author huffman Sun May 31 21:59:33 2009 -0700 (2009-05-31) changeset 31349 2261c8781f73 parent 31348 738eb25e1dd8 child 31350 f20a61cec3d4
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
 src/HOL/Lim.thy file | annotate | diff | revisions src/HOL/Limits.thy file | annotate | diff | revisions src/HOL/SEQ.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Lim.thy	Sun May 31 11:27:19 2009 -0700
1.2 +++ b/src/HOL/Lim.thy	Sun May 31 21:59:33 2009 -0700
1.3 @@ -13,6 +13,10 @@
1.4  text{*Standard Definitions*}
1.5
1.6  definition
1.7 +  at :: "'a::metric_space \<Rightarrow> 'a filter" where
1.8 +  "at a = Abs_filter (\<lambda>P. \<exists>r>0. \<forall>x. x \<noteq> a \<and> dist x a < r \<longrightarrow> P x)"
1.9 +
1.10 +definition
1.11    LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"
1.12          ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
1.13    [code del]: "f -- a --> L =
1.14 @@ -27,6 +31,20 @@
1.15    isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
1.16    [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
1.17
1.18 +subsection {* Neighborhood Filter *}
1.19 +
1.20 +lemma eventually_at:
1.21 +  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
1.22 +unfolding at_def
1.23 +apply (rule eventually_Abs_filter)
1.24 +apply (rule_tac x=1 in exI, simp)
1.25 +apply (clarify, rule_tac x=r in exI, simp)
1.26 +apply (clarify, rename_tac r s)
1.27 +apply (rule_tac x="min r s" in exI, simp)
1.28 +done
1.29 +
1.30 +lemma LIM_conv_tendsto: "(f -- a --> L) \<longleftrightarrow> tendsto f L (at a)"
1.31 +unfolding LIM_def tendsto_def eventually_at ..
1.32
1.33  subsection {* Limits of Functions *}
1.34
1.35 @@ -86,33 +104,7 @@
1.36    fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.37    assumes f: "f -- a --> L" and g: "g -- a --> M"
1.38    shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
1.39 -proof (rule metric_LIM_I)
1.40 -  fix r :: real
1.41 -  assume r: "0 < r"
1.42 -  from metric_LIM_D [OF f half_gt_zero [OF r]]
1.43 -  obtain fs
1.44 -    where fs:    "0 < fs"
1.45 -      and fs_lt: "\<forall>x. x \<noteq> a \<and> dist x a < fs \<longrightarrow> dist (f x) L < r/2"
1.46 -  by blast
1.47 -  from metric_LIM_D [OF g half_gt_zero [OF r]]
1.48 -  obtain gs
1.49 -    where gs:    "0 < gs"
1.50 -      and gs_lt: "\<forall>x. x \<noteq> a \<and> dist x a < gs \<longrightarrow> dist (g x) M < r/2"
1.51 -  by blast
1.52 -  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x + g x) (L + M) < r"
1.53 -  proof (intro exI conjI strip)
1.54 -    show "0 < min fs gs"  by (simp add: fs gs)
1.55 -    fix x :: 'a
1.56 -    assume "x \<noteq> a \<and> dist x a < min fs gs"
1.57 -    hence "x \<noteq> a \<and> dist x a < fs \<and> dist x a < gs" by simp
1.58 -    with fs_lt gs_lt
1.59 -    have "dist (f x) L < r/2" and "dist (g x) M < r/2" by blast+
1.60 -    hence "dist (f x) L + dist (g x) M < r" by arith
1.61 -    thus "dist (f x + g x) (L + M) < r"
1.62 -      unfolding dist_norm
1.63 -      by (blast intro: norm_diff_triangle_ineq order_le_less_trans)
1.64 -  qed
1.65 -qed
1.66 +using assms unfolding LIM_conv_tendsto by (rule tendsto_add)
1.67
1.69    fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.70 @@ -127,7 +119,7 @@
1.71  lemma LIM_minus:
1.72    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.73    shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
1.74 -by (simp only: LIM_def dist_norm minus_diff_minus norm_minus_cancel)
1.75 +unfolding LIM_conv_tendsto by (rule tendsto_minus)
1.76
1.77  (* TODO: delete *)
1.79 @@ -138,7 +130,7 @@
1.80  lemma LIM_diff:
1.81    fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.82    shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
1.83 -by (simp only: diff_minus LIM_add LIM_minus)
1.84 +unfolding LIM_conv_tendsto by (rule tendsto_diff)
1.85
1.86  lemma LIM_zero:
1.87    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.88 @@ -178,7 +170,7 @@
1.89  lemma LIM_norm:
1.90    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.91    shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
1.92 -by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)
1.93 +unfolding LIM_conv_tendsto by (rule tendsto_norm)
1.94
1.95  lemma LIM_norm_zero:
1.96    fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
1.97 @@ -369,26 +361,12 @@
1.98
1.99  text {* Bounded Linear Operators *}
1.100
1.101 -lemma (in bounded_linear) cont: "f -- a --> f a"
1.102 -proof (rule LIM_I)
1.103 -  fix r::real assume r: "0 < r"
1.104 -  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
1.105 -    using pos_bounded by fast
1.106 -  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
1.107 -  proof (rule exI, safe)
1.108 -    from r K show "0 < r / K" by (rule divide_pos_pos)
1.109 -  next
1.110 -    fix x assume x: "norm (x - a) < r / K"
1.111 -    have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
1.112 -    also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
1.113 -    also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
1.114 -    finally show "norm (f x - f a) < r" .
1.115 -  qed
1.116 -qed
1.117 -
1.118  lemma (in bounded_linear) LIM:
1.119    "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
1.120 -by (rule LIM_compose [OF cont])
1.121 +unfolding LIM_conv_tendsto by (rule tendsto)
1.122 +
1.123 +lemma (in bounded_linear) cont: "f -- a --> f a"
1.124 +by (rule LIM [OF LIM_ident])
1.125
1.126  lemma (in bounded_linear) LIM_zero:
1.127    "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
1.128 @@ -396,40 +374,16 @@
1.129
1.130  text {* Bounded Bilinear Operators *}
1.131
1.132 +lemma (in bounded_bilinear) LIM:
1.133 +  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
1.134 +unfolding LIM_conv_tendsto by (rule tendsto)
1.135 +
1.136  lemma (in bounded_bilinear) LIM_prod_zero:
1.137    fixes a :: "'d::metric_space"
1.138    assumes f: "f -- a --> 0"
1.139    assumes g: "g -- a --> 0"
1.140    shows "(\<lambda>x. f x ** g x) -- a --> 0"
1.141 -proof (rule metric_LIM_I, unfold dist_norm)
1.142 -  fix r::real assume r: "0 < r"
1.143 -  obtain K where K: "0 < K"
1.144 -    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
1.145 -    using pos_bounded by fast
1.146 -  from K have K': "0 < inverse K"
1.147 -    by (rule positive_imp_inverse_positive)
1.148 -  obtain s where s: "0 < s"
1.149 -    and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
1.150 -    using metric_LIM_D [OF f r, unfolded dist_norm] by auto
1.151 -  obtain t where t: "0 < t"
1.152 -    and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
1.153 -    using metric_LIM_D [OF g K', unfolded dist_norm] by auto
1.154 -  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> norm (f x ** g x - 0) < r"
1.155 -  proof (rule exI, safe)
1.156 -    from s t show "0 < min s t" by simp
1.157 -  next
1.158 -    fix x assume x: "x \<noteq> a"
1.159 -    assume "dist x a < min s t"
1.160 -    hence xs: "dist x a < s" and xt: "dist x a < t" by simp_all
1.161 -    from x xs have 1: "norm (f x) < r" by (rule norm_f)
1.162 -    from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
1.163 -    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
1.164 -    also from 1 2 K have "\<dots> < r * inverse K * K"
1.165 -      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
1.166 -    also from K have "r * inverse K * K = r" by simp
1.167 -    finally show "norm (f x ** g x - 0) < r" by simp
1.168 -  qed
1.169 -qed
1.170 +using LIM [OF f g] by (simp add: zero_left)
1.171
1.172  lemma (in bounded_bilinear) LIM_left_zero:
1.173    "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
1.174 @@ -439,19 +393,6 @@
1.175    "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
1.176  by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
1.177
1.178 -lemma (in bounded_bilinear) LIM:
1.179 -  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
1.180 -apply (drule LIM_zero)
1.181 -apply (drule LIM_zero)
1.182 -apply (rule LIM_zero_cancel)
1.183 -apply (subst prod_diff_prod)
1.186 -apply (erule (1) LIM_prod_zero)
1.187 -apply (erule LIM_left_zero)
1.188 -apply (erule LIM_right_zero)
1.189 -done
1.190 -
1.191  lemmas LIM_mult = mult.LIM
1.192
1.193  lemmas LIM_mult_zero = mult.LIM_prod_zero
```
```     2.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
2.2 +++ b/src/HOL/Limits.thy	Sun May 31 21:59:33 2009 -0700
2.3 @@ -0,0 +1,296 @@
2.4 +(*  Title       : Limits.thy
2.5 +    Author      : Brian Huffman
2.6 +*)
2.7 +
2.8 +header {* Filters and Limits *}
2.9 +
2.10 +theory Limits
2.11 +imports RealVector RComplete
2.12 +begin
2.13 +
2.14 +subsection {* Filters *}
2.15 +
2.16 +typedef (open) 'a filter =
2.17 +  "{f :: ('a \<Rightarrow> bool) \<Rightarrow> bool. f (\<lambda>x. True)
2.18 +    \<and> (\<forall>P Q. (\<forall>x. P x \<longrightarrow> Q x) \<longrightarrow> f P \<longrightarrow> f Q)
2.19 +    \<and> (\<forall>P Q. f P \<longrightarrow> f Q \<longrightarrow> f (\<lambda>x. P x \<and> Q x))}"
2.20 +proof
2.21 +  show "(\<lambda>P. True) \<in> ?filter" by simp
2.22 +qed
2.23 +
2.24 +definition
2.25 +  eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" where
2.26 +  "eventually P F \<longleftrightarrow> Rep_filter F P"
2.27 +
2.28 +lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
2.29 +unfolding eventually_def using Rep_filter [of F] by blast
2.30 +
2.31 +lemma eventually_mono:
2.32 +  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
2.33 +unfolding eventually_def using Rep_filter [of F] by blast
2.34 +
2.35 +lemma eventually_conj:
2.36 +  "\<lbrakk>eventually (\<lambda>x. P x) F; eventually (\<lambda>x. Q x) F\<rbrakk>
2.37 +    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) F"
2.38 +unfolding eventually_def using Rep_filter [of F] by blast
2.39 +
2.40 +lemma eventually_mp:
2.41 +  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
2.42 +  assumes "eventually (\<lambda>x. P x) F"
2.43 +  shows "eventually (\<lambda>x. Q x) F"
2.44 +proof (rule eventually_mono)
2.45 +  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
2.46 +  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
2.47 +    using assms by (rule eventually_conj)
2.48 +qed
2.49 +
2.50 +lemma eventually_rev_mp:
2.51 +  assumes "eventually (\<lambda>x. P x) F"
2.52 +  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
2.53 +  shows "eventually (\<lambda>x. Q x) F"
2.54 +using assms(2) assms(1) by (rule eventually_mp)
2.55 +
2.56 +lemma eventually_conj_iff:
2.57 +  "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
2.58 +by (auto intro: eventually_conj elim: eventually_rev_mp)
2.59 +
2.60 +lemma eventually_Abs_filter:
2.61 +  assumes "f (\<lambda>x. True)"
2.62 +  assumes "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> f P \<Longrightarrow> f Q"
2.63 +  assumes "\<And>P Q. f P \<Longrightarrow> f Q \<Longrightarrow> f (\<lambda>x. P x \<and> Q x)"
2.64 +  shows "eventually P (Abs_filter f) \<longleftrightarrow> f P"
2.65 +unfolding eventually_def using assms
2.66 +by (subst Abs_filter_inverse, auto)
2.67 +
2.68 +lemma filter_ext:
2.69 +  "(\<And>P. eventually P F \<longleftrightarrow> eventually P F') \<Longrightarrow> F = F'"
2.70 +unfolding eventually_def
2.71 +by (simp add: Rep_filter_inject [THEN iffD1] ext)
2.72 +
2.73 +lemma eventually_elim1:
2.74 +  assumes "eventually (\<lambda>i. P i) F"
2.75 +  assumes "\<And>i. P i \<Longrightarrow> Q i"
2.76 +  shows "eventually (\<lambda>i. Q i) F"
2.77 +using assms by (auto elim!: eventually_rev_mp)
2.78 +
2.79 +lemma eventually_elim2:
2.80 +  assumes "eventually (\<lambda>i. P i) F"
2.81 +  assumes "eventually (\<lambda>i. Q i) F"
2.82 +  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
2.83 +  shows "eventually (\<lambda>i. R i) F"
2.84 +using assms by (auto elim!: eventually_rev_mp)
2.85 +
2.86 +
2.87 +subsection {* Convergence to Zero *}
2.88 +
2.89 +definition
2.90 +  Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" where
2.91 +  "Zfun S F = (\<forall>r>0. eventually (\<lambda>i. norm (S i) < r) F)"
2.92 +
2.93 +lemma ZfunI:
2.94 +  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) F) \<Longrightarrow> Zfun S F"
2.95 +unfolding Zfun_def by simp
2.96 +
2.97 +lemma ZfunD:
2.98 +  "\<lbrakk>Zfun S F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) F"
2.99 +unfolding Zfun_def by simp
2.100 +
2.101 +lemma Zfun_zero: "Zfun (\<lambda>i. 0) F"
2.102 +unfolding Zfun_def by simp
2.103 +
2.104 +lemma Zfun_norm_iff: "Zfun (\<lambda>i. norm (S i)) F = Zfun (\<lambda>i. S i) F"
2.105 +unfolding Zfun_def by simp
2.106 +
2.107 +lemma Zfun_imp_Zfun:
2.108 +  assumes X: "Zfun X F"
2.109 +  assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
2.110 +  shows "Zfun (\<lambda>n. Y n) F"
2.111 +proof (cases)
2.112 +  assume K: "0 < K"
2.113 +  show ?thesis
2.114 +  proof (rule ZfunI)
2.115 +    fix r::real assume "0 < r"
2.116 +    hence "0 < r / K"
2.117 +      using K by (rule divide_pos_pos)
2.118 +    then have "eventually (\<lambda>i. norm (X i) < r / K) F"
2.119 +      using ZfunD [OF X] by fast
2.120 +    then show "eventually (\<lambda>i. norm (Y i) < r) F"
2.121 +    proof (rule eventually_elim1)
2.122 +      fix i assume "norm (X i) < r / K"
2.123 +      hence "norm (X i) * K < r"
2.124 +        by (simp add: pos_less_divide_eq K)
2.125 +      thus "norm (Y i) < r"
2.126 +        by (simp add: order_le_less_trans [OF Y])
2.127 +    qed
2.128 +  qed
2.129 +next
2.130 +  assume "\<not> 0 < K"
2.131 +  hence K: "K \<le> 0" by (simp only: not_less)
2.132 +  {
2.133 +    fix i
2.134 +    have "norm (Y i) \<le> norm (X i) * K" by (rule Y)
2.135 +    also have "\<dots> \<le> norm (X i) * 0"
2.136 +      using K norm_ge_zero by (rule mult_left_mono)
2.137 +    finally have "norm (Y i) = 0" by simp
2.138 +  }
2.139 +  thus ?thesis by (simp add: Zfun_zero)
2.140 +qed
2.141 +
2.142 +lemma Zfun_le: "\<lbrakk>Zfun Y F; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zfun X F"
2.143 +by (erule_tac K="1" in Zfun_imp_Zfun, simp)
2.144 +
2.146 +  assumes X: "Zfun X F" and Y: "Zfun Y F"
2.147 +  shows "Zfun (\<lambda>n. X n + Y n) F"
2.148 +proof (rule ZfunI)
2.149 +  fix r::real assume "0 < r"
2.150 +  hence r: "0 < r / 2" by simp
2.151 +  have "eventually (\<lambda>i. norm (X i) < r/2) F"
2.152 +    using X r by (rule ZfunD)
2.153 +  moreover
2.154 +  have "eventually (\<lambda>i. norm (Y i) < r/2) F"
2.155 +    using Y r by (rule ZfunD)
2.156 +  ultimately
2.157 +  show "eventually (\<lambda>i. norm (X i + Y i) < r) F"
2.158 +  proof (rule eventually_elim2)
2.159 +    fix i
2.160 +    assume *: "norm (X i) < r/2" "norm (Y i) < r/2"
2.161 +    have "norm (X i + Y i) \<le> norm (X i) + norm (Y i)"
2.162 +      by (rule norm_triangle_ineq)
2.163 +    also have "\<dots> < r/2 + r/2"
2.164 +      using * by (rule add_strict_mono)
2.165 +    finally show "norm (X i + Y i) < r"
2.166 +      by simp
2.167 +  qed
2.168 +qed
2.169 +
2.170 +lemma Zfun_minus: "Zfun X F \<Longrightarrow> Zfun (\<lambda>i. - X i) F"
2.171 +unfolding Zfun_def by simp
2.172 +
2.173 +lemma Zfun_diff: "\<lbrakk>Zfun X F; Zfun Y F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>i. X i - Y i) F"
2.174 +by (simp only: diff_minus Zfun_add Zfun_minus)
2.175 +
2.176 +lemma (in bounded_linear) Zfun:
2.177 +  assumes X: "Zfun X F"
2.178 +  shows "Zfun (\<lambda>n. f (X n)) F"
2.179 +proof -
2.180 +  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
2.181 +    using bounded by fast
2.182 +  with X show ?thesis
2.183 +    by (rule Zfun_imp_Zfun)
2.184 +qed
2.185 +
2.186 +lemma (in bounded_bilinear) Zfun:
2.187 +  assumes X: "Zfun X F"
2.188 +  assumes Y: "Zfun Y F"
2.189 +  shows "Zfun (\<lambda>n. X n ** Y n) F"
2.190 +proof (rule ZfunI)
2.191 +  fix r::real assume r: "0 < r"
2.192 +  obtain K where K: "0 < K"
2.193 +    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
2.194 +    using pos_bounded by fast
2.195 +  from K have K': "0 < inverse K"
2.196 +    by (rule positive_imp_inverse_positive)
2.197 +  have "eventually (\<lambda>i. norm (X i) < r) F"
2.198 +    using X r by (rule ZfunD)
2.199 +  moreover
2.200 +  have "eventually (\<lambda>i. norm (Y i) < inverse K) F"
2.201 +    using Y K' by (rule ZfunD)
2.202 +  ultimately
2.203 +  show "eventually (\<lambda>i. norm (X i ** Y i) < r) F"
2.204 +  proof (rule eventually_elim2)
2.205 +    fix i
2.206 +    assume *: "norm (X i) < r" "norm (Y i) < inverse K"
2.207 +    have "norm (X i ** Y i) \<le> norm (X i) * norm (Y i) * K"
2.208 +      by (rule norm_le)
2.209 +    also have "norm (X i) * norm (Y i) * K < r * inverse K * K"
2.210 +      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
2.211 +    also from K have "r * inverse K * K = r"
2.212 +      by simp
2.213 +    finally show "norm (X i ** Y i) < r" .
2.214 +  qed
2.215 +qed
2.216 +
2.217 +lemma (in bounded_bilinear) Zfun_left:
2.218 +  "Zfun X F \<Longrightarrow> Zfun (\<lambda>n. X n ** a) F"
2.219 +by (rule bounded_linear_left [THEN bounded_linear.Zfun])
2.220 +
2.221 +lemma (in bounded_bilinear) Zfun_right:
2.222 +  "Zfun X F \<Longrightarrow> Zfun (\<lambda>n. a ** X n) F"
2.223 +by (rule bounded_linear_right [THEN bounded_linear.Zfun])
2.224 +
2.225 +lemmas Zfun_mult = mult.Zfun
2.226 +lemmas Zfun_mult_right = mult.Zfun_right
2.227 +lemmas Zfun_mult_left = mult.Zfun_left
2.228 +
2.229 +
2.230 +subsection{* Limits *}
2.231 +
2.232 +definition
2.233 +  tendsto :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool" where
2.234 +  "tendsto f l net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
2.235 +
2.236 +lemma tendstoI:
2.237 +  "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net)
2.238 +    \<Longrightarrow> tendsto f l net"
2.239 +  unfolding tendsto_def by auto
2.240 +
2.241 +lemma tendstoD:
2.242 +  "tendsto f l net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
2.243 +  unfolding tendsto_def by auto
2.244 +
2.245 +lemma tendsto_Zfun_iff: "tendsto (\<lambda>n. X n) L F = Zfun (\<lambda>n. X n - L) F"
2.246 +by (simp only: tendsto_def Zfun_def dist_norm)
2.247 +
2.248 +lemma tendsto_const: "tendsto (\<lambda>n. k) k F"
2.250 +
2.251 +lemma tendsto_norm:
2.252 +  fixes a :: "'a::real_normed_vector"
2.253 +  shows "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. norm (X n)) (norm a) F"
2.254 +apply (simp add: tendsto_def dist_norm, safe)
2.255 +apply (drule_tac x="e" in spec, safe)
2.256 +apply (erule eventually_elim1)
2.257 +apply (erule order_le_less_trans [OF norm_triangle_ineq3])
2.258 +done
2.259 +
2.261 +  fixes a b c d :: "'a::ab_group_add"
2.262 +  shows "(a + c) - (b + d) = (a - b) + (c - d)"
2.263 +by simp
2.264 +
2.265 +lemma minus_diff_minus:
2.266 +  fixes a b :: "'a::ab_group_add"
2.267 +  shows "(- a) - (- b) = - (a - b)"
2.268 +by simp
2.269 +
2.271 +  fixes a b :: "'a::real_normed_vector"
2.272 +  shows "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n + Y n) (a + b) F"
2.274 +
2.275 +lemma tendsto_minus:
2.276 +  fixes a :: "'a::real_normed_vector"
2.277 +  shows "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. - X n) (- a) F"
2.278 +by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
2.279 +
2.280 +lemma tendsto_minus_cancel:
2.281 +  fixes a :: "'a::real_normed_vector"
2.282 +  shows "tendsto (\<lambda>n. - X n) (- a) F \<Longrightarrow> tendsto X a F"
2.283 +by (drule tendsto_minus, simp)
2.284 +
2.285 +lemma tendsto_diff:
2.286 +  fixes a b :: "'a::real_normed_vector"
2.287 +  shows "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n - Y n) (a - b) F"
2.289 +
2.290 +lemma (in bounded_linear) tendsto:
2.291 +  "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. f (X n)) (f a) F"
2.292 +by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
2.293 +
2.294 +lemma (in bounded_bilinear) tendsto:
2.295 +  "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n ** Y n) (a ** b) F"
2.296 +by (simp only: tendsto_Zfun_iff prod_diff_prod
2.297 +               Zfun_add Zfun Zfun_left Zfun_right)
2.298 +
2.299 +end
```
```     3.1 --- a/src/HOL/SEQ.thy	Sun May 31 11:27:19 2009 -0700
3.2 +++ b/src/HOL/SEQ.thy	Sun May 31 21:59:33 2009 -0700
3.3 @@ -9,10 +9,14 @@
3.4  header {* Sequences and Convergence *}
3.5
3.6  theory SEQ
3.7 -imports RealVector RComplete
3.8 +imports Limits
3.9  begin
3.10
3.11  definition
3.12 +  sequentially :: "nat filter" where
3.13 +  "sequentially = Abs_filter (\<lambda>P. \<exists>N. \<forall>n\<ge>N. P n)"
3.14 +
3.15 +definition
3.16    Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
3.17      --{*Standard definition of sequence converging to zero*}
3.18    [code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
3.19 @@ -67,6 +71,24 @@
3.20    [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
3.21
3.22
3.23 +subsection {* Sequentially *}
3.24 +
3.25 +lemma eventually_sequentially:
3.26 +  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
3.27 +unfolding sequentially_def
3.28 +apply (rule eventually_Abs_filter)
3.29 +apply simp
3.30 +apply (clarify, rule_tac x=N in exI, simp)
3.31 +apply (clarify, rename_tac M N)
3.32 +apply (rule_tac x="max M N" in exI, simp)
3.33 +done
3.34 +
3.35 +lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
3.36 +unfolding Zseq_def Zfun_def eventually_sequentially ..
3.37 +
3.38 +lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
3.39 +unfolding LIMSEQ_def tendsto_def eventually_sequentially ..
3.40 +
3.41  subsection {* Bounded Sequences *}
3.42
3.43  lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
3.44 @@ -134,61 +156,14 @@
3.45    assumes X: "Zseq X"
3.46    assumes Y: "\<And>n. norm (Y n) \<le> norm (X n) * K"
3.47    shows "Zseq (\<lambda>n. Y n)"
3.48 -proof (cases)
3.49 -  assume K: "0 < K"
3.50 -  show ?thesis
3.51 -  proof (rule ZseqI)
3.52 -    fix r::real assume "0 < r"
3.53 -    hence "0 < r / K"
3.54 -      using K by (rule divide_pos_pos)
3.55 -    then obtain N where "\<forall>n\<ge>N. norm (X n) < r / K"
3.56 -      using ZseqD [OF X] by fast
3.57 -    hence "\<forall>n\<ge>N. norm (X n) * K < r"
3.58 -      by (simp add: pos_less_divide_eq K)
3.59 -    hence "\<forall>n\<ge>N. norm (Y n) < r"
3.60 -      by (simp add: order_le_less_trans [OF Y])
3.61 -    thus "\<exists>N. \<forall>n\<ge>N. norm (Y n) < r" ..
3.62 -  qed
3.63 -next
3.64 -  assume "\<not> 0 < K"
3.65 -  hence K: "K \<le> 0" by (simp only: linorder_not_less)
3.66 -  {
3.67 -    fix n::nat
3.68 -    have "norm (Y n) \<le> norm (X n) * K" by (rule Y)
3.69 -    also have "\<dots> \<le> norm (X n) * 0"
3.70 -      using K norm_ge_zero by (rule mult_left_mono)
3.71 -    finally have "norm (Y n) = 0" by simp
3.72 -  }
3.73 -  thus ?thesis by (simp add: Zseq_zero)
3.74 -qed
3.75 +using assms unfolding Zseq_conv_Zfun by (rule Zfun_imp_Zfun)
3.76
3.77  lemma Zseq_le: "\<lbrakk>Zseq Y; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zseq X"
3.78  by (erule_tac K="1" in Zseq_imp_Zseq, simp)
3.79
3.81 -  assumes X: "Zseq X"
3.82 -  assumes Y: "Zseq Y"
3.83 -  shows "Zseq (\<lambda>n. X n + Y n)"
3.84 -proof (rule ZseqI)
3.85 -  fix r::real assume "0 < r"
3.86 -  hence r: "0 < r / 2" by simp
3.87 -  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r/2"
3.88 -    using ZseqD [OF X r] by fast
3.89 -  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < r/2"
3.90 -    using ZseqD [OF Y r] by fast
3.91 -  show "\<exists>N. \<forall>n\<ge>N. norm (X n + Y n) < r"
3.92 -  proof (intro exI allI impI)
3.93 -    fix n assume n: "max M N \<le> n"
3.94 -    have "norm (X n + Y n) \<le> norm (X n) + norm (Y n)"
3.95 -      by (rule norm_triangle_ineq)
3.96 -    also have "\<dots> < r/2 + r/2"
3.98 -      from M n show "norm (X n) < r/2" by simp
3.99 -      from N n show "norm (Y n) < r/2" by simp
3.100 -    qed
3.101 -    finally show "norm (X n + Y n) < r" by simp
3.102 -  qed
3.103 -qed
3.104 +  "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n + Y n)"
3.105 +unfolding Zseq_conv_Zfun by (rule Zfun_add)
3.106
3.107  lemma Zseq_minus: "Zseq X \<Longrightarrow> Zseq (\<lambda>n. - X n)"
3.108  unfolding Zseq_def by simp
3.109 @@ -197,44 +172,12 @@
3.110  by (simp only: diff_minus Zseq_add Zseq_minus)
3.111
3.112  lemma (in bounded_linear) Zseq:
3.113 -  assumes X: "Zseq X"
3.114 -  shows "Zseq (\<lambda>n. f (X n))"
3.115 -proof -
3.116 -  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
3.117 -    using bounded by fast
3.118 -  with X show ?thesis
3.119 -    by (rule Zseq_imp_Zseq)
3.120 -qed
3.121 +  "Zseq X \<Longrightarrow> Zseq (\<lambda>n. f (X n))"
3.122 +unfolding Zseq_conv_Zfun by (rule Zfun)
3.123
3.124  lemma (in bounded_bilinear) Zseq:
3.125 -  assumes X: "Zseq X"
3.126 -  assumes Y: "Zseq Y"
3.127 -  shows "Zseq (\<lambda>n. X n ** Y n)"
3.128 -proof (rule ZseqI)
3.129 -  fix r::real assume r: "0 < r"
3.130 -  obtain K where K: "0 < K"
3.131 -    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
3.132 -    using pos_bounded by fast
3.133 -  from K have K': "0 < inverse K"
3.134 -    by (rule positive_imp_inverse_positive)
3.135 -  obtain M where M: "\<forall>n\<ge>M. norm (X n) < r"
3.136 -    using ZseqD [OF X r] by fast
3.137 -  obtain N where N: "\<forall>n\<ge>N. norm (Y n) < inverse K"
3.138 -    using ZseqD [OF Y K'] by fast
3.139 -  show "\<exists>N. \<forall>n\<ge>N. norm (X n ** Y n) < r"
3.140 -  proof (intro exI allI impI)
3.141 -    fix n assume n: "max M N \<le> n"
3.142 -    have "norm (X n ** Y n) \<le> norm (X n) * norm (Y n) * K"
3.143 -      by (rule norm_le)
3.144 -    also have "norm (X n) * norm (Y n) * K < r * inverse K * K"
3.145 -    proof (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero K)
3.146 -      from M n show Xn: "norm (X n) < r" by simp
3.147 -      from N n show Yn: "norm (Y n) < inverse K" by simp
3.148 -    qed
3.149 -    also from K have "r * inverse K * K = r" by simp
3.150 -    finally show "norm (X n ** Y n) < r" .
3.151 -  qed
3.152 -qed
3.153 +  "Zseq X \<Longrightarrow> Zseq Y \<Longrightarrow> Zseq (\<lambda>n. X n ** Y n)"
3.154 +unfolding Zseq_conv_Zfun by (rule Zfun)
3.155
3.156  lemma (in bounded_bilinear) Zseq_prod_Bseq:
3.157    assumes X: "Zseq X"
3.158 @@ -341,12 +284,7 @@
3.159  lemma LIMSEQ_norm:
3.160    fixes a :: "'a::real_normed_vector"
3.161    shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
3.162 -apply (simp add: LIMSEQ_iff, safe)
3.163 -apply (drule_tac x="r" in spec, safe)
3.164 -apply (rule_tac x="no" in exI, safe)
3.165 -apply (drule_tac x="n" in spec, safe)
3.166 -apply (erule order_le_less_trans [OF norm_triangle_ineq3])
3.167 -done
3.168 +unfolding LIMSEQ_conv_tendsto by (rule tendsto_norm)
3.169
3.170  lemma LIMSEQ_ignore_initial_segment:
3.171    "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
3.172 @@ -381,26 +319,15 @@
3.173    unfolding LIMSEQ_def
3.174    by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
3.175
3.176 -
3.178 -  fixes a b c d :: "'a::ab_group_add"
3.179 -  shows "(a + c) - (b + d) = (a - b) + (c - d)"
3.180 -by simp
3.181 -
3.182 -lemma minus_diff_minus:
3.183 -  fixes a b :: "'a::ab_group_add"
3.184 -  shows "(- a) - (- b) = - (a - b)"
3.185 -by simp
3.186 -
3.188    fixes a b :: "'a::real_normed_vector"
3.189    shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
3.191 +unfolding LIMSEQ_conv_tendsto by (rule tendsto_add)
3.192
3.193  lemma LIMSEQ_minus:
3.194    fixes a :: "'a::real_normed_vector"
3.195    shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
3.196 -by (simp only: LIMSEQ_Zseq_iff minus_diff_minus Zseq_minus)
3.197 +unfolding LIMSEQ_conv_tendsto by (rule tendsto_minus)
3.198
3.199  lemma LIMSEQ_minus_cancel:
3.200    fixes a :: "'a::real_normed_vector"
3.201 @@ -410,7 +337,7 @@
3.202  lemma LIMSEQ_diff:
3.203    fixes a b :: "'a::real_normed_vector"
3.204    shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
3.206 +unfolding LIMSEQ_conv_tendsto by (rule tendsto_diff)
3.207
3.208  lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
3.209  apply (rule ccontr)
3.210 @@ -425,12 +352,11 @@
3.211
3.212  lemma (in bounded_linear) LIMSEQ:
3.213    "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
3.214 -by (simp only: LIMSEQ_Zseq_iff diff [symmetric] Zseq)
3.215 +unfolding LIMSEQ_conv_tendsto by (rule tendsto)
3.216
3.217  lemma (in bounded_bilinear) LIMSEQ:
3.218    "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
3.219 -by (simp only: LIMSEQ_Zseq_iff prod_diff_prod
3.220 -               Zseq_add Zseq Zseq_left Zseq_right)
3.221 +unfolding LIMSEQ_conv_tendsto by (rule tendsto)
3.222
3.223  lemma LIMSEQ_mult:
3.224    fixes a b :: "'a::real_normed_algebra"
```