src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy
author wenzelm
Sat, 07 Apr 2012 16:41:59 +0200
changeset 47389 e8552cba702d
parent 45051 c478d1876371
child 47761 dfe747e72fa8
permissions -rw-r--r--
explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions; tuned;
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(*  Title:      HOL/Multivariate_Analysis/Extended_Real_Limits.thy
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    Author:     Johannes Hölzl, TU München
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    Author:     Robert Himmelmann, TU München
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    Author:     Armin Heller, TU München
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    Author:     Bogdan Grechuk, University of Edinburgh
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*)
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header {* Limits on the Extended real number line *}
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theory Extended_Real_Limits
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  imports Topology_Euclidean_Space "~~/src/HOL/Library/Extended_Real"
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begin
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lemma continuous_on_ereal[intro, simp]: "continuous_on A ereal"
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  unfolding continuous_on_topological open_ereal_def by auto
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lemma continuous_at_ereal[intro, simp]: "continuous (at x) ereal"
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  using continuous_on_eq_continuous_at[of UNIV] by auto
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lemma continuous_within_ereal[intro, simp]: "x \<in> A \<Longrightarrow> continuous (at x within A) ereal"
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  using continuous_on_eq_continuous_within[of A] by auto
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lemma ereal_open_uminus:
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  fixes S :: "ereal set"
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  assumes "open S"
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  shows "open (uminus ` S)"
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  unfolding open_ereal_def
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proof (intro conjI impI)
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  obtain x y where S: "open (ereal -` S)"
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    "\<infinity> \<in> S \<Longrightarrow> {ereal x<..} \<subseteq> S" "-\<infinity> \<in> S \<Longrightarrow> {..< ereal y} \<subseteq> S"
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    using `open S` unfolding open_ereal_def by auto
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  have "ereal -` uminus ` S = uminus ` (ereal -` S)"
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  proof safe
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    fix x y assume "ereal x = - y" "y \<in> S"
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    then show "x \<in> uminus ` ereal -` S" by (cases y) auto
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  next
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    fix x assume "ereal x \<in> S"
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    then show "- x \<in> ereal -` uminus ` S"
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      by (auto intro: image_eqI[of _ _ "ereal x"])
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  qed
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  then show "open (ereal -` uminus ` S)"
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    using S by (auto intro: open_negations)
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  { assume "\<infinity> \<in> uminus ` S"
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    then have "-\<infinity> \<in> S" by (metis image_iff ereal_uminus_uminus)
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    then have "uminus ` {..<ereal y} \<subseteq> uminus ` S" using S by (intro image_mono) auto
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    then show "\<exists>x. {ereal x<..} \<subseteq> uminus ` S" using ereal_uminus_lessThan by auto }
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  { assume "-\<infinity> \<in> uminus ` S"
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    then have "\<infinity> : S" by (metis image_iff ereal_uminus_uminus)
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    then have "uminus ` {ereal x<..} <= uminus ` S" using S by (intro image_mono) auto
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    then show "\<exists>y. {..<ereal y} <= uminus ` S" using ereal_uminus_greaterThan by auto }
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qed
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lemma ereal_uminus_complement:
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  fixes S :: "ereal set"
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  shows "uminus ` (- S) = - uminus ` S"
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  by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)
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lemma ereal_closed_uminus:
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  fixes S :: "ereal set"
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  assumes "closed S"
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  shows "closed (uminus ` S)"
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using assms unfolding closed_def
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using ereal_open_uminus[of "- S"] ereal_uminus_complement by auto
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instance ereal :: perfect_space
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proof (default, rule)
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  fix a :: ereal assume a: "open {a}"
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  show False
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  proof (cases a)
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    case MInf
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    then obtain y where "{..<ereal y} <= {a}" using a open_MInfty2[of "{a}"] by auto
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    hence "ereal(y - 1):{a}" apply (subst subsetD[of "{..<ereal y}"]) by auto
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    then show False using `a=(-\<infinity>)` by auto
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  next
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    case PInf
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    then obtain y where "{ereal y<..} <= {a}" using a open_PInfty2[of "{a}"] by auto
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    hence "ereal(y+1):{a}" apply (subst subsetD[of "{ereal y<..}"]) by auto
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    then show False using `a=\<infinity>` by auto
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  next
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    case (real r) then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
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    from ereal_open_cont_interval[OF a singletonI this] guess e . note e = this
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    then obtain b where b_def: "a<b & b<a+e"
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      using fin ereal_between ereal_dense[of a "a+e"] by auto
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    then have "b: {a-e <..< a+e}" using fin ereal_between[of a e] e by auto
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    then show False using b_def e by auto
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  qed
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qed
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lemma ereal_closed_contains_Inf:
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  fixes S :: "ereal set"
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  assumes "closed S" "S ~= {}"
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  shows "Inf S : S"
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proof(rule ccontr)
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  assume "Inf S \<notin> S" hence a: "open (-S)" "Inf S:(- S)" using assms by auto
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  show False
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  proof (cases "Inf S")
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    case MInf hence "(-\<infinity>) : - S" using a by auto
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    then obtain y where "{..<ereal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
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    hence "ereal y <= Inf S" by (metis Compl_anti_mono Compl_lessThan atLeast_iff
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      complete_lattice_class.Inf_greatest double_complement set_rev_mp)
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    then show False using MInf by auto
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  next
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    case PInf then have "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
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    then show False using `Inf S ~: S` by (simp add: top_ereal_def)
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  next
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    case (real r) then have fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>" by simp
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    from ereal_open_cont_interval[OF a this] guess e . note e = this
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    { fix x assume "x:S" hence "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
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      hence *: "x>Inf S-e" using e by (metis fin ereal_between(1) order_less_le_trans)
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      { assume "x<Inf S+e" hence "x:{Inf S-e <..< Inf S+e}" using * by auto
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        hence False using e `x:S` by auto
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      } hence "x>=Inf S+e" by (metis linorder_le_less_linear)
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    } hence "Inf S + e <= Inf S" by (metis le_Inf_iff)
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    then show False using real e by (cases e) auto
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  qed
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qed
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lemma ereal_closed_contains_Sup:
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  fixes S :: "ereal set"
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  assumes "closed S" "S ~= {}"
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  shows "Sup S : S"
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proof-
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  have "closed (uminus ` S)" by (metis assms(1) ereal_closed_uminus)
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  hence "Inf (uminus ` S) : uminus ` S" using assms ereal_closed_contains_Inf[of "uminus ` S"] by auto
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  hence "- Sup S : uminus ` S" using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
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  thus ?thesis by (metis imageI ereal_uminus_uminus ereal_minus_minus_image)
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qed
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lemma ereal_open_closed_aux:
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  fixes S :: "ereal set"
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  assumes "open S" "closed S"
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  assumes S: "(-\<infinity>) ~: S"
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  shows "S = {}"
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proof(rule ccontr)
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  assume "S ~= {}"
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  hence *: "(Inf S):S" by (metis assms(2) ereal_closed_contains_Inf)
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  { assume "Inf S=(-\<infinity>)" hence False using * assms(3) by auto }
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  moreover
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diff changeset
   139
  { assume "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44170
diff changeset
   140
    hence False by (metis assms(1) not_open_singleton) }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   141
  moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   142
  { assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   143
    from ereal_open_cont_interval[OF assms(1) * fin] guess e . note e = this
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   144
    then obtain b where b_def: "Inf S-e<b & b<Inf S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   145
      using fin ereal_between[of "Inf S" e] ereal_dense[of "Inf S-e"] by auto
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44571
diff changeset
   146
    hence "b: {Inf S-e <..< Inf S+e}" using e fin ereal_between[of "Inf S" e]
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44571
diff changeset
   147
      by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   148
    hence "b:S" using e by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   149
    hence False using b_def by (metis complete_lattice_class.Inf_lower leD)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   150
  } ultimately show False by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   151
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   152
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   153
lemma ereal_open_closed:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   154
  fixes S :: "ereal set"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   155
  shows "(open S & closed S) <-> (S = {} | S = UNIV)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   156
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   157
{ assume lhs: "open S & closed S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   158
  { assume "(-\<infinity>) ~: S" hence "S={}" using lhs ereal_open_closed_aux by auto }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   159
  moreover
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   160
  { assume "(-\<infinity>) : S" hence "(- S)={}" using lhs ereal_open_closed_aux[of "-S"] by auto }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   161
  ultimately have "S = {} | S = UNIV" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   162
} thus ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   163
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   164
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   165
lemma ereal_open_affinity_pos:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   166
  fixes S :: "ereal set"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   167
  assumes "open S" and m: "m \<noteq> \<infinity>" "0 < m" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   168
  shows "open ((\<lambda>x. m * x + t) ` S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   169
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   170
  obtain r where r[simp]: "m = ereal r" using m by (cases m) auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   171
  obtain p where p[simp]: "t = ereal p" using t by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   172
  have "r \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" using m by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   173
  from `open S`[THEN ereal_openE] guess l u . note T = this
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   174
  let ?f = "(\<lambda>x. m * x + t)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   175
  show ?thesis unfolding open_ereal_def
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   176
  proof (intro conjI impI exI subsetI)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   177
    have "ereal -` ?f ` S = (\<lambda>x. r * x + p) ` (ereal -` S)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   178
    proof safe
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   179
      fix x y assume "ereal y = m * x + t" "x \<in> S"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   180
      then show "y \<in> (\<lambda>x. r * x + p) ` ereal -` S"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   181
        using `r \<noteq> 0` by (cases x) (auto intro!: image_eqI[of _ _ "real x"] split: split_if_asm)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   182
    qed force
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   183
    then show "open (ereal -` ?f ` S)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   184
      using open_affinity[OF T(1) `r \<noteq> 0`] by (auto simp: ac_simps)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   185
  next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   186
    assume "\<infinity> \<in> ?f`S" with `0 < r` have "\<infinity> \<in> S" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   187
    fix x assume "x \<in> {ereal (r * l + p)<..}"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   188
    then have [simp]: "ereal (r * l + p) < x" by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   189
    show "x \<in> ?f`S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   190
    proof (rule image_eqI)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   191
      show "x = m * ((x - t) / m) + t"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   192
        using m t by (cases rule: ereal3_cases[of m x t]) auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   193
      have "ereal l < (x - t)/m"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   194
        using m t by (simp add: ereal_less_divide_pos ereal_less_minus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   195
      then show "(x - t)/m \<in> S" using T(2)[OF `\<infinity> \<in> S`] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   196
    qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   197
  next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   198
    assume "-\<infinity> \<in> ?f`S" with `0 < r` have "-\<infinity> \<in> S" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   199
    fix x assume "x \<in> {..<ereal (r * u + p)}"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   200
    then have [simp]: "x < ereal (r * u + p)" by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   201
    show "x \<in> ?f`S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   202
    proof (rule image_eqI)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   203
      show "x = m * ((x - t) / m) + t"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   204
        using m t by (cases rule: ereal3_cases[of m x t]) auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   205
      have "(x - t)/m < ereal u"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   206
        using m t by (simp add: ereal_divide_less_pos ereal_minus_less)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   207
      then show "(x - t)/m \<in> S" using T(3)[OF `-\<infinity> \<in> S`] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   208
    qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   209
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   210
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   211
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   212
lemma ereal_open_affinity:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   213
  fixes S :: "ereal set"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   214
  assumes "open S" and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   215
  shows "open ((\<lambda>x. m * x + t) ` S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   216
proof cases
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   217
  assume "0 < m" then show ?thesis
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   218
    using ereal_open_affinity_pos[OF `open S` _ _ t, of m] m by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   219
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   220
  assume "\<not> 0 < m" then
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   221
  have "0 < -m" using `m \<noteq> 0` by (cases m) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   222
  then have m: "-m \<noteq> \<infinity>" "0 < -m" using `\<bar>m\<bar> \<noteq> \<infinity>`
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   223
    by (auto simp: ereal_uminus_eq_reorder)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   224
  from ereal_open_affinity_pos[OF ereal_open_uminus[OF `open S`] m t]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   225
  show ?thesis unfolding image_image by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   226
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   227
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   228
lemma ereal_lim_mult:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   229
  fixes X :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   230
  assumes lim: "(X ---> L) net" and a: "\<bar>a\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   231
  shows "((\<lambda>i. a * X i) ---> a * L) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   232
proof cases
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   233
  assume "a \<noteq> 0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   234
  show ?thesis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   235
  proof (rule topological_tendstoI)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   236
    fix S assume "open S" "a * L \<in> S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   237
    have "a * L / a = L"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   238
      using `a \<noteq> 0` a by (cases rule: ereal2_cases[of a L]) auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   239
    then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   240
      using `a * L \<in> S` by (force simp: image_iff)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   241
    moreover have "open ((\<lambda>x. x / a) ` S)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   242
      using ereal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 0` a
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   243
      by (auto simp: ereal_divide_eq ereal_inverse_eq_0 divide_ereal_def ac_simps)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   244
    note * = lim[THEN topological_tendstoD, OF this L]
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   245
    { fix x from a `a \<noteq> 0` have "a * (x / a) = x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   246
        by (cases rule: ereal2_cases[of a x]) auto }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   247
    note this[simp]
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   248
    show "eventually (\<lambda>x. a * X x \<in> S) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   249
      by (rule eventually_mono[OF _ *]) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   250
  qed
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44571
diff changeset
   251
qed auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   252
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   253
lemma ereal_lim_uminus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   254
  fixes X :: "'a \<Rightarrow> ereal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   255
  using ereal_lim_mult[of X L net "ereal (-1)"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   256
        ereal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "ereal (-1)"]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   257
  by (auto simp add: algebra_simps)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   258
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   259
lemma Lim_bounded2_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   260
  assumes lim:"f ----> (l :: ereal)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   261
  and ge: "ALL n>=N. f n >= C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   262
  shows "l>=C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   263
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   264
def g == "(%i. -(f i))"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   265
{ fix n assume "n>=N" hence "g n <= -C" using assms ereal_minus_le_minus g_def by auto }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   266
hence "ALL n>=N. g n <= -C" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   267
moreover have limg: "g ----> (-l)" using g_def ereal_lim_uminus lim by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   268
ultimately have "-l <= -C" using Lim_bounded_ereal[of g "-l" _ "-C"] by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   269
from this show ?thesis using ereal_minus_le_minus by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   270
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   271
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   272
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   273
lemma ereal_open_atLeast: fixes x :: ereal shows "open {x..} \<longleftrightarrow> x = -\<infinity>"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   274
proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   275
  assume "x = -\<infinity>" then have "{x..} = UNIV" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   276
  then show "open {x..}" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   277
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   278
  assume "open {x..}"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   279
  then have "open {x..} \<and> closed {x..}" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   280
  then have "{x..} = UNIV" unfolding ereal_open_closed by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   281
  then show "x = -\<infinity>" by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   282
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   283
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   284
lemma ereal_open_mono_set:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   285
  fixes S :: "ereal set"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   286
  defines "a \<equiv> Inf S"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   287
  shows "(open S \<and> mono_set S) \<longleftrightarrow> (S = UNIV \<or> S = {a <..})"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   288
  by (metis Inf_UNIV a_def atLeast_eq_UNIV_iff ereal_open_atLeast
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   289
            ereal_open_closed mono_set_iff open_ereal_greaterThan)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   290
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   291
lemma ereal_closed_mono_set:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   292
  fixes S :: "ereal set"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   293
  shows "(closed S \<and> mono_set S) \<longleftrightarrow> (S = {} \<or> S = {Inf S ..})"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   294
  by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   295
            ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   296
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   297
lemma ereal_Liminf_Sup_monoset:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   298
  fixes f :: "'a => ereal"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   299
  shows "Liminf net f = Sup {l. \<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   300
  unfolding Liminf_Sup
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   301
proof (intro arg_cong[where f="\<lambda>P. Sup (Collect P)"] ext iffI allI impI)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   302
  fix l S assume ev: "\<forall>y<l. eventually (\<lambda>x. y < f x) net" and "open S" "mono_set S" "l \<in> S"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   303
  then have "S = UNIV \<or> S = {Inf S <..}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   304
    using ereal_open_mono_set[of S] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   305
  then show "eventually (\<lambda>x. f x \<in> S) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   306
  proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   307
    assume S: "S = {Inf S<..}"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   308
    then have "Inf S < l" using `l \<in> S` by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   309
    then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
44918
6a80fbc4e72c tune simpset for Complete_Lattices
noschinl
parents: 44571
diff changeset
   310
    then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   311
  qed auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   312
next
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   313
  fix l y assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "y < l"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   314
  have "eventually  (\<lambda>x. f x \<in> {y <..}) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   315
    using `y < l` by (intro S[rule_format]) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   316
  then show "eventually (\<lambda>x. y < f x) net" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   317
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   318
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   319
lemma ereal_Limsup_Inf_monoset:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   320
  fixes f :: "'a => ereal"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   321
  shows "Limsup net f = Inf {l. \<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   322
  unfolding Limsup_Inf
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   323
proof (intro arg_cong[where f="\<lambda>P. Inf (Collect P)"] ext iffI allI impI)
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   324
  fix l S assume ev: "\<forall>y>l. eventually (\<lambda>x. f x < y) net" and "open S" "mono_set (uminus`S)" "l \<in> S"
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   325
  then have "open (uminus`S) \<and> mono_set (uminus`S)" by (simp add: ereal_open_uminus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   326
  then have "S = UNIV \<or> S = {..< Sup S}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   327
    unfolding ereal_open_mono_set ereal_Inf_uminus_image_eq ereal_image_uminus_shift by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   328
  then show "eventually (\<lambda>x. f x \<in> S) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   329
  proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   330
    assume S: "S = {..< Sup S}"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   331
    then have "l < Sup S" using `l \<in> S` by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   332
    then have "eventually (\<lambda>x. f x < Sup S) net" using ev by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   333
    then show "eventually (\<lambda>x. f x \<in> S) net"  by (subst S) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   334
  qed auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   335
next
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   336
  fix l y assume S: "\<forall>S. open S \<longrightarrow> mono_set (uminus`S) \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "l < y"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   337
  have "eventually  (\<lambda>x. f x \<in> {..< y}) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   338
    using `l < y` by (intro S[rule_format]) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   339
  then show "eventually (\<lambda>x. f x < y) net" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   340
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   341
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   342
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   343
lemma open_uminus_iff: "open (uminus ` S) \<longleftrightarrow> open (S::ereal set)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   344
  using ereal_open_uminus[of S] ereal_open_uminus[of "uminus`S"] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   345
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   346
lemma ereal_Limsup_uminus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   347
  fixes f :: "'a => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   348
  shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   349
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   350
  { fix P l have "(\<exists>x. (l::ereal) = -x \<and> P x) \<longleftrightarrow> P (-l)" by (auto intro!: exI[of _ "-l"]) }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   351
  note Ex_cancel = this
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   352
  { fix P :: "ereal set \<Rightarrow> bool" have "(\<forall>S. P S) \<longleftrightarrow> (\<forall>S. P (uminus`S))"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   353
      apply auto by (erule_tac x="uminus`S" in allE) (auto simp: image_image) }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   354
  note add_uminus_image = this
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   355
  { fix x S have "(x::ereal) \<in> uminus`S \<longleftrightarrow> -x\<in>S" by (auto intro!: image_eqI[of _ _ "-x"]) }
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   356
  note remove_uminus_image = this
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   357
  show ?thesis
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   358
    unfolding ereal_Limsup_Inf_monoset ereal_Liminf_Sup_monoset
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   359
    unfolding ereal_Inf_uminus_image_eq[symmetric] image_Collect Ex_cancel
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   360
    by (subst add_uminus_image) (simp add: open_uminus_iff remove_uminus_image)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   361
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   362
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   363
lemma ereal_Liminf_uminus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   364
  fixes f :: "'a => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   365
  shows "Liminf net (\<lambda>x. - (f x)) = -(Limsup net f)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   366
  using ereal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   367
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   368
lemma ereal_Lim_uminus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   369
  fixes f :: "'a \<Rightarrow> ereal" shows "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x) ---> - f0) net"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   370
  using
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   371
    ereal_lim_mult[of f f0 net "- 1"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   372
    ereal_lim_mult[of "\<lambda>x. - (f x)" "-f0" net "- 1"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   373
  by (auto simp: ereal_uminus_reorder)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   374
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   375
lemma lim_imp_Limsup:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   376
  fixes f :: "'a => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   377
  assumes "\<not> trivial_limit net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   378
  assumes lim: "(f ---> f0) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   379
  shows "Limsup net f = f0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   380
  using ereal_Lim_uminus[of f f0] lim_imp_Liminf[of net "(%x. -(f x))" "-f0"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   381
     ereal_Liminf_uminus[of net f] assms by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   382
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   383
lemma Liminf_PInfty:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   384
  fixes f :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   385
  assumes "\<not> trivial_limit net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   386
  shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   387
proof (intro lim_imp_Liminf iffI assms)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   388
  assume rhs: "Liminf net f = \<infinity>"
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   389
  { fix S :: "ereal set" assume "open S & \<infinity> : S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   390
    then obtain m where "{ereal m<..} <= S" using open_PInfty2 by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   391
    moreover have "eventually (\<lambda>x. f x \<in> {ereal m<..}) net"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   392
      using rhs unfolding Liminf_Sup top_ereal_def[symmetric] Sup_eq_top_iff
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   393
      by (auto elim!: allE[where x="ereal m"] simp: top_ereal_def)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   394
    ultimately have "eventually (%x. f x : S) net" apply (subst eventually_mono) by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   395
  } then show "(f ---> \<infinity>) net" unfolding tendsto_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   396
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   397
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   398
lemma Limsup_MInfty:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   399
  fixes f :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   400
  assumes "\<not> trivial_limit net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   401
  shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   402
  using assms ereal_Lim_uminus[of f "-\<infinity>"] Liminf_PInfty[of _ "\<lambda>x. - (f x)"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   403
        ereal_Liminf_uminus[of _ f] by (auto simp: ereal_uminus_eq_reorder)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   404
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   405
lemma ereal_Liminf_eq_Limsup:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   406
  fixes f :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   407
  assumes ntriv: "\<not> trivial_limit net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   408
  assumes lim: "Liminf net f = f0" "Limsup net f = f0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   409
  shows "(f ---> f0) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   410
proof (cases f0)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   411
  case PInf then show ?thesis using Liminf_PInfty[OF ntriv] lim by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   412
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   413
  case MInf then show ?thesis using Limsup_MInfty[OF ntriv] lim by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   414
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   415
  case (real r)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   416
  show "(f ---> f0) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   417
  proof (rule topological_tendstoI)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   418
    fix S assume "open S""f0 \<in> S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   419
    then obtain a b where "a < Liminf net f" "Limsup net f < b" "{a<..<b} \<subseteq> S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   420
      using ereal_open_cont_interval2[of S f0] real lim by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   421
    then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   422
      unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 44125
diff changeset
   423
      by (auto intro!: eventually_conj)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   424
    with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   425
      by (rule_tac eventually_mono) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   426
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   427
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   428
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   429
lemma ereal_Liminf_eq_Limsup_iff:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   430
  fixes f :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   431
  assumes "\<not> trivial_limit net"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   432
  shows "(f ---> f0) net \<longleftrightarrow> Liminf net f = f0 \<and> Limsup net f = f0"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   433
  by (metis assms ereal_Liminf_eq_Limsup lim_imp_Liminf lim_imp_Limsup)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   434
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   435
lemma limsup_INFI_SUPR:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   436
  fixes f :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   437
  shows "limsup f = (INF n. SUP m:{n..}. f m)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   438
  using ereal_Limsup_uminus[of sequentially "\<lambda>x. - f x"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   439
  by (simp add: liminf_SUPR_INFI ereal_INFI_uminus ereal_SUPR_uminus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   440
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   441
lemma liminf_PInfty:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   442
  fixes X :: "nat => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   443
  shows "X ----> \<infinity> <-> liminf X = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   444
by (metis Liminf_PInfty trivial_limit_sequentially)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   445
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   446
lemma limsup_MInfty:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   447
  fixes X :: "nat => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   448
  shows "X ----> (-\<infinity>) <-> limsup X = (-\<infinity>)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   449
by (metis Limsup_MInfty trivial_limit_sequentially)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   450
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   451
lemma ereal_lim_mono:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   452
  fixes X Y :: "nat => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   453
  assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   454
  assumes "X ----> x" "Y ----> y"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   455
  shows "x <= y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   456
  by (metis ereal_Liminf_eq_Limsup_iff[OF trivial_limit_sequentially] assms liminf_mono)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   457
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   458
lemma incseq_le_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   459
  fixes X :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   460
  assumes inc: "incseq X" and lim: "X ----> L"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   461
  shows "X N \<le> L"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   462
  using inc
44125
230a8665c919 mark some redundant theorems as legacy
huffman
parents: 43923
diff changeset
   463
  by (intro ereal_lim_mono[of N, OF _ tendsto_const lim]) (simp add: incseq_def)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   464
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   465
lemma decseq_ge_ereal: assumes dec: "decseq X"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   466
  and lim: "X ----> (L::ereal)" shows "X N >= L"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   467
  using dec
44125
230a8665c919 mark some redundant theorems as legacy
huffman
parents: 43923
diff changeset
   468
  by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   469
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   470
lemma liminf_bounded_open:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   471
  fixes x :: "nat \<Rightarrow> ereal"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   472
  shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))" 
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   473
  (is "_ \<longleftrightarrow> ?P x0")
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   474
proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   475
  assume "?P x0" then show "x0 \<le> liminf x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   476
    unfolding ereal_Liminf_Sup_monoset eventually_sequentially
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   477
    by (intro complete_lattice_class.Sup_upper) auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   478
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   479
  assume "x0 \<le> liminf x"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   480
  { fix S :: "ereal set" assume om: "open S & mono_set S & x0:S"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   481
    { assume "S = UNIV" hence "EX N. (ALL n>=N. x n : S)" by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   482
    moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   483
    { assume "~(S=UNIV)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   484
      then obtain B where B_def: "S = {B<..}" using om ereal_open_mono_set by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   485
      hence "B<x0" using om by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   486
      hence "EX N. ALL n>=N. x n : S" unfolding B_def using `x0 \<le> liminf x` liminf_bounded_iff by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   487
    } ultimately have "EX N. (ALL n>=N. x n : S)" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   488
  } then show "?P x0" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   489
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   490
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   491
lemma limsup_subseq_mono:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   492
  fixes X :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   493
  assumes "subseq r"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   494
  shows "limsup (X \<circ> r) \<le> limsup X"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   495
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   496
  have "(\<lambda>n. - X n) \<circ> r = (\<lambda>n. - (X \<circ> r) n)" by (simp add: fun_eq_iff)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   497
  then have "- limsup X \<le> - limsup (X \<circ> r)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   498
     using liminf_subseq_mono[of r "(%n. - X n)"]
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   499
       ereal_Liminf_uminus[of sequentially X]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   500
       ereal_Liminf_uminus[of sequentially "X o r"] assms by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   501
  then show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   502
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   503
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   504
lemma bounded_abs:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   505
  assumes "(a::real)<=x" "x<=b"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   506
  shows "abs x <= max (abs a) (abs b)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   507
by (metis abs_less_iff assms leI le_max_iff_disj less_eq_real_def less_le_not_le less_minus_iff minus_minus)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   508
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   509
lemma bounded_increasing_convergent2: fixes f::"nat => real"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   510
  assumes "ALL n. f n <= B"  "ALL n m. n>=m --> f n >= f m"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   511
  shows "EX l. (f ---> l) sequentially"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   512
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   513
def N == "max (abs (f 0)) (abs B)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   514
{ fix n have "abs (f n) <= N" unfolding N_def apply (subst bounded_abs) using assms by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   515
hence "bounded {f n| n::nat. True}" unfolding bounded_real by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   516
from this show ?thesis apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   517
   using assms by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   518
qed
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   519
lemma lim_ereal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   520
  obtains l where "f ----> (l::ereal)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   521
proof(cases "f = (\<lambda>x. - \<infinity>)")
44125
230a8665c919 mark some redundant theorems as legacy
huffman
parents: 43923
diff changeset
   522
  case True then show thesis using tendsto_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   523
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   524
  case False
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   525
  from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   526
  have "ALL n>=N. f n >= f N" using assms by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   527
  hence minf: "ALL n>=N. f n > (-\<infinity>)" using N_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   528
  def Y == "(%n. (if n>=N then f n else f N))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   529
  hence incy: "!!n m. n>=m ==> Y n >= Y m" using assms by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   530
  from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   531
  show thesis
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   532
  proof(cases "EX B. ALL n. f n < ereal B")
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   533
    case False thus thesis apply- apply(rule that[of \<infinity>]) unfolding Lim_PInfty not_ex not_all
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   534
    apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   535
    apply(rule order_trans[OF _ assms[rule_format]]) by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   536
  next case True then guess B ..
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   537
    hence "ALL n. Y n < ereal B" using Y_def by auto note B = this[rule_format]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   538
    { fix n have "Y n < \<infinity>" using B[of n] apply (subst less_le_trans) by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   539
      hence "Y n ~= \<infinity> & Y n ~= (-\<infinity>)" using minfy by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   540
    } hence *: "ALL n. \<bar>Y n\<bar> \<noteq> \<infinity>" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   541
    { fix n have "real (Y n) < B" proof- case goal1 thus ?case
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   542
        using B[of n] apply-apply(subst(asm) ereal_real'[THEN sym]) defer defer
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   543
        unfolding ereal_less using * by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   544
      qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   545
    }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   546
    hence B': "ALL n. (real (Y n) <= B)" using less_imp_le by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   547
    have "EX l. (%n. real (Y n)) ----> l"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   548
      apply(rule bounded_increasing_convergent2)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   549
    proof safe show "!!n. real (Y n) <= B" using B' by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   550
      fix n m::nat assume "n<=m"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   551
      hence "ereal (real (Y n)) <= ereal (real (Y m))"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   552
        using incy[rule_format,of n m] apply(subst ereal_real)+
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   553
        using *[rule_format, of n] *[rule_format, of m] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   554
      thus "real (Y n) <= real (Y m)" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   555
    qed then guess l .. note l=this
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   556
    have "Y ----> ereal l" using l apply-apply(subst(asm) lim_ereal[THEN sym])
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   557
    unfolding ereal_real using * by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   558
    thus thesis apply-apply(rule that[of "ereal l"])
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   559
       apply (subst tail_same_limit[of Y _ N]) using Y_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   560
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   561
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   562
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   563
lemma lim_ereal_decreasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n <= f m"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   564
  obtains l where "f ----> (l::ereal)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   565
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   566
  from lim_ereal_increasing[of "\<lambda>x. - f x"] assms
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   567
  obtain l where "(\<lambda>x. - f x) ----> l" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   568
  from ereal_lim_mult[OF this, of "- 1"] show thesis
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   569
    by (intro that[of "-l"]) (simp add: ereal_uminus_eq_reorder)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   570
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   571
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   572
lemma compact_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   573
  fixes X :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   574
  shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   575
proof -
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   576
  obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   577
    using seq_monosub[of X] unfolding comp_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   578
  then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   579
    by (auto simp add: monoseq_def)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   580
  then obtain l where "(X\<circ>r) ----> l"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   581
     using lim_ereal_increasing[of "X \<circ> r"] lim_ereal_decreasing[of "X \<circ> r"] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   582
  then show ?thesis using `subseq r` by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   583
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   584
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   585
lemma ereal_Sup_lim:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   586
  assumes "\<And>n. b n \<in> s" "b ----> (a::ereal)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   587
  shows "a \<le> Sup s"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   588
by (metis Lim_bounded_ereal assms complete_lattice_class.Sup_upper)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   589
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   590
lemma ereal_Inf_lim:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   591
  assumes "\<And>n. b n \<in> s" "b ----> (a::ereal)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   592
  shows "Inf s \<le> a"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   593
by (metis Lim_bounded2_ereal assms complete_lattice_class.Inf_lower)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   594
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   595
lemma SUP_Lim_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   596
  fixes X :: "nat \<Rightarrow> ereal" assumes "incseq X" "X ----> l" shows "(SUP n. X n) = l"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   597
proof (rule ereal_SUPI)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   598
  fix n from assms show "X n \<le> l"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   599
    by (intro incseq_le_ereal) (simp add: incseq_def)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   600
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   601
  fix y assume "\<And>n. n \<in> UNIV \<Longrightarrow> X n \<le> y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   602
  with ereal_Sup_lim[OF _ `X ----> l`, of "{..y}"]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   603
  show "l \<le> y" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   604
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   605
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   606
lemma LIMSEQ_ereal_SUPR:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   607
  fixes X :: "nat \<Rightarrow> ereal" assumes "incseq X" shows "X ----> (SUP n. X n)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   608
proof (rule lim_ereal_increasing)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   609
  fix n m :: nat assume "m \<le> n" then show "X m \<le> X n"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   610
    using `incseq X` by (simp add: incseq_def)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   611
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   612
  fix l assume "X ----> l"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   613
  with SUP_Lim_ereal[of X, OF assms this] show ?thesis by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   614
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   615
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   616
lemma INF_Lim_ereal: "decseq X \<Longrightarrow> X ----> l \<Longrightarrow> (INF n. X n) = (l::ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   617
  using SUP_Lim_ereal[of "\<lambda>i. - X i" "- l"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   618
  by (simp add: ereal_SUPR_uminus ereal_lim_uminus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   619
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   620
lemma LIMSEQ_ereal_INFI: "decseq X \<Longrightarrow> X ----> (INF n. X n :: ereal)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   621
  using LIMSEQ_ereal_SUPR[of "\<lambda>i. - X i"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   622
  by (simp add: ereal_SUPR_uminus ereal_lim_uminus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   623
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   624
lemma SUP_eq_LIMSEQ:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   625
  assumes "mono f"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   626
  shows "(SUP n. ereal (f n)) = ereal x \<longleftrightarrow> f ----> x"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   627
proof
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   628
  have inc: "incseq (\<lambda>i. ereal (f i))"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   629
    using `mono f` unfolding mono_def incseq_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   630
  { assume "f ----> x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   631
   then have "(\<lambda>i. ereal (f i)) ----> ereal x" by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   632
   from SUP_Lim_ereal[OF inc this]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   633
   show "(SUP n. ereal (f n)) = ereal x" . }
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   634
  { assume "(SUP n. ereal (f n)) = ereal x"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   635
    with LIMSEQ_ereal_SUPR[OF inc]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   636
    show "f ----> x" by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   637
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   638
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   639
lemma Liminf_within:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   640
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   641
  shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   642
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   643
let ?l="(SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   644
{ fix T assume T_def: "open T & mono_set T & ?l:T"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   645
  have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   646
  proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   647
  { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   648
  moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   649
  { assume "~(T=UNIV)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   650
    then obtain B where "T={B<..}" using T_def ereal_open_mono_set[of T] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   651
    hence "B<?l" using T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   652
    then obtain d where d_def: "0<d & B<(INF y:(S Int ball x d - {x}). f y)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   653
      unfolding less_SUP_iff by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   654
    { fix y assume "y:S & 0 < dist y x & dist y x < d"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   655
      hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   656
      hence "f y:T" using d_def INF_lower[of y "S Int ball x d - {x}" f] `T={B<..}` by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   657
    } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   658
  } ultimately show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   659
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   660
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   661
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   662
{ fix z
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   663
  assume a: "ALL T. open T --> mono_set T --> z : T -->
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   664
     (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   665
  { fix B assume "B<z"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   666
    then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> B < f y)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   667
       using a[rule_format, of "{B<..}"] mono_greaterThan by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   668
    { fix y assume "y:(S Int ball x d - {x})"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   669
      hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
45051
c478d1876371 discontinued legacy theorem names from RealDef.thy
huffman
parents: 45032
diff changeset
   670
         by (metis dist_eq_0_iff less_le zero_le_dist)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   671
      hence "B <= f y" using d_def by auto
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   672
    } hence "B <= INFI (S Int ball x d - {x}) f" apply (subst INF_greatest) by auto
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   673
    also have "...<=?l" apply (subst SUP_upper) using d_def by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   674
    finally have "B<=?l" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   675
  } hence "z <= ?l" using ereal_le_ereal[of z "?l"] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   676
}
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   677
ultimately show ?thesis unfolding ereal_Liminf_Sup_monoset eventually_within
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   678
   apply (subst ereal_SupI[of _ "(SUP e:{0<..}. INFI (S Int ball x e - {x}) f)"]) by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   679
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   680
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   681
lemma Limsup_within:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   682
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   683
  shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   684
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   685
let ?l="(INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   686
{ fix T assume T_def: "open T & mono_set (uminus ` T) & ?l:T"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   687
  have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   688
  proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   689
  { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   690
  moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   691
  { assume "~(T=UNIV)" hence "~(uminus ` T = UNIV)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   692
       by (metis Int_UNIV_right Int_absorb1 image_mono ereal_minus_minus_image subset_UNIV)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   693
    hence "uminus ` T = {Inf (uminus ` T)<..}" using T_def ereal_open_mono_set[of "uminus ` T"]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   694
       ereal_open_uminus[of T] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   695
    then obtain B where "T={..<B}"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   696
      unfolding ereal_Inf_uminus_image_eq ereal_uminus_lessThan[symmetric]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   697
      unfolding inj_image_eq_iff[OF ereal_inj_on_uminus] by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   698
    hence "?l<B" using T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   699
    then obtain d where d_def: "0<d & (SUP y:(S Int ball x d - {x}). f y)<B"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   700
      unfolding INF_less_iff by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   701
    { fix y assume "y:S & 0 < dist y x & dist y x < d"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   702
      hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   703
      hence "f y:T" using d_def SUP_upper[of y "S Int ball x d - {x}" f] `T={..<B}` by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   704
    } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   705
  } ultimately show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   706
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   707
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   708
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   709
{ fix z
44170
510ac30f44c0 make Multivariate_Analysis work with separate set type
huffman
parents: 44167
diff changeset
   710
  assume a: "ALL T. open T --> mono_set (uminus ` T) --> z : T -->
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   711
     (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   712
  { fix B assume "z<B"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   713
    then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> f y<B)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   714
       using a[rule_format, of "{..<B}"] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   715
    { fix y assume "y:(S Int ball x d - {x})"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   716
      hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
45051
c478d1876371 discontinued legacy theorem names from RealDef.thy
huffman
parents: 45032
diff changeset
   717
         by (metis dist_eq_0_iff less_le zero_le_dist)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   718
      hence "f y <= B" using d_def by auto
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   719
    } hence "SUPR (S Int ball x d - {x}) f <= B" apply (subst SUP_least) by auto
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
   720
    moreover have "?l<=SUPR (S Int ball x d - {x}) f" apply (subst INF_lower) using d_def by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   721
    ultimately have "?l<=B" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   722
  } hence "?l <= z" using ereal_ge_ereal[of z "?l"] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   723
}
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   724
ultimately show ?thesis unfolding ereal_Limsup_Inf_monoset eventually_within
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   725
   apply (subst ereal_InfI) by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   726
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   727
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   728
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   729
lemma Liminf_within_UNIV:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   730
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   731
  shows "Liminf (at x) f = Liminf (at x within UNIV) f"
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44928
diff changeset
   732
  by simp (* TODO: delete *)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   733
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   734
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   735
lemma Liminf_at:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   736
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   737
  shows "Liminf (at x) f = (SUP e:{0<..}. INF y:(ball x e - {x}). f y)"
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44928
diff changeset
   738
  using Liminf_within[of x UNIV f] by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   739
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   740
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   741
lemma Limsup_within_UNIV:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   742
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   743
  shows "Limsup (at x) f = Limsup (at x within UNIV) f"
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44928
diff changeset
   744
  by simp (* TODO: delete *)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   745
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   746
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   747
lemma Limsup_at:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   748
  fixes f :: "'a::metric_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   749
  shows "Limsup (at x) f = (INF e:{0<..}. SUP y:(ball x e - {x}). f y)"
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44928
diff changeset
   750
  using Limsup_within[of x UNIV f] by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   751
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   752
lemma Lim_within_constant:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   753
  assumes "ALL y:S. f y = C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   754
  shows "(f ---> C) (at x within S)"
45032
5a4d62f9e88d Extended_Real_Limits: generalize some lemmas
huffman
parents: 45031
diff changeset
   755
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
5a4d62f9e88d Extended_Real_Limits: generalize some lemmas
huffman
parents: 45031
diff changeset
   756
  using assms by simp (metis open_UNIV UNIV_I)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   757
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   758
lemma Liminf_within_constant:
45032
5a4d62f9e88d Extended_Real_Limits: generalize some lemmas
huffman
parents: 45031
diff changeset
   759
  fixes f :: "'a::topological_space \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   760
  assumes "ALL y:S. f y = C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   761
  assumes "~trivial_limit (at x within S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   762
  shows "Liminf (at x within S) f = C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   763
by (metis Lim_within_constant assms lim_imp_Liminf)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   764
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   765
lemma Limsup_within_constant:
45032
5a4d62f9e88d Extended_Real_Limits: generalize some lemmas
huffman
parents: 45031
diff changeset
   766
  fixes f :: "'a::topological_space \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   767
  assumes "ALL y:S. f y = C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   768
  assumes "~trivial_limit (at x within S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   769
  shows "Limsup (at x within S) f = C"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   770
by (metis Lim_within_constant assms lim_imp_Limsup)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   771
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   772
lemma islimpt_punctured:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   773
"x islimpt S = x islimpt (S-{x})"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   774
unfolding islimpt_def by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   775
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   776
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   777
lemma islimpt_in_closure:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   778
"(x islimpt S) = (x:closure(S-{x}))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   779
unfolding closure_def using islimpt_punctured by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   780
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   781
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   782
lemma not_trivial_limit_within:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   783
  "~trivial_limit (at x within S) = (x:closure(S-{x}))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   784
using islimpt_in_closure by (metis trivial_limit_within)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   785
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   786
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   787
lemma not_trivial_limit_within_ball:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   788
  "(~trivial_limit (at x within S)) = (ALL e>0. S Int ball x e - {x} ~= {})"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   789
  (is "?lhs = ?rhs")
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   790
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   791
{ assume "?lhs"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   792
  { fix e :: real assume "e>0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   793
    then obtain y where "y:(S-{x}) & dist y x < e"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   794
       using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   795
    hence "y : (S Int ball x e - {x})" unfolding ball_def by (simp add: dist_commute)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   796
    hence "S Int ball x e - {x} ~= {}" by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   797
  } hence "?rhs" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   798
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   799
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   800
{ assume "?rhs"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   801
  { fix e :: real assume "e>0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   802
    then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   803
    hence "y:(S-{x}) & dist y x < e" unfolding ball_def by (simp add: dist_commute)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   804
    hence "EX y:(S-{x}). dist y x < e" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   805
  } hence "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   806
} ultimately show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   807
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   808
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   809
lemma liminf_ereal_cminus:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   810
  fixes f :: "nat \<Rightarrow> ereal" assumes "c \<noteq> -\<infinity>"
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   811
  shows "liminf (\<lambda>x. c - f x) = c - limsup f"
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   812
proof (cases c)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   813
  case PInf then show ?thesis by (simp add: Liminf_const)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   814
next
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   815
  case (real r) then show ?thesis
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   816
    unfolding liminf_SUPR_INFI limsup_INFI_SUPR
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   817
    apply (subst INFI_ereal_cminus)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   818
    apply auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   819
    apply (subst SUPR_ereal_cminus)
42950
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   820
    apply auto
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   821
    done
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   822
qed (insert `c \<noteq> -\<infinity>`, simp)
6e5c2a3c69da move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents: 41983
diff changeset
   823
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   824
subsubsection {* Continuity *}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   825
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   826
lemma continuous_imp_tendsto:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   827
  assumes "continuous (at x0) f"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   828
  assumes "x ----> x0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   829
  shows "(f o x) ----> (f x0)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   830
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   831
{ fix S assume "open S & (f x0):S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   832
  from this obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   833
     using assms continuous_at_open by metis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   834
  hence "(EX N. ALL n>=N. x n : T)" using assms tendsto_explicit T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   835
  hence "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   836
} from this show ?thesis using tendsto_explicit[of "f o x" "f x0"] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   837
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   838
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   839
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   840
lemma continuous_at_sequentially2:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   841
fixes f :: "'a::metric_space => 'b:: topological_space"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   842
shows "continuous (at x0) f <-> (ALL x. (x ----> x0) --> (f o x) ----> (f x0))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   843
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   844
{ assume "~(continuous (at x0) f)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   845
  from this obtain T where T_def:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   846
     "open T & f x0 : T & (ALL S. (open S & x0 : S) --> (EX x':S. f x' ~: T))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   847
     using continuous_at_open[of x0 f] by metis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   848
  def X == "{x'. f x' ~: T}" hence "x0 islimpt X" unfolding islimpt_def using T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   849
  from this obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   850
     using islimpt_sequential[of x0 X] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   851
  hence "~(f o x) ----> (f x0)" unfolding tendsto_explicit using X_def T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   852
  hence "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   853
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   854
from this show ?thesis using continuous_imp_tendsto by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   855
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   856
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   857
lemma continuous_at_of_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   858
  fixes x0 :: ereal
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   859
  assumes "\<bar>x0\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   860
  shows "continuous (at x0) real"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   861
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   862
{ fix T assume T_def: "open T & real x0 : T"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   863
  def S == "ereal ` T"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   864
  hence "ereal (real x0) : S" using T_def by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   865
  hence "x0 : S" using assms ereal_real by auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   866
  moreover have "open S" using open_ereal S_def T_def by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   867
  moreover have "ALL y:S. real y : T" using S_def T_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   868
  ultimately have "EX S. x0 : S & open S & (ALL y:S. real y : T)" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   869
} from this show ?thesis unfolding continuous_at_open by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   870
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   871
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   872
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   873
lemma continuous_at_iff_ereal:
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   874
fixes f :: "'a::t2_space => real"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   875
shows "continuous (at x0) f <-> continuous (at x0) (ereal o f)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   876
proof-
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   877
{ assume "continuous (at x0) f" hence "continuous (at x0) (ereal o f)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   878
     using continuous_at_ereal continuous_at_compose[of x0 f ereal] by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   879
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   880
moreover
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   881
{ assume "continuous (at x0) (ereal o f)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   882
  hence "continuous (at x0) (real o (ereal o f))"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   883
     using continuous_at_of_ereal by (intro continuous_at_compose[of x0 "ereal o f"]) auto
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   884
  moreover have "real o (ereal o f) = f" using real_ereal_id by (simp add: o_assoc)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   885
  ultimately have "continuous (at x0) f" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   886
} ultimately show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   887
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   888
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   889
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   890
lemma continuous_on_iff_ereal:
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   891
fixes f :: "'a::t2_space => real"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   892
fixes A assumes "open A"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   893
shows "continuous_on A f <-> continuous_on A (ereal o f)"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   894
   using continuous_at_iff_ereal assms by (auto simp add: continuous_on_eq_continuous_at)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   895
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   896
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
   897
lemma continuous_on_real: "continuous_on (UNIV-{\<infinity>,(-\<infinity>::ereal)}) real"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   898
   using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   899
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   900
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   901
lemma continuous_on_iff_real:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   902
  fixes f :: "'a::t2_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   903
  assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   904
  shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   905
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   906
  have "f ` A <= UNIV-{\<infinity>,(-\<infinity>)}" using assms by force
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   907
  hence *: "continuous_on (f ` A) real"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   908
     using continuous_on_real by (simp add: continuous_on_subset)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   909
have **: "continuous_on ((real o f) ` A) ereal"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   910
   using continuous_on_ereal continuous_on_subset[of "UNIV" "ereal" "(real o f) ` A"] by blast
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   911
{ assume "continuous_on A f" hence "continuous_on A (real o f)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   912
  apply (subst continuous_on_compose) using * by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   913
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   914
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   915
{ assume "continuous_on A (real o f)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   916
  hence "continuous_on A (ereal o (real o f))"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   917
     apply (subst continuous_on_compose) using ** by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   918
  hence "continuous_on A f"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   919
     apply (subst continuous_on_eq[of A "ereal o (real o f)" f])
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   920
     using assms ereal_real by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   921
}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   922
ultimately show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   923
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   924
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   925
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   926
lemma continuous_at_const:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   927
  fixes f :: "'a::t2_space => ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   928
  assumes "ALL x. (f x = C)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   929
  shows "ALL x. continuous (at x) f"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   930
unfolding continuous_at_open using assms t1_space by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   931
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   932
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   933
lemma closure_contains_Inf:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   934
  fixes S :: "real set"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   935
  assumes "S ~= {}" "EX B. ALL x:S. B<=x"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   936
  shows "Inf S : closure S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   937
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   938
have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] assms by metis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   939
{ fix e assume "e>(0 :: real)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   940
  from this obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   941
  moreover hence "x > Inf S - e" using * by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   942
  ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   943
  hence "EX x:S. abs (x - Inf S) < e" using x_def by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   944
} from this show ?thesis apply (subst closure_approachable) unfolding dist_norm by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   945
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   946
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   947
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   948
lemma closed_contains_Inf:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   949
  fixes S :: "real set"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   950
  assumes "S ~= {}" "EX B. ALL x:S. B<=x"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   951
  assumes "closed S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   952
  shows "Inf S : S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   953
by (metis closure_contains_Inf closure_closed assms)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   954
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   955
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   956
lemma mono_closed_real:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   957
  fixes S :: "real set"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   958
  assumes mono: "ALL y z. y:S & y<=z --> z:S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   959
  assumes "closed S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   960
  shows "S = {} | S = UNIV | (EX a. S = {a ..})"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   961
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   962
{ assume "S ~= {}"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   963
  { assume ex: "EX B. ALL x:S. B<=x"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   964
    hence *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   965
    hence "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   966
    hence "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   967
    hence "S = {Inf S ..}" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   968
    hence "EX a. S = {a ..}" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   969
  }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   970
  moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   971
  { assume "~(EX B. ALL x:S. B<=x)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   972
    hence nex: "ALL B. EX x:S. x<B" by (simp add: not_le)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   973
    { fix y obtain x where "x:S & x < y" using nex by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   974
      hence "y:S" using mono[rule_format, of x y] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   975
    } hence "S = UNIV" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   976
  } ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   977
} from this show ?thesis by blast
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   978
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   979
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   980
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   981
lemma mono_closed_ereal:
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   982
  fixes S :: "real set"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   983
  assumes mono: "ALL y z. y:S & y<=z --> z:S"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   984
  assumes "closed S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   985
  shows "EX a. S = {x. a <= ereal x}"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   986
proof-
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   987
{ assume "S = {}" hence ?thesis apply(rule_tac x=PInfty in exI) by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   988
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   989
{ assume "S = UNIV" hence ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   990
moreover
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   991
{ assume "EX a. S = {a ..}"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   992
  from this obtain a where "S={a ..}" by auto
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   993
  hence ?thesis apply(rule_tac x="ereal a" in exI) by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   994
} ultimately show ?thesis using mono_closed_real[of S] assms by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   995
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   996
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   997
subsection {* Sums *}
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
   998
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
   999
lemma setsum_ereal[simp]:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1000
  "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. f x)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1001
proof cases
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1002
  assume "finite A" then show ?thesis by induct auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1003
qed simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1004
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1005
lemma setsum_Pinfty:
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1006
  fixes f :: "'a \<Rightarrow> ereal"
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1007
  shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1008
proof safe
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1009
  assume *: "setsum f P = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1010
  show "finite P"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1011
  proof (rule ccontr) assume "infinite P" with * show False by auto qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1012
  show "\<exists>i\<in>P. f i = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1013
  proof (rule ccontr)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1014
    assume "\<not> ?thesis" then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1015
    from `finite P` this have "setsum f P \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1016
      by induct auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1017
    with * show False by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1018
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1019
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1020
  fix i assume "finite P" "i \<in> P" "f i = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1021
  thus "setsum f P = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1022
  proof induct
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1023
    case (insert x A)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1024
    show ?case using insert by (cases "x = i") auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1025
  qed simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1026
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1027
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1028
lemma setsum_Inf:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1029
  fixes f :: "'a \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1030
  shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>))"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1031
proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1032
  assume *: "\<bar>setsum f A\<bar> = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1033
  have "finite A" by (rule ccontr) (insert *, auto)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1034
  moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1035
  proof (rule ccontr)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1036
    assume "\<not> ?thesis" then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1037
    from bchoice[OF this] guess r ..
44142
8e27e0177518 avoid warnings about duplicate rules
huffman
parents: 44125
diff changeset
  1038
    with * show False by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1039
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1040
  ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1041
next
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1042
  assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1043
  then obtain i where "finite A" "i \<in> A" "\<bar>f i\<bar> = \<infinity>" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1044
  then show "\<bar>setsum f A\<bar> = \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1045
  proof induct
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1046
    case (insert j A) then show ?case
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1047
      by (cases rule: ereal3_cases[of "f i" "f j" "setsum f A"]) auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1048
  qed simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1049
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1050
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1051
lemma setsum_real_of_ereal:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1052
  fixes f :: "'i \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1053
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1054
  shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1055
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1056
  have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1057
  proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1058
    fix x assume "x \<in> S"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1059
    from assms[OF this] show "\<exists>r. f x = ereal r" by (cases "f x") auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1060
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1061
  from bchoice[OF this] guess r ..
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1062
  then show ?thesis by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1063
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1064
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1065
lemma setsum_ereal_0:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1066
  fixes f :: "'a \<Rightarrow> ereal" assumes "finite A" "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1067
  shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1068
proof
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1069
  assume *: "(\<Sum>x\<in>A. f x) = 0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1070
  then have "(\<Sum>x\<in>A. f x) \<noteq> \<infinity>" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1071
  then have "\<forall>i\<in>A. \<bar>f i\<bar> \<noteq> \<infinity>" using assms by (force simp: setsum_Pinfty)
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1072
  then have "\<forall>i\<in>A. \<exists>r. f i = ereal r" by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1073
  from bchoice[OF this] * assms show "\<forall>i\<in>A. f i = 0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1074
    using setsum_nonneg_eq_0_iff[of A "\<lambda>i. real (f i)"] by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1075
qed (rule setsum_0')
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1076
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1077
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1078
lemma setsum_ereal_right_distrib:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1079
  fixes f :: "'a \<Rightarrow> ereal" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1080
  shows "r * setsum f A = (\<Sum>n\<in>A. r * f n)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1081
proof cases
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1082
  assume "finite A" then show ?thesis using assms
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1083
    by induct (auto simp: ereal_right_distrib setsum_nonneg)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1084
qed simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1085
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1086
lemma sums_ereal_positive:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1087
  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. \<Sum>i<n. f i)"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1088
proof -
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1089
  have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1090
    using ereal_add_mono[OF _ assms] by (auto intro!: incseq_SucI)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1091
  from LIMSEQ_ereal_SUPR[OF this]
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1092
  show ?thesis unfolding sums_def by (simp add: atLeast0LessThan)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1093
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1094
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1095
lemma summable_ereal_pos:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1096
  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" shows "summable f"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1097
  using sums_ereal_positive[of f, OF assms] unfolding summable_def by auto
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1098
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1099
lemma suminf_ereal_eq_SUPR:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1100
  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1101
  shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1102
  using sums_ereal_positive[of f, OF assms, THEN sums_unique] by simp
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1103
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1104
lemma sums_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1105
  "(\<lambda>x. ereal (f x)) sums ereal x \<longleftrightarrow> f sums x"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1106
  unfolding sums_def by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1107
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1108
lemma suminf_bound:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1109
  fixes f :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1110
  assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x" and pos: "\<And>n. 0 \<le> f n"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1111
  shows "suminf f \<le> x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1112
proof (rule Lim_bounded_ereal)
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1113
  have "summable f" using pos[THEN summable_ereal_pos] .
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1114
  then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1115
    by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1116
  show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1117
    using assms by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1118
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1119
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1120
lemma suminf_bound_add:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1121
  fixes f :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1122
  assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x" and pos: "\<And>n. 0 \<le> f n" and "y \<noteq> -\<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1123
  shows "suminf f + y \<le> x"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1124
proof (cases y)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1125
  case (real r) then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1126
    using assms by (simp add: ereal_le_minus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1127
  then have "(\<Sum> n. f n) \<le> x - y" using pos by (rule suminf_bound)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1128
  then show "(\<Sum> n. f n) + y \<le> x"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1129
    using assms real by (simp add: ereal_le_minus)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1130
qed (insert assms, auto)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1131
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1132
lemma sums_finite:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1133
  assumes "\<forall>N\<ge>n. f N = 0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1134
  shows "f sums (\<Sum>N<n. f N)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1135
proof -
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1136
  { fix i have "(\<Sum>N<i + n. f N) = (\<Sum>N<n. f N)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1137
      by (induct i) (insert assms, auto) }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1138
  note this[simp]
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1139
  show ?thesis unfolding sums_def
44125
230a8665c919 mark some redundant theorems as legacy
huffman
parents: 43923
diff changeset
  1140
    by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan intro: tendsto_const)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1141
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1142
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1143
lemma suminf_finite:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1144
  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}" assumes "\<forall>N\<ge>n. f N = 0"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1145
  shows "suminf f = (\<Sum>N<n. f N)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1146
  using sums_finite[OF assms, THEN sums_unique] by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1147
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1148
lemma suminf_upper:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1149
  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>n. 0 \<le> f n"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1150
  shows "(\<Sum>n<N. f n) \<le> (\<Sum>n. f n)"
44928
7ef6505bde7f renamed Complete_Lattices lemmas, removed legacy names
hoelzl
parents: 44918
diff changeset
  1151
  unfolding suminf_ereal_eq_SUPR[OF assms] SUP_def
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44928
diff changeset
  1152
  by (auto intro: complete_lattice_class.Sup_upper)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1153
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1154
lemma suminf_0_le:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1155
  fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>n. 0 \<le> f n"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1156
  shows "0 \<le> (\<Sum>n. f n)"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1157
  using suminf_upper[of f 0, OF assms] by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1158
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1159
lemma suminf_le_pos:
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1160
  fixes f g :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1161
  assumes "\<And>N. f N \<le> g N" "\<And>N. 0 \<le> f N"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1162
  shows "suminf f \<le> suminf g"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1163
proof (safe intro!: suminf_bound)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1164
  fix n { fix N have "0 \<le> g N" using assms(2,1)[of N] by auto }
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1165
  have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1166
  also have "... \<le> suminf g" using `\<And>N. 0 \<le> g N` by (rule suminf_upper)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1167
  finally show "setsum f {..<n} \<le> suminf g" .
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1168
qed (rule assms(2))
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1169
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1170
lemma suminf_half_series_ereal: "(\<Sum>n. (1/2 :: ereal)^Suc n) = 1"
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1171
  using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1172
  by (simp add: one_ereal_def)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1173
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1174
lemma suminf_add_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1175
  fixes f g :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1176
  assumes "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1177
  shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1178
  apply (subst (1 2 3) suminf_ereal_eq_SUPR)
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1179
  unfolding setsum_addf
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1180
  by (intro assms ereal_add_nonneg_nonneg SUPR_ereal_add_pos incseq_setsumI setsum_nonneg ballI)+
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1181
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1182
lemma suminf_cmult_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1183
  fixes f g :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1184
  assumes "\<And>i. 0 \<le> f i" "0 \<le> a"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1185
  shows "(\<Sum>i. a * f i) = a * suminf f"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1186
  by (auto simp: setsum_ereal_right_distrib[symmetric] assms
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1187
                 ereal_zero_le_0_iff setsum_nonneg suminf_ereal_eq_SUPR
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1188
           intro!: SUPR_ereal_cmult )
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1189
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1190
lemma suminf_PInfty:
43923
ab93d0190a5d add ereal to typeclass infinity
hoelzl
parents: 43920
diff changeset
  1191
  fixes f :: "nat \<Rightarrow> ereal"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1192
  assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1193
  shows "f i \<noteq> \<infinity>"
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1194
proof -
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1195
  from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1196
  have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>" by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1197
  then show ?thesis
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1198
    unfolding setsum_Pinfty by simp
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1199
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1200
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1201
lemma suminf_PInfty_fun:
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1202
  assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1203
  shows "\<exists>f'. f = (\<lambda>x. ereal (f' x))"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1204
proof -
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1205
  have "\<forall>i. \<exists>r. f i = ereal r"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1206
  proof
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1207
    fix i show "\<exists>r. f i = ereal r"
41980
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1208
      using suminf_PInfty[OF assms] assms(1)[of i] by (cases "f i") auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1209
  qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1210
  from choice[OF this] show ?thesis by auto
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1211
qed
28b51effc5ed split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
diff changeset
  1212
43920
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1213
lemma summable_ereal:
cedb5cb948fd Rename extreal => ereal
hoelzl
parents: 42950
diff changeset
  1214
  assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>"