--- a/src/HOL/Lim.thy Fri Mar 22 10:41:42 2013 +0100
+++ b/src/HOL/Lim.thy Fri Mar 22 10:41:43 2013 +0100
@@ -10,22 +10,8 @@
imports SEQ
begin
-definition
- isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
- "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
-
subsection {* Limits of Functions *}
-lemma metric_LIM_I:
- "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
- \<Longrightarrow> f -- a --> L"
-by (simp add: LIM_def)
-
-lemma metric_LIM_D:
- "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
- \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
-by (simp add: LIM_def)
-
lemma LIM_eq:
fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
shows "f -- a --> L =
@@ -80,13 +66,6 @@
shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
unfolding tendsto_iff dist_norm by simp
-lemma metric_LIM_imp_LIM:
- assumes f: "f -- a --> l"
- assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
- shows "g -- a --> m"
- by (rule metric_tendsto_imp_tendsto [OF f],
- auto simp add: eventually_at_topological le)
-
lemma LIM_imp_LIM:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
@@ -96,18 +75,6 @@
by (rule metric_LIM_imp_LIM [OF f],
simp add: dist_norm le)
-lemma metric_LIM_equal2:
- assumes 1: "0 < R"
- assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
- shows "g -- a --> l \<Longrightarrow> f -- a --> l"
-apply (rule topological_tendstoI)
-apply (drule (2) topological_tendstoD)
-apply (simp add: eventually_at, safe)
-apply (rule_tac x="min d R" in exI, safe)
-apply (simp add: 1)
-apply (simp add: 2)
-done
-
lemma LIM_equal2:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
assumes 1: "0 < R"
@@ -115,14 +82,6 @@
shows "g -- a --> l \<Longrightarrow> f -- a --> l"
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
-lemma metric_LIM_compose2:
- assumes f: "f -- a --> b"
- assumes g: "g -- b --> c"
- assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
- shows "(\<lambda>x. g (f x)) -- a --> c"
- using g f inj [folded eventually_at]
- by (rule tendsto_compose_eventually)
-
lemma LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f: "f -- a --> b"
@@ -199,13 +158,6 @@
shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
unfolding isCont_def by (rule tendsto_divide)
-lemma metric_isCont_LIM_compose2:
- assumes f [unfolded isCont_def]: "isCont f a"
- assumes g: "g -- f a --> l"
- assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
- shows "(\<lambda>x. g (f x)) -- a --> l"
-by (rule metric_LIM_compose2 [OF f g inj])
-
lemma isCont_LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f [unfolded isCont_def]: "isCont f a"
@@ -251,18 +203,6 @@
subsection {* Uniform Continuity *}
-lemma isUCont_isCont: "isUCont f ==> isCont f x"
-by (simp add: isUCont_def isCont_def LIM_def, force)
-
-lemma isUCont_Cauchy:
- "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
-unfolding isUCont_def
-apply (rule metric_CauchyI)
-apply (drule_tac x=e in spec, safe)
-apply (drule_tac e=s in metric_CauchyD, safe)
-apply (rule_tac x=M in exI, simp)
-done
-
lemma (in bounded_linear) isUCont: "isUCont f"
unfolding isUCont_def dist_norm
proof (intro allI impI)
--- a/src/HOL/Limits.thy Fri Mar 22 10:41:42 2013 +0100
+++ b/src/HOL/Limits.thy Fri Mar 22 10:41:43 2013 +0100
@@ -11,31 +11,6 @@
definition at_infinity :: "'a::real_normed_vector filter" where
"at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
-
-lemma eventually_nhds_metric:
- "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
-unfolding eventually_nhds open_dist
-apply safe
-apply fast
-apply (rule_tac x="{x. dist x a < d}" in exI, simp)
-apply clarsimp
-apply (rule_tac x="d - dist x a" in exI, clarsimp)
-apply (simp only: less_diff_eq)
-apply (erule le_less_trans [OF dist_triangle])
-done
-
-lemma eventually_at:
- fixes a :: "'a::metric_space"
- shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
-unfolding at_def eventually_within eventually_nhds_metric by auto
-lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
- "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
- unfolding eventually_within eventually_at dist_nz by auto
-
-lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
- "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
- unfolding eventually_within_less by auto (metis dense order_le_less_trans)
-
lemma eventually_at_infinity:
"eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
unfolding at_infinity_def
@@ -246,39 +221,8 @@
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
by (simp only: tendsto_iff Zfun_def dist_norm)
-
-lemma metric_tendsto_imp_tendsto:
- assumes f: "(f ---> a) F"
- assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
- shows "(g ---> b) F"
-proof (rule tendstoI)
- fix e :: real assume "0 < e"
- with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
- with le show "eventually (\<lambda>x. dist (g x) b < e) F"
- using le_less_trans by (rule eventually_elim2)
-qed
subsubsection {* Distance and norms *}
-lemma tendsto_dist [tendsto_intros]:
- assumes f: "(f ---> l) F" and g: "(g ---> m) F"
- shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
-proof (rule tendstoI)
- fix e :: real assume "0 < e"
- hence e2: "0 < e/2" by simp
- from tendstoD [OF f e2] tendstoD [OF g e2]
- show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
- proof (eventually_elim)
- case (elim x)
- then show "dist (dist (f x) (g x)) (dist l m) < e"
- unfolding dist_real_def
- using dist_triangle2 [of "f x" "g x" "l"]
- using dist_triangle2 [of "g x" "l" "m"]
- using dist_triangle3 [of "l" "m" "f x"]
- using dist_triangle [of "f x" "m" "g x"]
- by arith
- qed
-qed
-
lemma norm_conv_dist: "norm x = dist x 0"
unfolding dist_norm by simp
@@ -544,50 +488,6 @@
shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
unfolding sgn_div_norm by (simp add: tendsto_intros)
-lemma filterlim_at_bot_at_right:
- fixes f :: "real \<Rightarrow> 'b::linorder"
- assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
- assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
- assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
- assumes P: "eventually P at_bot"
- shows "filterlim f at_bot (at_right a)"
-proof -
- from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
- unfolding eventually_at_bot_linorder by auto
- show ?thesis
- proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
- fix z assume "z \<le> x"
- with x have "P z" by auto
- have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
- using bound[OF bij(2)[OF `P z`]]
- by (auto simp add: eventually_within_less dist_real_def intro!: exI[of _ "g z - a"])
- with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
- by eventually_elim (metis bij `P z` mono)
- qed
-qed
-
-lemma filterlim_at_top_at_left:
- fixes f :: "real \<Rightarrow> 'b::linorder"
- assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
- assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
- assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
- assumes P: "eventually P at_top"
- shows "filterlim f at_top (at_left a)"
-proof -
- from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
- unfolding eventually_at_top_linorder by auto
- show ?thesis
- proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
- fix z assume "x \<le> z"
- with x have "P z" by auto
- have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
- using bound[OF bij(2)[OF `P z`]]
- by (auto simp add: eventually_within_less dist_real_def intro!: exI[of _ "a - g z"])
- with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
- by eventually_elim (metis bij `P z` mono)
- qed
-qed
-
lemma filterlim_at_infinity:
fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
assumes "0 \<le> c"
@@ -607,13 +507,6 @@
qed
qed force
-lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
- unfolding filterlim_at_top
- apply (intro allI)
- apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
- apply (auto simp: natceiling_le_eq)
- done
-
subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
text {*
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Metric_Spaces.thy Fri Mar 22 10:41:43 2013 +0100
@@ -0,0 +1,573 @@
+(* Title: HOL/Metric_Spaces.thy
+ Author: Brian Huffman
+ Author: Johannes Hölzl
+*)
+
+header {* Metric Spaces *}
+
+theory Metric_Spaces
+imports RComplete Topological_Spaces
+begin
+
+
+subsection {* Metric spaces *}
+
+class dist =
+ fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
+
+class open_dist = "open" + dist +
+ assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+
+class metric_space = open_dist +
+ assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
+ assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
+begin
+
+lemma dist_self [simp]: "dist x x = 0"
+by simp
+
+lemma zero_le_dist [simp]: "0 \<le> dist x y"
+using dist_triangle2 [of x x y] by simp
+
+lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
+by (simp add: less_le)
+
+lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
+by (simp add: not_less)
+
+lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
+by (simp add: le_less)
+
+lemma dist_commute: "dist x y = dist y x"
+proof (rule order_antisym)
+ show "dist x y \<le> dist y x"
+ using dist_triangle2 [of x y x] by simp
+ show "dist y x \<le> dist x y"
+ using dist_triangle2 [of y x y] by simp
+qed
+
+lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
+using dist_triangle2 [of x z y] by (simp add: dist_commute)
+
+lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
+using dist_triangle2 [of x y a] by (simp add: dist_commute)
+
+lemma dist_triangle_alt:
+ shows "dist y z <= dist x y + dist x z"
+by (rule dist_triangle3)
+
+lemma dist_pos_lt:
+ shows "x \<noteq> y ==> 0 < dist x y"
+by (simp add: zero_less_dist_iff)
+
+lemma dist_nz:
+ shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
+by (simp add: zero_less_dist_iff)
+
+lemma dist_triangle_le:
+ shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
+by (rule order_trans [OF dist_triangle2])
+
+lemma dist_triangle_lt:
+ shows "dist x z + dist y z < e ==> dist x y < e"
+by (rule le_less_trans [OF dist_triangle2])
+
+lemma dist_triangle_half_l:
+ shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
+by (rule dist_triangle_lt [where z=y], simp)
+
+lemma dist_triangle_half_r:
+ shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
+by (rule dist_triangle_half_l, simp_all add: dist_commute)
+
+subclass topological_space
+proof
+ have "\<exists>e::real. 0 < e"
+ by (fast intro: zero_less_one)
+ then show "open UNIV"
+ unfolding open_dist by simp
+next
+ fix S T assume "open S" "open T"
+ then show "open (S \<inter> T)"
+ unfolding open_dist
+ apply clarify
+ apply (drule (1) bspec)+
+ apply (clarify, rename_tac r s)
+ apply (rule_tac x="min r s" in exI, simp)
+ done
+next
+ fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
+ unfolding open_dist by fast
+qed
+
+lemma (in metric_space) open_ball: "open {y. dist x y < d}"
+proof (unfold open_dist, intro ballI)
+ fix y assume *: "y \<in> {y. dist x y < d}"
+ then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
+ by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
+qed
+
+end
+
+instance metric_space \<subseteq> t2_space
+proof
+ fix x y :: "'a::metric_space"
+ assume xy: "x \<noteq> y"
+ let ?U = "{y'. dist x y' < dist x y / 2}"
+ let ?V = "{x'. dist y x' < dist x y / 2}"
+ have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
+ \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
+ have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
+ using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
+ using open_ball[of _ "dist x y / 2"] by auto
+ then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
+ by blast
+qed
+
+lemma eventually_nhds_metric:
+ fixes a :: "'a :: metric_space"
+ shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
+unfolding eventually_nhds open_dist
+apply safe
+apply fast
+apply (rule_tac x="{x. dist x a < d}" in exI, simp)
+apply clarsimp
+apply (rule_tac x="d - dist x a" in exI, clarsimp)
+apply (simp only: less_diff_eq)
+apply (erule le_less_trans [OF dist_triangle])
+done
+
+lemma eventually_at:
+ fixes a :: "'a::metric_space"
+ shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
+unfolding at_def eventually_within eventually_nhds_metric by auto
+
+lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
+ fixes a :: "'a :: metric_space"
+ shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
+ unfolding eventually_within eventually_at dist_nz by auto
+
+lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
+ fixes a :: "'a :: metric_space"
+ shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
+ unfolding eventually_within_less by auto (metis dense order_le_less_trans)
+
+lemma tendstoI:
+ fixes l :: "'a :: metric_space"
+ assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
+ shows "(f ---> l) F"
+ apply (rule topological_tendstoI)
+ apply (simp add: open_dist)
+ apply (drule (1) bspec, clarify)
+ apply (drule assms)
+ apply (erule eventually_elim1, simp)
+ done
+
+lemma tendstoD:
+ fixes l :: "'a :: metric_space"
+ shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
+ apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
+ apply (clarsimp simp add: open_dist)
+ apply (rule_tac x="e - dist x l" in exI, clarsimp)
+ apply (simp only: less_diff_eq)
+ apply (erule le_less_trans [OF dist_triangle])
+ apply simp
+ apply simp
+ done
+
+lemma tendsto_iff:
+ fixes l :: "'a :: metric_space"
+ shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
+ using tendstoI tendstoD by fast
+
+lemma metric_tendsto_imp_tendsto:
+ fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
+ assumes f: "(f ---> a) F"
+ assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
+ shows "(g ---> b) F"
+proof (rule tendstoI)
+ fix e :: real assume "0 < e"
+ with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
+ with le show "eventually (\<lambda>x. dist (g x) b < e) F"
+ using le_less_trans by (rule eventually_elim2)
+qed
+
+lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
+ unfolding filterlim_at_top
+ apply (intro allI)
+ apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
+ apply (auto simp: natceiling_le_eq)
+ done
+
+subsubsection {* Limits of Sequences *}
+
+lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
+ unfolding tendsto_iff eventually_sequentially ..
+
+lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
+ unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
+
+lemma metric_LIMSEQ_I:
+ "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
+by (simp add: LIMSEQ_def)
+
+lemma metric_LIMSEQ_D:
+ "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
+by (simp add: LIMSEQ_def)
+
+
+subsubsection {* Limits of Functions *}
+
+lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
+ (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
+ --> dist (f x) L < r)"
+unfolding tendsto_iff eventually_at ..
+
+lemma metric_LIM_I:
+ "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
+ \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
+by (simp add: LIM_def)
+
+lemma metric_LIM_D:
+ "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
+ \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
+by (simp add: LIM_def)
+
+lemma metric_LIM_imp_LIM:
+ assumes f: "f -- a --> (l::'a::metric_space)"
+ assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
+ shows "g -- a --> (m::'b::metric_space)"
+ by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
+
+lemma metric_LIM_equal2:
+ assumes 1: "0 < R"
+ assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
+ shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
+apply (rule topological_tendstoI)
+apply (drule (2) topological_tendstoD)
+apply (simp add: eventually_at, safe)
+apply (rule_tac x="min d R" in exI, safe)
+apply (simp add: 1)
+apply (simp add: 2)
+done
+
+lemma metric_LIM_compose2:
+ assumes f: "f -- (a::'a::metric_space) --> b"
+ assumes g: "g -- b --> c"
+ assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
+ shows "(\<lambda>x. g (f x)) -- a --> c"
+ using g f inj [folded eventually_at]
+ by (rule tendsto_compose_eventually)
+
+lemma metric_isCont_LIM_compose2:
+ fixes f :: "'a :: metric_space \<Rightarrow> _"
+ assumes f [unfolded isCont_def]: "isCont f a"
+ assumes g: "g -- f a --> l"
+ assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
+ shows "(\<lambda>x. g (f x)) -- a --> l"
+by (rule metric_LIM_compose2 [OF f g inj])
+
+subsection {* Complete metric spaces *}
+
+subsection {* Cauchy sequences *}
+
+definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
+ "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
+
+subsection {* Cauchy Sequences *}
+
+lemma metric_CauchyI:
+ "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
+ by (simp add: Cauchy_def)
+
+lemma metric_CauchyD:
+ "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
+ by (simp add: Cauchy_def)
+
+lemma metric_Cauchy_iff2:
+ "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
+apply (simp add: Cauchy_def, auto)
+apply (drule reals_Archimedean, safe)
+apply (drule_tac x = n in spec, auto)
+apply (rule_tac x = M in exI, auto)
+apply (drule_tac x = m in spec, simp)
+apply (drule_tac x = na in spec, auto)
+done
+
+lemma Cauchy_subseq_Cauchy:
+ "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
+apply (auto simp add: Cauchy_def)
+apply (drule_tac x=e in spec, clarify)
+apply (rule_tac x=M in exI, clarify)
+apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
+done
+
+
+subsubsection {* Cauchy Sequences are Convergent *}
+
+class complete_space = metric_space +
+ assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
+
+theorem LIMSEQ_imp_Cauchy:
+ assumes X: "X ----> a" shows "Cauchy X"
+proof (rule metric_CauchyI)
+ fix e::real assume "0 < e"
+ hence "0 < e/2" by simp
+ with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
+ then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
+ show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
+ proof (intro exI allI impI)
+ fix m assume "N \<le> m"
+ hence m: "dist (X m) a < e/2" using N by fast
+ fix n assume "N \<le> n"
+ hence n: "dist (X n) a < e/2" using N by fast
+ have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
+ by (rule dist_triangle2)
+ also from m n have "\<dots> < e" by simp
+ finally show "dist (X m) (X n) < e" .
+ qed
+qed
+
+lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
+unfolding convergent_def
+by (erule exE, erule LIMSEQ_imp_Cauchy)
+
+lemma Cauchy_convergent_iff:
+ fixes X :: "nat \<Rightarrow> 'a::complete_space"
+ shows "Cauchy X = convergent X"
+by (fast intro: Cauchy_convergent convergent_Cauchy)
+
+subsection {* Uniform Continuity *}
+
+definition
+ isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
+ "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
+
+lemma isUCont_isCont: "isUCont f ==> isCont f x"
+by (simp add: isUCont_def isCont_def LIM_def, force)
+
+lemma isUCont_Cauchy:
+ "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
+unfolding isUCont_def
+apply (rule metric_CauchyI)
+apply (drule_tac x=e in spec, safe)
+apply (drule_tac e=s in metric_CauchyD, safe)
+apply (rule_tac x=M in exI, simp)
+done
+
+subsection {* The set of real numbers is a complete metric space *}
+
+instantiation real :: metric_space
+begin
+
+definition dist_real_def:
+ "dist x y = \<bar>x - y\<bar>"
+
+definition open_real_def:
+ "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
+
+instance
+ by default (auto simp: open_real_def dist_real_def)
+end
+
+instance real :: linorder_topology
+proof
+ show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
+ proof (rule ext, safe)
+ fix S :: "real set" assume "open S"
+ then guess f unfolding open_real_def bchoice_iff ..
+ then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
+ by (fastforce simp: dist_real_def)
+ show "generate_topology (range lessThan \<union> range greaterThan) S"
+ apply (subst *)
+ apply (intro generate_topology_Union generate_topology.Int)
+ apply (auto intro: generate_topology.Basis)
+ done
+ next
+ fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
+ moreover have "\<And>a::real. open {..<a}"
+ unfolding open_real_def dist_real_def
+ proof clarify
+ fix x a :: real assume "x < a"
+ hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
+ thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
+ qed
+ moreover have "\<And>a::real. open {a <..}"
+ unfolding open_real_def dist_real_def
+ proof clarify
+ fix x a :: real assume "a < x"
+ hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
+ thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
+ qed
+ ultimately show "open S"
+ by induct auto
+ qed
+qed
+
+lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
+lemmas open_real_lessThan = open_lessThan[where 'a=real]
+lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
+lemmas closed_real_atMost = closed_atMost[where 'a=real]
+lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
+lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
+
+text {*
+Proof that Cauchy sequences converge based on the one from
+http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
+*}
+
+text {*
+ If sequence @{term "X"} is Cauchy, then its limit is the lub of
+ @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
+*}
+
+lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
+by (simp add: isUbI setleI)
+
+lemma increasing_LIMSEQ:
+ fixes f :: "nat \<Rightarrow> real"
+ assumes inc: "\<And>n. f n \<le> f (Suc n)"
+ and bdd: "\<And>n. f n \<le> l"
+ and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
+ shows "f ----> l"
+proof (rule increasing_tendsto)
+ fix x assume "x < l"
+ with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
+ by auto
+ from en[OF `0 < e`] obtain n where "l - e \<le> f n"
+ by (auto simp: field_simps)
+ with `e < l - x` `0 < e` have "x < f n" by simp
+ with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
+ by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
+qed (insert bdd, auto)
+
+lemma real_Cauchy_convergent:
+ fixes X :: "nat \<Rightarrow> real"
+ assumes X: "Cauchy X"
+ shows "convergent X"
+proof -
+ def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
+ then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
+
+ { fix N x assume N: "\<forall>n\<ge>N. X n < x"
+ have "isUb UNIV S x"
+ proof (rule isUb_UNIV_I)
+ fix y::real assume "y \<in> S"
+ hence "\<exists>M. \<forall>n\<ge>M. y < X n"
+ by (simp add: S_def)
+ then obtain M where "\<forall>n\<ge>M. y < X n" ..
+ hence "y < X (max M N)" by simp
+ also have "\<dots> < x" using N by simp
+ finally show "y \<le> x"
+ by (rule order_less_imp_le)
+ qed }
+ note bound_isUb = this
+
+ have "\<exists>u. isLub UNIV S u"
+ proof (rule reals_complete)
+ obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
+ using X[THEN metric_CauchyD, OF zero_less_one] by auto
+ hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
+ show "\<exists>x. x \<in> S"
+ proof
+ from N have "\<forall>n\<ge>N. X N - 1 < X n"
+ by (simp add: abs_diff_less_iff dist_real_def)
+ thus "X N - 1 \<in> S" by (rule mem_S)
+ qed
+ show "\<exists>u. isUb UNIV S u"
+ proof
+ from N have "\<forall>n\<ge>N. X n < X N + 1"
+ by (simp add: abs_diff_less_iff dist_real_def)
+ thus "isUb UNIV S (X N + 1)"
+ by (rule bound_isUb)
+ qed
+ qed
+ then obtain x where x: "isLub UNIV S x" ..
+ have "X ----> x"
+ proof (rule metric_LIMSEQ_I)
+ fix r::real assume "0 < r"
+ hence r: "0 < r/2" by simp
+ obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
+ using metric_CauchyD [OF X r] by auto
+ hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
+ hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
+ by (simp only: dist_real_def abs_diff_less_iff)
+
+ from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
+ hence "X N - r/2 \<in> S" by (rule mem_S)
+ hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
+
+ from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
+ hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
+ hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
+
+ show "\<exists>N. \<forall>n\<ge>N. dist (X n) x < r"
+ proof (intro exI allI impI)
+ fix n assume n: "N \<le> n"
+ from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
+ thus "dist (X n) x < r" using 1 2
+ by (simp add: abs_diff_less_iff dist_real_def)
+ qed
+ qed
+ then show ?thesis unfolding convergent_def by auto
+qed
+
+instance real :: complete_space
+ by intro_classes (rule real_Cauchy_convergent)
+
+lemma tendsto_dist [tendsto_intros]:
+ fixes l m :: "'a :: metric_space"
+ assumes f: "(f ---> l) F" and g: "(g ---> m) F"
+ shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
+proof (rule tendstoI)
+ fix e :: real assume "0 < e"
+ hence e2: "0 < e/2" by simp
+ from tendstoD [OF f e2] tendstoD [OF g e2]
+ show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
+ proof (eventually_elim)
+ case (elim x)
+ then show "dist (dist (f x) (g x)) (dist l m) < e"
+ unfolding dist_real_def
+ using dist_triangle2 [of "f x" "g x" "l"]
+ using dist_triangle2 [of "g x" "l" "m"]
+ using dist_triangle3 [of "l" "m" "f x"]
+ using dist_triangle [of "f x" "m" "g x"]
+ by arith
+ qed
+qed
+
+lemma tendsto_at_topI_sequentially:
+ fixes f :: "real \<Rightarrow> real"
+ assumes mono: "mono f"
+ assumes limseq: "(\<lambda>n. f (real n)) ----> y"
+ shows "(f ---> y) at_top"
+proof (rule tendstoI)
+ fix e :: real assume "0 < e"
+ with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
+ by (auto simp: LIMSEQ_def dist_real_def)
+ { fix x :: real
+ from ex_le_of_nat[of x] guess n ..
+ note monoD[OF mono this]
+ also have "f (real_of_nat n) \<le> y"
+ by (rule LIMSEQ_le_const[OF limseq])
+ (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
+ finally have "f x \<le> y" . }
+ note le = this
+ have "eventually (\<lambda>x. real N \<le> x) at_top"
+ by (rule eventually_ge_at_top)
+ then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
+ proof eventually_elim
+ fix x assume N': "real N \<le> x"
+ with N[of N] le have "y - f (real N) < e" by auto
+ moreover note monoD[OF mono N']
+ ultimately show "dist (f x) y < e"
+ using le[of x] by (auto simp: dist_real_def field_simps)
+ qed
+qed
+
+lemma Cauchy_iff2:
+ "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
+ unfolding metric_Cauchy_iff2 dist_real_def ..
+
+end
+
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri Mar 22 10:41:42 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri Mar 22 10:41:43 2013 +0100
@@ -1416,7 +1416,7 @@
unfolding tendsto_def Limits.eventually_within by simp
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
- unfolding tendsto_def Limits.eventually_within
+ unfolding tendsto_def Topological_Spaces.eventually_within
by (auto elim!: eventually_elim1)
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
@@ -4667,7 +4667,7 @@
hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
using `a \<notin> U` by (fast elim: eventually_mono [rotated])
thus ?thesis
- unfolding Limits.eventually_within Limits.eventually_at
+ unfolding Limits.eventually_within Metric_Spaces.eventually_at
by (rule ex_forward, cut_tac `f x \<noteq> a`, auto simp: dist_commute)
qed
--- a/src/HOL/RealVector.thy Fri Mar 22 10:41:42 2013 +0100
+++ b/src/HOL/RealVector.thy Fri Mar 22 10:41:43 2013 +0100
@@ -5,7 +5,7 @@
header {* Vector Spaces and Algebras over the Reals *}
theory RealVector
-imports RComplete
+imports RComplete Metric_Spaces
begin
subsection {* Locale for additive functions *}
@@ -434,106 +434,6 @@
by (rule Reals_cases) auto
-subsection {* Metric spaces *}
-
-class dist =
- fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
-
-class open_dist = "open" + dist +
- assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-
-class metric_space = open_dist +
- assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
- assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
-begin
-
-lemma dist_self [simp]: "dist x x = 0"
-by simp
-
-lemma zero_le_dist [simp]: "0 \<le> dist x y"
-using dist_triangle2 [of x x y] by simp
-
-lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
-by (simp add: less_le)
-
-lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
-by (simp add: not_less)
-
-lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
-by (simp add: le_less)
-
-lemma dist_commute: "dist x y = dist y x"
-proof (rule order_antisym)
- show "dist x y \<le> dist y x"
- using dist_triangle2 [of x y x] by simp
- show "dist y x \<le> dist x y"
- using dist_triangle2 [of y x y] by simp
-qed
-
-lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
-using dist_triangle2 [of x z y] by (simp add: dist_commute)
-
-lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
-using dist_triangle2 [of x y a] by (simp add: dist_commute)
-
-lemma dist_triangle_alt:
- shows "dist y z <= dist x y + dist x z"
-by (rule dist_triangle3)
-
-lemma dist_pos_lt:
- shows "x \<noteq> y ==> 0 < dist x y"
-by (simp add: zero_less_dist_iff)
-
-lemma dist_nz:
- shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
-by (simp add: zero_less_dist_iff)
-
-lemma dist_triangle_le:
- shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
-by (rule order_trans [OF dist_triangle2])
-
-lemma dist_triangle_lt:
- shows "dist x z + dist y z < e ==> dist x y < e"
-by (rule le_less_trans [OF dist_triangle2])
-
-lemma dist_triangle_half_l:
- shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_lt [where z=y], simp)
-
-lemma dist_triangle_half_r:
- shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
-by (rule dist_triangle_half_l, simp_all add: dist_commute)
-
-subclass topological_space
-proof
- have "\<exists>e::real. 0 < e"
- by (fast intro: zero_less_one)
- then show "open UNIV"
- unfolding open_dist by simp
-next
- fix S T assume "open S" "open T"
- then show "open (S \<inter> T)"
- unfolding open_dist
- apply clarify
- apply (drule (1) bspec)+
- apply (clarify, rename_tac r s)
- apply (rule_tac x="min r s" in exI, simp)
- done
-next
- fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
- unfolding open_dist by fast
-qed
-
-lemma (in metric_space) open_ball: "open {y. dist x y < d}"
-proof (unfold open_dist, intro ballI)
- fix y assume *: "y \<in> {y. dist x y < d}"
- then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
- by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
-qed
-
-end
-
-
subsection {* Real normed vector spaces *}
class norm =
@@ -774,16 +674,9 @@
definition real_norm_def [simp]:
"norm r = \<bar>r\<bar>"
-definition dist_real_def:
- "dist x y = \<bar>x - y\<bar>"
-
-definition open_real_def:
- "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
-
instance
apply (intro_classes, unfold real_norm_def real_scaleR_def)
apply (rule dist_real_def)
-apply (rule open_real_def)
apply (simp add: sgn_real_def)
apply (rule abs_eq_0)
apply (rule abs_triangle_ineq)
@@ -793,47 +686,6 @@
end
-instance real :: linorder_topology
-proof
- show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
- proof (rule ext, safe)
- fix S :: "real set" assume "open S"
- then guess f unfolding open_real_def bchoice_iff ..
- then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
- by (fastforce simp: dist_real_def)
- show "generate_topology (range lessThan \<union> range greaterThan) S"
- apply (subst *)
- apply (intro generate_topology_Union generate_topology.Int)
- apply (auto intro: generate_topology.Basis)
- done
- next
- fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
- moreover have "\<And>a::real. open {..<a}"
- unfolding open_real_def dist_real_def
- proof clarify
- fix x a :: real assume "x < a"
- hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
- thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
- qed
- moreover have "\<And>a::real. open {a <..}"
- unfolding open_real_def dist_real_def
- proof clarify
- fix x a :: real assume "a < x"
- hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
- thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
- qed
- ultimately show "open S"
- by induct auto
- qed
-qed
-
-lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
-lemmas open_real_lessThan = open_lessThan[where 'a=real]
-lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
-lemmas closed_real_atMost = closed_atMost[where 'a=real]
-lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
-lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
-
subsection {* Extra type constraints *}
text {* Only allow @{term "open"} in class @{text topological_space}. *}
@@ -851,7 +703,6 @@
setup {* Sign.add_const_constraint
(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
-
subsection {* Sign function *}
lemma norm_sgn:
@@ -1057,21 +908,6 @@
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
unfolding of_real_def by (rule bounded_linear_scaleR_left)
-
-instance metric_space \<subseteq> t2_space
-proof
- fix x y :: "'a::metric_space"
- assume xy: "x \<noteq> y"
- let ?U = "{y'. dist x y' < dist x y / 2}"
- let ?V = "{x'. dist y x' < dist x y / 2}"
- have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
- \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
- have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
- using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
- using open_ball[of _ "dist x y / 2"] by auto
- then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
- by blast
-qed
instance real_normed_algebra_1 \<subseteq> perfect_space
proof
fix x::'a
--- a/src/HOL/SEQ.thy Fri Mar 22 10:41:42 2013 +0100
+++ b/src/HOL/SEQ.thy Fri Mar 22 10:41:43 2013 +0100
@@ -20,10 +20,6 @@
--{*Standard definition for bounded sequence*}
"Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
-definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
- "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
-
-
subsection {* Bounded Sequences *}
lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
@@ -80,25 +76,11 @@
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
by simp
-lemma LIMSEQ_def: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
-unfolding tendsto_iff eventually_sequentially ..
-
lemma LIMSEQ_iff:
fixes L :: "'a::real_normed_vector"
shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
unfolding LIMSEQ_def dist_norm ..
-lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
- unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
-
-lemma metric_LIMSEQ_I:
- "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
-by (simp add: LIMSEQ_def)
-
-lemma metric_LIMSEQ_D:
- "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
-by (simp add: LIMSEQ_def)
-
lemma LIMSEQ_I:
fixes L :: "'a::real_normed_vector"
shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
@@ -344,31 +326,11 @@
by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
Bseq_mono_convergent)
-subsection {* Cauchy Sequences *}
-
-lemma metric_CauchyI:
- "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
- by (simp add: Cauchy_def)
-
-lemma metric_CauchyD:
- "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
- by (simp add: Cauchy_def)
-
lemma Cauchy_iff:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
unfolding Cauchy_def dist_norm ..
-lemma Cauchy_iff2:
- "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
-apply (simp add: Cauchy_iff, auto)
-apply (drule reals_Archimedean, safe)
-apply (drule_tac x = n in spec, auto)
-apply (rule_tac x = M in exI, auto)
-apply (drule_tac x = m in spec, simp)
-apply (drule_tac x = na in spec, auto)
-done
-
lemma CauchyI:
fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
@@ -379,14 +341,6 @@
shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
by (simp add: Cauchy_iff)
-lemma Cauchy_subseq_Cauchy:
- "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
-apply (auto simp add: Cauchy_def)
-apply (drule_tac x=e in spec, clarify)
-apply (rule_tac x=M in exI, clarify)
-apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
-done
-
subsubsection {* Cauchy Sequences are Bounded *}
text{*A Cauchy sequence is bounded -- this is the standard
@@ -412,129 +366,9 @@
apply (simp add: order_less_imp_le)
done
-subsubsection {* Cauchy Sequences are Convergent *}
-
-class complete_space = metric_space +
- assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
-
class banach = real_normed_vector + complete_space
-theorem LIMSEQ_imp_Cauchy:
- assumes X: "X ----> a" shows "Cauchy X"
-proof (rule metric_CauchyI)
- fix e::real assume "0 < e"
- hence "0 < e/2" by simp
- with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
- then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
- show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
- proof (intro exI allI impI)
- fix m assume "N \<le> m"
- hence m: "dist (X m) a < e/2" using N by fast
- fix n assume "N \<le> n"
- hence n: "dist (X n) a < e/2" using N by fast
- have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
- by (rule dist_triangle2)
- also from m n have "\<dots> < e" by simp
- finally show "dist (X m) (X n) < e" .
- qed
-qed
-
-lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
-unfolding convergent_def
-by (erule exE, erule LIMSEQ_imp_Cauchy)
-
-lemma Cauchy_convergent_iff:
- fixes X :: "nat \<Rightarrow> 'a::complete_space"
- shows "Cauchy X = convergent X"
-by (fast intro: Cauchy_convergent convergent_Cauchy)
-
-text {*
-Proof that Cauchy sequences converge based on the one from
-http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
-*}
-
-text {*
- If sequence @{term "X"} is Cauchy, then its limit is the lub of
- @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
-*}
-
-lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
-by (simp add: isUbI setleI)
-
-lemma real_Cauchy_convergent:
- fixes X :: "nat \<Rightarrow> real"
- assumes X: "Cauchy X"
- shows "convergent X"
-proof -
- def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
- then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
-
- { fix N x assume N: "\<forall>n\<ge>N. X n < x"
- have "isUb UNIV S x"
- proof (rule isUb_UNIV_I)
- fix y::real assume "y \<in> S"
- hence "\<exists>M. \<forall>n\<ge>M. y < X n"
- by (simp add: S_def)
- then obtain M where "\<forall>n\<ge>M. y < X n" ..
- hence "y < X (max M N)" by simp
- also have "\<dots> < x" using N by simp
- finally show "y \<le> x"
- by (rule order_less_imp_le)
- qed }
- note bound_isUb = this
-
- have "\<exists>u. isLub UNIV S u"
- proof (rule reals_complete)
- obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
- using CauchyD [OF X zero_less_one] by auto
- hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
- show "\<exists>x. x \<in> S"
- proof
- from N have "\<forall>n\<ge>N. X N - 1 < X n"
- by (simp add: abs_diff_less_iff)
- thus "X N - 1 \<in> S" by (rule mem_S)
- qed
- show "\<exists>u. isUb UNIV S u"
- proof
- from N have "\<forall>n\<ge>N. X n < X N + 1"
- by (simp add: abs_diff_less_iff)
- thus "isUb UNIV S (X N + 1)"
- by (rule bound_isUb)
- qed
- qed
- then obtain x where x: "isLub UNIV S x" ..
- have "X ----> x"
- proof (rule LIMSEQ_I)
- fix r::real assume "0 < r"
- hence r: "0 < r/2" by simp
- obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
- using CauchyD [OF X r] by auto
- hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
- hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
- by (simp only: real_norm_def abs_diff_less_iff)
-
- from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
- hence "X N - r/2 \<in> S" by (rule mem_S)
- hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
-
- from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
- hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
- hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
-
- show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
- proof (intro exI allI impI)
- fix n assume n: "N \<le> n"
- from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
- thus "norm (X n - x) < r" using 1 2
- by (simp add: abs_diff_less_iff)
- qed
- qed
- then show ?thesis unfolding convergent_def by auto
-qed
-
-instance real :: banach
- by intro_classes (rule real_Cauchy_convergent)
-
+instance real :: banach by default
subsection {* Power Sequences *}
@@ -593,33 +427,4 @@
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
by (rule LIMSEQ_power_zero) simp
-lemma tendsto_at_topI_sequentially:
- fixes f :: "real \<Rightarrow> real"
- assumes mono: "mono f"
- assumes limseq: "(\<lambda>n. f (real n)) ----> y"
- shows "(f ---> y) at_top"
-proof (rule tendstoI)
- fix e :: real assume "0 < e"
- with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
- by (auto simp: LIMSEQ_def dist_real_def)
- { fix x :: real
- from ex_le_of_nat[of x] guess n ..
- note monoD[OF mono this]
- also have "f (real_of_nat n) \<le> y"
- by (rule LIMSEQ_le_const[OF limseq])
- (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
- finally have "f x \<le> y" . }
- note le = this
- have "eventually (\<lambda>x. real N \<le> x) at_top"
- by (rule eventually_ge_at_top)
- then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
- proof eventually_elim
- fix x assume N': "real N \<le> x"
- with N[of N] le have "y - f (real N) < e" by auto
- moreover note monoD[OF mono N']
- ultimately show "dist (f x) y < e"
- using le[of x] by (auto simp: dist_real_def field_simps)
- qed
-qed
-
end