src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
changeset 44211 bd7c586b902e
parent 44210 eba74571833b
child 44212 4d10e7f342b1
     1.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Aug 15 10:49:48 2011 -0700
     1.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Aug 15 12:13:46 2011 -0700
     1.3 @@ -976,21 +976,10 @@
     1.4  
     1.5  text{* Combining theorems for "eventually" *}
     1.6  
     1.7 -lemma eventually_conjI:
     1.8 -  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
     1.9 -    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
    1.10 -by (rule eventually_conj) (* FIXME: delete *)
    1.11 -
    1.12  lemma eventually_rev_mono:
    1.13    "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
    1.14  using eventually_mono [of P Q] by fast
    1.15  
    1.16 -lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
    1.17 -  by (rule eventually_conj_iff) (* FIXME: delete *)
    1.18 -
    1.19 -lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
    1.20 -  by (rule eventually_False) (* FIXME: delete *)
    1.21 -
    1.22  lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
    1.23    by (simp add: eventually_False)
    1.24  
    1.25 @@ -1255,7 +1244,7 @@
    1.26      hence "dist (f x) 0 < e" by (simp add: dist_norm)
    1.27    }
    1.28    thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
    1.29 -    using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
    1.30 +    using eventually_conj_iff[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
    1.31      using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
    1.32      using assms `e>0` unfolding tendsto_iff by auto
    1.33  qed
    1.34 @@ -1271,7 +1260,7 @@
    1.35      assume "norm (f x) \<le> norm (g x)" "dist (g x) 0 < e"
    1.36      hence "dist (f x) 0 < e" by (simp add: dist_norm)}
    1.37    thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
    1.38 -    using eventually_and[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
    1.39 +    using eventually_conj_iff[of "\<lambda>x. norm (f x) \<le> norm (g x)" "\<lambda>x. dist (g x) 0 < e" net]
    1.40      using eventually_mono[of "\<lambda>x. norm (f x) \<le> norm (g x) \<and> dist (g x) 0 < e" "\<lambda>x. dist (f x) 0 < e" net]
    1.41      using assms `e>0` unfolding tendsto_iff by blast
    1.42  qed
    1.43 @@ -1304,7 +1293,7 @@
    1.44    with assms(2) have "eventually (\<lambda>x. dist (f x) l < dist a l - e) net"
    1.45      by (rule tendstoD)
    1.46    with assms(3) have "eventually (\<lambda>x. dist a (f x) \<le> e \<and> dist (f x) l < dist a l - e) net"
    1.47 -    by (rule eventually_conjI)
    1.48 +    by (rule eventually_conj)
    1.49    then obtain w where "dist a (f w) \<le> e" "dist (f w) l < dist a l - e"
    1.50      using assms(1) eventually_happens by auto
    1.51    hence "dist a (f w) + dist (f w) l < e + (dist a l - e)"
    1.52 @@ -1326,7 +1315,7 @@
    1.53    with assms(2) have "eventually (\<lambda>x. dist (f x) l < norm l - e) net"
    1.54      by (rule tendstoD)
    1.55    with assms(3) have "eventually (\<lambda>x. norm (f x) \<le> e \<and> dist (f x) l < norm l - e) net"
    1.56 -    by (rule eventually_conjI)
    1.57 +    by (rule eventually_conj)
    1.58    then obtain w where "norm (f w) \<le> e" "dist (f w) l < norm l - e"
    1.59      using assms(1) eventually_happens by auto
    1.60    hence "norm (f w - l) < norm l - e" "norm (f w) \<le> e" by (simp_all add: dist_norm)
    1.61 @@ -1345,7 +1334,7 @@
    1.62    with assms(2) have "eventually (\<lambda>x. dist (f x) l < e - norm l) net"
    1.63      by (rule tendstoD)
    1.64    with assms(3) have "eventually (\<lambda>x. e \<le> norm (f x) \<and> dist (f x) l < e - norm l) net"
    1.65 -    by (rule eventually_conjI)
    1.66 +    by (rule eventually_conj)
    1.67    then obtain w where "e \<le> norm (f w)" "dist (f w) l < e - norm l"
    1.68      using assms(1) eventually_happens by auto
    1.69    hence "norm (f w - l) + norm l < e" "e \<le> norm (f w)" by (simp_all add: dist_norm)
    1.70 @@ -4236,7 +4225,7 @@
    1.71    { fix x and e::real assume "x\<in>s" "e>0"
    1.72      have "eventually (\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3) net" using `e>0` assms(3)[THEN spec[where x="e/3"]] by auto
    1.73      then obtain n where n:"\<forall>xa\<in>s. norm (f n xa - g xa) < e / 3"  "continuous_on s (f n)"
    1.74 -      using eventually_and[of "(\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3)" "(\<lambda>n. continuous_on s (f n))" net] assms(1,2) eventually_happens by blast
    1.75 +      using eventually_conj_iff[of "(\<lambda>n. \<forall>x\<in>s. norm (f n x - g x) < e / 3)" "(\<lambda>n. continuous_on s (f n))" net] assms(1,2) eventually_happens by blast
    1.76      have "e / 3 > 0" using `e>0` by auto
    1.77      then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
    1.78        using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF `x\<in>s`, THEN spec[where x="e/3"]] by blast
    1.79 @@ -5305,7 +5294,7 @@
    1.80  lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
    1.81    assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$$i = b) net"
    1.82    shows "l$$i = b"
    1.83 -  using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
    1.84 +  using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
    1.85  text{* Limits relative to a union.                                               *}
    1.86  
    1.87  lemma eventually_within_Un: