src/HOL/Library/Extended_Reals.thy
 author bulwahn Tue Apr 05 09:38:28 2011 +0200 (2011-04-05) changeset 42231 bc1891226d00 parent 41983 2dc6e382a58b child 42600 604661fb94eb permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Extended_Reals.thy
```
```     2     Author:     Johannes Hölzl, TU München
```
```     3     Author:     Robert Himmelmann, TU München
```
```     4     Author:     Armin Heller, TU München
```
```     5     Author:     Bogdan Grechuk, University of Edinburgh
```
```     6 *)
```
```     7
```
```     8 header {* Extended real number line *}
```
```     9
```
```    10 theory Extended_Reals
```
```    11   imports Complex_Main
```
```    12 begin
```
```    13
```
```    14 text {*
```
```    15
```
```    16 For more lemmas about the extended real numbers go to
```
```    17   @{text "src/HOL/Multivaraite_Analysis/Extended_Real_Limits.thy"}
```
```    18
```
```    19 *}
```
```    20
```
```    21 lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
```
```    22 proof
```
```    23   assume "{x..} = UNIV"
```
```    24   show "x = bot"
```
```    25   proof (rule ccontr)
```
```    26     assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less)
```
```    27     then show False using `{x..} = UNIV` by simp
```
```    28   qed
```
```    29 qed auto
```
```    30
```
```    31 lemma SUPR_pair:
```
```    32   "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
```
```    33   by (rule antisym) (auto intro!: SUP_leI le_SUPI2)
```
```    34
```
```    35 lemma INFI_pair:
```
```    36   "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
```
```    37   by (rule antisym) (auto intro!: le_INFI INF_leI2)
```
```    38
```
```    39 subsection {* Definition and basic properties *}
```
```    40
```
```    41 datatype extreal = extreal real | PInfty | MInfty
```
```    42
```
```    43 notation (xsymbols)
```
```    44   PInfty  ("\<infinity>")
```
```    45
```
```    46 notation (HTML output)
```
```    47   PInfty  ("\<infinity>")
```
```    48
```
```    49 declare [[coercion "extreal :: real \<Rightarrow> extreal"]]
```
```    50
```
```    51 instantiation extreal :: uminus
```
```    52 begin
```
```    53   fun uminus_extreal where
```
```    54     "- (extreal r) = extreal (- r)"
```
```    55   | "- \<infinity> = MInfty"
```
```    56   | "- MInfty = \<infinity>"
```
```    57   instance ..
```
```    58 end
```
```    59
```
```    60 lemma inj_extreal[simp]: "inj_on extreal A"
```
```    61   unfolding inj_on_def by auto
```
```    62
```
```    63 lemma MInfty_neq_PInfty[simp]:
```
```    64   "\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all
```
```    65
```
```    66 lemma MInfty_neq_extreal[simp]:
```
```    67   "extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all
```
```    68
```
```    69 lemma MInfinity_cases[simp]:
```
```    70   "(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z"
```
```    71   by simp
```
```    72
```
```    73 lemma extreal_uminus_uminus[simp]:
```
```    74   fixes a :: extreal shows "- (- a) = a"
```
```    75   by (cases a) simp_all
```
```    76
```
```    77 lemma MInfty_eq[simp]:
```
```    78   "MInfty = - \<infinity>" by simp
```
```    79
```
```    80 declare uminus_extreal.simps(2)[simp del]
```
```    81
```
```    82 lemma extreal_cases[case_names real PInf MInf, cases type: extreal]:
```
```    83   assumes "\<And>r. x = extreal r \<Longrightarrow> P"
```
```    84   assumes "x = \<infinity> \<Longrightarrow> P"
```
```    85   assumes "x = -\<infinity> \<Longrightarrow> P"
```
```    86   shows P
```
```    87   using assms by (cases x) auto
```
```    88
```
```    89 lemmas extreal2_cases = extreal_cases[case_product extreal_cases]
```
```    90 lemmas extreal3_cases = extreal2_cases[case_product extreal_cases]
```
```    91
```
```    92 lemma extreal_uminus_eq_iff[simp]:
```
```    93   fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b"
```
```    94   by (cases rule: extreal2_cases[of a b]) simp_all
```
```    95
```
```    96 function of_extreal :: "extreal \<Rightarrow> real" where
```
```    97 "of_extreal (extreal r) = r" |
```
```    98 "of_extreal \<infinity> = 0" |
```
```    99 "of_extreal (-\<infinity>) = 0"
```
```   100   by (auto intro: extreal_cases)
```
```   101 termination proof qed (rule wf_empty)
```
```   102
```
```   103 defs (overloaded)
```
```   104   real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal"
```
```   105
```
```   106 lemma real_of_extreal[simp]:
```
```   107     "real (- x :: extreal) = - (real x)"
```
```   108     "real (extreal r) = r"
```
```   109     "real \<infinity> = 0"
```
```   110   by (cases x) (simp_all add: real_of_extreal_def)
```
```   111
```
```   112 lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}"
```
```   113 proof safe
```
```   114   fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>"
```
```   115   then show "x = -\<infinity>" by (cases x) auto
```
```   116 qed auto
```
```   117
```
```   118 lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)"
```
```   119 proof safe
```
```   120   fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
```
```   121 qed auto
```
```   122
```
```   123 instantiation extreal :: number
```
```   124 begin
```
```   125 definition [simp]: "number_of x = extreal (number_of x)"
```
```   126 instance proof qed
```
```   127 end
```
```   128
```
```   129 instantiation extreal :: abs
```
```   130 begin
```
```   131   function abs_extreal where
```
```   132     "\<bar>extreal r\<bar> = extreal \<bar>r\<bar>"
```
```   133   | "\<bar>-\<infinity>\<bar> = \<infinity>"
```
```   134   | "\<bar>\<infinity>\<bar> = \<infinity>"
```
```   135   by (auto intro: extreal_cases)
```
```   136   termination proof qed (rule wf_empty)
```
```   137   instance ..
```
```   138 end
```
```   139
```
```   140 lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
```
```   141   by (cases x) auto
```
```   142
```
```   143 lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
```
```   144   by (cases x) auto
```
```   145
```
```   146 lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>"
```
```   147   by (cases x) auto
```
```   148
```
```   149 subsubsection "Addition"
```
```   150
```
```   151 instantiation extreal :: comm_monoid_add
```
```   152 begin
```
```   153
```
```   154 definition "0 = extreal 0"
```
```   155
```
```   156 function plus_extreal where
```
```   157 "extreal r + extreal p = extreal (r + p)" |
```
```   158 "\<infinity> + a = \<infinity>" |
```
```   159 "a + \<infinity> = \<infinity>" |
```
```   160 "extreal r + -\<infinity> = - \<infinity>" |
```
```   161 "-\<infinity> + extreal p = -\<infinity>" |
```
```   162 "-\<infinity> + -\<infinity> = -\<infinity>"
```
```   163 proof -
```
```   164   case (goal1 P x)
```
```   165   moreover then obtain a b where "x = (a, b)" by (cases x) auto
```
```   166   ultimately show P
```
```   167    by (cases rule: extreal2_cases[of a b]) auto
```
```   168 qed auto
```
```   169 termination proof qed (rule wf_empty)
```
```   170
```
```   171 lemma Infty_neq_0[simp]:
```
```   172   "\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>"
```
```   173   "-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>"
```
```   174   by (simp_all add: zero_extreal_def)
```
```   175
```
```   176 lemma extreal_eq_0[simp]:
```
```   177   "extreal r = 0 \<longleftrightarrow> r = 0"
```
```   178   "0 = extreal r \<longleftrightarrow> r = 0"
```
```   179   unfolding zero_extreal_def by simp_all
```
```   180
```
```   181 instance
```
```   182 proof
```
```   183   fix a :: extreal show "0 + a = a"
```
```   184     by (cases a) (simp_all add: zero_extreal_def)
```
```   185   fix b :: extreal show "a + b = b + a"
```
```   186     by (cases rule: extreal2_cases[of a b]) simp_all
```
```   187   fix c :: extreal show "a + b + c = a + (b + c)"
```
```   188     by (cases rule: extreal3_cases[of a b c]) simp_all
```
```   189 qed
```
```   190 end
```
```   191
```
```   192 lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)"
```
```   193   unfolding zero_extreal_def abs_extreal.simps by simp
```
```   194
```
```   195 lemma extreal_uminus_zero[simp]:
```
```   196   "- 0 = (0::extreal)"
```
```   197   by (simp add: zero_extreal_def)
```
```   198
```
```   199 lemma extreal_uminus_zero_iff[simp]:
```
```   200   fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0"
```
```   201   by (cases a) simp_all
```
```   202
```
```   203 lemma extreal_plus_eq_PInfty[simp]:
```
```   204   shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
```
```   205   by (cases rule: extreal2_cases[of a b]) auto
```
```   206
```
```   207 lemma extreal_plus_eq_MInfty[simp]:
```
```   208   shows "a + b = -\<infinity> \<longleftrightarrow>
```
```   209     (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
```
```   210   by (cases rule: extreal2_cases[of a b]) auto
```
```   211
```
```   212 lemma extreal_add_cancel_left:
```
```   213   assumes "a \<noteq> -\<infinity>"
```
```   214   shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
```
```   215   using assms by (cases rule: extreal3_cases[of a b c]) auto
```
```   216
```
```   217 lemma extreal_add_cancel_right:
```
```   218   assumes "a \<noteq> -\<infinity>"
```
```   219   shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
```
```   220   using assms by (cases rule: extreal3_cases[of a b c]) auto
```
```   221
```
```   222 lemma extreal_real:
```
```   223   "extreal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
```
```   224   by (cases x) simp_all
```
```   225
```
```   226 lemma real_of_extreal_add:
```
```   227   fixes a b :: extreal
```
```   228   shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
```
```   229   by (cases rule: extreal2_cases[of a b]) auto
```
```   230
```
```   231 subsubsection "Linear order on @{typ extreal}"
```
```   232
```
```   233 instantiation extreal :: linorder
```
```   234 begin
```
```   235
```
```   236 function less_extreal where
```
```   237 "extreal x < extreal y \<longleftrightarrow> x < y" |
```
```   238 "        \<infinity> < a         \<longleftrightarrow> False" |
```
```   239 "        a < -\<infinity>        \<longleftrightarrow> False" |
```
```   240 "extreal x < \<infinity>         \<longleftrightarrow> True" |
```
```   241 "       -\<infinity> < extreal r \<longleftrightarrow> True" |
```
```   242 "       -\<infinity> < \<infinity>         \<longleftrightarrow> True"
```
```   243 proof -
```
```   244   case (goal1 P x)
```
```   245   moreover then obtain a b where "x = (a,b)" by (cases x) auto
```
```   246   ultimately show P by (cases rule: extreal2_cases[of a b]) auto
```
```   247 qed simp_all
```
```   248 termination by (relation "{}") simp
```
```   249
```
```   250 definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y"
```
```   251
```
```   252 lemma extreal_infty_less[simp]:
```
```   253   "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
```
```   254   "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
```
```   255   by (cases x, simp_all) (cases x, simp_all)
```
```   256
```
```   257 lemma extreal_infty_less_eq[simp]:
```
```   258   "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
```
```   259   "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
```
```   260   by (auto simp add: less_eq_extreal_def)
```
```   261
```
```   262 lemma extreal_less[simp]:
```
```   263   "extreal r < 0 \<longleftrightarrow> (r < 0)"
```
```   264   "0 < extreal r \<longleftrightarrow> (0 < r)"
```
```   265   "0 < \<infinity>"
```
```   266   "-\<infinity> < 0"
```
```   267   by (simp_all add: zero_extreal_def)
```
```   268
```
```   269 lemma extreal_less_eq[simp]:
```
```   270   "x \<le> \<infinity>"
```
```   271   "-\<infinity> \<le> x"
```
```   272   "extreal r \<le> extreal p \<longleftrightarrow> r \<le> p"
```
```   273   "extreal r \<le> 0 \<longleftrightarrow> r \<le> 0"
```
```   274   "0 \<le> extreal r \<longleftrightarrow> 0 \<le> r"
```
```   275   by (auto simp add: less_eq_extreal_def zero_extreal_def)
```
```   276
```
```   277 lemma extreal_infty_less_eq2:
```
```   278   "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>"
```
```   279   "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>"
```
```   280   by simp_all
```
```   281
```
```   282 instance
```
```   283 proof
```
```   284   fix x :: extreal show "x \<le> x"
```
```   285     by (cases x) simp_all
```
```   286   fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```   287     by (cases rule: extreal2_cases[of x y]) auto
```
```   288   show "x \<le> y \<or> y \<le> x "
```
```   289     by (cases rule: extreal2_cases[of x y]) auto
```
```   290   { assume "x \<le> y" "y \<le> x" then show "x = y"
```
```   291     by (cases rule: extreal2_cases[of x y]) auto }
```
```   292   { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
```
```   293     by (cases rule: extreal3_cases[of x y z]) auto }
```
```   294 qed
```
```   295 end
```
```   296
```
```   297 instance extreal :: ordered_ab_semigroup_add
```
```   298 proof
```
```   299   fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b"
```
```   300     by (cases rule: extreal3_cases[of a b c]) auto
```
```   301 qed
```
```   302
```
```   303 lemma extreal_MInfty_lessI[intro, simp]:
```
```   304   "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
```
```   305   by (cases a) auto
```
```   306
```
```   307 lemma extreal_less_PInfty[intro, simp]:
```
```   308   "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
```
```   309   by (cases a) auto
```
```   310
```
```   311 lemma extreal_less_extreal_Ex:
```
```   312   fixes a b :: extreal
```
```   313   shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)"
```
```   314   by (cases x) auto
```
```   315
```
```   316 lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))"
```
```   317 proof (cases x)
```
```   318   case (real r) then show ?thesis
```
```   319     using reals_Archimedean2[of r] by simp
```
```   320 qed simp_all
```
```   321
```
```   322 lemma extreal_add_mono:
```
```   323   fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
```
```   324   using assms
```
```   325   apply (cases a)
```
```   326   apply (cases rule: extreal3_cases[of b c d], auto)
```
```   327   apply (cases rule: extreal3_cases[of b c d], auto)
```
```   328   done
```
```   329
```
```   330 lemma extreal_minus_le_minus[simp]:
```
```   331   fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
```
```   332   by (cases rule: extreal2_cases[of a b]) auto
```
```   333
```
```   334 lemma extreal_minus_less_minus[simp]:
```
```   335   fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a"
```
```   336   by (cases rule: extreal2_cases[of a b]) auto
```
```   337
```
```   338 lemma extreal_le_real_iff:
```
```   339   "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
```
```   340   by (cases y) auto
```
```   341
```
```   342 lemma real_le_extreal_iff:
```
```   343   "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
```
```   344   by (cases y) auto
```
```   345
```
```   346 lemma extreal_less_real_iff:
```
```   347   "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
```
```   348   by (cases y) auto
```
```   349
```
```   350 lemma real_less_extreal_iff:
```
```   351   "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
```
```   352   by (cases y) auto
```
```   353
```
```   354 lemma real_of_extreal_positive_mono:
```
```   355   assumes "x \<noteq> \<infinity>" "y \<noteq> \<infinity>" "0 \<le> x" "x \<le> y"
```
```   356   shows "real x \<le> real y"
```
```   357   using assms by (cases rule: extreal2_cases[of x y]) auto
```
```   358
```
```   359 lemma real_of_extreal_pos:
```
```   360   fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
```
```   361
```
```   362 lemmas real_of_extreal_ord_simps =
```
```   363   extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff
```
```   364
```
```   365 lemma extreal_dense:
```
```   366   fixes x y :: extreal assumes "x < y"
```
```   367   shows "EX z. x < z & z < y"
```
```   368 proof -
```
```   369 { assume a: "x = (-\<infinity>)"
```
```   370   { assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) }
```
```   371   moreover
```
```   372   { assume "y ~= \<infinity>"
```
```   373     with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
```
```   374     hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"])
```
```   375   } ultimately have ?thesis by auto
```
```   376 }
```
```   377 moreover
```
```   378 { assume "x ~= (-\<infinity>)"
```
```   379   with `x < y` obtain p where p: "x = extreal p" by (cases x) auto
```
```   380   { assume "y = \<infinity>" hence ?thesis using `x < y` p
```
```   381        by (auto intro!: exI[of _ "extreal (p + 1)"]) }
```
```   382   moreover
```
```   383   { assume "y ~= \<infinity>"
```
```   384     with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
```
```   385     with p `x < y` have "p < r" by auto
```
```   386     with dense obtain z where "p < z" "z < r" by auto
```
```   387     hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"])
```
```   388   } ultimately have ?thesis by auto
```
```   389 } ultimately show ?thesis by auto
```
```   390 qed
```
```   391
```
```   392 lemma extreal_dense2:
```
```   393   fixes x y :: extreal assumes "x < y"
```
```   394   shows "EX z. x < extreal z & extreal z < y"
```
```   395   by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3))
```
```   396
```
```   397 lemma extreal_add_strict_mono:
```
```   398   fixes a b c d :: extreal
```
```   399   assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
```
```   400   shows "a + c < b + d"
```
```   401   using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto
```
```   402
```
```   403 lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
```
```   404   by (cases rule: extreal2_cases[of b c]) auto
```
```   405
```
```   406 lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto
```
```   407
```
```   408 lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)"
```
```   409   by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus)
```
```   410
```
```   411 lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)"
```
```   412   by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus)
```
```   413
```
```   414 lemmas extreal_uminus_reorder =
```
```   415   extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder
```
```   416
```
```   417 lemma extreal_bot:
```
```   418   fixes x :: extreal assumes "\<And>B. x \<le> extreal B" shows "x = - \<infinity>"
```
```   419 proof (cases x)
```
```   420   case (real r) with assms[of "r - 1"] show ?thesis by auto
```
```   421 next case PInf with assms[of 0] show ?thesis by auto
```
```   422 next case MInf then show ?thesis by simp
```
```   423 qed
```
```   424
```
```   425 lemma extreal_top:
```
```   426   fixes x :: extreal assumes "\<And>B. x \<ge> extreal B" shows "x = \<infinity>"
```
```   427 proof (cases x)
```
```   428   case (real r) with assms[of "r + 1"] show ?thesis by auto
```
```   429 next case MInf with assms[of 0] show ?thesis by auto
```
```   430 next case PInf then show ?thesis by simp
```
```   431 qed
```
```   432
```
```   433 lemma
```
```   434   shows extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)"
```
```   435     and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)"
```
```   436   by (simp_all add: min_def max_def)
```
```   437
```
```   438 lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)"
```
```   439   by (auto simp: zero_extreal_def)
```
```   440
```
```   441 lemma
```
```   442   fixes f :: "nat \<Rightarrow> extreal"
```
```   443   shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
```
```   444   and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
```
```   445   unfolding decseq_def incseq_def by auto
```
```   446
```
```   447 lemma extreal_add_nonneg_nonneg:
```
```   448   fixes a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
```
```   449   using add_mono[of 0 a 0 b] by simp
```
```   450
```
```   451 lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
```
```   452   by auto
```
```   453
```
```   454 lemma incseq_setsumI:
```
```   455   fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
```
```   456   assumes "\<And>i. 0 \<le> f i"
```
```   457   shows "incseq (\<lambda>i. setsum f {..< i})"
```
```   458 proof (intro incseq_SucI)
```
```   459   fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
```
```   460     using assms by (rule add_left_mono)
```
```   461   then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
```
```   462     by auto
```
```   463 qed
```
```   464
```
```   465 lemma incseq_setsumI2:
```
```   466   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
```
```   467   assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
```
```   468   shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
```
```   469   using assms unfolding incseq_def by (auto intro: setsum_mono)
```
```   470
```
```   471 subsubsection "Multiplication"
```
```   472
```
```   473 instantiation extreal :: "{comm_monoid_mult, sgn}"
```
```   474 begin
```
```   475
```
```   476 definition "1 = extreal 1"
```
```   477
```
```   478 function sgn_extreal where
```
```   479   "sgn (extreal r) = extreal (sgn r)"
```
```   480 | "sgn \<infinity> = 1"
```
```   481 | "sgn (-\<infinity>) = -1"
```
```   482 by (auto intro: extreal_cases)
```
```   483 termination proof qed (rule wf_empty)
```
```   484
```
```   485 function times_extreal where
```
```   486 "extreal r * extreal p = extreal (r * p)" |
```
```   487 "extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
```
```   488 "\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
```
```   489 "extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
```
```   490 "-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
```
```   491 "\<infinity> * \<infinity> = \<infinity>" |
```
```   492 "-\<infinity> * \<infinity> = -\<infinity>" |
```
```   493 "\<infinity> * -\<infinity> = -\<infinity>" |
```
```   494 "-\<infinity> * -\<infinity> = \<infinity>"
```
```   495 proof -
```
```   496   case (goal1 P x)
```
```   497   moreover then obtain a b where "x = (a, b)" by (cases x) auto
```
```   498   ultimately show P by (cases rule: extreal2_cases[of a b]) auto
```
```   499 qed simp_all
```
```   500 termination by (relation "{}") simp
```
```   501
```
```   502 instance
```
```   503 proof
```
```   504   fix a :: extreal show "1 * a = a"
```
```   505     by (cases a) (simp_all add: one_extreal_def)
```
```   506   fix b :: extreal show "a * b = b * a"
```
```   507     by (cases rule: extreal2_cases[of a b]) simp_all
```
```   508   fix c :: extreal show "a * b * c = a * (b * c)"
```
```   509     by (cases rule: extreal3_cases[of a b c])
```
```   510        (simp_all add: zero_extreal_def zero_less_mult_iff)
```
```   511 qed
```
```   512 end
```
```   513
```
```   514 lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)"
```
```   515   unfolding one_extreal_def by simp
```
```   516
```
```   517 lemma extreal_mult_zero[simp]:
```
```   518   fixes a :: extreal shows "a * 0 = 0"
```
```   519   by (cases a) (simp_all add: zero_extreal_def)
```
```   520
```
```   521 lemma extreal_zero_mult[simp]:
```
```   522   fixes a :: extreal shows "0 * a = 0"
```
```   523   by (cases a) (simp_all add: zero_extreal_def)
```
```   524
```
```   525 lemma extreal_m1_less_0[simp]:
```
```   526   "-(1::extreal) < 0"
```
```   527   by (simp add: zero_extreal_def one_extreal_def)
```
```   528
```
```   529 lemma extreal_zero_m1[simp]:
```
```   530   "1 \<noteq> (0::extreal)"
```
```   531   by (simp add: zero_extreal_def one_extreal_def)
```
```   532
```
```   533 lemma extreal_times_0[simp]:
```
```   534   fixes x :: extreal shows "0 * x = 0"
```
```   535   by (cases x) (auto simp: zero_extreal_def)
```
```   536
```
```   537 lemma extreal_times[simp]:
```
```   538   "1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1"
```
```   539   "1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1"
```
```   540   by (auto simp add: times_extreal_def one_extreal_def)
```
```   541
```
```   542 lemma extreal_plus_1[simp]:
```
```   543   "1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)"
```
```   544   "1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>"
```
```   545   unfolding one_extreal_def by auto
```
```   546
```
```   547 lemma extreal_zero_times[simp]:
```
```   548   fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   549   by (cases rule: extreal2_cases[of a b]) auto
```
```   550
```
```   551 lemma extreal_mult_eq_PInfty[simp]:
```
```   552   shows "a * b = \<infinity> \<longleftrightarrow>
```
```   553     (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
```
```   554   by (cases rule: extreal2_cases[of a b]) auto
```
```   555
```
```   556 lemma extreal_mult_eq_MInfty[simp]:
```
```   557   shows "a * b = -\<infinity> \<longleftrightarrow>
```
```   558     (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
```
```   559   by (cases rule: extreal2_cases[of a b]) auto
```
```   560
```
```   561 lemma extreal_0_less_1[simp]: "0 < (1::extreal)"
```
```   562   by (simp_all add: zero_extreal_def one_extreal_def)
```
```   563
```
```   564 lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)"
```
```   565   by (simp_all add: zero_extreal_def one_extreal_def)
```
```   566
```
```   567 lemma extreal_mult_minus_left[simp]:
```
```   568   fixes a b :: extreal shows "-a * b = - (a * b)"
```
```   569   by (cases rule: extreal2_cases[of a b]) auto
```
```   570
```
```   571 lemma extreal_mult_minus_right[simp]:
```
```   572   fixes a b :: extreal shows "a * -b = - (a * b)"
```
```   573   by (cases rule: extreal2_cases[of a b]) auto
```
```   574
```
```   575 lemma extreal_mult_infty[simp]:
```
```   576   "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
```
```   577   by (cases a) auto
```
```   578
```
```   579 lemma extreal_infty_mult[simp]:
```
```   580   "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
```
```   581   by (cases a) auto
```
```   582
```
```   583 lemma extreal_mult_strict_right_mono:
```
```   584   assumes "a < b" and "0 < c" "c < \<infinity>"
```
```   585   shows "a * c < b * c"
```
```   586   using assms
```
```   587   by (cases rule: extreal3_cases[of a b c])
```
```   588      (auto simp: zero_le_mult_iff extreal_less_PInfty)
```
```   589
```
```   590 lemma extreal_mult_strict_left_mono:
```
```   591   "\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b"
```
```   592   using extreal_mult_strict_right_mono by (simp add: mult_commute[of c])
```
```   593
```
```   594 lemma extreal_mult_right_mono:
```
```   595   fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
```
```   596   using assms
```
```   597   apply (cases "c = 0") apply simp
```
```   598   by (cases rule: extreal3_cases[of a b c])
```
```   599      (auto simp: zero_le_mult_iff extreal_less_PInfty)
```
```   600
```
```   601 lemma extreal_mult_left_mono:
```
```   602   fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
```
```   603   using extreal_mult_right_mono by (simp add: mult_commute[of c])
```
```   604
```
```   605 lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)"
```
```   606   by (simp add: one_extreal_def zero_extreal_def)
```
```   607
```
```   608 lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)"
```
```   609   by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
```
```   610
```
```   611 lemma extreal_right_distrib:
```
```   612   fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
```
```   613   by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
```
```   614
```
```   615 lemma extreal_left_distrib:
```
```   616   fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
```
```   617   by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
```
```   618
```
```   619 lemma extreal_mult_le_0_iff:
```
```   620   fixes a b :: extreal
```
```   621   shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
```
```   622   by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff)
```
```   623
```
```   624 lemma extreal_zero_le_0_iff:
```
```   625   fixes a b :: extreal
```
```   626   shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
```
```   627   by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
```
```   628
```
```   629 lemma extreal_mult_less_0_iff:
```
```   630   fixes a b :: extreal
```
```   631   shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
```
```   632   by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff)
```
```   633
```
```   634 lemma extreal_zero_less_0_iff:
```
```   635   fixes a b :: extreal
```
```   636   shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
```
```   637   by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
```
```   638
```
```   639 lemma extreal_distrib:
```
```   640   fixes a b c :: extreal
```
```   641   assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
```
```   642   shows "(a + b) * c = a * c + b * c"
```
```   643   using assms
```
```   644   by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
```
```   645
```
```   646 lemma extreal_le_epsilon:
```
```   647   fixes x y :: extreal
```
```   648   assumes "ALL e. 0 < e --> x <= y + e"
```
```   649   shows "x <= y"
```
```   650 proof-
```
```   651 { assume a: "EX r. y = extreal r"
```
```   652   from this obtain r where r_def: "y = extreal r" by auto
```
```   653   { assume "x=(-\<infinity>)" hence ?thesis by auto }
```
```   654   moreover
```
```   655   { assume "~(x=(-\<infinity>))"
```
```   656     from this obtain p where p_def: "x = extreal p"
```
```   657     using a assms[rule_format, of 1] by (cases x) auto
```
```   658     { fix e have "0 < e --> p <= r + e"
```
```   659       using assms[rule_format, of "extreal e"] p_def r_def by auto }
```
```   660     hence "p <= r" apply (subst field_le_epsilon) by auto
```
```   661     hence ?thesis using r_def p_def by auto
```
```   662   } ultimately have ?thesis by blast
```
```   663 }
```
```   664 moreover
```
```   665 { assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
```
```   666     using assms[rule_format, of 1] by (cases x) auto
```
```   667 } ultimately show ?thesis by (cases y) auto
```
```   668 qed
```
```   669
```
```   670
```
```   671 lemma extreal_le_epsilon2:
```
```   672   fixes x y :: extreal
```
```   673   assumes "ALL e. 0 < e --> x <= y + extreal e"
```
```   674   shows "x <= y"
```
```   675 proof-
```
```   676 { fix e :: extreal assume "e>0"
```
```   677   { assume "e=\<infinity>" hence "x<=y+e" by auto }
```
```   678   moreover
```
```   679   { assume "e~=\<infinity>"
```
```   680     from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto
```
```   681     hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
```
```   682   } ultimately have "x<=y+e" by blast
```
```   683 } from this show ?thesis using extreal_le_epsilon by auto
```
```   684 qed
```
```   685
```
```   686 lemma extreal_le_real:
```
```   687   fixes x y :: extreal
```
```   688   assumes "ALL z. x <= extreal z --> y <= extreal z"
```
```   689   shows "y <= x"
```
```   690 by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1)
```
```   691           extreal_less_eq(2) order_refl uminus_extreal.simps(2))
```
```   692
```
```   693 lemma extreal_le_extreal:
```
```   694   fixes x y :: extreal
```
```   695   assumes "\<And>B. B < x \<Longrightarrow> B <= y"
```
```   696   shows "x <= y"
```
```   697 by (metis assms extreal_dense leD linorder_le_less_linear)
```
```   698
```
```   699 lemma extreal_ge_extreal:
```
```   700   fixes x y :: extreal
```
```   701   assumes "ALL B. B>x --> B >= y"
```
```   702   shows "x >= y"
```
```   703 by (metis assms extreal_dense leD linorder_le_less_linear)
```
```   704
```
```   705 subsubsection {* Power *}
```
```   706
```
```   707 instantiation extreal :: power
```
```   708 begin
```
```   709 primrec power_extreal where
```
```   710   "power_extreal x 0 = 1" |
```
```   711   "power_extreal x (Suc n) = x * x ^ n"
```
```   712 instance ..
```
```   713 end
```
```   714
```
```   715 lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)"
```
```   716   by (induct n) (auto simp: one_extreal_def)
```
```   717
```
```   718 lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)"
```
```   719   by (induct n) (auto simp: one_extreal_def)
```
```   720
```
```   721 lemma extreal_power_uminus[simp]:
```
```   722   fixes x :: extreal
```
```   723   shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
```
```   724   by (induct n) (auto simp: one_extreal_def)
```
```   725
```
```   726 lemma extreal_power_number_of[simp]:
```
```   727   "(number_of num :: extreal) ^ n = extreal (number_of num ^ n)"
```
```   728   by (induct n) (auto simp: one_extreal_def)
```
```   729
```
```   730 lemma zero_le_power_extreal[simp]:
```
```   731   fixes a :: extreal assumes "0 \<le> a"
```
```   732   shows "0 \<le> a ^ n"
```
```   733   using assms by (induct n) (auto simp: extreal_zero_le_0_iff)
```
```   734
```
```   735 subsubsection {* Subtraction *}
```
```   736
```
```   737 lemma extreal_minus_minus_image[simp]:
```
```   738   fixes S :: "extreal set"
```
```   739   shows "uminus ` uminus ` S = S"
```
```   740   by (auto simp: image_iff)
```
```   741
```
```   742 lemma extreal_uminus_lessThan[simp]:
```
```   743   fixes a :: extreal shows "uminus ` {..<a} = {-a<..}"
```
```   744 proof (safe intro!: image_eqI)
```
```   745   fix x assume "-a < x"
```
```   746   then have "- x < - (- a)" by (simp del: extreal_uminus_uminus)
```
```   747   then show "- x < a" by simp
```
```   748 qed auto
```
```   749
```
```   750 lemma extreal_uminus_greaterThan[simp]:
```
```   751   "uminus ` {(a::extreal)<..} = {..<-a}"
```
```   752   by (metis extreal_uminus_lessThan extreal_uminus_uminus
```
```   753             extreal_minus_minus_image)
```
```   754
```
```   755 instantiation extreal :: minus
```
```   756 begin
```
```   757 definition "x - y = x + -(y::extreal)"
```
```   758 instance ..
```
```   759 end
```
```   760
```
```   761 lemma extreal_minus[simp]:
```
```   762   "extreal r - extreal p = extreal (r - p)"
```
```   763   "-\<infinity> - extreal r = -\<infinity>"
```
```   764   "extreal r - \<infinity> = -\<infinity>"
```
```   765   "\<infinity> - x = \<infinity>"
```
```   766   "-\<infinity> - \<infinity> = -\<infinity>"
```
```   767   "x - -y = x + y"
```
```   768   "x - 0 = x"
```
```   769   "0 - x = -x"
```
```   770   by (simp_all add: minus_extreal_def)
```
```   771
```
```   772 lemma extreal_x_minus_x[simp]:
```
```   773   "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)"
```
```   774   by (cases x) simp_all
```
```   775
```
```   776 lemma extreal_eq_minus_iff:
```
```   777   fixes x y z :: extreal
```
```   778   shows "x = z - y \<longleftrightarrow>
```
```   779     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
```
```   780     (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
```
```   781     (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
```
```   782     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
```
```   783   by (cases rule: extreal3_cases[of x y z]) auto
```
```   784
```
```   785 lemma extreal_eq_minus:
```
```   786   fixes x y z :: extreal
```
```   787   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
```
```   788   by (auto simp: extreal_eq_minus_iff)
```
```   789
```
```   790 lemma extreal_less_minus_iff:
```
```   791   fixes x y z :: extreal
```
```   792   shows "x < z - y \<longleftrightarrow>
```
```   793     (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
```
```   794     (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
```
```   795     (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
```
```   796   by (cases rule: extreal3_cases[of x y z]) auto
```
```   797
```
```   798 lemma extreal_less_minus:
```
```   799   fixes x y z :: extreal
```
```   800   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
```
```   801   by (auto simp: extreal_less_minus_iff)
```
```   802
```
```   803 lemma extreal_le_minus_iff:
```
```   804   fixes x y z :: extreal
```
```   805   shows "x \<le> z - y \<longleftrightarrow>
```
```   806     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
```
```   807     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
```
```   808   by (cases rule: extreal3_cases[of x y z]) auto
```
```   809
```
```   810 lemma extreal_le_minus:
```
```   811   fixes x y z :: extreal
```
```   812   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
```
```   813   by (auto simp: extreal_le_minus_iff)
```
```   814
```
```   815 lemma extreal_minus_less_iff:
```
```   816   fixes x y z :: extreal
```
```   817   shows "x - y < z \<longleftrightarrow>
```
```   818     y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
```
```   819     (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
```
```   820   by (cases rule: extreal3_cases[of x y z]) auto
```
```   821
```
```   822 lemma extreal_minus_less:
```
```   823   fixes x y z :: extreal
```
```   824   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
```
```   825   by (auto simp: extreal_minus_less_iff)
```
```   826
```
```   827 lemma extreal_minus_le_iff:
```
```   828   fixes x y z :: extreal
```
```   829   shows "x - y \<le> z \<longleftrightarrow>
```
```   830     (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
```
```   831     (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
```
```   832     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
```
```   833   by (cases rule: extreal3_cases[of x y z]) auto
```
```   834
```
```   835 lemma extreal_minus_le:
```
```   836   fixes x y z :: extreal
```
```   837   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
```
```   838   by (auto simp: extreal_minus_le_iff)
```
```   839
```
```   840 lemma extreal_minus_eq_minus_iff:
```
```   841   fixes a b c :: extreal
```
```   842   shows "a - b = a - c \<longleftrightarrow>
```
```   843     b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
```
```   844   by (cases rule: extreal3_cases[of a b c]) auto
```
```   845
```
```   846 lemma extreal_add_le_add_iff:
```
```   847   "c + a \<le> c + b \<longleftrightarrow>
```
```   848     a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
```
```   849   by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
```
```   850
```
```   851 lemma extreal_mult_le_mult_iff:
```
```   852   "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```   853   by (cases rule: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
```
```   854
```
```   855 lemma extreal_minus_mono:
```
```   856   fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C"
```
```   857   shows "A - C \<le> B - D"
```
```   858   using assms
```
```   859   by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all
```
```   860
```
```   861 lemma real_of_extreal_minus:
```
```   862   "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
```
```   863   by (cases rule: extreal2_cases[of a b]) auto
```
```   864
```
```   865 lemma extreal_diff_positive:
```
```   866   fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
```
```   867   by (cases rule: extreal2_cases[of a b]) auto
```
```   868
```
```   869 lemma extreal_between:
```
```   870   fixes x e :: extreal
```
```   871   assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
```
```   872   shows "x - e < x" "x < x + e"
```
```   873 using assms apply (cases x, cases e) apply auto
```
```   874 using assms by (cases x, cases e) auto
```
```   875
```
```   876 subsubsection {* Division *}
```
```   877
```
```   878 instantiation extreal :: inverse
```
```   879 begin
```
```   880
```
```   881 function inverse_extreal where
```
```   882 "inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" |
```
```   883 "inverse \<infinity> = 0" |
```
```   884 "inverse (-\<infinity>) = 0"
```
```   885   by (auto intro: extreal_cases)
```
```   886 termination by (relation "{}") simp
```
```   887
```
```   888 definition "x / y = x * inverse (y :: extreal)"
```
```   889
```
```   890 instance proof qed
```
```   891 end
```
```   892
```
```   893 lemma extreal_inverse[simp]:
```
```   894   "inverse 0 = \<infinity>"
```
```   895   "inverse (1::extreal) = 1"
```
```   896   by (simp_all add: one_extreal_def zero_extreal_def)
```
```   897
```
```   898 lemma extreal_divide[simp]:
```
```   899   "extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))"
```
```   900   unfolding divide_extreal_def by (auto simp: divide_real_def)
```
```   901
```
```   902 lemma extreal_divide_same[simp]:
```
```   903   "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
```
```   904   by (cases x)
```
```   905      (simp_all add: divide_real_def divide_extreal_def one_extreal_def)
```
```   906
```
```   907 lemma extreal_inv_inv[simp]:
```
```   908   "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
```
```   909   by (cases x) auto
```
```   910
```
```   911 lemma extreal_inverse_minus[simp]:
```
```   912   "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
```
```   913   by (cases x) simp_all
```
```   914
```
```   915 lemma extreal_uminus_divide[simp]:
```
```   916   fixes x y :: extreal shows "- x / y = - (x / y)"
```
```   917   unfolding divide_extreal_def by simp
```
```   918
```
```   919 lemma extreal_divide_Infty[simp]:
```
```   920   "x / \<infinity> = 0" "x / -\<infinity> = 0"
```
```   921   unfolding divide_extreal_def by simp_all
```
```   922
```
```   923 lemma extreal_divide_one[simp]:
```
```   924   "x / 1 = (x::extreal)"
```
```   925   unfolding divide_extreal_def by simp
```
```   926
```
```   927 lemma extreal_divide_extreal[simp]:
```
```   928   "\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
```
```   929   unfolding divide_extreal_def by simp
```
```   930
```
```   931 lemma zero_le_divide_extreal[simp]:
```
```   932   fixes a :: extreal assumes "0 \<le> a" "0 \<le> b"
```
```   933   shows "0 \<le> a / b"
```
```   934   using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff)
```
```   935
```
```   936 lemma extreal_le_divide_pos:
```
```   937   "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
```
```   938   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```   939
```
```   940 lemma extreal_divide_le_pos:
```
```   941   "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
```
```   942   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```   943
```
```   944 lemma extreal_le_divide_neg:
```
```   945   "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
```
```   946   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```   947
```
```   948 lemma extreal_divide_le_neg:
```
```   949   "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
```
```   950   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```   951
```
```   952 lemma extreal_inverse_antimono_strict:
```
```   953   fixes x y :: extreal
```
```   954   shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
```
```   955   by (cases rule: extreal2_cases[of x y]) auto
```
```   956
```
```   957 lemma extreal_inverse_antimono:
```
```   958   fixes x y :: extreal
```
```   959   shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
```
```   960   by (cases rule: extreal2_cases[of x y]) auto
```
```   961
```
```   962 lemma inverse_inverse_Pinfty_iff[simp]:
```
```   963   "inverse x = \<infinity> \<longleftrightarrow> x = 0"
```
```   964   by (cases x) auto
```
```   965
```
```   966 lemma extreal_inverse_eq_0:
```
```   967   "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
```
```   968   by (cases x) auto
```
```   969
```
```   970 lemma extreal_0_gt_inverse:
```
```   971   fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
```
```   972   by (cases x) auto
```
```   973
```
```   974 lemma extreal_mult_less_right:
```
```   975   assumes "b * a < c * a" "0 < a" "a < \<infinity>"
```
```   976   shows "b < c"
```
```   977   using assms
```
```   978   by (cases rule: extreal3_cases[of a b c])
```
```   979      (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
```
```   980
```
```   981 lemma extreal_power_divide:
```
```   982   "y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n"
```
```   983   by (cases rule: extreal2_cases[of x y])
```
```   984      (auto simp: one_extreal_def zero_extreal_def power_divide not_le
```
```   985                  power_less_zero_eq zero_le_power_iff)
```
```   986
```
```   987 lemma extreal_le_mult_one_interval:
```
```   988   fixes x y :: extreal
```
```   989   assumes y: "y \<noteq> -\<infinity>"
```
```   990   assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```   991   shows "x \<le> y"
```
```   992 proof (cases x)
```
```   993   case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_extreal_def)
```
```   994 next
```
```   995   case (real r) note r = this
```
```   996   show "x \<le> y"
```
```   997   proof (cases y)
```
```   998     case (real p) note p = this
```
```   999     have "r \<le> p"
```
```  1000     proof (rule field_le_mult_one_interval)
```
```  1001       fix z :: real assume "0 < z" and "z < 1"
```
```  1002       with z[of "extreal z"]
```
```  1003       show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def)
```
```  1004     qed
```
```  1005     then show "x \<le> y" using p r by simp
```
```  1006   qed (insert y, simp_all)
```
```  1007 qed simp
```
```  1008
```
```  1009 subsection "Complete lattice"
```
```  1010
```
```  1011 instantiation extreal :: lattice
```
```  1012 begin
```
```  1013 definition [simp]: "sup x y = (max x y :: extreal)"
```
```  1014 definition [simp]: "inf x y = (min x y :: extreal)"
```
```  1015 instance proof qed simp_all
```
```  1016 end
```
```  1017
```
```  1018 instantiation extreal :: complete_lattice
```
```  1019 begin
```
```  1020
```
```  1021 definition "bot = -\<infinity>"
```
```  1022 definition "top = \<infinity>"
```
```  1023
```
```  1024 definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)"
```
```  1025 definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)"
```
```  1026
```
```  1027 lemma extreal_complete_Sup:
```
```  1028   fixes S :: "extreal set" assumes "S \<noteq> {}"
```
```  1029   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
```
```  1030 proof cases
```
```  1031   assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x"
```
```  1032   then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto
```
```  1033   then have "\<infinity> \<notin> S" by force
```
```  1034   show ?thesis
```
```  1035   proof cases
```
```  1036     assume "S = {-\<infinity>}"
```
```  1037     then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
```
```  1038   next
```
```  1039     assume "S \<noteq> {-\<infinity>}"
```
```  1040     with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
```
```  1041     with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
```
```  1042       by (auto simp: real_of_extreal_ord_simps)
```
```  1043     with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
```
```  1044     obtain s where s:
```
```  1045        "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
```
```  1046        by auto
```
```  1047     show ?thesis
```
```  1048     proof (safe intro!: exI[of _ "extreal s"])
```
```  1049       fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s"
```
```  1050       proof (cases z)
```
```  1051         case (real r)
```
```  1052         then show ?thesis
```
```  1053           using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto
```
```  1054       qed auto
```
```  1055     next
```
```  1056       fix z assume *: "\<forall>y\<in>S. y \<le> z"
```
```  1057       with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z"
```
```  1058       proof (cases z)
```
```  1059         case (real u)
```
```  1060         with * have "s \<le> u"
```
```  1061           by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps)
```
```  1062         then show ?thesis using real by simp
```
```  1063       qed auto
```
```  1064     qed
```
```  1065   qed
```
```  1066 next
```
```  1067   assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)"
```
```  1068   show ?thesis
```
```  1069   proof (safe intro!: exI[of _ \<infinity>])
```
```  1070     fix y assume **: "\<forall>z\<in>S. z \<le> y"
```
```  1071     with * show "\<infinity> \<le> y"
```
```  1072     proof (cases y)
```
```  1073       case MInf with * ** show ?thesis by (force simp: not_le)
```
```  1074     qed auto
```
```  1075   qed simp
```
```  1076 qed
```
```  1077
```
```  1078 lemma extreal_complete_Inf:
```
```  1079   fixes S :: "extreal set" assumes "S ~= {}"
```
```  1080   shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
```
```  1081 proof-
```
```  1082 def S1 == "uminus ` S"
```
```  1083 hence "S1 ~= {}" using assms by auto
```
```  1084 from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
```
```  1085    using extreal_complete_Sup[of S1] by auto
```
```  1086 { fix z assume "ALL y:S. z <= y"
```
```  1087   hence "ALL y:S1. y <= -z" unfolding S1_def by auto
```
```  1088   hence "x <= -z" using x_def by auto
```
```  1089   hence "z <= -x"
```
```  1090     apply (subst extreal_uminus_uminus[symmetric])
```
```  1091     unfolding extreal_minus_le_minus . }
```
```  1092 moreover have "(ALL y:S. -x <= y)"
```
```  1093    using x_def unfolding S1_def
```
```  1094    apply simp
```
```  1095    apply (subst (3) extreal_uminus_uminus[symmetric])
```
```  1096    unfolding extreal_minus_le_minus by simp
```
```  1097 ultimately show ?thesis by auto
```
```  1098 qed
```
```  1099
```
```  1100 lemma extreal_complete_uminus_eq:
```
```  1101   fixes S :: "extreal set"
```
```  1102   shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
```
```  1103      \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
```
```  1104   by simp (metis extreal_minus_le_minus extreal_uminus_uminus)
```
```  1105
```
```  1106 lemma extreal_Sup_uminus_image_eq:
```
```  1107   fixes S :: "extreal set"
```
```  1108   shows "Sup (uminus ` S) = - Inf S"
```
```  1109 proof cases
```
```  1110   assume "S = {}"
```
```  1111   moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)"
```
```  1112     by (rule the_equality) (auto intro!: extreal_bot)
```
```  1113   moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)"
```
```  1114     by (rule some_equality) (auto intro!: extreal_top)
```
```  1115   ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def
```
```  1116     Least_def Greatest_def GreatestM_def by simp
```
```  1117 next
```
```  1118   assume "S \<noteq> {}"
```
```  1119   with extreal_complete_Sup[of "uminus`S"]
```
```  1120   obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
```
```  1121     unfolding extreal_complete_uminus_eq by auto
```
```  1122   show "Sup (uminus ` S) = - Inf S"
```
```  1123     unfolding Inf_extreal_def Greatest_def GreatestM_def
```
```  1124   proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
```
```  1125     show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
```
```  1126       using x .
```
```  1127     fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
```
```  1128     then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)"
```
```  1129       unfolding extreal_complete_uminus_eq by simp
```
```  1130     then show "Sup (uminus ` S) = -x'"
```
```  1131       unfolding Sup_extreal_def extreal_uminus_eq_iff
```
```  1132       by (intro Least_equality) auto
```
```  1133   qed
```
```  1134 qed
```
```  1135
```
```  1136 instance
```
```  1137 proof
```
```  1138   { fix x :: extreal and A
```
```  1139     show "bot <= x" by (cases x) (simp_all add: bot_extreal_def)
```
```  1140     show "x <= top" by (simp add: top_extreal_def) }
```
```  1141
```
```  1142   { fix x :: extreal and A assume "x : A"
```
```  1143     with extreal_complete_Sup[of A]
```
```  1144     obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
```
```  1145     hence "x <= s" using `x : A` by auto
```
```  1146     also have "... = Sup A" using s unfolding Sup_extreal_def
```
```  1147       by (auto intro!: Least_equality[symmetric])
```
```  1148     finally show "x <= Sup A" . }
```
```  1149   note le_Sup = this
```
```  1150
```
```  1151   { fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)"
```
```  1152     show "Sup A <= x"
```
```  1153     proof (cases "A = {}")
```
```  1154       case True
```
```  1155       hence "Sup A = -\<infinity>" unfolding Sup_extreal_def
```
```  1156         by (auto intro!: Least_equality)
```
```  1157       thus "Sup A <= x" by simp
```
```  1158     next
```
```  1159       case False
```
```  1160       with extreal_complete_Sup[of A]
```
```  1161       obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
```
```  1162       hence "Sup A = s"
```
```  1163         unfolding Sup_extreal_def by (auto intro!: Least_equality)
```
```  1164       also have "s <= x" using * s by auto
```
```  1165       finally show "Sup A <= x" .
```
```  1166     qed }
```
```  1167   note Sup_le = this
```
```  1168
```
```  1169   { fix x :: extreal and A assume "x \<in> A"
```
```  1170     with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
```
```  1171       unfolding extreal_Sup_uminus_image_eq by simp }
```
```  1172
```
```  1173   { fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)"
```
```  1174     with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
```
```  1175       unfolding extreal_Sup_uminus_image_eq by force }
```
```  1176 qed
```
```  1177 end
```
```  1178
```
```  1179 lemma extreal_SUPR_uminus:
```
```  1180   fixes f :: "'a => extreal"
```
```  1181   shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
```
```  1182   unfolding SUPR_def INFI_def
```
```  1183   using extreal_Sup_uminus_image_eq[of "f`R"]
```
```  1184   by (simp add: image_image)
```
```  1185
```
```  1186 lemma extreal_INFI_uminus:
```
```  1187   fixes f :: "'a => extreal"
```
```  1188   shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
```
```  1189   using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
```
```  1190
```
```  1191 lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)"
```
```  1192   using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
```
```  1193
```
```  1194 lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)"
```
```  1195   by (auto intro!: inj_onI)
```
```  1196
```
```  1197 lemma extreal_image_uminus_shift:
```
```  1198   fixes X Y :: "extreal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
```
```  1199 proof
```
```  1200   assume "uminus ` X = Y"
```
```  1201   then have "uminus ` uminus ` X = uminus ` Y"
```
```  1202     by (simp add: inj_image_eq_iff)
```
```  1203   then show "X = uminus ` Y" by (simp add: image_image)
```
```  1204 qed (simp add: image_image)
```
```  1205
```
```  1206 lemma Inf_extreal_iff:
```
```  1207   fixes z :: extreal
```
```  1208   shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
```
```  1209   by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
```
```  1210             order_less_le_trans)
```
```  1211
```
```  1212 lemma Sup_eq_MInfty:
```
```  1213   fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
```
```  1214 proof
```
```  1215   assume a: "Sup S = -\<infinity>"
```
```  1216   with complete_lattice_class.Sup_upper[of _ S]
```
```  1217   show "S={} \<or> S={-\<infinity>}" by auto
```
```  1218 next
```
```  1219   assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
```
```  1220     unfolding Sup_extreal_def by (auto intro!: Least_equality)
```
```  1221 qed
```
```  1222
```
```  1223 lemma Inf_eq_PInfty:
```
```  1224   fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
```
```  1225   using Sup_eq_MInfty[of "uminus`S"]
```
```  1226   unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp
```
```  1227
```
```  1228 lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>"
```
```  1229   unfolding Inf_extreal_def
```
```  1230   by (auto intro!: Greatest_equality)
```
```  1231
```
```  1232 lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>"
```
```  1233   unfolding Sup_extreal_def
```
```  1234   by (auto intro!: Least_equality)
```
```  1235
```
```  1236 lemma extreal_SUPI:
```
```  1237   fixes x :: extreal
```
```  1238   assumes "!!i. i : A ==> f i <= x"
```
```  1239   assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
```
```  1240   shows "(SUP i:A. f i) = x"
```
```  1241   unfolding SUPR_def Sup_extreal_def
```
```  1242   using assms by (auto intro!: Least_equality)
```
```  1243
```
```  1244 lemma extreal_INFI:
```
```  1245   fixes x :: extreal
```
```  1246   assumes "!!i. i : A ==> f i >= x"
```
```  1247   assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
```
```  1248   shows "(INF i:A. f i) = x"
```
```  1249   unfolding INFI_def Inf_extreal_def
```
```  1250   using assms by (auto intro!: Greatest_equality)
```
```  1251
```
```  1252 lemma Sup_extreal_close:
```
```  1253   fixes e :: extreal
```
```  1254   assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
```
```  1255   shows "\<exists>x\<in>S. Sup S - e < x"
```
```  1256   using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
```
```  1257
```
```  1258 lemma Inf_extreal_close:
```
```  1259   fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
```
```  1260   shows "\<exists>x\<in>X. x < Inf X + e"
```
```  1261 proof (rule Inf_less_iff[THEN iffD1])
```
```  1262   show "Inf X < Inf X + e" using assms
```
```  1263     by (cases e) auto
```
```  1264 qed
```
```  1265
```
```  1266 lemma Sup_eq_top_iff:
```
```  1267   fixes A :: "'a::{complete_lattice, linorder} set"
```
```  1268   shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
```
```  1269 proof
```
```  1270   assume *: "Sup A = top"
```
```  1271   show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
```
```  1272   proof (intro allI impI)
```
```  1273     fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
```
```  1274       unfolding less_Sup_iff by auto
```
```  1275   qed
```
```  1276 next
```
```  1277   assume *: "\<forall>x<top. \<exists>i\<in>A. x < i"
```
```  1278   show "Sup A = top"
```
```  1279   proof (rule ccontr)
```
```  1280     assume "Sup A \<noteq> top"
```
```  1281     with top_greatest[of "Sup A"]
```
```  1282     have "Sup A < top" unfolding le_less by auto
```
```  1283     then have "Sup A < Sup A"
```
```  1284       using * unfolding less_Sup_iff by auto
```
```  1285     then show False by auto
```
```  1286   qed
```
```  1287 qed
```
```  1288
```
```  1289 lemma SUP_eq_top_iff:
```
```  1290   fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
```
```  1291   shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
```
```  1292   unfolding SUPR_def Sup_eq_top_iff by auto
```
```  1293
```
```  1294 lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>"
```
```  1295 proof -
```
```  1296   { fix x assume "x \<noteq> \<infinity>"
```
```  1297     then have "\<exists>k::nat. x < extreal (real k)"
```
```  1298     proof (cases x)
```
```  1299       case MInf then show ?thesis by (intro exI[of _ 0]) auto
```
```  1300     next
```
```  1301       case (real r)
```
```  1302       moreover obtain k :: nat where "r < real k"
```
```  1303         using ex_less_of_nat by (auto simp: real_eq_of_nat)
```
```  1304       ultimately show ?thesis by auto
```
```  1305     qed simp }
```
```  1306   then show ?thesis
```
```  1307     using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"]
```
```  1308     by (auto simp: top_extreal_def)
```
```  1309 qed
```
```  1310
```
```  1311 lemma extreal_le_Sup:
```
```  1312   fixes x :: extreal
```
```  1313   shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
```
```  1314 (is "?lhs <-> ?rhs")
```
```  1315 proof-
```
```  1316 { assume "?rhs"
```
```  1317   { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
```
```  1318     from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto
```
```  1319     from this obtain i where "i : A & y <= f i" using `?rhs` by auto
```
```  1320     hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto
```
```  1321     hence False using y_def by auto
```
```  1322   } hence "?lhs" by auto
```
```  1323 }
```
```  1324 moreover
```
```  1325 { assume "?lhs" hence "?rhs"
```
```  1326   by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff
```
```  1327       inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
```
```  1328 } ultimately show ?thesis by auto
```
```  1329 qed
```
```  1330
```
```  1331 lemma extreal_Inf_le:
```
```  1332   fixes x :: extreal
```
```  1333   shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
```
```  1334 (is "?lhs <-> ?rhs")
```
```  1335 proof-
```
```  1336 { assume "?rhs"
```
```  1337   { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
```
```  1338     from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto
```
```  1339     from this obtain i where "i : A & f i <= y" using `?rhs` by auto
```
```  1340     hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto
```
```  1341     hence False using y_def by auto
```
```  1342   } hence "?lhs" by auto
```
```  1343 }
```
```  1344 moreover
```
```  1345 { assume "?lhs" hence "?rhs"
```
```  1346   by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff
```
```  1347       inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
```
```  1348 } ultimately show ?thesis by auto
```
```  1349 qed
```
```  1350
```
```  1351 lemma Inf_less:
```
```  1352   fixes x :: extreal
```
```  1353   assumes "(INF i:A. f i) < x"
```
```  1354   shows "EX i. i : A & f i <= x"
```
```  1355 proof(rule ccontr)
```
```  1356   assume "~ (EX i. i : A & f i <= x)"
```
```  1357   hence "ALL i:A. f i > x" by auto
```
```  1358   hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto
```
```  1359   thus False using assms by auto
```
```  1360 qed
```
```  1361
```
```  1362 lemma same_INF:
```
```  1363   assumes "ALL e:A. f e = g e"
```
```  1364   shows "(INF e:A. f e) = (INF e:A. g e)"
```
```  1365 proof-
```
```  1366 have "f ` A = g ` A" unfolding image_def using assms by auto
```
```  1367 thus ?thesis unfolding INFI_def by auto
```
```  1368 qed
```
```  1369
```
```  1370 lemma same_SUP:
```
```  1371   assumes "ALL e:A. f e = g e"
```
```  1372   shows "(SUP e:A. f e) = (SUP e:A. g e)"
```
```  1373 proof-
```
```  1374 have "f ` A = g ` A" unfolding image_def using assms by auto
```
```  1375 thus ?thesis unfolding SUPR_def by auto
```
```  1376 qed
```
```  1377
```
```  1378 lemma SUPR_eq:
```
```  1379   assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
```
```  1380   assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
```
```  1381   shows "(SUP i:A. f i) = (SUP j:B. g j)"
```
```  1382 proof (intro antisym)
```
```  1383   show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
```
```  1384     using assms by (metis SUP_leI le_SUPI2)
```
```  1385   show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
```
```  1386     using assms by (metis SUP_leI le_SUPI2)
```
```  1387 qed
```
```  1388
```
```  1389 lemma SUP_extreal_le_addI:
```
```  1390   assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
```
```  1391   shows "SUPR UNIV f + y \<le> z"
```
```  1392 proof (cases y)
```
```  1393   case (real r)
```
```  1394   then have "\<And>i. f i \<le> z - y" using assms by (simp add: extreal_le_minus_iff)
```
```  1395   then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI)
```
```  1396   then show ?thesis using real by (simp add: extreal_le_minus_iff)
```
```  1397 qed (insert assms, auto)
```
```  1398
```
```  1399 lemma SUPR_extreal_add:
```
```  1400   fixes f g :: "nat \<Rightarrow> extreal"
```
```  1401   assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
```
```  1402   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
```
```  1403 proof (rule extreal_SUPI)
```
```  1404   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
```
```  1405   have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
```
```  1406     unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD)
```
```  1407   { fix j
```
```  1408     { fix i
```
```  1409       have "f i + g j \<le> f i + g (max i j)"
```
```  1410         using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
```
```  1411       also have "\<dots> \<le> f (max i j) + g (max i j)"
```
```  1412         using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
```
```  1413       also have "\<dots> \<le> y" using * by auto
```
```  1414       finally have "f i + g j \<le> y" . }
```
```  1415     then have "SUPR UNIV f + g j \<le> y"
```
```  1416       using assms(4)[of j] by (intro SUP_extreal_le_addI) auto
```
```  1417     then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
```
```  1418   then have "SUPR UNIV g + SUPR UNIV f \<le> y"
```
```  1419     using f by (rule SUP_extreal_le_addI)
```
```  1420   then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
```
```  1421 qed (auto intro!: add_mono le_SUPI)
```
```  1422
```
```  1423 lemma SUPR_extreal_add_pos:
```
```  1424   fixes f g :: "nat \<Rightarrow> extreal"
```
```  1425   assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
```
```  1426   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
```
```  1427 proof (intro SUPR_extreal_add inc)
```
```  1428   fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
```
```  1429 qed
```
```  1430
```
```  1431 lemma SUPR_extreal_setsum:
```
```  1432   fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal"
```
```  1433   assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
```
```  1434   shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
```
```  1435 proof cases
```
```  1436   assume "finite A" then show ?thesis using assms
```
```  1437     by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos)
```
```  1438 qed simp
```
```  1439
```
```  1440 lemma SUPR_extreal_cmult:
```
```  1441   fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
```
```  1442   shows "(SUP i. c * f i) = c * SUPR UNIV f"
```
```  1443 proof (rule extreal_SUPI)
```
```  1444   fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto
```
```  1445   then show "c * f i \<le> c * SUPR UNIV f"
```
```  1446     using `0 \<le> c` by (rule extreal_mult_left_mono)
```
```  1447 next
```
```  1448   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
```
```  1449   show "c * SUPR UNIV f \<le> y"
```
```  1450   proof cases
```
```  1451     assume c: "0 < c \<and> c \<noteq> \<infinity>"
```
```  1452     with * have "SUPR UNIV f \<le> y / c"
```
```  1453       by (intro SUP_leI) (auto simp: extreal_le_divide_pos)
```
```  1454     with c show ?thesis
```
```  1455       by (auto simp: extreal_le_divide_pos)
```
```  1456   next
```
```  1457     { assume "c = \<infinity>" have ?thesis
```
```  1458       proof cases
```
```  1459         assume "\<forall>i. f i = 0"
```
```  1460         moreover then have "range f = {0}" by auto
```
```  1461         ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def)
```
```  1462       next
```
```  1463         assume "\<not> (\<forall>i. f i = 0)"
```
```  1464         then obtain i where "f i \<noteq> 0" by auto
```
```  1465         with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
```
```  1466       qed }
```
```  1467     moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
```
```  1468     ultimately show ?thesis using * `0 \<le> c` by auto
```
```  1469   qed
```
```  1470 qed
```
```  1471
```
```  1472 lemma SUP_PInfty:
```
```  1473   fixes f :: "'a \<Rightarrow> extreal"
```
```  1474   assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i"
```
```  1475   shows "(SUP i:A. f i) = \<infinity>"
```
```  1476   unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def]
```
```  1477   apply simp
```
```  1478 proof safe
```
```  1479   fix x assume "x \<noteq> \<infinity>"
```
```  1480   show "\<exists>i\<in>A. x < f i"
```
```  1481   proof (cases x)
```
```  1482     case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
```
```  1483   next
```
```  1484     case MInf with assms[of "0"] show ?thesis by force
```
```  1485   next
```
```  1486     case (real r)
```
```  1487     with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < extreal (real n)" by auto
```
```  1488     moreover from assms[of n] guess i ..
```
```  1489     ultimately show ?thesis
```
```  1490       by (auto intro!: bexI[of _ i])
```
```  1491   qed
```
```  1492 qed
```
```  1493
```
```  1494 lemma Sup_countable_SUPR:
```
```  1495   assumes "A \<noteq> {}"
```
```  1496   shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
```
```  1497 proof (cases "Sup A")
```
```  1498   case (real r)
```
```  1499   have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
```
```  1500   proof
```
```  1501     fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x"
```
```  1502       using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def)
```
```  1503     then guess x ..
```
```  1504     then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
```
```  1505       by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff)
```
```  1506   qed
```
```  1507   from choice[OF this] guess f .. note f = this
```
```  1508   have "SUPR UNIV f = Sup A"
```
```  1509   proof (rule extreal_SUPI)
```
```  1510     fix i show "f i \<le> Sup A" using f
```
```  1511       by (auto intro!: complete_lattice_class.Sup_upper)
```
```  1512   next
```
```  1513     fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
```
```  1514     show "Sup A \<le> y"
```
```  1515     proof (rule extreal_le_epsilon, intro allI impI)
```
```  1516       fix e :: extreal assume "0 < e"
```
```  1517       show "Sup A \<le> y + e"
```
```  1518       proof (cases e)
```
```  1519         case (real r)
```
```  1520         hence "0 < r" using `0 < e` by auto
```
```  1521         then obtain n ::nat where *: "1 / real n < r" "0 < n"
```
```  1522           using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
```
```  1523         have "Sup A \<le> f n + 1 / extreal (real n)" using f[THEN spec, of n] by auto
```
```  1524         also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def )
```
```  1525         with bound have "f n + 1 / extreal (real n) \<le> y + e" by (rule add_mono) simp
```
```  1526         finally show "Sup A \<le> y + e" .
```
```  1527       qed (insert `0 < e`, auto)
```
```  1528     qed
```
```  1529   qed
```
```  1530   with f show ?thesis by (auto intro!: exI[of _ f])
```
```  1531 next
```
```  1532   case PInf
```
```  1533   from `A \<noteq> {}` obtain x where "x \<in> A" by auto
```
```  1534   show ?thesis
```
```  1535   proof cases
```
```  1536     assume "\<infinity> \<in> A"
```
```  1537     moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
```
```  1538     ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
```
```  1539   next
```
```  1540     assume "\<infinity> \<notin> A"
```
```  1541     have "\<exists>x\<in>A. 0 \<le> x"
```
```  1542       by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear)
```
```  1543     then obtain x where "x \<in> A" "0 \<le> x" by auto
```
```  1544     have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f"
```
```  1545     proof (rule ccontr)
```
```  1546       assume "\<not> ?thesis"
```
```  1547       then have "\<exists>n::nat. Sup A \<le> x + extreal (real n)"
```
```  1548         by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
```
```  1549       then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
```
```  1550         by(cases x) auto
```
```  1551     qed
```
```  1552     from choice[OF this] guess f .. note f = this
```
```  1553     have "SUPR UNIV f = \<infinity>"
```
```  1554     proof (rule SUP_PInfty)
```
```  1555       fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i"
```
```  1556         using f[THEN spec, of n] `0 \<le> x`
```
```  1557         by (cases rule: extreal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
```
```  1558     qed
```
```  1559     then show ?thesis using f PInf by (auto intro!: exI[of _ f])
```
```  1560   qed
```
```  1561 next
```
```  1562   case MInf
```
```  1563   with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
```
```  1564   then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
```
```  1565 qed
```
```  1566
```
```  1567 lemma SUPR_countable_SUPR:
```
```  1568   "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
```
```  1569   using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def)
```
```  1570
```
```  1571
```
```  1572 lemma Sup_extreal_cadd:
```
```  1573   fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1574   shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
```
```  1575 proof (rule antisym)
```
```  1576   have *: "\<And>a::extreal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
```
```  1577     by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
```
```  1578   then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
```
```  1579   show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
```
```  1580   proof (cases a)
```
```  1581     case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant)
```
```  1582   next
```
```  1583     case (real r)
```
```  1584     then have **: "op + (- a) ` op + a ` A = A"
```
```  1585       by (auto simp: image_iff ac_simps zero_extreal_def[symmetric])
```
```  1586     from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
```
```  1587       by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
```
```  1588   qed (insert `a \<noteq> -\<infinity>`, auto)
```
```  1589 qed
```
```  1590
```
```  1591 lemma Sup_extreal_cminus:
```
```  1592   fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1593   shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
```
```  1594   using Sup_extreal_cadd[of "uminus ` A" a] assms
```
```  1595   by (simp add: comp_def image_image minus_extreal_def
```
```  1596                  extreal_Sup_uminus_image_eq)
```
```  1597
```
```  1598 lemma SUPR_extreal_cminus:
```
```  1599   fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1600   shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
```
```  1601   using Sup_extreal_cminus[of "f`A" a] assms
```
```  1602   unfolding SUPR_def INFI_def image_image by auto
```
```  1603
```
```  1604 lemma Inf_extreal_cminus:
```
```  1605   fixes A :: "extreal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
```
```  1606   shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
```
```  1607 proof -
```
```  1608   { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto }
```
```  1609   moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
```
```  1610     by (auto simp: image_image)
```
```  1611   ultimately show ?thesis
```
```  1612     using Sup_extreal_cminus[of "uminus ` A" "-a"] assms
```
```  1613     by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq)
```
```  1614 qed
```
```  1615
```
```  1616 lemma INFI_extreal_cminus:
```
```  1617   fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
```
```  1618   shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
```
```  1619   using Inf_extreal_cminus[of "f`A" a] assms
```
```  1620   unfolding SUPR_def INFI_def image_image
```
```  1621   by auto
```
```  1622
```
```  1623 subsection "Limits on @{typ extreal}"
```
```  1624
```
```  1625 subsubsection "Topological space"
```
```  1626
```
```  1627 instantiation extreal :: topological_space
```
```  1628 begin
```
```  1629
```
```  1630 definition "open A \<longleftrightarrow> open (extreal -` A)
```
```  1631        \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A))
```
```  1632        \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))"
```
```  1633
```
```  1634 lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)"
```
```  1635   unfolding open_extreal_def by auto
```
```  1636
```
```  1637 lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)"
```
```  1638   unfolding open_extreal_def by auto
```
```  1639
```
```  1640 lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A"
```
```  1641   using open_PInfty[OF assms] by auto
```
```  1642
```
```  1643 lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A"
```
```  1644   using open_MInfty[OF assms] by auto
```
```  1645
```
```  1646 lemma extreal_openE: assumes "open A" obtains x y where
```
```  1647   "open (extreal -` A)"
```
```  1648   "\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A"
```
```  1649   "-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A"
```
```  1650   using assms open_extreal_def by auto
```
```  1651
```
```  1652 instance
```
```  1653 proof
```
```  1654   let ?U = "UNIV::extreal set"
```
```  1655   show "open ?U" unfolding open_extreal_def
```
```  1656     by (auto intro!: exI[of _ 0])
```
```  1657 next
```
```  1658   fix S T::"extreal set" assume "open S" and "open T"
```
```  1659   from `open S`[THEN extreal_openE] guess xS yS .
```
```  1660   moreover from `open T`[THEN extreal_openE] guess xT yT .
```
```  1661   ultimately have
```
```  1662     "open (extreal -` (S \<inter> T))"
```
```  1663     "\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T"
```
```  1664     "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T"
```
```  1665     by auto
```
```  1666   then show "open (S Int T)" unfolding open_extreal_def by blast
```
```  1667 next
```
```  1668   fix K :: "extreal set set" assume "\<forall>S\<in>K. open S"
```
```  1669   then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and>
```
```  1670     (\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)"
```
```  1671     by (auto simp: open_extreal_def)
```
```  1672   then show "open (Union K)" unfolding open_extreal_def
```
```  1673   proof (intro conjI impI)
```
```  1674     show "open (extreal -` \<Union>K)"
```
```  1675       using *[THEN choice] by (auto simp: vimage_Union)
```
```  1676   qed ((metis UnionE Union_upper subset_trans *)+)
```
```  1677 qed
```
```  1678 end
```
```  1679
```
```  1680 lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)"
```
```  1681   by (auto simp: inj_vimage_image_eq open_extreal_def)
```
```  1682
```
```  1683 lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)"
```
```  1684   unfolding open_extreal_def by auto
```
```  1685
```
```  1686 lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}"
```
```  1687 proof -
```
```  1688   have "\<And>x. extreal -` {..<extreal x} = {..< x}"
```
```  1689     "extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto
```
```  1690   then show ?thesis by (cases a) (auto simp: open_extreal_def)
```
```  1691 qed
```
```  1692
```
```  1693 lemma open_extreal_greaterThan[intro, simp]:
```
```  1694   "open {a :: extreal <..}"
```
```  1695 proof -
```
```  1696   have "\<And>x. extreal -` {extreal x<..} = {x<..}"
```
```  1697     "extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto
```
```  1698   then show ?thesis by (cases a) (auto simp: open_extreal_def)
```
```  1699 qed
```
```  1700
```
```  1701 lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}"
```
```  1702   unfolding greaterThanLessThan_def by auto
```
```  1703
```
```  1704 lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}"
```
```  1705 proof -
```
```  1706   have "- {a ..} = {..< a}" by auto
```
```  1707   then show "closed {a ..}"
```
```  1708     unfolding closed_def using open_extreal_lessThan by auto
```
```  1709 qed
```
```  1710
```
```  1711 lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}"
```
```  1712 proof -
```
```  1713   have "- {.. b} = {b <..}" by auto
```
```  1714   then show "closed {.. b}"
```
```  1715     unfolding closed_def using open_extreal_greaterThan by auto
```
```  1716 qed
```
```  1717
```
```  1718 lemma closed_extreal_atLeastAtMost[simp, intro]:
```
```  1719   shows "closed {a :: extreal .. b}"
```
```  1720   unfolding atLeastAtMost_def by auto
```
```  1721
```
```  1722 lemma closed_extreal_singleton:
```
```  1723   "closed {a :: extreal}"
```
```  1724 by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost)
```
```  1725
```
```  1726 lemma extreal_open_cont_interval:
```
```  1727   assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  1728   obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
```
```  1729 proof-
```
```  1730   from `open S` have "open (extreal -` S)" by (rule extreal_openE)
```
```  1731   then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S"
```
```  1732     using assms unfolding open_dist by force
```
```  1733   show thesis
```
```  1734   proof (intro that subsetI)
```
```  1735     show "0 < extreal e" using `0 < e` by auto
```
```  1736     fix y assume "y \<in> {x - extreal e<..<x + extreal e}"
```
```  1737     with assms obtain t where "y = extreal t" "dist t (real x) < e"
```
```  1738       apply (cases y) by (auto simp: dist_real_def)
```
```  1739     then show "y \<in> S" using e[of t] by auto
```
```  1740   qed
```
```  1741 qed
```
```  1742
```
```  1743 lemma extreal_open_cont_interval2:
```
```  1744   assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  1745   obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
```
```  1746 proof-
```
```  1747   guess e using extreal_open_cont_interval[OF assms] .
```
```  1748   with that[of "x-e" "x+e"] extreal_between[OF x, of e]
```
```  1749   show thesis by auto
```
```  1750 qed
```
```  1751
```
```  1752 instance extreal :: t2_space
```
```  1753 proof
```
```  1754   fix x y :: extreal assume "x ~= y"
```
```  1755   let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
```
```  1756
```
```  1757   { fix x y :: extreal assume "x < y"
```
```  1758     from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
```
```  1759     have "?P x y"
```
```  1760       apply (rule exI[of _ "{..<z}"])
```
```  1761       apply (rule exI[of _ "{z<..}"])
```
```  1762       using z by auto }
```
```  1763   note * = this
```
```  1764
```
```  1765   from `x ~= y`
```
```  1766   show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
```
```  1767   proof (cases rule: linorder_cases)
```
```  1768     assume "x = y" with `x ~= y` show ?thesis by simp
```
```  1769   next assume "x < y" from *[OF this] show ?thesis by auto
```
```  1770   next assume "y < x" from *[OF this] show ?thesis by auto
```
```  1771   qed
```
```  1772 qed
```
```  1773
```
```  1774 subsubsection {* Convergent sequences *}
```
```  1775
```
```  1776 lemma lim_extreal[simp]:
```
```  1777   "((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
```
```  1778 proof (intro iffI topological_tendstoI)
```
```  1779   fix S assume "?l" "open S" "x \<in> S"
```
```  1780   then show "eventually (\<lambda>x. f x \<in> S) net"
```
```  1781     using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`]
```
```  1782     by (simp add: inj_image_mem_iff)
```
```  1783 next
```
```  1784   fix S assume "?r" "open S" "extreal x \<in> S"
```
```  1785   show "eventually (\<lambda>x. extreal (f x) \<in> S) net"
```
```  1786     using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`]
```
```  1787     using `extreal x \<in> S` by auto
```
```  1788 qed
```
```  1789
```
```  1790 lemma lim_real_of_extreal[simp]:
```
```  1791   assumes lim: "(f ---> extreal x) net"
```
```  1792   shows "((\<lambda>x. real (f x)) ---> x) net"
```
```  1793 proof (intro topological_tendstoI)
```
```  1794   fix S assume "open S" "x \<in> S"
```
```  1795   then have S: "open S" "extreal x \<in> extreal ` S"
```
```  1796     by (simp_all add: inj_image_mem_iff)
```
```  1797   have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto
```
```  1798   from this lim[THEN topological_tendstoD, OF open_extreal, OF S]
```
```  1799   show "eventually (\<lambda>x. real (f x) \<in> S) net"
```
```  1800     by (rule eventually_mono)
```
```  1801 qed
```
```  1802
```
```  1803 lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r")
```
```  1804 proof assume ?r show ?l apply(rule topological_tendstoI)
```
```  1805     unfolding eventually_sequentially
```
```  1806   proof- fix S assume "open S" "\<infinity> : S"
```
```  1807     from open_PInfty[OF this] guess B .. note B=this
```
```  1808     from `?r`[rule_format,of "B+1"] guess N .. note N=this
```
```  1809     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
```
```  1810     proof safe case goal1
```
```  1811       have "extreal B < extreal (B + 1)" by auto
```
```  1812       also have "... <= f n" using goal1 N by auto
```
```  1813       finally show ?case using B by fastsimp
```
```  1814     qed
```
```  1815   qed
```
```  1816 next assume ?l show ?r
```
```  1817   proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto
```
```  1818     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
```
```  1819     guess N .. note N=this
```
```  1820     show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto
```
```  1821   qed
```
```  1822 qed
```
```  1823
```
```  1824
```
```  1825 lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r")
```
```  1826 proof assume ?r show ?l apply(rule topological_tendstoI)
```
```  1827     unfolding eventually_sequentially
```
```  1828   proof- fix S assume "open S" "(-\<infinity>) : S"
```
```  1829     from open_MInfty[OF this] guess B .. note B=this
```
```  1830     from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
```
```  1831     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
```
```  1832     proof safe case goal1
```
```  1833       have "extreal (B - 1) >= f n" using goal1 N by auto
```
```  1834       also have "... < extreal B" by auto
```
```  1835       finally show ?case using B by fastsimp
```
```  1836     qed
```
```  1837   qed
```
```  1838 next assume ?l show ?r
```
```  1839   proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto
```
```  1840     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
```
```  1841     guess N .. note N=this
```
```  1842     show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto
```
```  1843   qed
```
```  1844 qed
```
```  1845
```
```  1846
```
```  1847 lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>"
```
```  1848 proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
```
```  1849   from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
```
```  1850   guess N .. note N=this[rule_format,OF le_refl]
```
```  1851   hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans)
```
```  1852   hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto
```
```  1853   thus False by auto
```
```  1854 qed
```
```  1855
```
```  1856
```
```  1857 lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)"
```
```  1858 proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
```
```  1859   from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
```
```  1860   guess N .. note N=this[rule_format,OF le_refl]
```
```  1861   hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast
```
```  1862   thus False by auto
```
```  1863 qed
```
```  1864
```
```  1865
```
```  1866 lemma tendsto_explicit:
```
```  1867   "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
```
```  1868   unfolding tendsto_def eventually_sequentially by auto
```
```  1869
```
```  1870
```
```  1871 lemma tendsto_obtains_N:
```
```  1872   assumes "f ----> f0"
```
```  1873   assumes "open S" "f0 : S"
```
```  1874   obtains N where "ALL n>=N. f n : S"
```
```  1875   using tendsto_explicit[of f f0] assms by auto
```
```  1876
```
```  1877
```
```  1878 lemma tail_same_limit:
```
```  1879   fixes X Y N
```
```  1880   assumes "X ----> L" "ALL n>=N. X n = Y n"
```
```  1881   shows "Y ----> L"
```
```  1882 proof-
```
```  1883 { fix S assume "open S" and "L:S"
```
```  1884   from this obtain N1 where "ALL n>=N1. X n : S"
```
```  1885      using assms unfolding tendsto_def eventually_sequentially by auto
```
```  1886   hence "ALL n>=max N N1. Y n : S" using assms by auto
```
```  1887   hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
```
```  1888 }
```
```  1889 thus ?thesis using tendsto_explicit by auto
```
```  1890 qed
```
```  1891
```
```  1892
```
```  1893 lemma Lim_bounded_PInfty2:
```
```  1894 assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B"
```
```  1895 shows "l ~= \<infinity>"
```
```  1896 proof-
```
```  1897   def g == "(%n. if n>=N then f n else extreal B)"
```
```  1898   hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
```
```  1899   moreover have "!!n. g n <= extreal B" using g_def assms by auto
```
```  1900   ultimately show ?thesis using  Lim_bounded_PInfty by auto
```
```  1901 qed
```
```  1902
```
```  1903 lemma Lim_bounded_extreal:
```
```  1904   assumes lim:"f ----> (l :: extreal)"
```
```  1905   and "ALL n>=M. f n <= C"
```
```  1906   shows "l<=C"
```
```  1907 proof-
```
```  1908 { assume "l=(-\<infinity>)" hence ?thesis by auto }
```
```  1909 moreover
```
```  1910 { assume "~(l=(-\<infinity>))"
```
```  1911   { assume "C=\<infinity>" hence ?thesis by auto }
```
```  1912   moreover
```
```  1913   { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
```
```  1914     hence "l=(-\<infinity>)" using assms
```
```  1915        tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
```
```  1916     hence ?thesis by auto }
```
```  1917   moreover
```
```  1918   { assume "EX B. C = extreal B"
```
```  1919     from this obtain B where B_def: "C=extreal B" by auto
```
```  1920     hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
```
```  1921     from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
```
```  1922     from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}"
```
```  1923        apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto
```
```  1924     { fix n assume "n>=N"
```
```  1925       hence "EX r. extreal r = f n" using N_def by (cases "f n") auto
```
```  1926     } from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis
```
```  1927     hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
```
```  1928     hence *: "(%n. g n) ----> m" using m_def by auto
```
```  1929     { fix n assume "n>=max N M"
```
```  1930       hence "extreal (g n) <= extreal B" using assms g_def B_def by auto
```
```  1931       hence "g n <= B" by auto
```
```  1932     } hence "EX N. ALL n>=N. g n <= B" by blast
```
```  1933     hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
```
```  1934     hence ?thesis using m_def B_def by auto
```
```  1935   } ultimately have ?thesis by (cases C) auto
```
```  1936 } ultimately show ?thesis by blast
```
```  1937 qed
```
```  1938
```
```  1939 lemma real_of_extreal_0[simp]: "real (0::extreal) = 0"
```
```  1940   unfolding real_of_extreal_def zero_extreal_def by simp
```
```  1941
```
```  1942 lemma real_of_extreal_mult[simp]:
```
```  1943   fixes a b :: extreal shows "real (a * b) = real a * real b"
```
```  1944   by (cases rule: extreal2_cases[of a b]) auto
```
```  1945
```
```  1946 lemma real_of_extreal_eq_0:
```
```  1947   "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
```
```  1948   by (cases x) auto
```
```  1949
```
```  1950 lemma tendsto_extreal_realD:
```
```  1951   fixes f :: "'a \<Rightarrow> extreal"
```
```  1952   assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net"
```
```  1953   shows "(f ---> x) net"
```
```  1954 proof (intro topological_tendstoI)
```
```  1955   fix S assume S: "open S" "x \<in> S"
```
```  1956   with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
```
```  1957   from tendsto[THEN topological_tendstoD, OF this]
```
```  1958   show "eventually (\<lambda>x. f x \<in> S) net"
```
```  1959     by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0)
```
```  1960 qed
```
```  1961
```
```  1962 lemma tendsto_extreal_realI:
```
```  1963   fixes f :: "'a \<Rightarrow> extreal"
```
```  1964   assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
```
```  1965   shows "((\<lambda>x. extreal (real (f x))) ---> x) net"
```
```  1966 proof (intro topological_tendstoI)
```
```  1967   fix S assume "open S" "x \<in> S"
```
```  1968   with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
```
```  1969   from tendsto[THEN topological_tendstoD, OF this]
```
```  1970   show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net"
```
```  1971     by (elim eventually_elim1) (auto simp: extreal_real)
```
```  1972 qed
```
```  1973
```
```  1974 lemma extreal_mult_cancel_left:
```
```  1975   fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow>
```
```  1976     ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
```
```  1977   by (cases rule: extreal3_cases[of a b c])
```
```  1978      (simp_all add: zero_less_mult_iff)
```
```  1979
```
```  1980 lemma extreal_inj_affinity:
```
```  1981   assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
```
```  1982   shows "inj_on (\<lambda>x. m * x + t) A"
```
```  1983   using assms
```
```  1984   by (cases rule: extreal2_cases[of m t])
```
```  1985      (auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left)
```
```  1986
```
```  1987 lemma extreal_PInfty_eq_plus[simp]:
```
```  1988   shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
```
```  1989   by (cases rule: extreal2_cases[of a b]) auto
```
```  1990
```
```  1991 lemma extreal_MInfty_eq_plus[simp]:
```
```  1992   shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
```
```  1993   by (cases rule: extreal2_cases[of a b]) auto
```
```  1994
```
```  1995 lemma extreal_less_divide_pos:
```
```  1996   "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
```
```  1997   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```  1998
```
```  1999 lemma extreal_divide_less_pos:
```
```  2000   "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
```
```  2001   by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
```
```  2002
```
```  2003 lemma extreal_divide_eq:
```
```  2004   "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
```
```  2005   by (cases rule: extreal3_cases[of a b c])
```
```  2006      (simp_all add: field_simps)
```
```  2007
```
```  2008 lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
```
```  2009   by (cases a) auto
```
```  2010
```
```  2011 lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
```
```  2012   by (cases x) auto
```
```  2013
```
```  2014 lemma extreal_LimI_finite:
```
```  2015   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  2016   assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
```
```  2017   shows "u ----> x"
```
```  2018 proof (rule topological_tendstoI, unfold eventually_sequentially)
```
```  2019   obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto
```
```  2020   fix S assume "open S" "x : S"
```
```  2021   then have "open (extreal -` S)" unfolding open_extreal_def by auto
```
```  2022   with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S"
```
```  2023     unfolding open_real_def rx_def by auto
```
```  2024   then obtain n where
```
```  2025     upper: "!!N. n <= N ==> u N < x + extreal r" and
```
```  2026     lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto
```
```  2027   show "EX N. ALL n>=N. u n : S"
```
```  2028   proof (safe intro!: exI[of _ n])
```
```  2029     fix N assume "n <= N"
```
```  2030     from upper[OF this] lower[OF this] assms `0 < r`
```
```  2031     have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
```
```  2032     from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto
```
```  2033     hence "rx < ra + r" and "ra < rx + r"
```
```  2034        using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
```
```  2035     hence "dist (real (u N)) rx < r"
```
```  2036       using rx_def ra_def
```
```  2037       by (auto simp: dist_real_def abs_diff_less_iff field_simps)
```
```  2038     from dist[OF this] show "u N : S" using `u N  ~: {\<infinity>, -\<infinity>}`
```
```  2039       by (auto simp: extreal_real split: split_if_asm)
```
```  2040   qed
```
```  2041 qed
```
```  2042
```
```  2043 lemma extreal_LimI_finite_iff:
```
```  2044   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  2045   shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
```
```  2046   (is "?lhs <-> ?rhs")
```
```  2047 proof
```
```  2048   assume lim: "u ----> x"
```
```  2049   { fix r assume "(r::extreal)>0"
```
```  2050     from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
```
```  2051        apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
```
```  2052        using lim extreal_between[of x r] assms `r>0` by auto
```
```  2053     hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
```
```  2054       using extreal_minus_less[of r x] by (cases r) auto
```
```  2055   } then show "?rhs" by auto
```
```  2056 next
```
```  2057   assume ?rhs then show "u ----> x"
```
```  2058     using extreal_LimI_finite[of x] assms by auto
```
```  2059 qed
```
```  2060
```
```  2061
```
```  2062 subsubsection {* @{text Liminf} and @{text Limsup} *}
```
```  2063
```
```  2064 definition
```
```  2065   "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
```
```  2066
```
```  2067 definition
```
```  2068   "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
```
```  2069
```
```  2070 lemma Liminf_Sup:
```
```  2071   fixes f :: "'a => 'b::{complete_lattice, linorder}"
```
```  2072   shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
```
```  2073   by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
```
```  2074
```
```  2075 lemma Limsup_Inf:
```
```  2076   fixes f :: "'a => 'b::{complete_lattice, linorder}"
```
```  2077   shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
```
```  2078   by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
```
```  2079
```
```  2080 lemma extreal_SupI:
```
```  2081   fixes x :: extreal
```
```  2082   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  2083   assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
```
```  2084   shows "Sup A = x"
```
```  2085   unfolding Sup_extreal_def
```
```  2086   using assms by (auto intro!: Least_equality)
```
```  2087
```
```  2088 lemma extreal_InfI:
```
```  2089   fixes x :: extreal
```
```  2090   assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
```
```  2091   assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
```
```  2092   shows "Inf A = x"
```
```  2093   unfolding Inf_extreal_def
```
```  2094   using assms by (auto intro!: Greatest_equality)
```
```  2095
```
```  2096 lemma Limsup_const:
```
```  2097   fixes c :: "'a::{complete_lattice, linorder}"
```
```  2098   assumes ntriv: "\<not> trivial_limit net"
```
```  2099   shows "Limsup net (\<lambda>x. c) = c"
```
```  2100   unfolding Limsup_Inf
```
```  2101 proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
```
```  2102   fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
```
```  2103   show "c \<le> x"
```
```  2104   proof (rule ccontr)
```
```  2105     assume "\<not> c \<le> x" then have "x < c" by auto
```
```  2106     then show False using ntriv * by (auto simp: trivial_limit_def)
```
```  2107   qed
```
```  2108 qed auto
```
```  2109
```
```  2110 lemma Liminf_const:
```
```  2111   fixes c :: "'a::{complete_lattice, linorder}"
```
```  2112   assumes ntriv: "\<not> trivial_limit net"
```
```  2113   shows "Liminf net (\<lambda>x. c) = c"
```
```  2114   unfolding Liminf_Sup
```
```  2115 proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
```
```  2116   fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
```
```  2117   show "x \<le> c"
```
```  2118   proof (rule ccontr)
```
```  2119     assume "\<not> x \<le> c" then have "c < x" by auto
```
```  2120     then show False using ntriv * by (auto simp: trivial_limit_def)
```
```  2121   qed
```
```  2122 qed auto
```
```  2123
```
```  2124 lemma mono_set:
```
```  2125   fixes S :: "('a::order) set"
```
```  2126   shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
```
```  2127   by (auto simp: mono_def mem_def)
```
```  2128
```
```  2129 lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
```
```  2130 lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
```
```  2131 lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
```
```  2132 lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
```
```  2133
```
```  2134 lemma mono_set_iff:
```
```  2135   fixes S :: "'a::{linorder,complete_lattice} set"
```
```  2136   defines "a \<equiv> Inf S"
```
```  2137   shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
```
```  2138 proof
```
```  2139   assume "mono S"
```
```  2140   then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
```
```  2141   show ?c
```
```  2142   proof cases
```
```  2143     assume "a \<in> S"
```
```  2144     show ?c
```
```  2145       using mono[OF _ `a \<in> S`]
```
```  2146       by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
```
```  2147   next
```
```  2148     assume "a \<notin> S"
```
```  2149     have "S = {a <..}"
```
```  2150     proof safe
```
```  2151       fix x assume "x \<in> S"
```
```  2152       then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
```
```  2153       then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
```
```  2154     next
```
```  2155       fix x assume "a < x"
```
```  2156       then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
```
```  2157       with mono[of y x] show "x \<in> S" by auto
```
```  2158     qed
```
```  2159     then show ?c ..
```
```  2160   qed
```
```  2161 qed auto
```
```  2162
```
```  2163 lemma lim_imp_Liminf:
```
```  2164   fixes f :: "'a \<Rightarrow> extreal"
```
```  2165   assumes ntriv: "\<not> trivial_limit net"
```
```  2166   assumes lim: "(f ---> f0) net"
```
```  2167   shows "Liminf net f = f0"
```
```  2168   unfolding Liminf_Sup
```
```  2169 proof (safe intro!: extreal_SupI)
```
```  2170   fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
```
```  2171   show "y \<le> f0"
```
```  2172   proof (rule extreal_le_extreal)
```
```  2173     fix B assume "B < y"
```
```  2174     { assume "f0 < B"
```
```  2175       then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
```
```  2176          using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
```
```  2177          by (auto intro: eventually_conj)
```
```  2178       also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
```
```  2179       finally have False using ntriv[unfolded trivial_limit_def] by auto
```
```  2180     } then show "B \<le> f0" by (metis linorder_le_less_linear)
```
```  2181   qed
```
```  2182 next
```
```  2183   fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
```
```  2184   show "f0 \<le> y"
```
```  2185   proof (safe intro!: *[rule_format])
```
```  2186     fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
```
```  2187       using lim[THEN topological_tendstoD, of "{y <..}"] by auto
```
```  2188   qed
```
```  2189 qed
```
```  2190
```
```  2191 lemma extreal_Liminf_le_Limsup:
```
```  2192   fixes f :: "'a \<Rightarrow> extreal"
```
```  2193   assumes ntriv: "\<not> trivial_limit net"
```
```  2194   shows "Liminf net f \<le> Limsup net f"
```
```  2195   unfolding Limsup_Inf Liminf_Sup
```
```  2196 proof (safe intro!: complete_lattice_class.Inf_greatest  complete_lattice_class.Sup_least)
```
```  2197   fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
```
```  2198   show "u \<le> v"
```
```  2199   proof (rule ccontr)
```
```  2200     assume "\<not> u \<le> v"
```
```  2201     then obtain t where "t < u" "v < t"
```
```  2202       using extreal_dense[of v u] by (auto simp: not_le)
```
```  2203     then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
```
```  2204       using * by (auto intro: eventually_conj)
```
```  2205     also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
```
```  2206     finally show False using ntriv by (auto simp: trivial_limit_def)
```
```  2207   qed
```
```  2208 qed
```
```  2209
```
```  2210 lemma Liminf_mono:
```
```  2211   fixes f g :: "'a => extreal"
```
```  2212   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
```
```  2213   shows "Liminf net f \<le> Liminf net g"
```
```  2214   unfolding Liminf_Sup
```
```  2215 proof (safe intro!: Sup_mono bexI)
```
```  2216   fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
```
```  2217   then have "eventually (\<lambda>x. y < f x) net" by auto
```
```  2218   then show "eventually (\<lambda>x. y < g x) net"
```
```  2219     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
```
```  2220 qed simp
```
```  2221
```
```  2222 lemma Liminf_eq:
```
```  2223   fixes f g :: "'a \<Rightarrow> extreal"
```
```  2224   assumes "eventually (\<lambda>x. f x = g x) net"
```
```  2225   shows "Liminf net f = Liminf net g"
```
```  2226   by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
```
```  2227
```
```  2228 lemma Liminf_mono_all:
```
```  2229   fixes f g :: "'a \<Rightarrow> extreal"
```
```  2230   assumes "\<And>x. f x \<le> g x"
```
```  2231   shows "Liminf net f \<le> Liminf net g"
```
```  2232   using assms by (intro Liminf_mono always_eventually) auto
```
```  2233
```
```  2234 lemma Limsup_mono:
```
```  2235   fixes f g :: "'a \<Rightarrow> extreal"
```
```  2236   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
```
```  2237   shows "Limsup net f \<le> Limsup net g"
```
```  2238   unfolding Limsup_Inf
```
```  2239 proof (safe intro!: Inf_mono bexI)
```
```  2240   fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
```
```  2241   then have "eventually (\<lambda>x. g x < y) net" by auto
```
```  2242   then show "eventually (\<lambda>x. f x < y) net"
```
```  2243     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
```
```  2244 qed simp
```
```  2245
```
```  2246 lemma Limsup_mono_all:
```
```  2247   fixes f g :: "'a \<Rightarrow> extreal"
```
```  2248   assumes "\<And>x. f x \<le> g x"
```
```  2249   shows "Limsup net f \<le> Limsup net g"
```
```  2250   using assms by (intro Limsup_mono always_eventually) auto
```
```  2251
```
```  2252 lemma Limsup_eq:
```
```  2253   fixes f g :: "'a \<Rightarrow> extreal"
```
```  2254   assumes "eventually (\<lambda>x. f x = g x) net"
```
```  2255   shows "Limsup net f = Limsup net g"
```
```  2256   by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
```
```  2257
```
```  2258 abbreviation "liminf \<equiv> Liminf sequentially"
```
```  2259
```
```  2260 abbreviation "limsup \<equiv> Limsup sequentially"
```
```  2261
```
```  2262 lemma (in complete_lattice) less_INFD:
```
```  2263   assumes "y < INFI A f"" i \<in> A" shows "y < f i"
```
```  2264 proof -
```
```  2265   note `y < INFI A f`
```
```  2266   also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI)
```
```  2267   finally show "y < f i" .
```
```  2268 qed
```
```  2269
```
```  2270 lemma liminf_SUPR_INFI:
```
```  2271   fixes f :: "nat \<Rightarrow> extreal"
```
```  2272   shows "liminf f = (SUP n. INF m:{n..}. f m)"
```
```  2273   unfolding Liminf_Sup eventually_sequentially
```
```  2274 proof (safe intro!: antisym complete_lattice_class.Sup_least)
```
```  2275   fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)"
```
```  2276   proof (rule extreal_le_extreal)
```
```  2277     fix y assume "y < x"
```
```  2278     with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
```
```  2279     then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
```
```  2280     also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto
```
```  2281     finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
```
```  2282   qed
```
```  2283 next
```
```  2284   show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
```
```  2285   proof (unfold SUPR_def, safe intro!: Sup_mono bexI)
```
```  2286     fix y n assume "y < INFI {n..} f"
```
```  2287     from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
```
```  2288   qed (rule order_refl)
```
```  2289 qed
```
```  2290
```
```  2291 lemma tail_same_limsup:
```
```  2292   fixes X Y :: "nat => extreal"
```
```  2293   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
```
```  2294   shows "limsup X = limsup Y"
```
```  2295   using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
```
```  2296
```
```  2297 lemma tail_same_liminf:
```
```  2298   fixes X Y :: "nat => extreal"
```
```  2299   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
```
```  2300   shows "liminf X = liminf Y"
```
```  2301   using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
```
```  2302
```
```  2303 lemma liminf_mono:
```
```  2304   fixes X Y :: "nat \<Rightarrow> extreal"
```
```  2305   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
```
```  2306   shows "liminf X \<le> liminf Y"
```
```  2307   using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
```
```  2308
```
```  2309 lemma limsup_mono:
```
```  2310   fixes X Y :: "nat => extreal"
```
```  2311   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
```
```  2312   shows "limsup X \<le> limsup Y"
```
```  2313   using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
```
```  2314
```
```  2315 declare trivial_limit_sequentially[simp]
```
```  2316
```
```  2317 lemma
```
```  2318   fixes X :: "nat \<Rightarrow> extreal"
```
```  2319   shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
```
```  2320     and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
```
```  2321   unfolding incseq_def decseq_def by auto
```
```  2322
```
```  2323 lemma liminf_bounded:
```
```  2324   fixes X Y :: "nat \<Rightarrow> extreal"
```
```  2325   assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
```
```  2326   shows "C \<le> liminf X"
```
```  2327   using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
```
```  2328
```
```  2329 lemma limsup_bounded:
```
```  2330   fixes X Y :: "nat => extreal"
```
```  2331   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
```
```  2332   shows "limsup X \<le> C"
```
```  2333   using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
```
```  2334
```
```  2335 lemma liminf_bounded_iff:
```
```  2336   fixes x :: "nat \<Rightarrow> extreal"
```
```  2337   shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
```
```  2338 proof safe
```
```  2339   fix B assume "B < C" "C \<le> liminf x"
```
```  2340   then have "B < liminf x" by auto
```
```  2341   then obtain N where "B < (INF m:{N..}. x m)"
```
```  2342     unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto
```
```  2343   from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
```
```  2344 next
```
```  2345   assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
```
```  2346   { fix B assume "B<C"
```
```  2347     then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
```
```  2348     hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto
```
```  2349     also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp
```
```  2350     finally have "B \<le> liminf x" .
```
```  2351   } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
```
```  2352 qed
```
```  2353
```
```  2354 lemma liminf_subseq_mono:
```
```  2355   fixes X :: "nat \<Rightarrow> extreal"
```
```  2356   assumes "subseq r"
```
```  2357   shows "liminf X \<le> liminf (X \<circ> r) "
```
```  2358 proof-
```
```  2359   have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
```
```  2360   proof (safe intro!: INF_mono)
```
```  2361     fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
```
```  2362       using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
```
```  2363   qed
```
```  2364   then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
```
```  2365 qed
```
```  2366
```
```  2367 lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x"
```
```  2368   using assms by auto
```
```  2369
```
```  2370 lemma extreal_le_extreal_bounded:
```
```  2371   fixes x y z :: extreal
```
```  2372   assumes "z \<le> y"
```
```  2373   assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
```
```  2374   shows "x \<le> y"
```
```  2375 proof (rule extreal_le_extreal)
```
```  2376   fix B assume "B < x"
```
```  2377   show "B \<le> y"
```
```  2378   proof cases
```
```  2379     assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
```
```  2380   next
```
```  2381     assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
```
```  2382   qed
```
```  2383 qed
```
```  2384
```
```  2385 lemma fixes x y :: extreal
```
```  2386   shows Sup_atMost[simp]: "Sup {.. y} = y"
```
```  2387     and Sup_lessThan[simp]: "Sup {..< y} = y"
```
```  2388     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
```
```  2389     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
```
```  2390     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
```
```  2391   by (auto simp: Sup_extreal_def intro!: Least_equality
```
```  2392            intro: extreal_le_extreal extreal_le_extreal_bounded[of x])
```
```  2393
```
```  2394 lemma Sup_greaterThanlessThan[simp]:
```
```  2395   fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y"
```
```  2396   unfolding Sup_extreal_def
```
```  2397 proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y])
```
```  2398   fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
```
```  2399   from extreal_dense[OF `x < y`] guess w .. note w = this
```
```  2400   with z[THEN bspec, of w] show "x \<le> z" by auto
```
```  2401 qed auto
```
```  2402
```
```  2403 lemma real_extreal_id: "real o extreal = id"
```
```  2404 proof-
```
```  2405 { fix x have "(real o extreal) x = id x" by auto }
```
```  2406 from this show ?thesis using ext by blast
```
```  2407 qed
```
```  2408
```
```  2409
```
```  2410 lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
```
```  2411 by (metis range_extreal open_extreal open_UNIV)
```
```  2412
```
```  2413 lemma extreal_le_distrib:
```
```  2414   fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
```
```  2415   by (cases rule: extreal3_cases[of a b c])
```
```  2416      (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
```
```  2417
```
```  2418 lemma extreal_pos_distrib:
```
```  2419   fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
```
```  2420   using assms by (cases rule: extreal3_cases[of a b c])
```
```  2421                  (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
```
```  2422
```
```  2423 lemma extreal_pos_le_distrib:
```
```  2424 fixes a b c :: extreal
```
```  2425 assumes "c>=0"
```
```  2426 shows "c * (a + b) <= c * a + c * b"
```
```  2427   using assms by (cases rule: extreal3_cases[of a b c])
```
```  2428                  (auto simp add: field_simps)
```
```  2429
```
```  2430 lemma extreal_max_mono:
```
```  2431   "[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d"
```
```  2432   by (metis sup_extreal_def sup_mono)
```
```  2433
```
```  2434
```
```  2435 lemma extreal_max_least:
```
```  2436   "[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
```
```  2437   by (metis sup_extreal_def sup_least)
```
```  2438
```
```  2439 end
```