# HG changeset patch
# User hoelzl
# Date 1364296861 -3600
# Node ID f415febf42347e9c4b4e7c087be2bd89635356bb
# Parent  609914f0934a8409376c0022841b263f0571fab9
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces

diff -r 609914f0934a -r f415febf4234 src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Tue Mar 26 12:21:00 2013 +0100
+++ b/src/HOL/Limits.thy	Tue Mar 26 12:21:01 2013 +0100
@@ -57,7 +57,21 @@
   "at_bot \<le> (at_infinity :: real filter)"
   unfolding at_infinity_eq_at_top_bot by simp
 
-subsection {* Boundedness *}
+subsubsection {* Boundedness *}
+
+definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
+  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
+
+abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
+  "Bseq X \<equiv> Bfun X sequentially"
+
+lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
+
+lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
+  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
+
+lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
+  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
 
 lemma Bfun_def:
   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
@@ -87,6 +101,154 @@
   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
 using assms unfolding Bfun_def by fast
 
+lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
+  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
+  apply (erule_tac x=1 in allE)
+  apply simp
+  apply safe
+  apply (rule_tac x="X M" in exI)
+  apply (rule_tac x=1 in exI)
+  apply (erule_tac x=M in allE)
+  apply simp
+  apply (rule_tac x=M in exI)
+  apply (auto simp: dist_commute)
+  done
+
+
+subsubsection {* Bounded Sequences *}
+
+lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
+  by (intro BfunI) (auto simp: eventually_sequentially)
+
+lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
+  by (intro BfunI) (auto simp: eventually_sequentially)
+
+lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
+  unfolding Bfun_def eventually_sequentially
+proof safe
+  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
+  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
+    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
+       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
+qed auto
+
+lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+unfolding Bseq_def by auto
+
+lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
+by (simp add: Bseq_def)
+
+lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
+by (auto simp add: Bseq_def)
+
+lemma lemma_NBseq_def:
+  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
+proof safe
+  fix K :: real
+  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
+  then have "K \<le> real (Suc n)" by auto
+  moreover assume "\<forall>m. norm (X m) \<le> K"
+  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
+    by (blast intro: order_trans)
+  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
+qed (force simp add: real_of_nat_Suc)
+
+text{* alternative definition for Bseq *}
+lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
+apply (simp add: Bseq_def)
+apply (simp (no_asm) add: lemma_NBseq_def)
+done
+
+lemma lemma_NBseq_def2:
+     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
+apply (subst lemma_NBseq_def, auto)
+apply (rule_tac x = "Suc N" in exI)
+apply (rule_tac [2] x = N in exI)
+apply (auto simp add: real_of_nat_Suc)
+ prefer 2 apply (blast intro: order_less_imp_le)
+apply (drule_tac x = n in spec, simp)
+done
+
+(* yet another definition for Bseq *)
+lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
+by (simp add: Bseq_def lemma_NBseq_def2)
+
+subsubsection{*A Few More Equivalence Theorems for Boundedness*}
+
+text{*alternative formulation for boundedness*}
+lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
+apply (unfold Bseq_def, safe)
+apply (rule_tac [2] x = "k + norm x" in exI)
+apply (rule_tac x = K in exI, simp)
+apply (rule exI [where x = 0], auto)
+apply (erule order_less_le_trans, simp)
+apply (drule_tac x=n in spec, fold diff_minus)
+apply (drule order_trans [OF norm_triangle_ineq2])
+apply simp
+done
+
+text{*alternative formulation for boundedness*}
+lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
+apply safe
+apply (simp add: Bseq_def, safe)
+apply (rule_tac x = "K + norm (X N)" in exI)
+apply auto
+apply (erule order_less_le_trans, simp)
+apply (rule_tac x = N in exI, safe)
+apply (drule_tac x = n in spec)
+apply (rule order_trans [OF norm_triangle_ineq], simp)
+apply (auto simp add: Bseq_iff2)
+done
+
+lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
+apply (simp add: Bseq_def)
+apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
+apply (drule_tac x = n in spec, arith)
+done
+
+subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
+
+lemma Bseq_isUb:
+  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
+by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
+
+text{* Use completeness of reals (supremum property)
+   to show that any bounded sequence has a least upper bound*}
+
+lemma Bseq_isLub:
+  "!!(X::nat=>real). Bseq X ==>
+   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
+by (blast intro: reals_complete Bseq_isUb)
+
+lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
+  by (simp add: Bseq_def)
+
+lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
+  apply (simp add: subset_eq)
+  apply (rule BseqI'[where K="max (norm a) (norm b)"])
+  apply (erule_tac x=n in allE)
+  apply auto
+  done
+
+lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
+  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
+
+lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
+  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
+
+subsection {* Bounded Monotonic Sequences *}
+
+subsubsection{*A Bounded and Monotonic Sequence Converges*}
+
+(* TODO: delete *)
+(* FIXME: one use in NSA/HSEQ.thy *)
+lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
+  apply (rule_tac x="X m" in exI)
+  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
+  unfolding eventually_sequentially
+  apply blast
+  done
+
 subsection {* Convergence to Zero *}
 
 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
@@ -242,6 +404,37 @@
 
 subsubsection {* Distance and norms *}
 
+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 continuous_dist[continuous_intros]:
+  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
+  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
+  unfolding continuous_def by (rule tendsto_dist)
+
+lemma continuous_on_dist[continuous_on_intros]:
+  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
+  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
+  unfolding continuous_on_def by (auto intro: tendsto_dist)
+
 lemma tendsto_norm [tendsto_intros]:
   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
   unfolding norm_conv_dist by (intro tendsto_intros)
@@ -1152,123 +1345,6 @@
 apply (drule tendsto_minus, auto)
 done
 
-subsection {* Bounded Monotonic Sequences *}
-
-subsubsection {* Bounded Sequences *}
-
-lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
-  by (intro BfunI) (auto simp: eventually_sequentially)
-
-lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
-  by (intro BfunI) (auto simp: eventually_sequentially)
-
-lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
-  unfolding Bfun_def eventually_sequentially
-proof safe
-  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
-  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
-    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
-       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
-qed auto
-
-lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-unfolding Bseq_def by auto
-
-lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
-by (simp add: Bseq_def)
-
-lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
-by (auto simp add: Bseq_def)
-
-lemma lemma_NBseq_def:
-  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
-proof safe
-  fix K :: real
-  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
-  then have "K \<le> real (Suc n)" by auto
-  moreover assume "\<forall>m. norm (X m) \<le> K"
-  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
-    by (blast intro: order_trans)
-  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
-qed (force simp add: real_of_nat_Suc)
-
-text{* alternative definition for Bseq *}
-lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
-apply (simp add: Bseq_def)
-apply (simp (no_asm) add: lemma_NBseq_def)
-done
-
-lemma lemma_NBseq_def2:
-     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
-apply (subst lemma_NBseq_def, auto)
-apply (rule_tac x = "Suc N" in exI)
-apply (rule_tac [2] x = N in exI)
-apply (auto simp add: real_of_nat_Suc)
- prefer 2 apply (blast intro: order_less_imp_le)
-apply (drule_tac x = n in spec, simp)
-done
-
-(* yet another definition for Bseq *)
-lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
-by (simp add: Bseq_def lemma_NBseq_def2)
-
-subsubsection{*A Few More Equivalence Theorems for Boundedness*}
-
-text{*alternative formulation for boundedness*}
-lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
-apply (unfold Bseq_def, safe)
-apply (rule_tac [2] x = "k + norm x" in exI)
-apply (rule_tac x = K in exI, simp)
-apply (rule exI [where x = 0], auto)
-apply (erule order_less_le_trans, simp)
-apply (drule_tac x=n in spec, fold diff_minus)
-apply (drule order_trans [OF norm_triangle_ineq2])
-apply simp
-done
-
-text{*alternative formulation for boundedness*}
-lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
-apply safe
-apply (simp add: Bseq_def, safe)
-apply (rule_tac x = "K + norm (X N)" in exI)
-apply auto
-apply (erule order_less_le_trans, simp)
-apply (rule_tac x = N in exI, safe)
-apply (drule_tac x = n in spec)
-apply (rule order_trans [OF norm_triangle_ineq], simp)
-apply (auto simp add: Bseq_iff2)
-done
-
-lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
-apply (simp add: Bseq_def)
-apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
-apply (drule_tac x = n in spec, arith)
-done
-
-subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
-
-lemma Bseq_isUb:
-  "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
-by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
-
-text{* Use completeness of reals (supremum property)
-   to show that any bounded sequence has a least upper bound*}
-
-lemma Bseq_isLub:
-  "!!(X::nat=>real). Bseq X ==>
-   \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
-by (blast intro: reals_complete Bseq_isUb)
-
-subsubsection{*A Bounded and Monotonic Sequence Converges*}
-
-(* TODO: delete *)
-(* FIXME: one use in NSA/HSEQ.thy *)
-lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
-  apply (rule_tac x="X m" in exI)
-  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
-  unfolding eventually_sequentially
-  apply blast
-  done
 
 text {* A monotone sequence converges to its least upper bound. *}
 
@@ -1301,9 +1377,6 @@
    "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
   by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
 
-lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
-  by (simp add: Bseq_def)
-
 text{*Main monotonicity theorem*}
 
 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
@@ -1325,19 +1398,6 @@
   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 Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
-  apply (simp add: subset_eq)
-  apply (rule BseqI'[where K="max (norm a) (norm b)"])
-  apply (erule_tac x=n in allE)
-  apply auto
-  done
-
-lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
-  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
-
-lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
-  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
-
 lemma incseq_convergent:
   fixes X :: "nat \<Rightarrow> real"
   assumes "incseq X" and "\<forall>i. X i \<le> B"
@@ -1375,10 +1435,6 @@
 apply simp
 done
 
-class banach = real_normed_vector + complete_space
-
-instance real :: banach by default
-
 subsection {* Power Sequences *}
 
 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
@@ -1615,6 +1671,22 @@
 
 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
+
 lemma (in bounded_linear) isUCont: "isUCont f"
 unfolding isUCont_def dist_norm
 proof (intro allI impI)
@@ -1637,7 +1709,6 @@
 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
 by (rule isUCont [THEN isUCont_Cauchy])
 
-
 lemma LIM_less_bound: 
   fixes f :: "real \<Rightarrow> real"
   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
@@ -1878,5 +1949,6 @@
   fixes f :: "real \<Rightarrow> real"
   shows "f -- c --> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
+
 end
 
diff -r 609914f0934a -r f415febf4234 src/HOL/Metric_Spaces.thy
--- a/src/HOL/Metric_Spaces.thy	Tue Mar 26 12:21:00 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,629 +0,0 @@
-(*  Title:      HOL/Metric_Spaces.thy
-    Author:     Brian Huffman
-    Author:     Johannes Hölzl
-*)
-
-header {* Metric Spaces *}
-
-theory Metric_Spaces
-imports Real 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 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
-
-subclass first_countable_topology
-proof
-  fix x 
-  show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
-  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
-    fix S assume "open S" "x \<in> S"
-    then obtain e where "0 < e" "{y. dist x y < e} \<subseteq> S"
-      by (auto simp: open_dist subset_eq dist_commute)
-    moreover
-    then obtain i where "inverse (Suc i) < e"
-      by (auto dest!: reals_Archimedean)
-    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
-      by auto
-    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
-      by blast
-  qed (auto intro: open_ball)
-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])
-
-subsubsection {* Boundedness *}
-
-definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
-  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
-
-abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
-  "Bseq X \<equiv> Bfun X sequentially"
-
-lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
-
-lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
-  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
-
-lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
-  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
-
-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
-
-lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
-  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
-  apply (erule_tac x=1 in allE)
-  apply simp
-  apply safe
-  apply (rule_tac x="X M" in exI)
-  apply (rule_tac x=1 in exI)
-  apply (erule_tac x=M in allE)
-  apply simp
-  apply (rule_tac x=M in exI)
-  apply (auto simp: dist_commute)
-  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 continuous_dist[continuous_intros]:
-  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
-  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
-  unfolding continuous_def by (rule tendsto_dist)
-
-lemma continuous_on_dist[continuous_on_intros]:
-  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
-  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
-  unfolding continuous_on_def by (auto intro: tendsto_dist)
-
-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
-
diff -r 609914f0934a -r f415febf4234 src/HOL/Real_Vector_Spaces.thy
--- a/src/HOL/Real_Vector_Spaces.thy	Tue Mar 26 12:21:00 2013 +0100
+++ b/src/HOL/Real_Vector_Spaces.thy	Tue Mar 26 12:21:01 2013 +0100
@@ -1,11 +1,12 @@
 (*  Title:      HOL/Real_Vector_Spaces.thy
     Author:     Brian Huffman
+    Author:     Johannes Hölzl
 *)
 
 header {* Vector Spaces and Algebras over the Reals *}
 
 theory Real_Vector_Spaces
-imports Metric_Spaces
+imports Real Topological_Spaces
 begin
 
 subsection {* Locale for additive functions *}
@@ -436,6 +437,9 @@
 
 subsection {* Real normed vector spaces *}
 
+class dist =
+  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
+
 class norm =
   fixes norm :: "'a \<Rightarrow> real"
 
@@ -445,6 +449,9 @@
 class dist_norm = dist + norm + minus +
   assumes dist_norm: "dist x y = norm (x - y)"
 
+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 real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
@@ -653,6 +660,133 @@
   shows "norm (x ^ n) = norm x ^ n"
 by (induct n) (simp_all add: norm_mult)
 
+
+subsection {* Metric spaces *}
+
+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 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
+
+subclass first_countable_topology
+proof
+  fix x 
+  show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
+  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
+    fix S assume "open S" "x \<in> S"
+    then obtain e where "0 < e" "{y. dist x y < e} \<subseteq> S"
+      by (auto simp: open_dist subset_eq dist_commute)
+    moreover
+    then obtain i where "inverse (Suc i) < e"
+      by (auto dest!: reals_Archimedean)
+    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
+      by auto
+    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
+      by blast
+  qed (auto intro: open_ball)
+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
+
 text {* Every normed vector space is a metric space. *}
 
 instance real_normed_vector < metric_space
@@ -670,12 +804,19 @@
 instantiation real :: real_normed_field
 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)"
+
 definition real_norm_def [simp]:
   "norm r = \<bar>r\<bar>"
 
 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)
@@ -685,8 +826,49 @@
 
 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
+
 instance real :: linear_continuum_topology ..
 
+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}. *}
@@ -919,4 +1101,361 @@
     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
 qed
 
+subsection {* Filters and Limits on Metric Space *}
+
+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:
+  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:
+  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 \<le> 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_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 ..
+
+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
+
+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)
+
+subsubsection {* Cauchy Sequences are Convergent *}
+
+class complete_space = metric_space +
+  assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
+
+lemma Cauchy_convergent_iff:
+  fixes X :: "nat \<Rightarrow> 'a::complete_space"
+  shows "Cauchy X = convergent X"
+by (fast intro: Cauchy_convergent convergent_Cauchy)
+
+subsection {* The set of real numbers is a complete metric space *}
+
+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)
+
+class banach = real_normed_vector + complete_space
+
+instance real :: banach by default
+
+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