iff by blast
+next
+ assume ?rhs then show ?lhs
+ unfolding prod_defs
+ by (rule_tac x="0" in exI) (auto simp: )
+qed
+
+lemma convergent_prod_iff_convergent:
+ fixes f :: "nat \<Rightarrow> 'a :: {topological_semigroup_mult,t2_space,idom}"
+ assumes "\<And>i. f i \<noteq> 0"
+ shows "convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. f i) \<and> lim (\<lambda>n. \<Prod>i\<le>n. f i) \<noteq> 0"
+ by (force simp add: convergent_prod_iff_nz_lim assms convergent_def limI)
+
+
lemma abs_convergent_prod_altdef:
- "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
+ fixes f :: "nat \<Rightarrow> 'a :: {one,real_normed_vector}"
+ shows "abs_convergent_prod f \<longleftrightarrow> convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
proof
assume "abs_convergent_prod f"
thus "convergent (\<lambda>n. \<Prod>i\<le>n. 1 + norm (f i - 1))"
@@ -180,7 +220,7 @@
also have "norm (1::'a) = 1" by simp
also note insert.IH
also have "(\<Prod>n\<in>A. 1 + norm (f n)) - 1 + norm (f x) * (\<Prod>x\<in>A. 1 + norm (f x)) =
- (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
+ (\<Prod>n\<in>insert x A. 1 + norm (f n)) - 1"
using insert.hyps by (simp add: algebra_simps)
finally show ?case by - (simp_all add: mult_left_mono)
qed simp_all
@@ -297,8 +337,9 @@
shows "convergent_prod f"
proof -
from assms obtain M L where "(\<lambda>n. \<Prod>k\<le>n. f (k + (M + m))) \<longlonglongrightarrow> L" "L \<noteq> 0"
- by (auto simp: convergent_prod_def add.assoc)
- thus "convergent_prod f" unfolding convergent_prod_def by blast
+ by (auto simp: prod_defs add.assoc)
+ thus "convergent_prod f"
+ unfolding prod_defs by blast
qed
lemma abs_convergent_prod_offset:
@@ -334,7 +375,8 @@
by (intro tendsto_divide tendsto_const) auto
hence "(\<lambda>n. \<Prod>k\<le>n. f (k + M + m)) \<longlonglongrightarrow> L / C" by simp
moreover from \<open>L \<noteq> 0\<close> have "L / C \<noteq> 0" by simp
- ultimately show ?thesis unfolding convergent_prod_def by blast
+ ultimately show ?thesis
+ unfolding prod_defs by blast
qed
lemma abs_convergent_prod_ignore_initial_segment:
@@ -343,11 +385,6 @@
using assms unfolding abs_convergent_prod_def
by (rule convergent_prod_ignore_initial_segment)
-lemma summable_LIMSEQ':
- assumes "summable (f::nat\<Rightarrow>'a::{t2_space,comm_monoid_add})"
- shows "(\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> suminf f"
- using assms sums_def_le by blast
-
lemma abs_convergent_prod_imp_convergent_prod:
fixes f :: "nat \<Rightarrow> 'a :: {real_normed_div_algebra,complete_space,comm_ring_1}"
assumes "abs_convergent_prod f"
@@ -473,7 +510,98 @@
qed simp_all
thus False by simp
qed
- with L show ?thesis by (auto simp: convergent_prod_def)
+ with L show ?thesis by (auto simp: prod_defs)
+qed
+
+lemma convergent_prod_offset_0:
+ fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
+ assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
+ shows "\<exists>p. gen_has_prod f 0 p"
+ using assms
+ unfolding convergent_prod_def
+proof (clarsimp simp: prod_defs)
+ fix M p
+ assume "(\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p" "p \<noteq> 0"
+ then have "(\<lambda>n. prod f {..<M} * (\<Prod>i\<le>n. f (i + M))) \<longlonglongrightarrow> prod f {..<M} * p"
+ by (metis tendsto_mult_left)
+ moreover have "prod f {..<M} * (\<Prod>i\<le>n. f (i + M)) = prod f {..n+M}" for n
+ proof -
+ have "{..n+M} = {..<M} \<union> {M..n+M}"
+ by auto
+ then have "prod f {..n+M} = prod f {..<M} * prod f {M..n+M}"
+ by simp (subst prod.union_disjoint; force)
+ also have "... = prod f {..<M} * (\<Prod>i\<le>n. f (i + M))"
+ by (metis (mono_tags, lifting) add.left_neutral atMost_atLeast0 prod_shift_bounds_cl_nat_ivl)
+ finally show ?thesis by metis
+ qed
+ ultimately have "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prod f {..<M} * p"
+ by (auto intro: LIMSEQ_offset [where k=M])
+ then show "\<exists>p. (\<lambda>n. prod f {..n}) \<longlonglongrightarrow> p \<and> p \<noteq> 0"
+ using \<open>p \<noteq> 0\<close> assms
+ by (rule_tac x="prod f {..<M} * p" in exI) auto
+qed
+
+lemma prodinf_eq_lim:
+ fixes f :: "nat \<Rightarrow> 'a :: {idom,topological_semigroup_mult,t2_space}"
+ assumes "convergent_prod f" "\<And>i. f i \<noteq> 0"
+ shows "prodinf f = lim (\<lambda>n. \<Prod>i\<le>n. f i)"
+ using assms convergent_prod_offset_0 [OF assms]
+ by (simp add: prod_defs lim_def) (metis (no_types) assms(1) convergent_prod_to_zero_iff)
+
+lemma has_prod_one[simp, intro]: "(\<lambda>n. 1) has_prod 1"
+ unfolding prod_defs by auto
+
+lemma convergent_prod_one[simp, intro]: "convergent_prod (\<lambda>n. 1)"
+ unfolding prod_defs by auto
+
+lemma prodinf_cong: "(\<And>n. f n = g n) \<Longrightarrow> prodinf f = prodinf g"
+ by presburger
+
+lemma convergent_prod_cong:
+ fixes f g :: "nat \<Rightarrow> 'a::{field,topological_semigroup_mult,t2_space}"
+ assumes ev: "eventually (\<lambda>x. f x = g x) sequentially" and f: "\<And>i. f i \<noteq> 0" and g: "\<And>i. g i \<noteq> 0"
+ shows "convergent_prod f = convergent_prod g"
+proof -
+ from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
+ by (auto simp: eventually_at_top_linorder)
+ define C where "C = (\<Prod>k<N. f k / g k)"
+ with g have "C \<noteq> 0"
+ by (simp add: f)
+ have *: "eventually (\<lambda>n. prod f {..n} = C * prod g {..n}) sequentially"
+ using eventually_ge_at_top[of N]
+ proof eventually_elim
+ case (elim n)
+ then have "{..n} = {..<N} \<union> {N..n}"
+ by auto
+ also have "prod f ... = prod f {..<N} * prod f {N..n}"
+ by (intro prod.union_disjoint) auto
+ also from N have "prod f {N..n} = prod g {N..n}"
+ by (intro prod.cong) simp_all
+ also have "prod f {..<N} * prod g {N..n} = C * (prod g {..<N} * prod g {N..n})"
+ unfolding C_def by (simp add: g prod_dividef)
+ also have "prod g {..<N} * prod g {N..n} = prod g ({..<N} \<union> {N..n})"
+ by (intro prod.union_disjoint [symmetric]) auto
+ also from elim have "{..<N} \<union> {N..n} = {..n}"
+ by auto
+ finally show "prod f {..n} = C * prod g {..n}" .
+ qed
+ then have cong: "convergent (\<lambda>n. prod f {..n}) = convergent (\<lambda>n. C * prod g {..n})"
+ by (rule convergent_cong)
+ show ?thesis
+ proof
+ assume cf: "convergent_prod f"
+ then have "\<not> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> 0"
+ using tendsto_mult_left * convergent_prod_to_zero_iff f filterlim_cong by fastforce
+ then show "convergent_prod g"
+ by (metis convergent_mult_const_iff \<open>C \<noteq> 0\<close> cong cf convergent_LIMSEQ_iff convergent_prod_iff_convergent convergent_prod_imp_convergent g)
+ next
+ assume cg: "convergent_prod g"
+ have "\<exists>a. C * a \<noteq> 0 \<and> (\<lambda>n. prod g {..n}) \<longlonglongrightarrow> a"
+ by (metis (no_types) \<open>C \<noteq> 0\<close> cg convergent_prod_iff_nz_lim divide_eq_0_iff g nonzero_mult_div_cancel_right)
+ then show "convergent_prod f"
+ using "*" tendsto_mult_left filterlim_cong
+ by (fastforce simp add: convergent_prod_iff_nz_lim f)
+ qed
qed
end