author huffman Sun Aug 14 10:25:43 2011 -0700 (2011-08-14) changeset 44205 18da2a87421c parent 44195 f5363511b212 child 44206 5e4a1664106e
generalize constant 'lim' and limit uniqueness theorems to class t2_space
 src/HOL/Lim.thy file | annotate | diff | revisions src/HOL/Limits.thy file | annotate | diff | revisions src/HOL/SEQ.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Lim.thy	Sun Aug 14 08:45:38 2011 -0700
1.2 +++ b/src/HOL/Lim.thy	Sun Aug 14 10:25:43 2011 -0700
1.3 @@ -181,32 +181,32 @@
1.4  lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
1.5    by (rule tendsto_rabs_zero_iff)
1.6
1.7 -lemma at_neq_bot:
1.8 +lemma trivial_limit_at:
1.9    fixes a :: "'a::real_normed_algebra_1"
1.10 -  shows "at a \<noteq> bot"  -- {* TODO: find a more appropriate class *}
1.11 -unfolding eventually_False [symmetric]
1.12 +  shows "\<not> trivial_limit (at a)"  -- {* TODO: find a more appropriate class *}
1.13 +unfolding trivial_limit_def
1.14  unfolding eventually_at dist_norm
1.15  by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
1.16
1.17  lemma LIM_const_not_eq:
1.18    fixes a :: "'a::real_normed_algebra_1"
1.19 -  fixes k L :: "'b::metric_space"
1.20 +  fixes k L :: "'b::t2_space"
1.21    shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
1.22 -by (simp add: tendsto_const_iff at_neq_bot)
1.23 +by (simp add: tendsto_const_iff trivial_limit_at)
1.24
1.25  lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
1.26
1.27  lemma LIM_const_eq:
1.28    fixes a :: "'a::real_normed_algebra_1"
1.29 -  fixes k L :: "'b::metric_space"
1.30 +  fixes k L :: "'b::t2_space"
1.31    shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
1.32 -by (simp add: tendsto_const_iff at_neq_bot)
1.33 +  by (simp add: tendsto_const_iff trivial_limit_at)
1.34
1.35  lemma LIM_unique:
1.36    fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
1.37 -  fixes L M :: "'b::metric_space"
1.38 +  fixes L M :: "'b::t2_space"
1.39    shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
1.40 -by (drule (1) tendsto_dist, simp add: tendsto_const_iff at_neq_bot)
1.41 +  using trivial_limit_at by (rule tendsto_unique)
1.42
1.43  lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
1.44  by (rule tendsto_ident_at)
```
```     2.1 --- a/src/HOL/Limits.thy	Sun Aug 14 08:45:38 2011 -0700
2.2 +++ b/src/HOL/Limits.thy	Sun Aug 14 10:25:43 2011 -0700
2.3 @@ -581,15 +581,37 @@
2.4  lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
2.6
2.7 +lemma tendsto_unique:
2.8 +  fixes f :: "'a \<Rightarrow> 'b::t2_space"
2.9 +  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
2.10 +  shows "a = b"
2.11 +proof (rule ccontr)
2.12 +  assume "a \<noteq> b"
2.13 +  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
2.14 +    using hausdorff [OF `a \<noteq> b`] by fast
2.15 +  have "eventually (\<lambda>x. f x \<in> U) F"
2.16 +    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
2.17 +  moreover
2.18 +  have "eventually (\<lambda>x. f x \<in> V) F"
2.19 +    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
2.20 +  ultimately
2.21 +  have "eventually (\<lambda>x. False) F"
2.22 +  proof (rule eventually_elim2)
2.23 +    fix x
2.24 +    assume "f x \<in> U" "f x \<in> V"
2.25 +    hence "f x \<in> U \<inter> V" by simp
2.26 +    with `U \<inter> V = {}` show "False" by simp
2.27 +  qed
2.28 +  with `\<not> trivial_limit F` show "False"
2.29 +    by (simp add: trivial_limit_def)
2.30 +qed
2.31 +
2.32  lemma tendsto_const_iff:
2.33 -  fixes k l :: "'a::metric_space"
2.34 -  assumes "F \<noteq> bot" shows "((\<lambda>n. k) ---> l) F \<longleftrightarrow> k = l"
2.35 -  apply (safe intro!: tendsto_const)
2.36 -  apply (rule ccontr)
2.37 -  apply (drule_tac e="dist k l" in tendstoD)
2.38 -  apply (simp add: zero_less_dist_iff)
2.39 -  apply (simp add: eventually_False assms)
2.40 -  done
2.41 +  fixes a b :: "'a::t2_space"
2.42 +  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
2.43 +  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
2.44 +
2.45 +subsubsection {* Distance and norms *}
2.46
2.47  lemma tendsto_dist [tendsto_intros]:
2.48    assumes f: "(f ---> l) F" and g: "(g ---> m) F"
2.49 @@ -611,8 +633,6 @@
2.50    qed
2.51  qed
2.52
2.53 -subsubsection {* Norms *}
2.54 -
2.55  lemma norm_conv_dist: "norm x = dist x 0"
2.56    unfolding dist_norm by simp
2.57
2.58 @@ -865,31 +885,4 @@
2.59    shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
2.60    unfolding sgn_div_norm by (simp add: tendsto_intros)
2.61
2.62 -subsubsection {* Uniqueness *}
2.63 -
2.64 -lemma tendsto_unique:
2.65 -  fixes f :: "'a \<Rightarrow> 'b::t2_space"
2.66 -  assumes "\<not> trivial_limit F"  "(f ---> l) F"  "(f ---> l') F"
2.67 -  shows "l = l'"
2.68 -proof (rule ccontr)
2.69 -  assume "l \<noteq> l'"
2.70 -  obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
2.71 -    using hausdorff [OF `l \<noteq> l'`] by fast
2.72 -  have "eventually (\<lambda>x. f x \<in> U) F"
2.73 -    using `(f ---> l) F` `open U` `l \<in> U` by (rule topological_tendstoD)
2.74 -  moreover
2.75 -  have "eventually (\<lambda>x. f x \<in> V) F"
2.76 -    using `(f ---> l') F` `open V` `l' \<in> V` by (rule topological_tendstoD)
2.77 -  ultimately
2.78 -  have "eventually (\<lambda>x. False) F"
2.79 -  proof (rule eventually_elim2)
2.80 -    fix x
2.81 -    assume "f x \<in> U" "f x \<in> V"
2.82 -    hence "f x \<in> U \<inter> V" by simp
2.83 -    with `U \<inter> V = {}` show "False" by simp
2.84 -  qed
2.85 -  with `\<not> trivial_limit F` show "False"
2.86 -    by (simp add: trivial_limit_def)
2.87 -qed
2.88 -
2.89  end
```
```     3.1 --- a/src/HOL/SEQ.thy	Sun Aug 14 08:45:38 2011 -0700
3.2 +++ b/src/HOL/SEQ.thy	Sun Aug 14 10:25:43 2011 -0700
3.3 @@ -189,7 +189,7 @@
3.4    "X ----> L \<equiv> (X ---> L) sequentially"
3.5
3.6  definition
3.7 -  lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
3.8 +  lim :: "(nat \<Rightarrow> 'a::t2_space) \<Rightarrow> 'a" where
3.9      --{*Standard definition of limit using choice operator*}
3.10    "lim X = (THE L. X ----> L)"
3.11
3.12 @@ -301,9 +301,9 @@
3.13  by (rule tendsto_const)
3.14
3.15  lemma LIMSEQ_const_iff:
3.16 -  fixes k l :: "'a::metric_space"
3.17 +  fixes k l :: "'a::t2_space"
3.18    shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
3.19 -by (rule tendsto_const_iff, rule sequentially_bot)
3.20 +  using trivial_limit_sequentially by (rule tendsto_const_iff)
3.21
3.22  lemma LIMSEQ_norm:
3.23    fixes a :: "'a::real_normed_vector"
3.24 @@ -366,9 +366,9 @@
3.25  by (rule tendsto_diff)
3.26
3.27  lemma LIMSEQ_unique:
3.28 -  fixes a b :: "'a::metric_space"
3.29 +  fixes a b :: "'a::t2_space"
3.30    shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
3.31 -by (drule (1) tendsto_dist, simp add: LIMSEQ_const_iff)
3.32 +  using trivial_limit_sequentially by (rule tendsto_unique)
3.33
3.34  lemma (in bounded_linear) LIMSEQ:
3.35    "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
```