src/HOL/Limits.thy
 changeset 44205 18da2a87421c parent 44195 f5363511b212 child 44206 5e4a1664106e
```     1.1 --- a/src/HOL/Limits.thy	Sun Aug 14 08:45:38 2011 -0700
1.2 +++ b/src/HOL/Limits.thy	Sun Aug 14 10:25:43 2011 -0700
1.3 @@ -581,15 +581,37 @@
1.4  lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
1.6
1.7 +lemma tendsto_unique:
1.8 +  fixes f :: "'a \<Rightarrow> 'b::t2_space"
1.9 +  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
1.10 +  shows "a = b"
1.11 +proof (rule ccontr)
1.12 +  assume "a \<noteq> b"
1.13 +  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
1.14 +    using hausdorff [OF `a \<noteq> b`] by fast
1.15 +  have "eventually (\<lambda>x. f x \<in> U) F"
1.16 +    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
1.17 +  moreover
1.18 +  have "eventually (\<lambda>x. f x \<in> V) F"
1.19 +    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
1.20 +  ultimately
1.21 +  have "eventually (\<lambda>x. False) F"
1.22 +  proof (rule eventually_elim2)
1.23 +    fix x
1.24 +    assume "f x \<in> U" "f x \<in> V"
1.25 +    hence "f x \<in> U \<inter> V" by simp
1.26 +    with `U \<inter> V = {}` show "False" by simp
1.27 +  qed
1.28 +  with `\<not> trivial_limit F` show "False"
1.29 +    by (simp add: trivial_limit_def)
1.30 +qed
1.31 +
1.32  lemma tendsto_const_iff:
1.33 -  fixes k l :: "'a::metric_space"
1.34 -  assumes "F \<noteq> bot" shows "((\<lambda>n. k) ---> l) F \<longleftrightarrow> k = l"
1.35 -  apply (safe intro!: tendsto_const)
1.36 -  apply (rule ccontr)
1.37 -  apply (drule_tac e="dist k l" in tendstoD)
1.38 -  apply (simp add: zero_less_dist_iff)
1.39 -  apply (simp add: eventually_False assms)
1.40 -  done
1.41 +  fixes a b :: "'a::t2_space"
1.42 +  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
1.43 +  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
1.44 +
1.45 +subsubsection {* Distance and norms *}
1.46
1.47  lemma tendsto_dist [tendsto_intros]:
1.48    assumes f: "(f ---> l) F" and g: "(g ---> m) F"
1.49 @@ -611,8 +633,6 @@
1.50    qed
1.51  qed
1.52
1.53 -subsubsection {* Norms *}
1.54 -
1.55  lemma norm_conv_dist: "norm x = dist x 0"
1.56    unfolding dist_norm by simp
1.57
1.58 @@ -865,31 +885,4 @@
1.59    shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
1.60    unfolding sgn_div_norm by (simp add: tendsto_intros)
1.61
1.62 -subsubsection {* Uniqueness *}
1.63 -
1.64 -lemma tendsto_unique:
1.65 -  fixes f :: "'a \<Rightarrow> 'b::t2_space"
1.66 -  assumes "\<not> trivial_limit F"  "(f ---> l) F"  "(f ---> l') F"
1.67 -  shows "l = l'"
1.68 -proof (rule ccontr)
1.69 -  assume "l \<noteq> l'"
1.70 -  obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
1.71 -    using hausdorff [OF `l \<noteq> l'`] by fast
1.72 -  have "eventually (\<lambda>x. f x \<in> U) F"
1.73 -    using `(f ---> l) F` `open U` `l \<in> U` by (rule topological_tendstoD)
1.74 -  moreover
1.75 -  have "eventually (\<lambda>x. f x \<in> V) F"
1.76 -    using `(f ---> l') F` `open V` `l' \<in> V` by (rule topological_tendstoD)
1.77 -  ultimately
1.78 -  have "eventually (\<lambda>x. False) F"
1.79 -  proof (rule eventually_elim2)
1.80 -    fix x
1.81 -    assume "f x \<in> U" "f x \<in> V"
1.82 -    hence "f x \<in> U \<inter> V" by simp
1.83 -    with `U \<inter> V = {}` show "False" by simp
1.84 -  qed
1.85 -  with `\<not> trivial_limit F` show "False"
1.86 -    by (simp add: trivial_limit_def)
1.87 -qed
1.88 -
1.89  end
```