src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 changeset 53640 3170b5eb9f5a parent 53597 ea99a7964174 child 53813 0a62ad289c4d
```     1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sat Sep 14 22:50:15 2013 +0200
1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sat Sep 14 23:52:36 2013 +0200
1.3 @@ -25,7 +25,7 @@
1.4    using dist_triangle[of y z x] by (simp add: dist_commute)
1.5
1.6  (* LEGACY *)
1.7 -lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"
1.8 +lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"
1.9    by (rule LIMSEQ_subseq_LIMSEQ)
1.10
1.11  lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
1.12 @@ -85,7 +85,7 @@
1.13    show "topological_basis B"
1.14      using assms unfolding topological_basis_def
1.15    proof safe
1.16 -    fix O'::"'a set"
1.17 +    fix O' :: "'a set"
1.18      assume "open O'"
1.19      with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
1.20        by (force intro: bchoice simp: Bex_def)
1.21 @@ -138,14 +138,14 @@
1.22  qed
1.23
1.24  lemma basis_dense:
1.25 -  fixes B::"'a set set"
1.26 -    and f::"'a set \<Rightarrow> 'a"
1.27 +  fixes B :: "'a set set"
1.28 +    and f :: "'a set \<Rightarrow> 'a"
1.29    assumes "topological_basis B"
1.30      and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
1.31    shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"
1.32  proof (intro allI impI)
1.33 -  fix X::"'a set"
1.34 -  assume "open X" "X \<noteq> {}"
1.35 +  fix X :: "'a set"
1.36 +  assume "open X" and "X \<noteq> {}"
1.37    from topological_basisE[OF `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
1.38    guess B' . note B' = this
1.39    then show "\<exists>B'\<in>B. f B' \<in> X"
1.40 @@ -180,7 +180,7 @@
1.41  subsection {* Countable Basis *}
1.42
1.43  locale countable_basis =
1.44 -  fixes B::"'a::topological_space set set"
1.45 +  fixes B :: "'a::topological_space set set"
1.46    assumes is_basis: "topological_basis B"
1.47      and countable_basis: "countable B"
1.48  begin
1.49 @@ -283,8 +283,9 @@
1.50    proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
1.51      fix a b
1.52      assume x: "a \<in> A" "b \<in> B"
1.53 -    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
1.54 -      unfolding mem_Times_iff by (auto intro: open_Times)
1.55 +    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
1.56 +      unfolding mem_Times_iff
1.57 +      by (auto intro: open_Times)
1.58    next
1.59      fix S
1.60      assume "open S" "x \<in> S"
1.61 @@ -418,7 +419,7 @@
1.62
1.63  text{* Infer the "universe" from union of all sets in the topology. *}
1.64
1.65 -definition "topspace T =  \<Union>{S. openin T S}"
1.66 +definition "topspace T = \<Union>{S. openin T S}"
1.67
1.68  subsubsection {* Main properties of open sets *}
1.69
1.70 @@ -1007,7 +1008,7 @@
1.71
1.72  lemma islimpt_approachable_le:
1.73    fixes x :: "'a::metric_space"
1.74 -  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
1.75 +  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
1.76    unfolding islimpt_approachable
1.77    using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
1.78      THEN arg_cong [where f=Not]]
1.79 @@ -1043,7 +1044,7 @@
1.80  lemma finite_set_avoid:
1.81    fixes a :: "'a::metric_space"
1.82    assumes fS: "finite S"
1.83 -  shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
1.84 +  shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
1.85  proof (induct rule: finite_induct[OF fS])
1.86    case 1
1.87    then show ?case by (auto intro: zero_less_one)
1.88 @@ -1423,8 +1424,9 @@
1.89    apply (drule_tac x=UNIV in spec, simp)
1.90    done
1.91
1.92 -lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"
1.93 -  using islimpt_in_closure by (metis trivial_limit_within)
1.94 +lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
1.95 +  using islimpt_in_closure
1.96 +  by (metis trivial_limit_within)
1.97
1.98  text {* Some property holds "sufficiently close" to the limit point. *}
1.99
1.100 @@ -1463,19 +1465,19 @@
1.101  text{* Show that they yield usual definitions in the various cases. *}
1.102
1.103  lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
1.104 -           (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
1.105 +    (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
1.106    by (auto simp add: tendsto_iff eventually_at_le dist_nz)
1.107
1.108  lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
1.109 -        (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
1.110 +    (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a  < d \<longrightarrow> dist (f x) l < e)"
1.111    by (auto simp add: tendsto_iff eventually_at dist_nz)
1.112
1.113  lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
1.114 -        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
1.115 +    (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"
1.116    by (auto simp add: tendsto_iff eventually_at2)
1.117
1.118  lemma Lim_at_infinity:
1.119 -  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
1.120 +  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
1.121    by (auto simp add: tendsto_iff eventually_at_infinity)
1.122
1.123  lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
1.124 @@ -1489,7 +1491,8 @@
1.125  lemma Lim_Un:
1.126    assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
1.127    shows "(f ---> l) (at x within (S \<union> T))"
1.128 -  using assms unfolding tendsto_def eventually_at_filter
1.129 +  using assms
1.130 +  unfolding tendsto_def eventually_at_filter
1.131    apply clarify
1.132    apply (drule spec, drule (1) mp, drule (1) mp)
1.133    apply (drule spec, drule (1) mp, drule (1) mp)
1.134 @@ -1515,10 +1518,10 @@
1.135    from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
1.136    {
1.137      assume "?lhs"
1.138 -    then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1.139 +    then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
1.140        unfolding eventually_at_topological
1.141        by auto
1.142 -    with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
1.143 +    with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
1.144        by auto
1.145      then show "?rhs"
1.146        unfolding eventually_at_topological by auto
1.147 @@ -1546,11 +1549,11 @@
1.148    assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
1.149      and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
1.150    shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
1.151 -proof cases
1.152 -  assume "{x<..} \<inter> I = {}"
1.153 +proof (cases "{x<..} \<inter> I = {}")
1.154 +  case True
1.155    then show ?thesis by (simp add: Lim_within_empty)
1.156  next
1.157 -  assume e: "{x<..} \<inter> I \<noteq> {}"
1.158 +  case False
1.159    show ?thesis
1.160    proof (rule order_tendstoI)
1.161      fix a
1.162 @@ -1558,7 +1561,7 @@
1.163      {
1.164        fix y
1.165        assume "y \<in> {x<..} \<inter> I"
1.166 -      with e bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
1.167 +      with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
1.168          by (auto intro: cInf_lower)
1.169        with a have "a < f y"
1.170          by (blast intro: less_le_trans)
1.171 @@ -1568,7 +1571,7 @@
1.172    next
1.173      fix a
1.174      assume "Inf (f ` ({x<..} \<inter> I)) < a"
1.175 -    from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a"
1.176 +    from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
1.177        by auto
1.178      then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
1.179        unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)
1.180 @@ -1625,7 +1628,7 @@
1.181      and A :: "'a filter"
1.182    assumes "(f ---> l) A"
1.183      and "l \<noteq> 0"
1.184 -  shows "((inverse o f) ---> inverse l) A"
1.185 +  shows "((inverse \<circ> f) ---> inverse l) A"
1.186    unfolding o_def using assms by (rule tendsto_inverse)
1.187
1.188  lemma Lim_null:
1.189 @@ -1646,7 +1649,7 @@
1.190  lemma Lim_transform_bound:
1.191    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.192      and g :: "'a \<Rightarrow> 'c::real_normed_vector"
1.193 -  assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"
1.194 +  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
1.195      and "(g ---> 0) net"
1.196    shows "(f ---> 0) net"
1.197    using assms(1) tendsto_norm_zero [OF assms(2)]
1.198 @@ -1657,7 +1660,7 @@
1.199  lemma Lim_in_closed_set:
1.200    assumes "closed S"
1.201      and "eventually (\<lambda>x. f(x) \<in> S) net"
1.202 -    and "\<not>(trivial_limit net)" "(f ---> l) net"
1.203 +    and "\<not> trivial_limit net" "(f ---> l) net"
1.204    shows "l \<in> S"
1.205  proof (rule ccontr)
1.206    assume "l \<notin> S"
1.207 @@ -1676,8 +1679,8 @@
1.208  lemma Lim_dist_ubound:
1.209    assumes "\<not>(trivial_limit net)"
1.210      and "(f ---> l) net"
1.211 -    and "eventually (\<lambda>x. dist a (f x) <= e) net"
1.212 -  shows "dist a l <= e"
1.213 +    and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
1.214 +  shows "dist a l \<le> e"
1.215  proof -
1.216    have "dist a l \<in> {..e}"
1.217    proof (rule Lim_in_closed_set)
1.218 @@ -1714,7 +1717,9 @@
1.219
1.220  lemma Lim_norm_lbound:
1.221    fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
1.222 -  assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
1.223 +  assumes "\<not> trivial_limit net"
1.224 +    and "(f ---> l) net"
1.225 +    and "eventually (\<lambda>x. e \<le> norm (f x)) net"
1.226    shows "e \<le> norm l"
1.227  proof -
1.228    have "norm l \<in> {e..}"
1.229 @@ -1946,7 +1951,7 @@
1.230
1.231
1.232  lemma not_trivial_limit_within_ball:
1.233 -  "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
1.234 +  "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
1.235    (is "?lhs = ?rhs")
1.236  proof -
1.237    {
1.238 @@ -1954,12 +1959,12 @@
1.239      {
1.240        fix e :: real
1.241        assume "e > 0"
1.242 -      then obtain y where "y:(S-{x}) & dist y x < e"
1.243 +      then obtain y where "y \<in> S - {x}" and "dist y x < e"
1.244          using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
1.245          by auto
1.246 -      then have "y : (S Int ball x e - {x})"
1.247 +      then have "y \<in> S \<inter> ball x e - {x}"
1.248          unfolding ball_def by (simp add: dist_commute)
1.249 -      then have "S Int ball x e - {x} ~= {}" by blast
1.250 +      then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
1.251      }
1.252      then have "?rhs" by auto
1.253    }
1.254 @@ -1969,11 +1974,11 @@
1.255      {
1.256        fix e :: real
1.257        assume "e > 0"
1.258 -      then obtain y where "y : (S Int ball x e - {x})"
1.259 +      then obtain y where "y \<in> S \<inter> ball x e - {x}"
1.260          using `?rhs` by blast
1.261 -      then have "y:(S-{x}) & dist y x < e"
1.262 -        unfolding ball_def by (simp add: dist_commute)
1.263 -      then have "EX y:(S-{x}). dist y x < e"
1.264 +      then have "y \<in> S - {x}" and "dist y x < e"
1.265 +        unfolding ball_def by (simp_all add: dist_commute)
1.266 +      then have "\<exists>y \<in> S - {x}. dist y x < e"
1.267          by auto
1.268      }
1.269      then have "?lhs"
1.270 @@ -2004,16 +2009,18 @@
1.271    assumes "a \<in> A"
1.272    shows "infdist a A = 0"
1.273  proof -
1.274 -  from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
1.275 -  with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
1.276 +  from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0"
1.277 +    by auto
1.278 +  with infdist_nonneg[of a A] assms show "infdist a A = 0"
1.279 +    by auto
1.280  qed
1.281
1.282  lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
1.283 -proof cases
1.284 -  assume "A = {}"
1.285 +proof (cases "A = {}")
1.286 +  case True
1.287    then show ?thesis by (simp add: infdist_def)
1.288  next
1.289 -  assume "A \<noteq> {}"
1.290 +  case False
1.291    then obtain a where "a \<in> A" by auto
1.292    have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
1.293    proof (rule cInf_greatest)
1.294 @@ -2202,7 +2209,7 @@
1.295    ultimately show "?rhs" by auto
1.296  next
1.297    assume "?rhs"
1.298 -  then have "e>0" by auto
1.299 +  then have "e > 0" by auto
1.300    {
1.301      fix d :: real
1.302      assume "d > 0"
1.303 @@ -2340,7 +2347,7 @@
1.304  lemma interior_cball:
1.305    fixes x :: "'a::{real_normed_vector, perfect_space}"
1.306    shows "interior (cball x e) = ball x e"
1.307 -proof (cases "e\<ge>0")
1.308 +proof (cases "e \<ge> 0")
1.309    case False note cs = this
1.310    from cs have "ball x e = {}"
1.311      using ball_empty[of e x] by auto
1.312 @@ -2409,7 +2416,9 @@
1.313    then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
1.314      by auto
1.315    ultimately show ?thesis
1.316 -    using interior_unique[of "ball x e" "cball x e"] using open_ball[of x e] by auto
1.317 +    using interior_unique[of "ball x e" "cball x e"]
1.318 +    using open_ball[of x e]
1.319 +    by auto
1.320  qed
1.321
1.322  lemma frontier_ball:
1.323 @@ -2422,13 +2431,13 @@
1.324
1.325  lemma frontier_cball:
1.326    fixes a :: "'a::{real_normed_vector, perfect_space}"
1.327 -  shows "frontier(cball a e) = {x. dist a x = e}"
1.328 +  shows "frontier (cball a e) = {x. dist a x = e}"
1.329    apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
1.331    apply arith
1.332    done
1.333
1.334 -lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
1.335 +lemma cball_eq_empty: "cball x e = {} \<longleftrightarrow> e < 0"
1.336    apply (simp add: set_eq_iff not_le)
1.337    apply (metis zero_le_dist dist_self order_less_le_trans)
1.338    done
1.339 @@ -2438,7 +2447,7 @@
1.340
1.341  lemma cball_eq_sing:
1.342    fixes x :: "'a::{metric_space,perfect_space}"
1.343 -  shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
1.344 +  shows "cball x e = {x} \<longleftrightarrow> e = 0"
1.345  proof (rule linorder_cases)
1.346    assume e: "0 < e"
1.347    obtain a where "a \<noteq> x" "dist a x < e"
1.348 @@ -2466,7 +2475,8 @@
1.349  lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
1.350    unfolding bounded_def
1.351    apply safe
1.352 -  apply (rule_tac x="dist a x + e" in exI, clarify)
1.353 +  apply (rule_tac x="dist a x + e" in exI)
1.354 +  apply clarify
1.355    apply (drule (1) bspec)
1.356    apply (erule order_trans [OF dist_triangle add_left_mono])
1.357    apply auto
1.358 @@ -2526,7 +2536,7 @@
1.359    apply auto
1.360    done
1.361
1.362 -lemma bounded_ball[simp,intro]: "bounded(ball x e)"
1.363 +lemma bounded_ball[simp,intro]: "bounded (ball x e)"
1.364    by (metis ball_subset_cball bounded_cball bounded_subset)
1.365
1.366  lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
1.367 @@ -2534,14 +2544,16 @@
1.368    apply (rename_tac x y r s)
1.369    apply (rule_tac x=x in exI)
1.370    apply (rule_tac x="max r (dist x y + s)" in exI)
1.371 -  apply (rule ballI, rename_tac z, safe)
1.372 -  apply (drule (1) bspec, simp)
1.373 +  apply (rule ballI)
1.374 +  apply safe
1.375 +  apply (drule (1) bspec)
1.376 +  apply simp
1.377    apply (drule (1) bspec)
1.378    apply (rule min_max.le_supI2)
1.379    apply (erule order_trans [OF dist_triangle add_left_mono])
1.380    done
1.381
1.382 -lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
1.383 +lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
1.384    by (induct rule: finite_induct[of F]) auto
1.385
1.386  lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
1.387 @@ -2549,22 +2561,27 @@
1.388
1.389  lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
1.390  proof -
1.391 -  have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
1.392 -  then have "bounded {x}" unfolding bounded_def by fast
1.393 -  then show ?thesis by (metis insert_is_Un bounded_Un)
1.394 +  have "\<forall>y\<in>{x}. dist x y \<le> 0"
1.395 +    by simp
1.396 +  then have "bounded {x}"
1.397 +    unfolding bounded_def by fast
1.398 +  then show ?thesis
1.399 +    by (metis insert_is_Un bounded_Un)
1.400  qed
1.401
1.402  lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
1.403    by (induct set: finite) simp_all
1.404
1.405 -lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
1.406 +lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
1.408 -  apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
1.409 +  apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x \<le> y \<longrightarrow> x \<le> 1 + abs y)")
1.410    apply metis
1.411    apply arith
1.412    done
1.413
1.414 -lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
1.415 +lemma Bseq_eq_bounded:
1.416 +  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
1.417 +  shows "Bseq f \<longleftrightarrow> bounded (range f)"
1.418    unfolding Bseq_def bounded_pos by auto
1.419
1.420  lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
1.421 @@ -2575,11 +2592,13 @@
1.422
1.423  lemma not_bounded_UNIV[simp, intro]:
1.424    "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
1.425 -proof(auto simp add: bounded_pos not_le)
1.426 +proof (auto simp add: bounded_pos not_le)
1.427    obtain x :: 'a where "x \<noteq> 0"
1.428      using perfect_choose_dist [OF zero_less_one] by fast
1.429 -  fix b::real  assume b: "b >0"
1.430 -  have b1: "b +1 \<ge> 0" using b by simp
1.431 +  fix b :: real
1.432 +  assume b: "b >0"
1.433 +  have b1: "b +1 \<ge> 0"
1.434 +    using b by simp
1.435    with `x \<noteq> 0` have "b < norm (scaleR (b + 1) (sgn x))"
1.437    then show "\<exists>x::'a. b < norm x" ..
1.438 @@ -2590,15 +2609,17 @@
1.439      and "bounded_linear f"
1.440    shows "bounded (f ` S)"
1.441  proof -
1.442 -  from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"
1.443 +  from assms(1) obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
1.444      unfolding bounded_pos by auto
1.445 -  from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x"
1.446 +  from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
1.447      using bounded_linear.pos_bounded by (auto simp add: mult_ac)
1.448    {
1.449      fix x
1.450 -    assume "x\<in>S"
1.451 -    then have "norm x \<le> b" using b by auto
1.452 -    then have "norm (f x) \<le> B * b" using B(2)
1.453 +    assume "x \<in> S"
1.454 +    then have "norm x \<le> b"
1.455 +      using b by auto
1.456 +    then have "norm (f x) \<le> B * b"
1.457 +      using B(2)
1.458        apply (erule_tac x=x in allE)
1.459        apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
1.460        done
1.461 @@ -2624,11 +2645,11 @@
1.462    assumes "bounded S"
1.463    shows "bounded ((\<lambda>x. a + x) ` S)"
1.464  proof -
1.465 -  from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b"
1.466 +  from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
1.467      unfolding bounded_pos by auto
1.468    {
1.469      fix x
1.470 -    assume "x\<in>S"
1.471 +    assume "x \<in> S"
1.472      then have "norm (a + x) \<le> b + norm a"
1.473        using norm_triangle_ineq[of a x] b by auto
1.474    }
1.475 @@ -2648,7 +2669,8 @@
1.476
1.477  lemma bounded_has_Sup:
1.478    fixes S :: "real set"
1.479 -  assumes "bounded S" "S \<noteq> {}"
1.480 +  assumes "bounded S"
1.481 +    and "S \<noteq> {}"
1.482    shows "\<forall>x\<in>S. x \<le> Sup S"
1.483      and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
1.484  proof
1.485 @@ -2679,18 +2701,19 @@
1.486
1.487  lemma bounded_has_Inf:
1.488    fixes S :: "real set"
1.489 -  assumes "bounded S"  "S \<noteq> {}"
1.490 +  assumes "bounded S"
1.491 +    and "S \<noteq> {}"
1.492    shows "\<forall>x\<in>S. x \<ge> Inf S"
1.493      and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
1.494  proof
1.495    fix x
1.496 -  assume "x\<in>S"
1.497 +  assume "x \<in> S"
1.498    from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"
1.499      unfolding bounded_real by auto
1.500 -  then show "x \<ge> Inf S" using `x\<in>S`
1.501 +  then show "x \<ge> Inf S" using `x \<in> S`
1.502      by (metis cInf_lower_EX abs_le_D2 minus_le_iff)
1.503  next
1.504 -  show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b"
1.505 +  show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
1.506      using assms by (metis cInf_greatest)
1.507  qed
1.508
1.509 @@ -2715,25 +2738,29 @@
1.510  subsubsection {* Bolzano-Weierstrass property *}
1.511
1.512  lemma heine_borel_imp_bolzano_weierstrass:
1.513 -  assumes "compact s" and "infinite t" and "t \<subseteq> s"
1.514 +  assumes "compact s"
1.515 +    and "infinite t"
1.516 +    and "t \<subseteq> s"
1.517    shows "\<exists>x \<in> s. x islimpt t"
1.518  proof (rule ccontr)
1.519    assume "\<not> (\<exists>x \<in> s. x islimpt t)"
1.520 -  then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
1.521 +  then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
1.522      unfolding islimpt_def
1.523      using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
1.524      by auto
1.525 -  obtain g where g: "g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
1.526 +  obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
1.527      using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
1.528      using f by auto
1.529 -  from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
1.530 +  from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
1.531 +    by auto
1.532    {
1.533      fix x y
1.534 -    assume "x\<in>t" "y\<in>t" "f x = f y"
1.535 +    assume "x \<in> t" "y \<in> t" "f x = f y"
1.536      then have "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x"
1.537 -      using f[THEN bspec[where x=x]] and `t\<subseteq>s` by auto
1.538 +      using f[THEN bspec[where x=x]] and `t \<subseteq> s` by auto
1.539      then have "x = y"
1.540 -      using `f x = f y` and f[THEN bspec[where x=y]] and `y\<in>t` and `t\<subseteq>s` by auto
1.541 +      using `f x = f y` and f[THEN bspec[where x=y]] and `y \<in> t` and `t \<subseteq> s`
1.542 +      by auto
1.543    }
1.544    then have "inj_on f t"
1.545      unfolding inj_on_def by simp
1.546 @@ -2742,14 +2769,17 @@
1.547    moreover
1.548    {
1.549      fix x
1.550 -    assume "x\<in>t" "f x \<notin> g"
1.551 -    from g(3) assms(3) `x\<in>t` obtain h where "h\<in>g" and "x\<in>h" by auto
1.552 -    then obtain y where "y\<in>s" "h = f y"
1.553 +    assume "x \<in> t" "f x \<notin> g"
1.554 +    from g(3) assms(3) `x \<in> t` obtain h where "h \<in> g" and "x \<in> h"
1.555 +      by auto
1.556 +    then obtain y where "y \<in> s" "h = f y"
1.557        using g'[THEN bspec[where x=h]] by auto
1.558      then have "y = x"
1.559 -      using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`] by auto
1.560 +      using f[THEN bspec[where x=y]] and `x\<in>t` and `x\<in>h`[unfolded `h = f y`]
1.561 +      by auto
1.562      then have False
1.563 -      using `f x \<notin> g` `h\<in>g` unfolding `h = f y` by auto
1.564 +      using `f x \<notin> g` `h \<in> g` unfolding `h = f y`
1.565 +      by auto
1.566    }
1.567    then have "f ` t \<subseteq> g" by auto
1.568    ultimately show False
1.569 @@ -2786,7 +2816,8 @@
1.570    proof (rule topological_tendstoI)
1.571      fix S
1.572      assume "open S" "l \<in> S"
1.573 -    with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
1.574 +    with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
1.575 +      by auto
1.576      moreover
1.577      {
1.578        fix i
1.579 @@ -2810,12 +2841,18 @@
1.580    shows "infinite (range f)"
1.581  proof
1.582    assume "finite (range f)"
1.583 -  then have "closed (range f)" by (rule finite_imp_closed)
1.584 -  then have "open (- range f)" by (rule open_Compl)
1.585 -  from assms(1) have "l \<in> - range f" by auto
1.586 +  then have "closed (range f)"
1.587 +    by (rule finite_imp_closed)
1.588 +  then have "open (- range f)"
1.589 +    by (rule open_Compl)
1.590 +  from assms(1) have "l \<in> - range f"
1.591 +    by auto
1.592    from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
1.593 -    using `open (- range f)` `l \<in> - range f` by (rule topological_tendstoD)
1.594 -  then show False unfolding eventually_sequentially by auto
1.595 +    using `open (- range f)` `l \<in> - range f`
1.596 +    by (rule topological_tendstoD)
1.597 +  then show False
1.598 +    unfolding eventually_sequentially
1.599 +    by auto
1.600  qed
1.601
1.602  lemma closure_insert:
1.603 @@ -2928,7 +2965,7 @@
1.604  qed
1.605
1.606  lemma bolzano_weierstrass_imp_closed:
1.607 -  fixes s :: "'a::{first_countable_topology, t2_space} set"
1.608 +  fixes s :: "'a::{first_countable_topology,t2_space} set"
1.609    assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
1.610    shows "closed s"
1.611  proof -
1.612 @@ -3276,7 +3313,7 @@
1.613
1.614  definition seq_compact :: "'a::topological_space set \<Rightarrow> bool"
1.615    where "seq_compact S \<longleftrightarrow>
1.616 -    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
1.617 +    (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially))"
1.618
1.619  lemma seq_compact_imp_countably_compact:
1.620    fixes U :: "'a :: first_countable_topology set"
1.621 @@ -3391,7 +3428,7 @@
1.622  qed
1.623
1.624  lemma seq_compactI:
1.625 -  assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
1.626 +  assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
1.627    shows "seq_compact S"
1.628    unfolding seq_compact_def using assms by fast
1.629
1.630 @@ -3611,7 +3648,7 @@
1.631
1.632  lemma compact_def:
1.633    "compact (S :: 'a::metric_space set) \<longleftrightarrow>
1.634 -   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"
1.635 +   (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) ----> l))"
1.636    unfolding compact_eq_seq_compact_metric seq_compact_def by auto
1.637
1.638  subsubsection {* Complete the chain of compactness variants *}
1.639 @@ -4514,7 +4551,7 @@
1.640    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
1.641    shows "continuous (at a within s) f \<longleftrightarrow>
1.642      (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
1.643 -         \<longrightarrow> ((f o x) ---> f a) sequentially)"
1.644 +         \<longrightarrow> ((f \<circ> x) ---> f a) sequentially)"
1.645    (is "?lhs = ?rhs")
1.646  proof
1.647    assume ?lhs
1.648 @@ -4546,14 +4583,14 @@
1.649  lemma continuous_at_sequentially:
1.650    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
1.651    shows "continuous (at a) f \<longleftrightarrow>
1.652 -    (\<forall>x. (x ---> a) sequentially --> ((f o x) ---> f a) sequentially)"
1.653 +    (\<forall>x. (x ---> a) sequentially --> ((f \<circ> x) ---> f a) sequentially)"
1.654    using continuous_within_sequentially[of a UNIV f] by simp
1.655
1.656  lemma continuous_on_sequentially:
1.657    fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
1.658    shows "continuous_on s f \<longleftrightarrow>
1.659      (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
1.660 -      --> ((f o x) ---> f(a)) sequentially)"
1.661 +      --> ((f \<circ> x) ---> f a) sequentially)"
1.662    (is "?lhs = ?rhs")
1.663  proof
1.664    assume ?rhs
1.665 @@ -4804,8 +4841,8 @@
1.666
1.667  lemma uniformly_continuous_on_compose[continuous_on_intros]:
1.668    assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f ` s) g"
1.669 -  shows "uniformly_continuous_on s (g o f)"
1.670 -proof-
1.671 +  shows "uniformly_continuous_on s (g \<circ> f)"
1.672 +proof -
1.673    {
1.674      fix e :: real
1.675      assume "e > 0"
1.676 @@ -6818,7 +6855,7 @@
1.677
1.678  lemma Lim_component_eq:
1.679    fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
1.680 -  assumes net: "(f ---> l) net" "~(trivial_limit net)"
1.681 +  assumes net: "(f ---> l) net" "\<not> trivial_limit net"
1.682      and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
1.683    shows "l\<bullet>i = b"
1.684    using ev[unfolded order_eq_iff eventually_conj_iff]
1.685 @@ -6887,8 +6924,7 @@
1.686    (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
1.687    (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
1.688
1.689 -definition
1.690 -  homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
1.691 +definition homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
1.692      (infixr "homeomorphic" 60)
1.693    where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
1.694
1.695 @@ -7099,12 +7135,12 @@
1.696
1.697  lemma cauchy_isometric:
1.698    fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
1.699 -  assumes e: "0 < e"
1.700 +  assumes e: "e > 0"
1.701      and s: "subspace s"
1.702      and f: "bounded_linear f"
1.703 -    and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
1.704 -    and xs: "\<forall>n::nat. x n \<in> s"
1.705 -    and cf: "Cauchy(f o x)"
1.706 +    and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
1.707 +    and xs: "\<forall>n. x n \<in> s"
1.708 +    and cf: "Cauchy (f \<circ> x)"
1.709    shows "Cauchy x"
1.710  proof -
1.711    interpret f: bounded_linear f by fact
1.712 @@ -7145,24 +7181,31 @@
1.713      fix g
1.714      assume as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g"
1.715      then obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
1.716 -      using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
1.717 -    then have x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
1.718 -    then have "f \<circ> x = g" unfolding fun_eq_iff by auto
1.719 +      using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"]
1.720 +      by auto
1.721 +    then have x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)"
1.722 +      by auto
1.723 +    then have "f \<circ> x = g"
1.724 +      unfolding fun_eq_iff
1.725 +      by auto
1.726      then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
1.727        using cs[unfolded complete_def, THEN spec[where x="x"]]
1.728 -      using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1) by auto
1.729 +      using cauchy_isometric[OF `0<e` s f normf] and cfg and x(1)
1.730 +      by auto
1.731      then have "\<exists>l\<in>f ` s. (g ---> l) sequentially"
1.732        using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
1.733 -      unfolding `f \<circ> x = g` by auto
1.734 +      unfolding `f \<circ> x = g`
1.735 +      by auto
1.736    }
1.737 -  then show ?thesis unfolding complete_def by auto
1.738 +  then show ?thesis
1.739 +    unfolding complete_def by auto
1.740  qed
1.741
1.742  lemma injective_imp_isometric:
1.743    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
1.744    assumes s: "closed s" "subspace s"
1.745 -    and f: "bounded_linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
1.746 -  shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
1.747 +    and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
1.748 +  shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
1.749  proof (cases "s \<subseteq> {0::'a}")
1.750    case True
1.751    {
1.752 @@ -7175,8 +7218,10 @@
1.753  next
1.754    interpret f: bounded_linear f by fact
1.755    case False
1.756 -  then obtain a where a:"a\<noteq>0" "a\<in>s" by auto
1.757 -  from False have "s \<noteq> {}" by auto
1.758 +  then obtain a where a: "a \<noteq> 0" "a \<in> s"
1.759 +    by auto
1.760 +  from False have "s \<noteq> {}"
1.761 +    by auto
1.762    let ?S = "{f x| x. (x \<in> s \<and> norm x = norm a)}"
1.763    let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
1.764    let ?S'' = "{x::'a. norm x = norm a}"
```