src/HOL/Library/Liminf_Limsup.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (4 months ago) changeset 69946 494934c30f38 parent 69861 62e47f06d22c child 70378 ebd108578ab1 permissions -rw-r--r--
improved code equations taken over from AFP
```     1 (*  Title:      HOL/Library/Liminf_Limsup.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Manuel Eberl, TU München
```
```     4 *)
```
```     5
```
```     6 section \<open>Liminf and Limsup on conditionally complete lattices\<close>
```
```     7
```
```     8 theory Liminf_Limsup
```
```     9 imports Complex_Main
```
```    10 begin
```
```    11
```
```    12 lemma (in conditionally_complete_linorder) le_cSup_iff:
```
```    13   assumes "A \<noteq> {}" "bdd_above A"
```
```    14   shows "x \<le> Sup A \<longleftrightarrow> (\<forall>y<x. \<exists>a\<in>A. y < a)"
```
```    15 proof safe
```
```    16   fix y assume "x \<le> Sup A" "y < x"
```
```    17   then have "y < Sup A" by auto
```
```    18   then show "\<exists>a\<in>A. y < a"
```
```    19     unfolding less_cSup_iff[OF assms] .
```
```    20 qed (auto elim!: allE[of _ "Sup A"] simp add: not_le[symmetric] cSup_upper assms)
```
```    21
```
```    22 lemma (in conditionally_complete_linorder) le_cSUP_iff:
```
```    23   "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> x \<le> Sup (f ` A) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y < f i)"
```
```    24   using le_cSup_iff [of "f ` A"] by simp
```
```    25
```
```    26 lemma le_cSup_iff_less:
```
```    27   fixes x :: "'a :: {conditionally_complete_linorder, dense_linorder}"
```
```    28   shows "A \<noteq> {} \<Longrightarrow> bdd_above (f`A) \<Longrightarrow> x \<le> (SUP i\<in>A. f i) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y \<le> f i)"
```
```    29   by (simp add: le_cSUP_iff)
```
```    30      (blast intro: less_imp_le less_trans less_le_trans dest: dense)
```
```    31
```
```    32 lemma le_Sup_iff_less:
```
```    33   fixes x :: "'a :: {complete_linorder, dense_linorder}"
```
```    34   shows "x \<le> (SUP i\<in>A. f i) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y \<le> f i)" (is "?lhs = ?rhs")
```
```    35   unfolding le_SUP_iff
```
```    36   by (blast intro: less_imp_le less_trans less_le_trans dest: dense)
```
```    37
```
```    38 lemma (in conditionally_complete_linorder) cInf_le_iff:
```
```    39   assumes "A \<noteq> {}" "bdd_below A"
```
```    40   shows "Inf A \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>a\<in>A. y > a)"
```
```    41 proof safe
```
```    42   fix y assume "x \<ge> Inf A" "y > x"
```
```    43   then have "y > Inf A" by auto
```
```    44   then show "\<exists>a\<in>A. y > a"
```
```    45     unfolding cInf_less_iff[OF assms] .
```
```    46 qed (auto elim!: allE[of _ "Inf A"] simp add: not_le[symmetric] cInf_lower assms)
```
```    47
```
```    48 lemma (in conditionally_complete_linorder) cINF_le_iff:
```
```    49   "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> Inf (f ` A) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. y > f i)"
```
```    50   using cInf_le_iff [of "f ` A"] by simp
```
```    51
```
```    52 lemma cInf_le_iff_less:
```
```    53   fixes x :: "'a :: {conditionally_complete_linorder, dense_linorder}"
```
```    54   shows "A \<noteq> {} \<Longrightarrow> bdd_below (f`A) \<Longrightarrow> (INF i\<in>A. f i) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
```
```    55   by (simp add: cINF_le_iff)
```
```    56      (blast intro: less_imp_le less_trans le_less_trans dest: dense)
```
```    57
```
```    58 lemma Inf_le_iff_less:
```
```    59   fixes x :: "'a :: {complete_linorder, dense_linorder}"
```
```    60   shows "(INF i\<in>A. f i) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
```
```    61   unfolding INF_le_iff
```
```    62   by (blast intro: less_imp_le less_trans le_less_trans dest: dense)
```
```    63
```
```    64 lemma SUP_pair:
```
```    65   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: complete_lattice"
```
```    66   shows "(SUP i \<in> A. SUP j \<in> B. f i j) = (SUP p \<in> A \<times> B. f (fst p) (snd p))"
```
```    67   by (rule antisym) (auto intro!: SUP_least SUP_upper2)
```
```    68
```
```    69 lemma INF_pair:
```
```    70   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: complete_lattice"
```
```    71   shows "(INF i \<in> A. INF j \<in> B. f i j) = (INF p \<in> A \<times> B. f (fst p) (snd p))"
```
```    72   by (rule antisym) (auto intro!: INF_greatest INF_lower2)
```
```    73
```
```    74 lemma INF_Sigma:
```
```    75   fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: complete_lattice"
```
```    76   shows "(INF i \<in> A. INF j \<in> B i. f i j) = (INF p \<in> Sigma A B. f (fst p) (snd p))"
```
```    77   by (rule antisym) (auto intro!: INF_greatest INF_lower2)
```
```    78
```
```    79 subsubsection \<open>\<open>Liminf\<close> and \<open>Limsup\<close>\<close>
```
```    80
```
```    81 definition Liminf :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b :: complete_lattice" where
```
```    82   "Liminf F f = (SUP P\<in>{P. eventually P F}. INF x\<in>{x. P x}. f x)"
```
```    83
```
```    84 definition Limsup :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b :: complete_lattice" where
```
```    85   "Limsup F f = (INF P\<in>{P. eventually P F}. SUP x\<in>{x. P x}. f x)"
```
```    86
```
```    87 abbreviation "liminf \<equiv> Liminf sequentially"
```
```    88
```
```    89 abbreviation "limsup \<equiv> Limsup sequentially"
```
```    90
```
```    91 lemma Liminf_eqI:
```
```    92   "(\<And>P. eventually P F \<Longrightarrow> Inf (f ` (Collect P)) \<le> x) \<Longrightarrow>
```
```    93     (\<And>y. (\<And>P. eventually P F \<Longrightarrow> Inf (f ` (Collect P)) \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> Liminf F f = x"
```
```    94   unfolding Liminf_def by (auto intro!: SUP_eqI)
```
```    95
```
```    96 lemma Limsup_eqI:
```
```    97   "(\<And>P. eventually P F \<Longrightarrow> x \<le> Sup (f ` (Collect P))) \<Longrightarrow>
```
```    98     (\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> Sup (f ` (Collect P))) \<Longrightarrow> y \<le> x) \<Longrightarrow> Limsup F f = x"
```
```    99   unfolding Limsup_def by (auto intro!: INF_eqI)
```
```   100
```
```   101 lemma liminf_SUP_INF: "liminf f = (SUP n. INF m\<in>{n..}. f m)"
```
```   102   unfolding Liminf_def eventually_sequentially
```
```   103   by (rule SUP_eq) (auto simp: atLeast_def intro!: INF_mono)
```
```   104
```
```   105 lemma limsup_INF_SUP: "limsup f = (INF n. SUP m\<in>{n..}. f m)"
```
```   106   unfolding Limsup_def eventually_sequentially
```
```   107   by (rule INF_eq) (auto simp: atLeast_def intro!: SUP_mono)
```
```   108
```
```   109 lemma Limsup_const:
```
```   110   assumes ntriv: "\<not> trivial_limit F"
```
```   111   shows "Limsup F (\<lambda>x. c) = c"
```
```   112 proof -
```
```   113   have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
```
```   114   have "\<And>P. eventually P F \<Longrightarrow> (SUP x \<in> {x. P x}. c) = c"
```
```   115     using ntriv by (intro SUP_const) (auto simp: eventually_False *)
```
```   116   then show ?thesis
```
```   117     apply (auto simp add: Limsup_def)
```
```   118     apply (rule INF_const)
```
```   119     apply auto
```
```   120     using eventually_True apply blast
```
```   121     done
```
```   122 qed
```
```   123
```
```   124 lemma Liminf_const:
```
```   125   assumes ntriv: "\<not> trivial_limit F"
```
```   126   shows "Liminf F (\<lambda>x. c) = c"
```
```   127 proof -
```
```   128   have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
```
```   129   have "\<And>P. eventually P F \<Longrightarrow> (INF x \<in> {x. P x}. c) = c"
```
```   130     using ntriv by (intro INF_const) (auto simp: eventually_False *)
```
```   131   then show ?thesis
```
```   132     apply (auto simp add: Liminf_def)
```
```   133     apply (rule SUP_const)
```
```   134     apply auto
```
```   135     using eventually_True apply blast
```
```   136     done
```
```   137 qed
```
```   138
```
```   139 lemma Liminf_mono:
```
```   140   assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
```
```   141   shows "Liminf F f \<le> Liminf F g"
```
```   142   unfolding Liminf_def
```
```   143 proof (safe intro!: SUP_mono)
```
```   144   fix P assume "eventually P F"
```
```   145   with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
```
```   146   then show "\<exists>Q\<in>{P. eventually P F}. Inf (f ` (Collect P)) \<le> Inf (g ` (Collect Q))"
```
```   147     by (intro bexI[of _ ?Q]) (auto intro!: INF_mono)
```
```   148 qed
```
```   149
```
```   150 lemma Liminf_eq:
```
```   151   assumes "eventually (\<lambda>x. f x = g x) F"
```
```   152   shows "Liminf F f = Liminf F g"
```
```   153   by (intro antisym Liminf_mono eventually_mono[OF assms]) auto
```
```   154
```
```   155 lemma Limsup_mono:
```
```   156   assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
```
```   157   shows "Limsup F f \<le> Limsup F g"
```
```   158   unfolding Limsup_def
```
```   159 proof (safe intro!: INF_mono)
```
```   160   fix P assume "eventually P F"
```
```   161   with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
```
```   162   then show "\<exists>Q\<in>{P. eventually P F}. Sup (f ` (Collect Q)) \<le> Sup (g ` (Collect P))"
```
```   163     by (intro bexI[of _ ?Q]) (auto intro!: SUP_mono)
```
```   164 qed
```
```   165
```
```   166 lemma Limsup_eq:
```
```   167   assumes "eventually (\<lambda>x. f x = g x) net"
```
```   168   shows "Limsup net f = Limsup net g"
```
```   169   by (intro antisym Limsup_mono eventually_mono[OF assms]) auto
```
```   170
```
```   171 lemma Liminf_bot[simp]: "Liminf bot f = top"
```
```   172   unfolding Liminf_def top_unique[symmetric]
```
```   173   by (rule SUP_upper2[where i="\<lambda>x. False"]) simp_all
```
```   174
```
```   175 lemma Limsup_bot[simp]: "Limsup bot f = bot"
```
```   176   unfolding Limsup_def bot_unique[symmetric]
```
```   177   by (rule INF_lower2[where i="\<lambda>x. False"]) simp_all
```
```   178
```
```   179 lemma Liminf_le_Limsup:
```
```   180   assumes ntriv: "\<not> trivial_limit F"
```
```   181   shows "Liminf F f \<le> Limsup F f"
```
```   182   unfolding Limsup_def Liminf_def
```
```   183   apply (rule SUP_least)
```
```   184   apply (rule INF_greatest)
```
```   185 proof safe
```
```   186   fix P Q assume "eventually P F" "eventually Q F"
```
```   187   then have "eventually (\<lambda>x. P x \<and> Q x) F" (is "eventually ?C F") by (rule eventually_conj)
```
```   188   then have not_False: "(\<lambda>x. P x \<and> Q x) \<noteq> (\<lambda>x. False)"
```
```   189     using ntriv by (auto simp add: eventually_False)
```
```   190   have "Inf (f ` (Collect P)) \<le> Inf (f ` (Collect ?C))"
```
```   191     by (rule INF_mono) auto
```
```   192   also have "\<dots> \<le> Sup (f ` (Collect ?C))"
```
```   193     using not_False by (intro INF_le_SUP) auto
```
```   194   also have "\<dots> \<le> Sup (f ` (Collect Q))"
```
```   195     by (rule SUP_mono) auto
```
```   196   finally show "Inf (f ` (Collect P)) \<le> Sup (f ` (Collect Q))" .
```
```   197 qed
```
```   198
```
```   199 lemma Liminf_bounded:
```
```   200   assumes le: "eventually (\<lambda>n. C \<le> X n) F"
```
```   201   shows "C \<le> Liminf F X"
```
```   202   using Liminf_mono[OF le] Liminf_const[of F C]
```
```   203   by (cases "F = bot") simp_all
```
```   204
```
```   205 lemma Limsup_bounded:
```
```   206   assumes le: "eventually (\<lambda>n. X n \<le> C) F"
```
```   207   shows "Limsup F X \<le> C"
```
```   208   using Limsup_mono[OF le] Limsup_const[of F C]
```
```   209   by (cases "F = bot") simp_all
```
```   210
```
```   211 lemma le_Limsup:
```
```   212   assumes F: "F \<noteq> bot" and x: "\<forall>\<^sub>F x in F. l \<le> f x"
```
```   213   shows "l \<le> Limsup F f"
```
```   214   using F Liminf_bounded[of l f F] Liminf_le_Limsup[of F f] order.trans x by blast
```
```   215
```
```   216 lemma Liminf_le:
```
```   217   assumes F: "F \<noteq> bot" and x: "\<forall>\<^sub>F x in F. f x \<le> l"
```
```   218   shows "Liminf F f \<le> l"
```
```   219   using F Liminf_le_Limsup Limsup_bounded order.trans x by blast
```
```   220
```
```   221 lemma le_Liminf_iff:
```
```   222   fixes X :: "_ \<Rightarrow> _ :: complete_linorder"
```
```   223   shows "C \<le> Liminf F X \<longleftrightarrow> (\<forall>y<C. eventually (\<lambda>x. y < X x) F)"
```
```   224 proof -
```
```   225   have "eventually (\<lambda>x. y < X x) F"
```
```   226     if "eventually P F" "y < Inf (X ` (Collect P))" for y P
```
```   227     using that by (auto elim!: eventually_mono dest: less_INF_D)
```
```   228   moreover
```
```   229   have "\<exists>P. eventually P F \<and> y < Inf (X ` (Collect P))"
```
```   230     if "y < C" and y: "\<forall>y<C. eventually (\<lambda>x. y < X x) F" for y P
```
```   231   proof (cases "\<exists>z. y < z \<and> z < C")
```
```   232     case True
```
```   233     then obtain z where z: "y < z \<and> z < C" ..
```
```   234     moreover from z have "z \<le> Inf (X ` {x. z < X x})"
```
```   235       by (auto intro!: INF_greatest)
```
```   236     ultimately show ?thesis
```
```   237       using y by (intro exI[of _ "\<lambda>x. z < X x"]) auto
```
```   238   next
```
```   239     case False
```
```   240     then have "C \<le> Inf (X ` {x. y < X x})"
```
```   241       by (intro INF_greatest) auto
```
```   242     with \<open>y < C\<close> show ?thesis
```
```   243       using y by (intro exI[of _ "\<lambda>x. y < X x"]) auto
```
```   244   qed
```
```   245   ultimately show ?thesis
```
```   246     unfolding Liminf_def le_SUP_iff by auto
```
```   247 qed
```
```   248
```
```   249 lemma Limsup_le_iff:
```
```   250   fixes X :: "_ \<Rightarrow> _ :: complete_linorder"
```
```   251   shows "C \<ge> Limsup F X \<longleftrightarrow> (\<forall>y>C. eventually (\<lambda>x. y > X x) F)"
```
```   252 proof -
```
```   253   { fix y P assume "eventually P F" "y > Sup (X ` (Collect P))"
```
```   254     then have "eventually (\<lambda>x. y > X x) F"
```
```   255       by (auto elim!: eventually_mono dest: SUP_lessD) }
```
```   256   moreover
```
```   257   { fix y P assume "y > C" and y: "\<forall>y>C. eventually (\<lambda>x. y > X x) F"
```
```   258     have "\<exists>P. eventually P F \<and> y > Sup (X ` (Collect P))"
```
```   259     proof (cases "\<exists>z. C < z \<and> z < y")
```
```   260       case True
```
```   261       then obtain z where z: "C < z \<and> z < y" ..
```
```   262       moreover from z have "z \<ge> Sup (X ` {x. X x < z})"
```
```   263         by (auto intro!: SUP_least)
```
```   264       ultimately show ?thesis
```
```   265         using y by (intro exI[of _ "\<lambda>x. z > X x"]) auto
```
```   266     next
```
```   267       case False
```
```   268       then have "C \<ge> Sup (X ` {x. X x < y})"
```
```   269         by (intro SUP_least) (auto simp: not_less)
```
```   270       with \<open>y > C\<close> show ?thesis
```
```   271         using y by (intro exI[of _ "\<lambda>x. y > X x"]) auto
```
```   272     qed }
```
```   273   ultimately show ?thesis
```
```   274     unfolding Limsup_def INF_le_iff by auto
```
```   275 qed
```
```   276
```
```   277 lemma less_LiminfD:
```
```   278   "y < Liminf F (f :: _ \<Rightarrow> 'a :: complete_linorder) \<Longrightarrow> eventually (\<lambda>x. f x > y) F"
```
```   279   using le_Liminf_iff[of "Liminf F f" F f] by simp
```
```   280
```
```   281 lemma Limsup_lessD:
```
```   282   "y > Limsup F (f :: _ \<Rightarrow> 'a :: complete_linorder) \<Longrightarrow> eventually (\<lambda>x. f x < y) F"
```
```   283   using Limsup_le_iff[of F f "Limsup F f"] by simp
```
```   284
```
```   285 lemma lim_imp_Liminf:
```
```   286   fixes f :: "'a \<Rightarrow> _ :: {complete_linorder,linorder_topology}"
```
```   287   assumes ntriv: "\<not> trivial_limit F"
```
```   288   assumes lim: "(f \<longlongrightarrow> f0) F"
```
```   289   shows "Liminf F f = f0"
```
```   290 proof (intro Liminf_eqI)
```
```   291   fix P assume P: "eventually P F"
```
```   292   then have "eventually (\<lambda>x. Inf (f ` (Collect P)) \<le> f x) F"
```
```   293     by eventually_elim (auto intro!: INF_lower)
```
```   294   then show "Inf (f ` (Collect P)) \<le> f0"
```
```   295     by (rule tendsto_le[OF ntriv lim tendsto_const])
```
```   296 next
```
```   297   fix y assume upper: "\<And>P. eventually P F \<Longrightarrow> Inf (f ` (Collect P)) \<le> y"
```
```   298   show "f0 \<le> y"
```
```   299   proof cases
```
```   300     assume "\<exists>z. y < z \<and> z < f0"
```
```   301     then obtain z where "y < z \<and> z < f0" ..
```
```   302     moreover have "z \<le> Inf (f ` {x. z < f x})"
```
```   303       by (rule INF_greatest) simp
```
```   304     ultimately show ?thesis
```
```   305       using lim[THEN topological_tendstoD, THEN upper, of "{z <..}"] by auto
```
```   306   next
```
```   307     assume discrete: "\<not> (\<exists>z. y < z \<and> z < f0)"
```
```   308     show ?thesis
```
```   309     proof (rule classical)
```
```   310       assume "\<not> f0 \<le> y"
```
```   311       then have "eventually (\<lambda>x. y < f x) F"
```
```   312         using lim[THEN topological_tendstoD, of "{y <..}"] by auto
```
```   313       then have "eventually (\<lambda>x. f0 \<le> f x) F"
```
```   314         using discrete by (auto elim!: eventually_mono)
```
```   315       then have "Inf (f ` {x. f0 \<le> f x}) \<le> y"
```
```   316         by (rule upper)
```
```   317       moreover have "f0 \<le> Inf (f ` {x. f0 \<le> f x})"
```
```   318         by (intro INF_greatest) simp
```
```   319       ultimately show "f0 \<le> y" by simp
```
```   320     qed
```
```   321   qed
```
```   322 qed
```
```   323
```
```   324 lemma lim_imp_Limsup:
```
```   325   fixes f :: "'a \<Rightarrow> _ :: {complete_linorder,linorder_topology}"
```
```   326   assumes ntriv: "\<not> trivial_limit F"
```
```   327   assumes lim: "(f \<longlongrightarrow> f0) F"
```
```   328   shows "Limsup F f = f0"
```
```   329 proof (intro Limsup_eqI)
```
```   330   fix P assume P: "eventually P F"
```
```   331   then have "eventually (\<lambda>x. f x \<le> Sup (f ` (Collect P))) F"
```
```   332     by eventually_elim (auto intro!: SUP_upper)
```
```   333   then show "f0 \<le> Sup (f ` (Collect P))"
```
```   334     by (rule tendsto_le[OF ntriv tendsto_const lim])
```
```   335 next
```
```   336   fix y assume lower: "\<And>P. eventually P F \<Longrightarrow> y \<le> Sup (f ` (Collect P))"
```
```   337   show "y \<le> f0"
```
```   338   proof (cases "\<exists>z. f0 < z \<and> z < y")
```
```   339     case True
```
```   340     then obtain z where "f0 < z \<and> z < y" ..
```
```   341     moreover have "Sup (f ` {x. f x < z}) \<le> z"
```
```   342       by (rule SUP_least) simp
```
```   343     ultimately show ?thesis
```
```   344       using lim[THEN topological_tendstoD, THEN lower, of "{..< z}"] by auto
```
```   345   next
```
```   346     case False
```
```   347     show ?thesis
```
```   348     proof (rule classical)
```
```   349       assume "\<not> y \<le> f0"
```
```   350       then have "eventually (\<lambda>x. f x < y) F"
```
```   351         using lim[THEN topological_tendstoD, of "{..< y}"] by auto
```
```   352       then have "eventually (\<lambda>x. f x \<le> f0) F"
```
```   353         using False by (auto elim!: eventually_mono simp: not_less)
```
```   354       then have "y \<le> Sup (f ` {x. f x \<le> f0})"
```
```   355         by (rule lower)
```
```   356       moreover have "Sup (f ` {x. f x \<le> f0}) \<le> f0"
```
```   357         by (intro SUP_least) simp
```
```   358       ultimately show "y \<le> f0" by simp
```
```   359     qed
```
```   360   qed
```
```   361 qed
```
```   362
```
```   363 lemma Liminf_eq_Limsup:
```
```   364   fixes f0 :: "'a :: {complete_linorder,linorder_topology}"
```
```   365   assumes ntriv: "\<not> trivial_limit F"
```
```   366     and lim: "Liminf F f = f0" "Limsup F f = f0"
```
```   367   shows "(f \<longlongrightarrow> f0) F"
```
```   368 proof (rule order_tendstoI)
```
```   369   fix a assume "f0 < a"
```
```   370   with assms have "Limsup F f < a" by simp
```
```   371   then obtain P where "eventually P F" "Sup (f ` (Collect P)) < a"
```
```   372     unfolding Limsup_def INF_less_iff by auto
```
```   373   then show "eventually (\<lambda>x. f x < a) F"
```
```   374     by (auto elim!: eventually_mono dest: SUP_lessD)
```
```   375 next
```
```   376   fix a assume "a < f0"
```
```   377   with assms have "a < Liminf F f" by simp
```
```   378   then obtain P where "eventually P F" "a < Inf (f ` (Collect P))"
```
```   379     unfolding Liminf_def less_SUP_iff by auto
```
```   380   then show "eventually (\<lambda>x. a < f x) F"
```
```   381     by (auto elim!: eventually_mono dest: less_INF_D)
```
```   382 qed
```
```   383
```
```   384 lemma tendsto_iff_Liminf_eq_Limsup:
```
```   385   fixes f0 :: "'a :: {complete_linorder,linorder_topology}"
```
```   386   shows "\<not> trivial_limit F \<Longrightarrow> (f \<longlongrightarrow> f0) F \<longleftrightarrow> (Liminf F f = f0 \<and> Limsup F f = f0)"
```
```   387   by (metis Liminf_eq_Limsup lim_imp_Limsup lim_imp_Liminf)
```
```   388
```
```   389 lemma liminf_subseq_mono:
```
```   390   fixes X :: "nat \<Rightarrow> 'a :: complete_linorder"
```
```   391   assumes "strict_mono r"
```
```   392   shows "liminf X \<le> liminf (X \<circ> r) "
```
```   393 proof-
```
```   394   have "\<And>n. (INF m\<in>{n..}. X m) \<le> (INF m\<in>{n..}. (X \<circ> r) m)"
```
```   395   proof (safe intro!: INF_mono)
```
```   396     fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
```
```   397       using seq_suble[OF \<open>strict_mono r\<close>, of m] by (intro bexI[of _ "r m"]) auto
```
```   398   qed
```
```   399   then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUP_INF comp_def)
```
```   400 qed
```
```   401
```
```   402 lemma limsup_subseq_mono:
```
```   403   fixes X :: "nat \<Rightarrow> 'a :: complete_linorder"
```
```   404   assumes "strict_mono r"
```
```   405   shows "limsup (X \<circ> r) \<le> limsup X"
```
```   406 proof-
```
```   407   have "(SUP m\<in>{n..}. (X \<circ> r) m) \<le> (SUP m\<in>{n..}. X m)" for n
```
```   408   proof (safe intro!: SUP_mono)
```
```   409     fix m :: nat
```
```   410     assume "n \<le> m"
```
```   411     then show "\<exists>ma\<in>{n..}. (X \<circ> r) m \<le> X ma"
```
```   412       using seq_suble[OF \<open>strict_mono r\<close>, of m] by (intro bexI[of _ "r m"]) auto
```
```   413   qed
```
```   414   then show ?thesis
```
```   415     by (auto intro!: INF_mono simp: limsup_INF_SUP comp_def)
```
```   416 qed
```
```   417
```
```   418 lemma continuous_on_imp_continuous_within:
```
```   419   "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> x \<in> s \<Longrightarrow> continuous (at x within t) f"
```
```   420   unfolding continuous_on_eq_continuous_within
```
```   421   by (auto simp: continuous_within intro: tendsto_within_subset)
```
```   422
```
```   423 lemma Liminf_compose_continuous_mono:
```
```   424   fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
```
```   425   assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot"
```
```   426   shows "Liminf F (\<lambda>n. f (g n)) = f (Liminf F g)"
```
```   427 proof -
```
```   428   { fix P assume "eventually P F"
```
```   429     have "\<exists>x. P x"
```
```   430     proof (rule ccontr)
```
```   431       assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
```
```   432         by auto
```
```   433       with \<open>eventually P F\<close> F show False
```
```   434         by auto
```
```   435     qed }
```
```   436   note * = this
```
```   437
```
```   438   have "f (SUP P\<in>{P. eventually P F}. Inf (g ` Collect P)) =
```
```   439     Sup (f ` (\<lambda>P. Inf (g ` Collect P)) ` {P. eventually P F})"
```
```   440     using am continuous_on_imp_continuous_within [OF c]
```
```   441     by (rule continuous_at_Sup_mono) (auto intro: eventually_True)
```
```   442   then have "f (Liminf F g) = (SUP P \<in> {P. eventually P F}. f (Inf (g ` Collect P)))"
```
```   443     by (simp add: Liminf_def image_comp)
```
```   444   also have "\<dots> = (SUP P \<in> {P. eventually P F}. Inf (f ` (g ` Collect P)))"
```
```   445     using * continuous_at_Inf_mono [OF am continuous_on_imp_continuous_within [OF c]]
```
```   446     by auto
```
```   447   finally show ?thesis by (auto simp: Liminf_def image_comp)
```
```   448 qed
```
```   449
```
```   450 lemma Limsup_compose_continuous_mono:
```
```   451   fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
```
```   452   assumes c: "continuous_on UNIV f" and am: "mono f" and F: "F \<noteq> bot"
```
```   453   shows "Limsup F (\<lambda>n. f (g n)) = f (Limsup F g)"
```
```   454 proof -
```
```   455   { fix P assume "eventually P F"
```
```   456     have "\<exists>x. P x"
```
```   457     proof (rule ccontr)
```
```   458       assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
```
```   459         by auto
```
```   460       with \<open>eventually P F\<close> F show False
```
```   461         by auto
```
```   462     qed }
```
```   463   note * = this
```
```   464
```
```   465   have "f (INF P\<in>{P. eventually P F}. Sup (g ` Collect P)) =
```
```   466     Inf (f ` (\<lambda>P. Sup (g ` Collect P)) ` {P. eventually P F})"
```
```   467     using am continuous_on_imp_continuous_within [OF c]
```
```   468     by (rule continuous_at_Inf_mono) (auto intro: eventually_True)
```
```   469   then have "f (Limsup F g) = (INF P \<in> {P. eventually P F}. f (Sup (g ` Collect P)))"
```
```   470     by (simp add: Limsup_def image_comp)
```
```   471   also have "\<dots> = (INF P \<in> {P. eventually P F}. Sup (f ` (g ` Collect P)))"
```
```   472     using * continuous_at_Sup_mono [OF am continuous_on_imp_continuous_within [OF c]]
```
```   473     by auto
```
```   474   finally show ?thesis by (auto simp: Limsup_def image_comp)
```
```   475 qed
```
```   476
```
```   477 lemma Liminf_compose_continuous_antimono:
```
```   478   fixes f :: "'a::{complete_linorder,linorder_topology} \<Rightarrow> 'b::{complete_linorder,linorder_topology}"
```
```   479   assumes c: "continuous_on UNIV f"
```
```   480     and am: "antimono f"
```
```   481     and F: "F \<noteq> bot"
```
```   482   shows "Liminf F (\<lambda>n. f (g n)) = f (Limsup F g)"
```
```   483 proof -
```
```   484   have *: "\<exists>x. P x" if "eventually P F" for P
```
```   485   proof (rule ccontr)
```
```   486     assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
```
```   487       by auto
```
```   488     with \<open>eventually P F\<close> F show False
```
```   489       by auto
```
```   490   qed
```
```   491
```
```   492   have "f (INF P\<in>{P. eventually P F}. Sup (g ` Collect P)) =
```
```   493     Sup (f ` (\<lambda>P. Sup (g ` Collect P)) ` {P. eventually P F})"
```
```   494     using am continuous_on_imp_continuous_within [OF c]
```
```   495     by (rule continuous_at_Inf_antimono) (auto intro: eventually_True)
```
```   496   then have "f (Limsup F g) = (SUP P \<in> {P. eventually P F}. f (Sup (g ` Collect P)))"
```
```   497     by (simp add: Limsup_def image_comp)
```
```   498   also have "\<dots> = (SUP P \<in> {P. eventually P F}. Inf (f ` (g ` Collect P)))"
```
```   499     using * continuous_at_Sup_antimono [OF am continuous_on_imp_continuous_within [OF c]]
```
```   500     by auto
```
```   501   finally show ?thesis
```
```   502     by (auto simp: Liminf_def image_comp)
```
```   503 qed
```
```   504
```
```   505 lemma Limsup_compose_continuous_antimono:
```
```   506   fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
```
```   507   assumes c: "continuous_on UNIV f" and am: "antimono f" and F: "F \<noteq> bot"
```
```   508   shows "Limsup F (\<lambda>n. f (g n)) = f (Liminf F g)"
```
```   509 proof -
```
```   510   { fix P assume "eventually P F"
```
```   511     have "\<exists>x. P x"
```
```   512     proof (rule ccontr)
```
```   513       assume "\<not> (\<exists>x. P x)" then have "P = (\<lambda>x. False)"
```
```   514         by auto
```
```   515       with \<open>eventually P F\<close> F show False
```
```   516         by auto
```
```   517     qed }
```
```   518   note * = this
```
```   519
```
```   520   have "f (SUP P\<in>{P. eventually P F}. Inf (g ` Collect P)) =
```
```   521     Inf (f ` (\<lambda>P. Inf (g ` Collect P)) ` {P. eventually P F})"
```
```   522     using am continuous_on_imp_continuous_within [OF c]
```
```   523     by (rule continuous_at_Sup_antimono) (auto intro: eventually_True)
```
```   524   then have "f (Liminf F g) = (INF P \<in> {P. eventually P F}. f (Inf (g ` Collect P)))"
```
```   525     by (simp add: Liminf_def image_comp)
```
```   526   also have "\<dots> = (INF P \<in> {P. eventually P F}. Sup (f ` (g ` Collect P)))"
```
```   527     using * continuous_at_Inf_antimono [OF am continuous_on_imp_continuous_within [OF c]]
```
```   528     by auto
```
```   529   finally show ?thesis
```
```   530     by (auto simp: Limsup_def image_comp)
```
```   531 qed
```
```   532
```
```   533 lemma Liminf_filtermap_le: "Liminf (filtermap f F) g \<le> Liminf F (\<lambda>x. g (f x))"
```
```   534   apply (cases "F = bot", simp)
```
```   535   by (subst Liminf_def)
```
```   536     (auto simp add: INF_lower Liminf_bounded eventually_filtermap eventually_mono intro!: SUP_least)
```
```   537
```
```   538 lemma Limsup_filtermap_ge: "Limsup (filtermap f F) g \<ge> Limsup F (\<lambda>x. g (f x))"
```
```   539   apply (cases "F = bot", simp)
```
```   540   by (subst Limsup_def)
```
```   541     (auto simp add: SUP_upper Limsup_bounded eventually_filtermap eventually_mono intro!: INF_greatest)
```
```   542
```
```   543 lemma Liminf_least: "(\<And>P. eventually P F \<Longrightarrow> (INF x\<in>Collect P. f x) \<le> x) \<Longrightarrow> Liminf F f \<le> x"
```
```   544   by (auto intro!: SUP_least simp: Liminf_def)
```
```   545
```
```   546 lemma Limsup_greatest: "(\<And>P. eventually P F \<Longrightarrow> x \<le> (SUP x\<in>Collect P. f x)) \<Longrightarrow> Limsup F f \<ge> x"
```
```   547   by (auto intro!: INF_greatest simp: Limsup_def)
```
```   548
```
```   549 lemma Liminf_filtermap_ge: "inj f \<Longrightarrow> Liminf (filtermap f F) g \<ge> Liminf F (\<lambda>x. g (f x))"
```
```   550   apply (cases "F = bot", simp)
```
```   551   apply (rule Liminf_least)
```
```   552   subgoal for P
```
```   553     by (auto simp: eventually_filtermap the_inv_f_f
```
```   554         intro!: Liminf_bounded INF_lower2 eventually_mono[of P])
```
```   555   done
```
```   556
```
```   557 lemma Limsup_filtermap_le: "inj f \<Longrightarrow> Limsup (filtermap f F) g \<le> Limsup F (\<lambda>x. g (f x))"
```
```   558   apply (cases "F = bot", simp)
```
```   559   apply (rule Limsup_greatest)
```
```   560   subgoal for P
```
```   561     by (auto simp: eventually_filtermap the_inv_f_f
```
```   562         intro!: Limsup_bounded SUP_upper2 eventually_mono[of P])
```
```   563   done
```
```   564
```
```   565 lemma Liminf_filtermap_eq: "inj f \<Longrightarrow> Liminf (filtermap f F) g = Liminf F (\<lambda>x. g (f x))"
```
```   566   using Liminf_filtermap_le[of f F g] Liminf_filtermap_ge[of f F g]
```
```   567   by simp
```
```   568
```
```   569 lemma Limsup_filtermap_eq: "inj f \<Longrightarrow> Limsup (filtermap f F) g = Limsup F (\<lambda>x. g (f x))"
```
```   570   using Limsup_filtermap_le[of f F g] Limsup_filtermap_ge[of F g f]
```
```   571   by simp
```
```   572
```
```   573
```
```   574 subsection \<open>More Limits\<close>
```
```   575
```
```   576 lemma convergent_limsup_cl:
```
```   577   fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
```
```   578   shows "convergent X \<Longrightarrow> limsup X = lim X"
```
```   579   by (auto simp: convergent_def limI lim_imp_Limsup)
```
```   580
```
```   581 lemma convergent_liminf_cl:
```
```   582   fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
```
```   583   shows "convergent X \<Longrightarrow> liminf X = lim X"
```
```   584   by (auto simp: convergent_def limI lim_imp_Liminf)
```
```   585
```
```   586 lemma lim_increasing_cl:
```
```   587   assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<ge> f m"
```
```   588   obtains l where "f \<longlonglongrightarrow> (l::'a::{complete_linorder,linorder_topology})"
```
```   589 proof
```
```   590   show "f \<longlonglongrightarrow> (SUP n. f n)"
```
```   591     using assms
```
```   592     by (intro increasing_tendsto)
```
```   593        (auto simp: SUP_upper eventually_sequentially less_SUP_iff intro: less_le_trans)
```
```   594 qed
```
```   595
```
```   596 lemma lim_decreasing_cl:
```
```   597   assumes "\<And>n m. n \<ge> m \<Longrightarrow> f n \<le> f m"
```
```   598   obtains l where "f \<longlonglongrightarrow> (l::'a::{complete_linorder,linorder_topology})"
```
```   599 proof
```
```   600   show "f \<longlonglongrightarrow> (INF n. f n)"
```
```   601     using assms
```
```   602     by (intro decreasing_tendsto)
```
```   603        (auto simp: INF_lower eventually_sequentially INF_less_iff intro: le_less_trans)
```
```   604 qed
```
```   605
```
```   606 lemma compact_complete_linorder:
```
```   607   fixes X :: "nat \<Rightarrow> 'a::{complete_linorder,linorder_topology}"
```
```   608   shows "\<exists>l r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> l"
```
```   609 proof -
```
```   610   obtain r where "strict_mono r" and mono: "monoseq (X \<circ> r)"
```
```   611     using seq_monosub[of X]
```
```   612     unfolding comp_def
```
```   613     by auto
```
```   614   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)"
```
```   615     by (auto simp add: monoseq_def)
```
```   616   then obtain l where "(X \<circ> r) \<longlonglongrightarrow> l"
```
```   617      using lim_increasing_cl[of "X \<circ> r"] lim_decreasing_cl[of "X \<circ> r"]
```
```   618      by auto
```
```   619   then show ?thesis
```
```   620     using \<open>strict_mono r\<close> by auto
```
```   621 qed
```
```   622
```
```   623 lemma tendsto_Limsup:
```
```   624   fixes f :: "_ \<Rightarrow> 'a :: {complete_linorder,linorder_topology}"
```
```   625   shows "F \<noteq> bot \<Longrightarrow> Limsup F f = Liminf F f \<Longrightarrow> (f \<longlongrightarrow> Limsup F f) F"
```
```   626   by (subst tendsto_iff_Liminf_eq_Limsup) auto
```
```   627
```
```   628 lemma tendsto_Liminf:
```
```   629   fixes f :: "_ \<Rightarrow> 'a :: {complete_linorder,linorder_topology}"
```
```   630   shows "F \<noteq> bot \<Longrightarrow> Limsup F f = Liminf F f \<Longrightarrow> (f \<longlongrightarrow> Liminf F f) F"
```
```   631   by (subst tendsto_iff_Liminf_eq_Limsup) auto
```
```   632
```
```   633 end
```