author | hoelzl |
Mon, 03 Dec 2012 18:19:08 +0100 | |
changeset 50328 | 25b1e8686ce0 |
parent 50327 | bbea2e82871c |
child 50330 | d0b12171118e |
permissions | -rw-r--r-- |
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(* Title : Limits.thy |
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Author : Brian Huffman |
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*) |
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header {* Filters and Limits *} |
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theory Limits |
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imports RealVector |
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begin |
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subsection {* Filters *} |
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text {* |
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This definition also allows non-proper filters. |
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*} |
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locale is_filter = |
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fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" |
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assumes True: "F (\<lambda>x. True)" |
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assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" |
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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" |
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" |
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proof |
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show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) |
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qed |
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" |
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using Rep_filter [of F] by simp |
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lemma Abs_filter_inverse': |
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assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" |
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using assms by (simp add: Abs_filter_inverse) |
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subsection {* Eventually *} |
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
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where "eventually P F \<longleftrightarrow> Rep_filter F P" |
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lemma eventually_Abs_filter: |
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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" |
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unfolding eventually_def using assms by (simp add: Abs_filter_inverse) |
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lemma filter_eq_iff: |
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shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" |
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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. |
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" |
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unfolding eventually_def |
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by (rule is_filter.True [OF is_filter_Rep_filter]) |
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" |
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proof - |
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assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) |
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thus "eventually P F" by simp |
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qed |
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lemma eventually_mono: |
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"(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F" |
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unfolding eventually_def |
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by (rule is_filter.mono [OF is_filter_Rep_filter]) |
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lemma eventually_conj: |
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assumes P: "eventually (\<lambda>x. P x) F" |
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assumes Q: "eventually (\<lambda>x. Q x) F" |
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shows "eventually (\<lambda>x. P x \<and> Q x) F" |
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using assms unfolding eventually_def |
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by (rule is_filter.conj [OF is_filter_Rep_filter]) |
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lemma eventually_mp: |
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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assumes "eventually (\<lambda>x. P x) F" |
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shows "eventually (\<lambda>x. Q x) F" |
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proof (rule eventually_mono) |
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show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp |
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show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" |
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using assms by (rule eventually_conj) |
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qed |
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lemma eventually_rev_mp: |
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assumes "eventually (\<lambda>x. P x) F" |
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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shows "eventually (\<lambda>x. Q x) F" |
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using assms(2) assms(1) by (rule eventually_mp) |
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lemma eventually_conj_iff: |
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"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" |
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by (auto intro: eventually_conj elim: eventually_rev_mp) |
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lemma eventually_elim1: |
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assumes "eventually (\<lambda>i. P i) F" |
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assumes "\<And>i. P i \<Longrightarrow> Q i" |
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shows "eventually (\<lambda>i. Q i) F" |
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using assms by (auto elim!: eventually_rev_mp) |
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lemma eventually_elim2: |
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assumes "eventually (\<lambda>i. P i) F" |
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assumes "eventually (\<lambda>i. Q i) F" |
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assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" |
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shows "eventually (\<lambda>i. R i) F" |
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using assms by (auto elim!: eventually_rev_mp) |
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lemma eventually_subst: |
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assumes "eventually (\<lambda>n. P n = Q n) F" |
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shows "eventually P F = eventually Q F" (is "?L = ?R") |
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proof - |
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from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" |
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by (auto elim: eventually_elim1) |
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then show ?thesis by (auto elim: eventually_elim2) |
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qed |
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ML {* |
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fun eventually_elim_tac ctxt thms thm = |
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let |
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val thy = Proof_Context.theory_of ctxt |
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val mp_thms = thms RL [@{thm eventually_rev_mp}] |
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val raw_elim_thm = |
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(@{thm allI} RS @{thm always_eventually}) |
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|> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms |
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|> fold (fn _ => fn thm => @{thm impI} RS thm) thms |
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val cases_prop = prop_of (raw_elim_thm RS thm) |
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val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])]) |
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in |
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CASES cases (rtac raw_elim_thm 1) thm |
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end |
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*} |
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method_setup eventually_elim = {* |
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Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt)) |
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*} "elimination of eventually quantifiers" |
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subsection {* Finer-than relation *} |
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text {* @{term "F \<le> F'"} means that filter @{term F} is finer than |
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filter @{term F'}. *} |
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instantiation filter :: (type) complete_lattice |
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begin |
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definition le_filter_def: |
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"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" |
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definition |
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"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
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definition |
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"top = Abs_filter (\<lambda>P. \<forall>x. P x)" |
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definition |
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"bot = Abs_filter (\<lambda>P. True)" |
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definition |
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"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" |
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definition |
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"inf F F' = Abs_filter |
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(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
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definition |
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"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" |
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definition |
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"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" |
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lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" |
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unfolding top_filter_def |
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by (rule eventually_Abs_filter, rule is_filter.intro, auto) |
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lemma eventually_bot [simp]: "eventually P bot" |
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unfolding bot_filter_def |
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by (subst eventually_Abs_filter, rule is_filter.intro, auto) |
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lemma eventually_sup: |
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"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" |
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unfolding sup_filter_def |
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by (rule eventually_Abs_filter, rule is_filter.intro) |
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(auto elim!: eventually_rev_mp) |
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lemma eventually_inf: |
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"eventually P (inf F F') \<longleftrightarrow> |
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(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
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unfolding inf_filter_def |
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apply (rule eventually_Abs_filter, rule is_filter.intro) |
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apply (fast intro: eventually_True) |
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apply clarify |
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apply (intro exI conjI) |
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apply (erule (1) eventually_conj) |
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191 |
apply (erule (1) eventually_conj) |
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apply simp |
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apply auto |
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194 |
done |
36630 | 195 |
|
196 |
lemma eventually_Sup: |
|
44195 | 197 |
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" |
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198 |
unfolding Sup_filter_def |
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199 |
apply (rule eventually_Abs_filter, rule is_filter.intro) |
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apply (auto intro: eventually_conj elim!: eventually_rev_mp) |
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201 |
done |
36630 | 202 |
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203 |
instance proof |
44195 | 204 |
fix F F' F'' :: "'a filter" and S :: "'a filter set" |
205 |
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
206 |
by (rule less_filter_def) } |
|
207 |
{ show "F \<le> F" |
|
208 |
unfolding le_filter_def by simp } |
|
209 |
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" |
|
210 |
unfolding le_filter_def by simp } |
|
211 |
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'" |
|
212 |
unfolding le_filter_def filter_eq_iff by fast } |
|
213 |
{ show "F \<le> top" |
|
214 |
unfolding le_filter_def eventually_top by (simp add: always_eventually) } |
|
215 |
{ show "bot \<le> F" |
|
216 |
unfolding le_filter_def by simp } |
|
217 |
{ show "F \<le> sup F F'" and "F' \<le> sup F F'" |
|
218 |
unfolding le_filter_def eventually_sup by simp_all } |
|
219 |
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" |
|
220 |
unfolding le_filter_def eventually_sup by simp } |
|
221 |
{ show "inf F F' \<le> F" and "inf F F' \<le> F'" |
|
222 |
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } |
|
223 |
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" |
|
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unfolding le_filter_def eventually_inf |
44195 | 225 |
by (auto elim!: eventually_mono intro: eventually_conj) } |
226 |
{ assume "F \<in> S" thus "F \<le> Sup S" |
|
227 |
unfolding le_filter_def eventually_Sup by simp } |
|
228 |
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" |
|
229 |
unfolding le_filter_def eventually_Sup by simp } |
|
230 |
{ assume "F'' \<in> S" thus "Inf S \<le> F''" |
|
231 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
232 |
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" |
|
233 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
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234 |
qed |
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235 |
|
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236 |
end |
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237 |
|
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lemma filter_leD: |
44195 | 239 |
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" |
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240 |
unfolding le_filter_def by simp |
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241 |
|
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lemma filter_leI: |
44195 | 243 |
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" |
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244 |
unfolding le_filter_def by simp |
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245 |
|
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246 |
lemma eventually_False: |
44195 | 247 |
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" |
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248 |
unfolding filter_eq_iff by (auto elim: eventually_rev_mp) |
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|
249 |
|
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250 |
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" |
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251 |
where "trivial_limit F \<equiv> F = bot" |
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252 |
|
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lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" |
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|
254 |
by (rule eventually_False [symmetric]) |
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|
255 |
|
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|
256 |
|
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257 |
subsection {* Map function for filters *} |
36654 | 258 |
|
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259 |
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" |
44195 | 260 |
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" |
36654 | 261 |
|
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lemma eventually_filtermap: |
44195 | 263 |
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" |
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|
264 |
unfolding filtermap_def |
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|
265 |
apply (rule eventually_Abs_filter) |
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|
266 |
apply (rule is_filter.intro) |
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|
267 |
apply (auto elim!: eventually_rev_mp) |
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|
268 |
done |
36654 | 269 |
|
44195 | 270 |
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" |
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271 |
by (simp add: filter_eq_iff eventually_filtermap) |
36654 | 272 |
|
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273 |
lemma filtermap_filtermap: |
44195 | 274 |
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" |
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|
275 |
by (simp add: filter_eq_iff eventually_filtermap) |
36654 | 276 |
|
44195 | 277 |
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" |
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changeset
|
278 |
unfolding le_filter_def eventually_filtermap by simp |
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|
279 |
|
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|
280 |
lemma filtermap_bot [simp]: "filtermap f bot = bot" |
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281 |
by (simp add: filter_eq_iff eventually_filtermap) |
36654 | 282 |
|
50247
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283 |
subsection {* Order filters *} |
31392 | 284 |
|
50247
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285 |
definition at_top :: "('a::order) filter" |
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286 |
where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" |
31392 | 287 |
|
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|
288 |
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" |
50247
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|
289 |
unfolding at_top_def |
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|
290 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
50247
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|
291 |
fix P Q :: "'a \<Rightarrow> bool" |
36662
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changeset
|
292 |
assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n" |
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|
293 |
then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto |
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|
294 |
then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp |
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|
295 |
then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" .. |
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|
296 |
qed auto |
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changeset
|
297 |
|
50324
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|
298 |
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)" |
50247
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|
299 |
unfolding eventually_at_top_linorder |
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|
300 |
proof safe |
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|
301 |
fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N]) |
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|
302 |
next |
50324
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changeset
|
303 |
fix N assume "\<forall>n>N. P n" |
50247
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|
304 |
moreover from gt_ex[of N] guess y .. |
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|
305 |
ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y]) |
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|
306 |
qed |
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changeset
|
307 |
|
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|
308 |
definition at_bot :: "('a::order) filter" |
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|
309 |
where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)" |
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|
310 |
|
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311 |
lemma eventually_at_bot_linorder: |
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|
312 |
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" |
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changeset
|
313 |
unfolding at_bot_def |
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|
314 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
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changeset
|
315 |
fix P Q :: "'a \<Rightarrow> bool" |
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changeset
|
316 |
assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n" |
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|
317 |
then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto |
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changeset
|
318 |
then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp |
491c5c81c2e8
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changeset
|
319 |
then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" .. |
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|
320 |
qed auto |
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|
321 |
|
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|
322 |
lemma eventually_at_bot_dense: |
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|
323 |
fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)" |
491c5c81c2e8
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hoelzl
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changeset
|
324 |
unfolding eventually_at_bot_linorder |
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|
325 |
proof safe |
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changeset
|
326 |
fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N]) |
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|
327 |
next |
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changeset
|
328 |
fix N assume "\<forall>n<N. P n" |
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changeset
|
329 |
moreover from lt_ex[of N] guess y .. |
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changeset
|
330 |
ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y]) |
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|
331 |
qed |
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parents:
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changeset
|
332 |
|
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introduce filter_lim as a generatlization of tendsto
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diff
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|
333 |
subsection {* Sequentially *} |
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introduce filter_lim as a generatlization of tendsto
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diff
changeset
|
334 |
|
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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parents:
49834
diff
changeset
|
335 |
abbreviation sequentially :: "nat filter" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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parents:
49834
diff
changeset
|
336 |
where "sequentially == at_top" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
337 |
|
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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parents:
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diff
changeset
|
338 |
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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diff
changeset
|
339 |
unfolding at_top_def by simp |
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introduce filter_lim as a generatlization of tendsto
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parents:
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diff
changeset
|
340 |
|
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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parents:
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diff
changeset
|
341 |
lemma eventually_sequentially: |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
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parents:
49834
diff
changeset
|
342 |
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" |
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parents:
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diff
changeset
|
343 |
by (rule eventually_at_top_linorder) |
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diff
changeset
|
344 |
|
44342
8321948340ea
redefine constant 'trivial_limit' as an abbreviation
huffman
parents:
44282
diff
changeset
|
345 |
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" |
44081
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parents:
44079
diff
changeset
|
346 |
unfolding filter_eq_iff eventually_sequentially by auto |
36662
621122eeb138
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huffman
parents:
36656
diff
changeset
|
347 |
|
44342
8321948340ea
redefine constant 'trivial_limit' as an abbreviation
huffman
parents:
44282
diff
changeset
|
348 |
lemmas trivial_limit_sequentially = sequentially_bot |
8321948340ea
redefine constant 'trivial_limit' as an abbreviation
huffman
parents:
44282
diff
changeset
|
349 |
|
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
350 |
lemma eventually_False_sequentially [simp]: |
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
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parents:
36656
diff
changeset
|
351 |
"\<not> eventually (\<lambda>n. False) sequentially" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset
|
352 |
by (simp add: eventually_False) |
36662
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huffman
parents:
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diff
changeset
|
353 |
|
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
354 |
lemma le_sequentially: |
44195 | 355 |
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
356 |
unfolding le_filter_def eventually_sequentially |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
357 |
by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp) |
36662
621122eeb138
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huffman
parents:
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diff
changeset
|
358 |
|
45892 | 359 |
lemma eventually_sequentiallyI: |
360 |
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" |
|
361 |
shows "eventually P sequentially" |
|
362 |
using assms by (auto simp: eventually_sequentially) |
|
363 |
||
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
364 |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
365 |
subsection {* Standard filters *} |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
366 |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
367 |
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70) |
44195 | 368 |
where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)" |
31392 | 369 |
|
44206
5e4a1664106e
locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents:
44205
diff
changeset
|
370 |
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
371 |
where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" |
36654 | 372 |
|
44206
5e4a1664106e
locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents:
44205
diff
changeset
|
373 |
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
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diff
changeset
|
374 |
where "at a = nhds a within - {a}" |
31447
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generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
375 |
|
50326 | 376 |
abbreviation at_right :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where |
377 |
"at_right x \<equiv> at x within {x <..}" |
|
378 |
||
379 |
abbreviation at_left :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where |
|
380 |
"at_left x \<equiv> at x within {..< x}" |
|
381 |
||
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
382 |
definition at_infinity :: "'a::real_normed_vector filter" where |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
383 |
"at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
384 |
|
31392 | 385 |
lemma eventually_within: |
44195 | 386 |
"eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F" |
44081
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rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
387 |
unfolding within_def |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
388 |
by (rule eventually_Abs_filter, rule is_filter.intro) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
389 |
(auto elim!: eventually_rev_mp) |
31392 | 390 |
|
45031 | 391 |
lemma within_UNIV [simp]: "F within UNIV = F" |
392 |
unfolding filter_eq_iff eventually_within by simp |
|
393 |
||
394 |
lemma within_empty [simp]: "F within {} = bot" |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
395 |
unfolding filter_eq_iff eventually_within by simp |
36360
9d8f7efd9289
define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents:
36358
diff
changeset
|
396 |
|
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
397 |
lemma within_le: "F within S \<le> F" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
398 |
unfolding le_filter_def eventually_within by (auto elim: eventually_elim1) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
399 |
|
50323 | 400 |
lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S" |
401 |
unfolding le_filter_def eventually_within by (auto elim: eventually_elim2) |
|
402 |
||
403 |
lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'" |
|
404 |
by (blast intro: within_le le_withinI order_trans) |
|
405 |
||
36654 | 406 |
lemma eventually_nhds: |
407 |
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" |
|
408 |
unfolding nhds_def |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
409 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
36654 | 410 |
have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp |
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
411 |
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" .. |
36358
246493d61204
define nets directly as filters, instead of as filter bases
huffman
parents:
31902
diff
changeset
|
412 |
next |
246493d61204
define nets directly as filters, instead of as filter bases
huffman
parents:
31902
diff
changeset
|
413 |
fix P Q |
36654 | 414 |
assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" |
415 |
and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" |
|
36358
246493d61204
define nets directly as filters, instead of as filter bases
huffman
parents:
31902
diff
changeset
|
416 |
then obtain S T where |
36654 | 417 |
"open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" |
418 |
"open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto |
|
419 |
hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)" |
|
36358
246493d61204
define nets directly as filters, instead of as filter bases
huffman
parents:
31902
diff
changeset
|
420 |
by (simp add: open_Int) |
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
421 |
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" .. |
36358
246493d61204
define nets directly as filters, instead of as filter bases
huffman
parents:
31902
diff
changeset
|
422 |
qed auto |
31447
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
423 |
|
36656
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
424 |
lemma eventually_nhds_metric: |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
425 |
"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)" |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
426 |
unfolding eventually_nhds open_dist |
31447
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
427 |
apply safe |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
428 |
apply fast |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31488
diff
changeset
|
429 |
apply (rule_tac x="{x. dist x a < d}" in exI, simp) |
31447
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
430 |
apply clarsimp |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
431 |
apply (rule_tac x="d - dist x a" in exI, clarsimp) |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
432 |
apply (simp only: less_diff_eq) |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
433 |
apply (erule le_less_trans [OF dist_triangle]) |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
434 |
done |
97bab1ac463e
generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents:
31392
diff
changeset
|
435 |
|
44571 | 436 |
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot" |
437 |
unfolding trivial_limit_def eventually_nhds by simp |
|
438 |
||
36656
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
439 |
lemma eventually_at_topological: |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
440 |
"eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))" |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
441 |
unfolding at_def eventually_within eventually_nhds by simp |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
442 |
|
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
443 |
lemma eventually_at: |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
444 |
fixes a :: "'a::metric_space" |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
445 |
shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)" |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
446 |
unfolding at_def eventually_within eventually_nhds_metric by auto |
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
447 |
|
50327 | 448 |
lemma eventually_within_less: (* COPY FROM Topo/eventually_within *) |
449 |
"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)" |
|
450 |
unfolding eventually_within eventually_at dist_nz by auto |
|
451 |
||
452 |
lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *) |
|
453 |
"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)" |
|
454 |
unfolding eventually_within_less by auto (metis dense order_le_less_trans) |
|
455 |
||
44571 | 456 |
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}" |
457 |
unfolding trivial_limit_def eventually_at_topological |
|
458 |
by (safe, case_tac "S = {a}", simp, fast, fast) |
|
459 |
||
460 |
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot" |
|
461 |
by (simp add: at_eq_bot_iff not_open_singleton) |
|
462 |
||
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
463 |
lemma eventually_at_infinity: |
50325 | 464 |
"eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)" |
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
465 |
unfolding at_infinity_def |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
466 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
467 |
fix P Q :: "'a \<Rightarrow> bool" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
468 |
assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
469 |
then obtain r s where |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
470 |
"\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
471 |
then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
472 |
then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" .. |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
473 |
qed auto |
31392 | 474 |
|
50325 | 475 |
lemma at_infinity_eq_at_top_bot: |
476 |
"(at_infinity \<Colon> real filter) = sup at_top at_bot" |
|
477 |
unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder |
|
478 |
proof (intro arg_cong[where f=Abs_filter] ext iffI) |
|
479 |
fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" |
|
480 |
then guess r .. |
|
481 |
then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto |
|
482 |
then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto |
|
483 |
next |
|
484 |
fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" |
|
485 |
then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto |
|
486 |
then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" |
|
487 |
by (intro exI[of _ "max p (-q)"]) |
|
488 |
(auto simp: abs_real_def) |
|
489 |
qed |
|
490 |
||
491 |
lemma at_top_le_at_infinity: |
|
492 |
"at_top \<le> (at_infinity :: real filter)" |
|
493 |
unfolding at_infinity_eq_at_top_bot by simp |
|
494 |
||
495 |
lemma at_bot_le_at_infinity: |
|
496 |
"at_bot \<le> (at_infinity :: real filter)" |
|
497 |
unfolding at_infinity_eq_at_top_bot by simp |
|
498 |
||
31355 | 499 |
subsection {* Boundedness *} |
500 |
||
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
501 |
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" |
44195 | 502 |
where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)" |
31355 | 503 |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
504 |
lemma BfunI: |
44195 | 505 |
assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F" |
31355 | 506 |
unfolding Bfun_def |
507 |
proof (intro exI conjI allI) |
|
508 |
show "0 < max K 1" by simp |
|
509 |
next |
|
44195 | 510 |
show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F" |
31355 | 511 |
using K by (rule eventually_elim1, simp) |
512 |
qed |
|
513 |
||
514 |
lemma BfunE: |
|
44195 | 515 |
assumes "Bfun f F" |
516 |
obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F" |
|
31355 | 517 |
using assms unfolding Bfun_def by fast |
518 |
||
519 |
||
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
520 |
subsection {* Convergence to Zero *} |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
521 |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
522 |
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" |
44195 | 523 |
where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)" |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
524 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
525 |
lemma ZfunI: |
44195 | 526 |
"(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
527 |
unfolding Zfun_def by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
528 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
529 |
lemma ZfunD: |
44195 | 530 |
"\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
531 |
unfolding Zfun_def by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
532 |
|
31355 | 533 |
lemma Zfun_ssubst: |
44195 | 534 |
"eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
535 |
unfolding Zfun_def by (auto elim!: eventually_rev_mp) |
31355 | 536 |
|
44195 | 537 |
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
538 |
unfolding Zfun_def by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
539 |
|
44195 | 540 |
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
541 |
unfolding Zfun_def by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
542 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
543 |
lemma Zfun_imp_Zfun: |
44195 | 544 |
assumes f: "Zfun f F" |
545 |
assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F" |
|
546 |
shows "Zfun (\<lambda>x. g x) F" |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
547 |
proof (cases) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
548 |
assume K: "0 < K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
549 |
show ?thesis |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
550 |
proof (rule ZfunI) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
551 |
fix r::real assume "0 < r" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
552 |
hence "0 < r / K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
553 |
using K by (rule divide_pos_pos) |
44195 | 554 |
then have "eventually (\<lambda>x. norm (f x) < r / K) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
555 |
using ZfunD [OF f] by fast |
44195 | 556 |
with g show "eventually (\<lambda>x. norm (g x) < r) F" |
46887 | 557 |
proof eventually_elim |
558 |
case (elim x) |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
559 |
hence "norm (f x) * K < r" |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
560 |
by (simp add: pos_less_divide_eq K) |
46887 | 561 |
thus ?case |
562 |
by (simp add: order_le_less_trans [OF elim(1)]) |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
563 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
564 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
565 |
next |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
566 |
assume "\<not> 0 < K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
567 |
hence K: "K \<le> 0" by (simp only: not_less) |
31355 | 568 |
show ?thesis |
569 |
proof (rule ZfunI) |
|
570 |
fix r :: real |
|
571 |
assume "0 < r" |
|
44195 | 572 |
from g show "eventually (\<lambda>x. norm (g x) < r) F" |
46887 | 573 |
proof eventually_elim |
574 |
case (elim x) |
|
575 |
also have "norm (f x) * K \<le> norm (f x) * 0" |
|
31355 | 576 |
using K norm_ge_zero by (rule mult_left_mono) |
46887 | 577 |
finally show ?case |
31355 | 578 |
using `0 < r` by simp |
579 |
qed |
|
580 |
qed |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
581 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
582 |
|
44195 | 583 |
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
584 |
by (erule_tac K="1" in Zfun_imp_Zfun, simp) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
585 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
586 |
lemma Zfun_add: |
44195 | 587 |
assumes f: "Zfun f F" and g: "Zfun g F" |
588 |
shows "Zfun (\<lambda>x. f x + g x) F" |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
589 |
proof (rule ZfunI) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
590 |
fix r::real assume "0 < r" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
591 |
hence r: "0 < r / 2" by simp |
44195 | 592 |
have "eventually (\<lambda>x. norm (f x) < r/2) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
593 |
using f r by (rule ZfunD) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
594 |
moreover |
44195 | 595 |
have "eventually (\<lambda>x. norm (g x) < r/2) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
596 |
using g r by (rule ZfunD) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
597 |
ultimately |
44195 | 598 |
show "eventually (\<lambda>x. norm (f x + g x) < r) F" |
46887 | 599 |
proof eventually_elim |
600 |
case (elim x) |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
601 |
have "norm (f x + g x) \<le> norm (f x) + norm (g x)" |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
602 |
by (rule norm_triangle_ineq) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
603 |
also have "\<dots> < r/2 + r/2" |
46887 | 604 |
using elim by (rule add_strict_mono) |
605 |
finally show ?case |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
606 |
by simp |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
607 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
608 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
609 |
|
44195 | 610 |
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
611 |
unfolding Zfun_def by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
612 |
|
44195 | 613 |
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
614 |
by (simp only: diff_minus Zfun_add Zfun_minus) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
615 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
616 |
lemma (in bounded_linear) Zfun: |
44195 | 617 |
assumes g: "Zfun g F" |
618 |
shows "Zfun (\<lambda>x. f (g x)) F" |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
619 |
proof - |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
620 |
obtain K where "\<And>x. norm (f x) \<le> norm x * K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
621 |
using bounded by fast |
44195 | 622 |
then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F" |
31355 | 623 |
by simp |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
624 |
with g show ?thesis |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
625 |
by (rule Zfun_imp_Zfun) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
626 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
627 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
628 |
lemma (in bounded_bilinear) Zfun: |
44195 | 629 |
assumes f: "Zfun f F" |
630 |
assumes g: "Zfun g F" |
|
631 |
shows "Zfun (\<lambda>x. f x ** g x) F" |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
632 |
proof (rule ZfunI) |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
633 |
fix r::real assume r: "0 < r" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
634 |
obtain K where K: "0 < K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
635 |
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
636 |
using pos_bounded by fast |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
637 |
from K have K': "0 < inverse K" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
638 |
by (rule positive_imp_inverse_positive) |
44195 | 639 |
have "eventually (\<lambda>x. norm (f x) < r) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
640 |
using f r by (rule ZfunD) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
641 |
moreover |
44195 | 642 |
have "eventually (\<lambda>x. norm (g x) < inverse K) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
643 |
using g K' by (rule ZfunD) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
644 |
ultimately |
44195 | 645 |
show "eventually (\<lambda>x. norm (f x ** g x) < r) F" |
46887 | 646 |
proof eventually_elim |
647 |
case (elim x) |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
648 |
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
649 |
by (rule norm_le) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
650 |
also have "norm (f x) * norm (g x) * K < r * inverse K * K" |
46887 | 651 |
by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
652 |
also from K have "r * inverse K * K = r" |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
653 |
by simp |
46887 | 654 |
finally show ?case . |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
655 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
656 |
qed |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
657 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
658 |
lemma (in bounded_bilinear) Zfun_left: |
44195 | 659 |
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
660 |
by (rule bounded_linear_left [THEN bounded_linear.Zfun]) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
661 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
662 |
lemma (in bounded_bilinear) Zfun_right: |
44195 | 663 |
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
664 |
by (rule bounded_linear_right [THEN bounded_linear.Zfun]) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
665 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
666 |
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult] |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
667 |
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult] |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
668 |
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult] |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
669 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
670 |
|
31902 | 671 |
subsection {* Limits *} |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
672 |
|
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
673 |
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where |
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
674 |
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
675 |
|
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
676 |
syntax |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
677 |
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
678 |
|
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
679 |
translations |
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
680 |
"LIM x F1. f :> F2" == "CONST filterlim (%x. f) F2 F1" |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
681 |
|
50325 | 682 |
lemma filterlim_iff: |
683 |
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" |
|
684 |
unfolding filterlim_def le_filter_def eventually_filtermap .. |
|
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
685 |
|
50327 | 686 |
lemma filterlim_compose: |
50323 | 687 |
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" |
688 |
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) |
|
689 |
||
50327 | 690 |
lemma filterlim_mono: |
50323 | 691 |
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" |
692 |
unfolding filterlim_def by (metis filtermap_mono order_trans) |
|
693 |
||
50327 | 694 |
lemma filterlim_cong: |
695 |
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" |
|
696 |
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) |
|
697 |
||
50325 | 698 |
lemma filterlim_within: |
699 |
"(LIM x F1. f x :> F2 within S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F1 \<and> (LIM x F1. f x :> F2))" |
|
700 |
unfolding filterlim_def |
|
701 |
proof safe |
|
702 |
assume "filtermap f F1 \<le> F2 within S" then show "eventually (\<lambda>x. f x \<in> S) F1" |
|
703 |
by (auto simp: le_filter_def eventually_filtermap eventually_within elim!: allE[of _ "\<lambda>x. x \<in> S"]) |
|
704 |
qed (auto intro: within_le order_trans simp: le_within_iff eventually_filtermap) |
|
705 |
||
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
706 |
abbreviation (in topological_space) |
44206
5e4a1664106e
locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents:
44205
diff
changeset
|
707 |
tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where |
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
708 |
"(f ---> l) F \<equiv> filterlim f (nhds l) F" |
45892 | 709 |
|
31902 | 710 |
ML {* |
711 |
structure Tendsto_Intros = Named_Thms |
|
712 |
( |
|
45294 | 713 |
val name = @{binding tendsto_intros} |
31902 | 714 |
val description = "introduction rules for tendsto" |
715 |
) |
|
31565 | 716 |
*} |
717 |
||
31902 | 718 |
setup Tendsto_Intros.setup |
31565 | 719 |
|
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
720 |
lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)" |
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
721 |
unfolding filterlim_def |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
722 |
proof safe |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
723 |
fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
724 |
then show "eventually (\<lambda>x. f x \<in> S) F" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
725 |
unfolding eventually_nhds eventually_filtermap le_filter_def |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
726 |
by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
727 |
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
728 |
|
50325 | 729 |
lemma filterlim_at: |
730 |
"(LIM x F. f x :> at b) \<longleftrightarrow> (eventually (\<lambda>x. f x \<noteq> b) F \<and> (f ---> b) F)" |
|
731 |
by (simp add: at_def filterlim_within) |
|
732 |
||
44195 | 733 |
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
734 |
unfolding tendsto_def le_filter_def by fast |
36656
fec55067ae9b
add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents:
36655
diff
changeset
|
735 |
|
31488 | 736 |
lemma topological_tendstoI: |
44195 | 737 |
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) |
738 |
\<Longrightarrow> (f ---> l) F" |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
739 |
unfolding tendsto_def by auto |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
740 |
|
31488 | 741 |
lemma topological_tendstoD: |
44195 | 742 |
"(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F" |
31488 | 743 |
unfolding tendsto_def by auto |
744 |
||
745 |
lemma tendstoI: |
|
44195 | 746 |
assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F" |
747 |
shows "(f ---> l) F" |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
748 |
apply (rule topological_tendstoI) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
749 |
apply (simp add: open_dist) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
750 |
apply (drule (1) bspec, clarify) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
751 |
apply (drule assms) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
752 |
apply (erule eventually_elim1, simp) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
753 |
done |
31488 | 754 |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
755 |
lemma tendstoD: |
44195 | 756 |
"(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
757 |
apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
758 |
apply (clarsimp simp add: open_dist) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
759 |
apply (rule_tac x="e - dist x l" in exI, clarsimp) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
760 |
apply (simp only: less_diff_eq) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
761 |
apply (erule le_less_trans [OF dist_triangle]) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
762 |
apply simp |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
763 |
apply simp |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
764 |
done |
31488 | 765 |
|
766 |
lemma tendsto_iff: |
|
44195 | 767 |
"(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
768 |
using tendstoI tendstoD by fast |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
769 |
|
44195 | 770 |
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
771 |
by (simp only: tendsto_iff Zfun_def dist_norm) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
772 |
|
45031 | 773 |
lemma tendsto_bot [simp]: "(f ---> a) bot" |
774 |
unfolding tendsto_def by simp |
|
775 |
||
31565 | 776 |
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
777 |
unfolding tendsto_def eventually_at_topological by auto |
31565 | 778 |
|
779 |
lemma tendsto_ident_at_within [tendsto_intros]: |
|
36655 | 780 |
"((\<lambda>x. x) ---> a) (at a within S)" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
781 |
unfolding tendsto_def eventually_within eventually_at_topological by auto |
31565 | 782 |
|
44195 | 783 |
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
784 |
by (simp add: tendsto_def) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
785 |
|
44205
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
786 |
lemma tendsto_unique: |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
787 |
fixes f :: "'a \<Rightarrow> 'b::t2_space" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
788 |
assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
789 |
shows "a = b" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
790 |
proof (rule ccontr) |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
791 |
assume "a \<noteq> b" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
792 |
obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
793 |
using hausdorff [OF `a \<noteq> b`] by fast |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
794 |
have "eventually (\<lambda>x. f x \<in> U) F" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
795 |
using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD) |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
796 |
moreover |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
797 |
have "eventually (\<lambda>x. f x \<in> V) F" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
798 |
using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD) |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
799 |
ultimately |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
800 |
have "eventually (\<lambda>x. False) F" |
46887 | 801 |
proof eventually_elim |
802 |
case (elim x) |
|
44205
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
803 |
hence "f x \<in> U \<inter> V" by simp |
46887 | 804 |
with `U \<inter> V = {}` show ?case by simp |
44205
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
805 |
qed |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
806 |
with `\<not> trivial_limit F` show "False" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
807 |
by (simp add: trivial_limit_def) |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
808 |
qed |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
809 |
|
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
810 |
lemma tendsto_const_iff: |
44205
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
811 |
fixes a b :: "'a::t2_space" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
812 |
assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b" |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
813 |
by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const]) |
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
814 |
|
50323 | 815 |
lemma tendsto_at_iff_tendsto_nhds: |
816 |
"(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)" |
|
817 |
unfolding tendsto_def at_def eventually_within |
|
818 |
by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1) |
|
819 |
||
44218 | 820 |
lemma tendsto_compose: |
50323 | 821 |
"(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F" |
822 |
unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g]) |
|
44218 | 823 |
|
44253
c073a0bd8458
add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents:
44251
diff
changeset
|
824 |
lemma tendsto_compose_eventually: |
50325 | 825 |
"(g ---> m) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F" |
826 |
by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at) |
|
44253
c073a0bd8458
add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents:
44251
diff
changeset
|
827 |
|
44251 | 828 |
lemma metric_tendsto_imp_tendsto: |
829 |
assumes f: "(f ---> a) F" |
|
830 |
assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F" |
|
831 |
shows "(g ---> b) F" |
|
832 |
proof (rule tendstoI) |
|
833 |
fix e :: real assume "0 < e" |
|
834 |
with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD) |
|
835 |
with le show "eventually (\<lambda>x. dist (g x) b < e) F" |
|
836 |
using le_less_trans by (rule eventually_elim2) |
|
837 |
qed |
|
838 |
||
44205
18da2a87421c
generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents:
44195
diff
changeset
|
839 |
subsubsection {* Distance and norms *} |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
840 |
|
31565 | 841 |
lemma tendsto_dist [tendsto_intros]: |
44195 | 842 |
assumes f: "(f ---> l) F" and g: "(g ---> m) F" |
843 |
shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F" |
|
31565 | 844 |
proof (rule tendstoI) |
845 |
fix e :: real assume "0 < e" |
|
846 |
hence e2: "0 < e/2" by simp |
|
847 |
from tendstoD [OF f e2] tendstoD [OF g e2] |
|
44195 | 848 |
show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F" |
46887 | 849 |
proof (eventually_elim) |
850 |
case (elim x) |
|
31565 | 851 |
then show "dist (dist (f x) (g x)) (dist l m) < e" |
852 |
unfolding dist_real_def |
|
853 |
using dist_triangle2 [of "f x" "g x" "l"] |
|
854 |
using dist_triangle2 [of "g x" "l" "m"] |
|
855 |
using dist_triangle3 [of "l" "m" "f x"] |
|
856 |
using dist_triangle [of "f x" "m" "g x"] |
|
857 |
by arith |
|
858 |
qed |
|
859 |
qed |
|
860 |
||
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
861 |
lemma norm_conv_dist: "norm x = dist x 0" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
862 |
unfolding dist_norm by simp |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
863 |
|
31565 | 864 |
lemma tendsto_norm [tendsto_intros]: |
44195 | 865 |
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
866 |
unfolding norm_conv_dist by (intro tendsto_intros) |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
867 |
|
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
868 |
lemma tendsto_norm_zero: |
44195 | 869 |
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
870 |
by (drule tendsto_norm, simp) |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
871 |
|
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
872 |
lemma tendsto_norm_zero_cancel: |
44195 | 873 |
"((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
874 |
unfolding tendsto_iff dist_norm by simp |
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
875 |
|
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36656
diff
changeset
|
876 |
lemma tendsto_norm_zero_iff: |
44195 | 877 |
"((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
878 |
unfolding tendsto_iff dist_norm by simp |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
879 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
880 |
lemma tendsto_rabs [tendsto_intros]: |
44195 | 881 |
"(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
882 |
by (fold real_norm_def, rule tendsto_norm) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
883 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
884 |
lemma tendsto_rabs_zero: |
44195 | 885 |
"(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
886 |
by (fold real_norm_def, rule tendsto_norm_zero) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
887 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
888 |
lemma tendsto_rabs_zero_cancel: |
44195 | 889 |
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
890 |
by (fold real_norm_def, rule tendsto_norm_zero_cancel) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
891 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
892 |
lemma tendsto_rabs_zero_iff: |
44195 | 893 |
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
894 |
by (fold real_norm_def, rule tendsto_norm_zero_iff) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
895 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
896 |
subsubsection {* Addition and subtraction *} |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
897 |
|
31565 | 898 |
lemma tendsto_add [tendsto_intros]: |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
899 |
fixes a b :: "'a::real_normed_vector" |
44195 | 900 |
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
901 |
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
902 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
903 |
lemma tendsto_add_zero: |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
904 |
fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector" |
44195 | 905 |
shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
906 |
by (drule (1) tendsto_add, simp) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
907 |
|
31565 | 908 |
lemma tendsto_minus [tendsto_intros]: |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
909 |
fixes a :: "'a::real_normed_vector" |
44195 | 910 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
911 |
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
912 |
|
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
913 |
lemma tendsto_minus_cancel: |
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
914 |
fixes a :: "'a::real_normed_vector" |
44195 | 915 |
shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
916 |
by (drule tendsto_minus, simp) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
917 |
|
31565 | 918 |
lemma tendsto_diff [tendsto_intros]: |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
919 |
fixes a b :: "'a::real_normed_vector" |
44195 | 920 |
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
921 |
by (simp add: diff_minus tendsto_add tendsto_minus) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
922 |
|
31588 | 923 |
lemma tendsto_setsum [tendsto_intros]: |
924 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector" |
|
44195 | 925 |
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F" |
926 |
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F" |
|
31588 | 927 |
proof (cases "finite S") |
928 |
assume "finite S" thus ?thesis using assms |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
929 |
by (induct, simp add: tendsto_const, simp add: tendsto_add) |
31588 | 930 |
next |
931 |
assume "\<not> finite S" thus ?thesis |
|
932 |
by (simp add: tendsto_const) |
|
933 |
qed |
|
934 |
||
45892 | 935 |
lemma real_tendsto_sandwich: |
936 |
fixes f g h :: "'a \<Rightarrow> real" |
|
937 |
assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net" |
|
938 |
assumes lim: "(f ---> c) net" "(h ---> c) net" |
|
939 |
shows "(g ---> c) net" |
|
940 |
proof - |
|
941 |
have "((\<lambda>n. g n - f n) ---> 0) net" |
|
942 |
proof (rule metric_tendsto_imp_tendsto) |
|
943 |
show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net" |
|
944 |
using ev by (rule eventually_elim2) (simp add: dist_real_def) |
|
945 |
show "((\<lambda>n. h n - f n) ---> 0) net" |
|
946 |
using tendsto_diff[OF lim(2,1)] by simp |
|
947 |
qed |
|
948 |
from tendsto_add[OF this lim(1)] show ?thesis by simp |
|
949 |
qed |
|
950 |
||
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
951 |
subsubsection {* Linear operators and multiplication *} |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
952 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
953 |
lemma (in bounded_linear) tendsto: |
44195 | 954 |
"(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
955 |
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
956 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
957 |
lemma (in bounded_linear) tendsto_zero: |
44195 | 958 |
"(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
959 |
by (drule tendsto, simp only: zero) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
960 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
961 |
lemma (in bounded_bilinear) tendsto: |
44195 | 962 |
"\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F" |
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
963 |
by (simp only: tendsto_Zfun_iff prod_diff_prod |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
964 |
Zfun_add Zfun Zfun_left Zfun_right) |
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
965 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
966 |
lemma (in bounded_bilinear) tendsto_zero: |
44195 | 967 |
assumes f: "(f ---> 0) F" |
968 |
assumes g: "(g ---> 0) F" |
|
969 |
shows "((\<lambda>x. f x ** g x) ---> 0) F" |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
970 |
using tendsto [OF f g] by (simp add: zero_left) |
31355 | 971 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
972 |
lemma (in bounded_bilinear) tendsto_left_zero: |
44195 | 973 |
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
974 |
by (rule bounded_linear.tendsto_zero [OF bounded_linear_left]) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
975 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
976 |
lemma (in bounded_bilinear) tendsto_right_zero: |
44195 | 977 |
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
978 |
by (rule bounded_linear.tendsto_zero [OF bounded_linear_right]) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
979 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
980 |
lemmas tendsto_of_real [tendsto_intros] = |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
981 |
bounded_linear.tendsto [OF bounded_linear_of_real] |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
982 |
|
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
983 |
lemmas tendsto_scaleR [tendsto_intros] = |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
984 |
bounded_bilinear.tendsto [OF bounded_bilinear_scaleR] |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
985 |
|
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
986 |
lemmas tendsto_mult [tendsto_intros] = |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
987 |
bounded_bilinear.tendsto [OF bounded_bilinear_mult] |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
988 |
|
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
989 |
lemmas tendsto_mult_zero = |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
990 |
bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult] |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
991 |
|
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
992 |
lemmas tendsto_mult_left_zero = |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
993 |
bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult] |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
994 |
|
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
995 |
lemmas tendsto_mult_right_zero = |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
996 |
bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult] |
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44342
diff
changeset
|
997 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
998 |
lemma tendsto_power [tendsto_intros]: |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
999 |
fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}" |
44195 | 1000 |
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1001 |
by (induct n) (simp_all add: tendsto_const tendsto_mult) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1002 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1003 |
lemma tendsto_setprod [tendsto_intros]: |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1004 |
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}" |
44195 | 1005 |
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F" |
1006 |
shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F" |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1007 |
proof (cases "finite S") |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1008 |
assume "finite S" thus ?thesis using assms |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1009 |
by (induct, simp add: tendsto_const, simp add: tendsto_mult) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1010 |
next |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1011 |
assume "\<not> finite S" thus ?thesis |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1012 |
by (simp add: tendsto_const) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1013 |
qed |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1014 |
|
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1015 |
subsubsection {* Inverse and division *} |
31355 | 1016 |
|
1017 |
lemma (in bounded_bilinear) Zfun_prod_Bfun: |
|
44195 | 1018 |
assumes f: "Zfun f F" |
1019 |
assumes g: "Bfun g F" |
|
1020 |
shows "Zfun (\<lambda>x. f x ** g x) F" |
|
31355 | 1021 |
proof - |
1022 |
obtain K where K: "0 \<le> K" |
|
1023 |
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K" |
|
1024 |
using nonneg_bounded by fast |
|
1025 |
obtain B where B: "0 < B" |
|
44195 | 1026 |
and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1027 |
using g by (rule BfunE) |
44195 | 1028 |
have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F" |
46887 | 1029 |
using norm_g proof eventually_elim |
1030 |
case (elim x) |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1031 |
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" |
31355 | 1032 |
by (rule norm_le) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1033 |
also have "\<dots> \<le> norm (f x) * B * K" |
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1034 |
by (intro mult_mono' order_refl norm_g norm_ge_zero |
46887 | 1035 |
mult_nonneg_nonneg K elim) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1036 |
also have "\<dots> = norm (f x) * (B * K)" |
31355 | 1037 |
by (rule mult_assoc) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1038 |
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" . |
31355 | 1039 |
qed |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1040 |
with f show ?thesis |
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1041 |
by (rule Zfun_imp_Zfun) |
31355 | 1042 |
qed |
1043 |
||
1044 |
lemma (in bounded_bilinear) flip: |
|
1045 |
"bounded_bilinear (\<lambda>x y. y ** x)" |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1046 |
apply default |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1047 |
apply (rule add_right) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1048 |
apply (rule add_left) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1049 |
apply (rule scaleR_right) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1050 |
apply (rule scaleR_left) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1051 |
apply (subst mult_commute) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1052 |
using bounded by fast |
31355 | 1053 |
|
1054 |
lemma (in bounded_bilinear) Bfun_prod_Zfun: |
|
44195 | 1055 |
assumes f: "Bfun f F" |
1056 |
assumes g: "Zfun g F" |
|
1057 |
shows "Zfun (\<lambda>x. f x ** g x) F" |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1058 |
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun) |
31355 | 1059 |
|
1060 |
lemma Bfun_inverse_lemma: |
|
1061 |
fixes x :: "'a::real_normed_div_algebra" |
|
1062 |
shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r" |
|
44081
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1063 |
apply (subst nonzero_norm_inverse, clarsimp) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1064 |
apply (erule (1) le_imp_inverse_le) |
730f7cced3a6
rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents:
44079
diff
changeset
|
1065 |
done |
31355 | 1066 |
|
1067 |
lemma Bfun_inverse: |
|
1068 |
fixes a :: "'a::real_normed_div_algebra" |
|
44195 | 1069 |
assumes f: "(f ---> a) F" |
31355 | 1070 |
assumes a: "a \<noteq> 0" |
44195 | 1071 |
shows "Bfun (\<lambda>x. inverse (f x)) F" |
31355 | 1072 |
proof - |
1073 |
from a have "0 < norm a" by simp |
|
1074 |
hence "\<exists>r>0. r < norm a" by (rule dense) |
|
1075 |
then obtain r where r1: "0 < r" and r2: "r < norm a" by fast |
|
44195 | 1076 |
have "eventually (\<lambda>x. dist (f x) a < r) F" |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1077 |
using tendstoD [OF f r1] by fast |
44195 | 1078 |
hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F" |
46887 | 1079 |
proof eventually_elim |
1080 |
case (elim x) |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1081 |
hence 1: "norm (f x - a) < r" |
31355 | 1082 |
by (simp add: dist_norm) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1083 |
hence 2: "f x \<noteq> 0" using r2 by auto |
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1084 |
hence "norm (inverse (f x)) = inverse (norm (f x))" |
31355 | 1085 |
by (rule nonzero_norm_inverse) |
1086 |
also have "\<dots> \<le> inverse (norm a - r)" |
|
1087 |
proof (rule le_imp_inverse_le) |
|
1088 |
show "0 < norm a - r" using r2 by simp |
|
1089 |
next |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1090 |
have "norm a - norm (f x) \<le> norm (a - f x)" |
31355 | 1091 |
by (rule norm_triangle_ineq2) |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1092 |
also have "\<dots> = norm (f x - a)" |
31355 | 1093 |
by (rule norm_minus_commute) |
1094 |
also have "\<dots> < r" using 1 . |
|
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1095 |
finally show "norm a - r \<le> norm (f x)" by simp |
31355 | 1096 |
qed |
31487
93938cafc0e6
put syntax for tendsto in Limits.thy; rename variables
huffman
parents:
31447
diff
changeset
|
1097 |
finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" . |
31355 | 1098 |
qed |
1099 |
thus ?thesis by (rule BfunI) |
|
1100 |
qed |
|
1101 |
||
31565 | 1102 |
lemma tendsto_inverse [tendsto_intros]: |
31355 | 1103 |
fixes a :: "'a::real_normed_div_algebra" |
44195 | 1104 |
assumes f: "(f ---> a) F" |
31355 | 1105 |
assumes a: "a \<noteq> 0" |
44195 | 1106 |
shows "((\<lambda>x. inverse (f x)) ---> inverse a) F" |
31355 | 1107 |
proof - |
1108 |
from a have "0 < norm a" by simp |
|
44195 | 1109 |
with f have "eventually (\<lambda>x. dist (f x) a < norm a) F" |
31355 | 1110 |
by (rule tendstoD) |
44195 | 1111 |
then have "eventually (\<lambda>x. f x \<noteq> 0) F" |
31355 | 1112 |
unfolding dist_norm by (auto elim!: eventually_elim1) |
44627 | 1113 |
with a have "eventually (\<lambda>x. inverse (f x) - inverse a = |
1114 |
- (inverse (f x) * (f x - a) * inverse a)) F" |
|
1115 |
by (auto elim!: eventually_elim1 simp: inverse_diff_inverse) |
|
1116 |
moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F" |
|
1117 |
by (intro Zfun_minus Zfun_mult_left |
|
1118 |
bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult] |
|
1119 |
Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff]) |
|
1120 |
ultimately show ?thesis |
|
1121 |
unfolding tendsto_Zfun_iff by (rule Zfun_ssubst) |
|
31355 | 1122 |
qed |
1123 |
||
31565 | 1124 |
lemma tendsto_divide [tendsto_intros]: |
31355 | 1125 |
fixes a b :: "'a::real_normed_field" |
44195 | 1126 |
shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk> |
1127 |
\<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44253
diff
changeset
|
1128 |
by (simp add: tendsto_mult tendsto_inverse divide_inverse) |
31355 | 1129 |
|
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1130 |
lemma tendsto_sgn [tendsto_intros]: |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1131 |
fixes l :: "'a::real_normed_vector" |
44195 | 1132 |
shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F" |
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1133 |
unfolding sgn_div_norm by (simp add: tendsto_intros) |
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
44081
diff
changeset
|
1134 |
|
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1135 |
subsection {* Limits to @{const at_top} and @{const at_bot} *} |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1136 |
|
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
1137 |
lemma filterlim_at_top: |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1138 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1139 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" |
50325 | 1140 |
by (auto simp: filterlim_iff eventually_at_top_dense elim!: eventually_elim1) |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1141 |
|
50323 | 1142 |
lemma filterlim_at_top_gt: |
1143 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b" |
|
1144 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z < f x) F)" |
|
1145 |
unfolding filterlim_at_top |
|
1146 |
proof safe |
|
1147 |
fix Z assume *: "\<forall>Z>c. eventually (\<lambda>x. Z < f x) F" |
|
1148 |
from gt_ex[of "max Z c"] guess x .. |
|
1149 |
with *[THEN spec, of x] show "eventually (\<lambda>x. Z < f x) F" |
|
1150 |
by (auto elim!: eventually_elim1) |
|
1151 |
qed simp |
|
1152 |
||
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
1153 |
lemma filterlim_at_bot: |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1154 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1155 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)" |
50325 | 1156 |
by (auto simp: filterlim_iff eventually_at_bot_dense elim!: eventually_elim1) |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1157 |
|
50323 | 1158 |
lemma filterlim_at_bot_lt: |
1159 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b" |
|
1160 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z > f x) F)" |
|
1161 |
unfolding filterlim_at_bot |
|
1162 |
proof safe |
|
1163 |
fix Z assume *: "\<forall>Z<c. eventually (\<lambda>x. Z > f x) F" |
|
1164 |
from lt_ex[of "min Z c"] guess x .. |
|
1165 |
with *[THEN spec, of x] show "eventually (\<lambda>x. Z > f x) F" |
|
1166 |
by (auto elim!: eventually_elim1) |
|
1167 |
qed simp |
|
1168 |
||
50325 | 1169 |
lemma filterlim_at_infinity: |
1170 |
fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector" |
|
1171 |
assumes "0 \<le> c" |
|
1172 |
shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)" |
|
1173 |
unfolding filterlim_iff eventually_at_infinity |
|
1174 |
proof safe |
|
1175 |
fix P :: "'a \<Rightarrow> bool" and b |
|
1176 |
assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F" |
|
1177 |
and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x" |
|
1178 |
have "max b (c + 1) > c" by auto |
|
1179 |
with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F" |
|
1180 |
by auto |
|
1181 |
then show "eventually (\<lambda>x. P (f x)) F" |
|
1182 |
proof eventually_elim |
|
1183 |
fix x assume "max b (c + 1) \<le> norm (f x)" |
|
1184 |
with P show "P (f x)" by auto |
|
1185 |
qed |
|
1186 |
qed force |
|
1187 |
||
50322
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
1188 |
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" |
b06b95a5fda2
rename filter_lim to filterlim to be consistent with filtermap
hoelzl
parents:
50247
diff
changeset
|
1189 |
unfolding filterlim_at_top |
50247
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1190 |
apply (intro allI) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1191 |
apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1192 |
apply (auto simp: natceiling_le_eq) |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1193 |
done |
491c5c81c2e8
introduce filter_lim as a generatlization of tendsto
hoelzl
parents:
49834
diff
changeset
|
1194 |
|
50323 | 1195 |
lemma filterlim_inverse_at_top_pos: |
1196 |
"LIM x (nhds 0 within {0::real <..}). inverse x :> at_top" |
|
1197 |
unfolding filterlim_at_top_gt[where c=0] eventually_within |
|
1198 |
proof safe |
|
1199 |
fix Z :: real assume [arith]: "0 < Z" |
|
1200 |
then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)" |
|
1201 |
by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"]) |
|
1202 |
then show "eventually (\<lambda>x. x \<in> {0<..} \<longrightarrow> Z < inverse x) (nhds 0)" |
|
1203 |
by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps) |
|
1204 |
qed |
|
1205 |
||
1206 |
lemma filterlim_inverse_at_top: |
|
1207 |
"(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top" |
|
1208 |
by (intro filterlim_compose[OF filterlim_inverse_at_top_pos]) |
|
1209 |
(simp add: filterlim_def eventually_filtermap le_within_iff) |
|
1210 |
||
1211 |
lemma filterlim_inverse_at_bot_neg: |
|
1212 |
"LIM x (nhds 0 within {..< 0::real}). inverse x :> at_bot" |
|
1213 |
unfolding filterlim_at_bot_lt[where c=0] eventually_within |
|
1214 |
proof safe |
|
1215 |
fix Z :: real assume [arith]: "Z < 0" |
|
1216 |
have "eventually (\<lambda>x. inverse Z < x) (nhds 0)" |
|
1217 |
by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"]) |
|
1218 |
then show "eventually (\<lambda>x. x \<in> {..< 0} \<longrightarrow> inverse x < Z) (nhds 0)" |
|
1219 |
by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps) |
|
1220 |
qed |
|
1221 |
||
1222 |
lemma filterlim_inverse_at_bot: |
|
1223 |
"(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot" |
|
1224 |
by (intro filterlim_compose[OF filterlim_inverse_at_bot_neg]) |
|
1225 |
(simp add: filterlim_def eventually_filtermap le_within_iff) |
|
1226 |
||
1227 |
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top" |
|
1228 |
unfolding filterlim_at_top eventually_at_bot_dense |
|
1229 |
by (blast intro: less_minus_iff[THEN iffD1]) |
|
1230 |
||
1231 |
lemma filterlim_uminus_at_top: "LIM x F. f x :> at_bot \<Longrightarrow> LIM x F. - (f x) :: real :> at_top" |
|
1232 |
by (rule filterlim_compose[OF filterlim_uminus_at_top_at_bot]) |
|
1233 |
||
1234 |
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot" |
|
1235 |
unfolding filterlim_at_bot eventually_at_top_dense |
|
1236 |
by (blast intro: minus_less_iff[THEN iffD1]) |
|
1237 |
||
1238 |
lemma filterlim_uminus_at_bot: "LIM x F. f x :> at_top \<Longrightarrow> LIM x F. - (f x) :: real :> at_bot" |
|
1239 |
by (rule filterlim_compose[OF filterlim_uminus_at_bot_at_top]) |
|
1240 |
||
50325 | 1241 |
lemma tendsto_inverse_0: |
1242 |
fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra" |
|
1243 |
shows "(inverse ---> (0::'a)) at_infinity" |
|
1244 |
unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity |
|
1245 |
proof safe |
|
1246 |
fix r :: real assume "0 < r" |
|
1247 |
show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r" |
|
1248 |
proof (intro exI[of _ "inverse (r / 2)"] allI impI) |
|
1249 |
fix x :: 'a |
|
1250 |
from `0 < r` have "0 < inverse (r / 2)" by simp |
|
1251 |
also assume *: "inverse (r / 2) \<le> norm x" |
|
1252 |
finally show "norm (inverse x) < r" |
|
1253 |
using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps) |
|
1254 |
qed |
|
1255 |
qed |
|
1256 |
||
1257 |
lemma filterlim_inverse_at_infinity: |
|
1258 |
fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}" |
|
1259 |
shows "filterlim inverse at_infinity (at (0::'a))" |
|
1260 |
unfolding filterlim_at_infinity[OF order_refl] |
|
1261 |
proof safe |
|
1262 |
fix r :: real assume "0 < r" |
|
1263 |
then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)" |
|
1264 |
unfolding eventually_at norm_inverse |
|
1265 |
by (intro exI[of _ "inverse r"]) |
|
1266 |
(auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide) |
|
1267 |
qed |
|
1268 |
||
1269 |
lemma filterlim_inverse_at_iff: |
|
1270 |
fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}" |
|
1271 |
shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)" |
|
1272 |
unfolding filterlim_def filtermap_filtermap[symmetric] |
|
1273 |
proof |
|
1274 |
assume "filtermap g F \<le> at_infinity" |
|
1275 |
then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity" |
|
1276 |
by (rule filtermap_mono) |
|
1277 |
also have "\<dots> \<le> at 0" |
|
1278 |
using tendsto_inverse_0 |
|
1279 |
by (auto intro!: le_withinI exI[of _ 1] |
|
1280 |
simp: eventually_filtermap eventually_at_infinity filterlim_def at_def) |
|
1281 |
finally show "filtermap inverse (filtermap g F) \<le> at 0" . |
|
1282 |
next |
|
1283 |
assume "filtermap inverse (filtermap g F) \<le> at 0" |
|
1284 |
then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)" |
|
1285 |
by (rule filtermap_mono) |
|
1286 |
with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity" |
|
1287 |
by (auto intro: order_trans simp: filterlim_def filtermap_filtermap) |
|
1288 |
qed |
|
1289 |
||
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1290 |
text {* |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1291 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1292 |
We only show rules for multiplication and addition when the functions are either against a real |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1293 |
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}. |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1294 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1295 |
*} |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1296 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1297 |
lemma filterlim_tendsto_pos_mult_at_top: |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1298 |
assumes f: "(f ---> c) F" and c: "0 < c" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1299 |
assumes g: "LIM x F. g x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1300 |
shows "LIM x F. (f x * g x :: real) :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1301 |
unfolding filterlim_at_top_gt[where c=0] |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1302 |
proof safe |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1303 |
fix Z :: real assume "0 < Z" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1304 |
from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1305 |
by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1 |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1306 |
simp: dist_real_def abs_real_def split: split_if_asm) |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1307 |
moreover from g have "eventually (\<lambda>x. (Z / c * 2) < g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1308 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1309 |
ultimately show "eventually (\<lambda>x. Z < f x * g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1310 |
proof eventually_elim |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1311 |
fix x assume "c / 2 < f x" "Z / c * 2 < g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1312 |
with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) < f x * g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1313 |
by (intro mult_strict_mono) (auto simp: zero_le_divide_iff) |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1314 |
with `0 < c` show "Z < f x * g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1315 |
by simp |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1316 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1317 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1318 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1319 |
lemma filterlim_at_top_mult_at_top: |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1320 |
assumes f: "LIM x F. f x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1321 |
assumes g: "LIM x F. g x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1322 |
shows "LIM x F. (f x * g x :: real) :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1323 |
unfolding filterlim_at_top_gt[where c=0] |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1324 |
proof safe |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1325 |
fix Z :: real assume "0 < Z" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1326 |
from f have "eventually (\<lambda>x. 1 < f x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1327 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1328 |
moreover from g have "eventually (\<lambda>x. Z < g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1329 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1330 |
ultimately show "eventually (\<lambda>x. Z < f x * g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1331 |
proof eventually_elim |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1332 |
fix x assume "1 < f x" "Z < g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1333 |
with `0 < Z` have "1 * Z < f x * g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1334 |
by (intro mult_strict_mono) (auto simp: zero_le_divide_iff) |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1335 |
then show "Z < f x * g x" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1336 |
by simp |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1337 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1338 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1339 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1340 |
lemma filterlim_tendsto_add_at_top: |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1341 |
assumes f: "(f ---> c) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1342 |
assumes g: "LIM x F. g x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1343 |
shows "LIM x F. (f x + g x :: real) :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1344 |
unfolding filterlim_at_top_gt[where c=0] |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1345 |
proof safe |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1346 |
fix Z :: real assume "0 < Z" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1347 |
from f have "eventually (\<lambda>x. c - 1 < f x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1348 |
by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def) |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1349 |
moreover from g have "eventually (\<lambda>x. Z - (c - 1) < g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1350 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1351 |
ultimately show "eventually (\<lambda>x. Z < f x + g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1352 |
by eventually_elim simp |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1353 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1354 |
|
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1355 |
lemma filterlim_at_top_add_at_top: |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1356 |
assumes f: "LIM x F. f x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1357 |
assumes g: "LIM x F. g x :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1358 |
shows "LIM x F. (f x + g x :: real) :> at_top" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1359 |
unfolding filterlim_at_top_gt[where c=0] |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1360 |
proof safe |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1361 |
fix Z :: real assume "0 < Z" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1362 |
from f have "eventually (\<lambda>x. 0 < f x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1363 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1364 |
moreover from g have "eventually (\<lambda>x. Z < g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1365 |
unfolding filterlim_at_top by auto |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1366 |
ultimately show "eventually (\<lambda>x. Z < f x + g x) F" |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1367 |
by eventually_elim simp |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1368 |
qed |
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1369 |
|
31349
2261c8781f73
new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff
changeset
|
1370 |
end |
50324
0a1242d5e7d4
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
hoelzl
parents:
50323
diff
changeset
|
1371 |