src/HOL/Library/Extended_Real.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 47108 2a1953f0d20d child 50104 de19856feb54 permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Extended_Real.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_Real
```
```    11 imports Complex_Main Extended_Nat
```
```    12 begin
```
```    13
```
```    14 text {*
```
```    15
```
```    16 For more lemmas about the extended real numbers go to
```
```    17   @{file "~~/src/HOL/Multivariate_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_least SUP_upper2)
```
```    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!: INF_greatest INF_lower2)
```
```    38
```
```    39 subsection {* Definition and basic properties *}
```
```    40
```
```    41 datatype ereal = ereal real | PInfty | MInfty
```
```    42
```
```    43 instantiation ereal :: uminus
```
```    44 begin
```
```    45   fun uminus_ereal where
```
```    46     "- (ereal r) = ereal (- r)"
```
```    47   | "- PInfty = MInfty"
```
```    48   | "- MInfty = PInfty"
```
```    49   instance ..
```
```    50 end
```
```    51
```
```    52 instantiation ereal :: infinity
```
```    53 begin
```
```    54   definition "(\<infinity>::ereal) = PInfty"
```
```    55   instance ..
```
```    56 end
```
```    57
```
```    58 declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
```
```    59
```
```    60 lemma ereal_uminus_uminus[simp]:
```
```    61   fixes a :: ereal shows "- (- a) = a"
```
```    62   by (cases a) simp_all
```
```    63
```
```    64 lemma
```
```    65   shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>"
```
```    66     and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>"
```
```    67     and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)"
```
```    68     and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r"
```
```    69     and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r"
```
```    70     and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y"
```
```    71     and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z"
```
```    72   by (simp_all add: infinity_ereal_def)
```
```    73
```
```    74 declare
```
```    75   PInfty_eq_infinity[code_post]
```
```    76   MInfty_eq_minfinity[code_post]
```
```    77
```
```    78 lemma [code_unfold]:
```
```    79   "\<infinity> = PInfty"
```
```    80   "-PInfty = MInfty"
```
```    81   by simp_all
```
```    82
```
```    83 lemma inj_ereal[simp]: "inj_on ereal A"
```
```    84   unfolding inj_on_def by auto
```
```    85
```
```    86 lemma ereal_cases[case_names real PInf MInf, cases type: ereal]:
```
```    87   assumes "\<And>r. x = ereal r \<Longrightarrow> P"
```
```    88   assumes "x = \<infinity> \<Longrightarrow> P"
```
```    89   assumes "x = -\<infinity> \<Longrightarrow> P"
```
```    90   shows P
```
```    91   using assms by (cases x) auto
```
```    92
```
```    93 lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
```
```    94 lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]
```
```    95
```
```    96 lemma ereal_uminus_eq_iff[simp]:
```
```    97   fixes a b :: ereal shows "-a = -b \<longleftrightarrow> a = b"
```
```    98   by (cases rule: ereal2_cases[of a b]) simp_all
```
```    99
```
```   100 function of_ereal :: "ereal \<Rightarrow> real" where
```
```   101 "of_ereal (ereal r) = r" |
```
```   102 "of_ereal \<infinity> = 0" |
```
```   103 "of_ereal (-\<infinity>) = 0"
```
```   104   by (auto intro: ereal_cases)
```
```   105 termination proof qed (rule wf_empty)
```
```   106
```
```   107 defs (overloaded)
```
```   108   real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal"
```
```   109
```
```   110 lemma real_of_ereal[simp]:
```
```   111     "real (- x :: ereal) = - (real x)"
```
```   112     "real (ereal r) = r"
```
```   113     "real (\<infinity>::ereal) = 0"
```
```   114   by (cases x) (simp_all add: real_of_ereal_def)
```
```   115
```
```   116 lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
```
```   117 proof safe
```
```   118   fix x assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
```
```   119   then show "x = -\<infinity>" by (cases x) auto
```
```   120 qed auto
```
```   121
```
```   122 lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
```
```   123 proof safe
```
```   124   fix x :: ereal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
```
```   125 qed auto
```
```   126
```
```   127
```
```   128 instantiation ereal :: abs
```
```   129 begin
```
```   130   function abs_ereal where
```
```   131     "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
```
```   132   | "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
```
```   133   | "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
```
```   134   by (auto intro: ereal_cases)
```
```   135   termination proof qed (rule wf_empty)
```
```   136   instance ..
```
```   137 end
```
```   138
```
```   139 lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
```
```   140   by (cases x) auto
```
```   141
```
```   142 lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> \<noteq> \<infinity> ; \<And>r. x = ereal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
```
```   143   by (cases x) auto
```
```   144
```
```   145 lemma abs_ereal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::ereal\<bar>"
```
```   146   by (cases x) auto
```
```   147
```
```   148 subsubsection "Addition"
```
```   149
```
```   150 instantiation ereal :: comm_monoid_add
```
```   151 begin
```
```   152
```
```   153 definition "0 = ereal 0"
```
```   154
```
```   155 function plus_ereal where
```
```   156 "ereal r + ereal p = ereal (r + p)" |
```
```   157 "\<infinity> + a = (\<infinity>::ereal)" |
```
```   158 "a + \<infinity> = (\<infinity>::ereal)" |
```
```   159 "ereal r + -\<infinity> = - \<infinity>" |
```
```   160 "-\<infinity> + ereal p = -(\<infinity>::ereal)" |
```
```   161 "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
```
```   162 proof -
```
```   163   case (goal1 P x)
```
```   164   moreover then obtain a b where "x = (a, b)" by (cases x) auto
```
```   165   ultimately show P
```
```   166    by (cases rule: ereal2_cases[of a b]) auto
```
```   167 qed auto
```
```   168 termination proof qed (rule wf_empty)
```
```   169
```
```   170 lemma Infty_neq_0[simp]:
```
```   171   "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
```
```   172   "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
```
```   173   by (simp_all add: zero_ereal_def)
```
```   174
```
```   175 lemma ereal_eq_0[simp]:
```
```   176   "ereal r = 0 \<longleftrightarrow> r = 0"
```
```   177   "0 = ereal r \<longleftrightarrow> r = 0"
```
```   178   unfolding zero_ereal_def by simp_all
```
```   179
```
```   180 instance
```
```   181 proof
```
```   182   fix a b c :: ereal
```
```   183   show "0 + a = a"
```
```   184     by (cases a) (simp_all add: zero_ereal_def)
```
```   185   show "a + b = b + a"
```
```   186     by (cases rule: ereal2_cases[of a b]) simp_all
```
```   187   show "a + b + c = a + (b + c)"
```
```   188     by (cases rule: ereal3_cases[of a b c]) simp_all
```
```   189 qed
```
```   190 end
```
```   191
```
```   192 lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
```
```   193   unfolding real_of_ereal_def zero_ereal_def by simp
```
```   194
```
```   195 lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
```
```   196   unfolding zero_ereal_def abs_ereal.simps by simp
```
```   197
```
```   198 lemma ereal_uminus_zero[simp]:
```
```   199   "- 0 = (0::ereal)"
```
```   200   by (simp add: zero_ereal_def)
```
```   201
```
```   202 lemma ereal_uminus_zero_iff[simp]:
```
```   203   fixes a :: ereal shows "-a = 0 \<longleftrightarrow> a = 0"
```
```   204   by (cases a) simp_all
```
```   205
```
```   206 lemma ereal_plus_eq_PInfty[simp]:
```
```   207   fixes a b :: ereal shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
```
```   208   by (cases rule: ereal2_cases[of a b]) auto
```
```   209
```
```   210 lemma ereal_plus_eq_MInfty[simp]:
```
```   211   fixes a b :: ereal shows "a + b = -\<infinity> \<longleftrightarrow>
```
```   212     (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
```
```   213   by (cases rule: ereal2_cases[of a b]) auto
```
```   214
```
```   215 lemma ereal_add_cancel_left:
```
```   216   fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
```
```   217   shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
```
```   218   using assms by (cases rule: ereal3_cases[of a b c]) auto
```
```   219
```
```   220 lemma ereal_add_cancel_right:
```
```   221   fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
```
```   222   shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
```
```   223   using assms by (cases rule: ereal3_cases[of a b c]) auto
```
```   224
```
```   225 lemma ereal_real:
```
```   226   "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
```
```   227   by (cases x) simp_all
```
```   228
```
```   229 lemma real_of_ereal_add:
```
```   230   fixes a b :: ereal
```
```   231   shows "real (a + b) =
```
```   232     (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)"
```
```   233   by (cases rule: ereal2_cases[of a b]) auto
```
```   234
```
```   235 subsubsection "Linear order on @{typ ereal}"
```
```   236
```
```   237 instantiation ereal :: linorder
```
```   238 begin
```
```   239
```
```   240 function less_ereal
```
```   241 where
```
```   242   "   ereal x < ereal y     \<longleftrightarrow> x < y"
```
```   243 | "(\<infinity>::ereal) < a           \<longleftrightarrow> False"
```
```   244 | "         a < -(\<infinity>::ereal) \<longleftrightarrow> False"
```
```   245 | "ereal x    < \<infinity>           \<longleftrightarrow> True"
```
```   246 | "        -\<infinity> < ereal r     \<longleftrightarrow> True"
```
```   247 | "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
```
```   248 proof -
```
```   249   case (goal1 P x)
```
```   250   moreover then obtain a b where "x = (a,b)" by (cases x) auto
```
```   251   ultimately show P by (cases rule: ereal2_cases[of a b]) auto
```
```   252 qed simp_all
```
```   253 termination by (relation "{}") simp
```
```   254
```
```   255 definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
```
```   256
```
```   257 lemma ereal_infty_less[simp]:
```
```   258   fixes x :: ereal
```
```   259   shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
```
```   260     "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
```
```   261   by (cases x, simp_all) (cases x, simp_all)
```
```   262
```
```   263 lemma ereal_infty_less_eq[simp]:
```
```   264   fixes x :: ereal
```
```   265   shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
```
```   266   "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
```
```   267   by (auto simp add: less_eq_ereal_def)
```
```   268
```
```   269 lemma ereal_less[simp]:
```
```   270   "ereal r < 0 \<longleftrightarrow> (r < 0)"
```
```   271   "0 < ereal r \<longleftrightarrow> (0 < r)"
```
```   272   "0 < (\<infinity>::ereal)"
```
```   273   "-(\<infinity>::ereal) < 0"
```
```   274   by (simp_all add: zero_ereal_def)
```
```   275
```
```   276 lemma ereal_less_eq[simp]:
```
```   277   "x \<le> (\<infinity>::ereal)"
```
```   278   "-(\<infinity>::ereal) \<le> x"
```
```   279   "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
```
```   280   "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
```
```   281   "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
```
```   282   by (auto simp add: less_eq_ereal_def zero_ereal_def)
```
```   283
```
```   284 lemma ereal_infty_less_eq2:
```
```   285   "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
```
```   286   "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
```
```   287   by simp_all
```
```   288
```
```   289 instance
```
```   290 proof
```
```   291   fix x y z :: ereal
```
```   292   show "x \<le> x"
```
```   293     by (cases x) simp_all
```
```   294   show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```   295     by (cases rule: ereal2_cases[of x y]) auto
```
```   296   show "x \<le> y \<or> y \<le> x "
```
```   297     by (cases rule: ereal2_cases[of x y]) auto
```
```   298   { assume "x \<le> y" "y \<le> x" then show "x = y"
```
```   299     by (cases rule: ereal2_cases[of x y]) auto }
```
```   300   { assume "x \<le> y" "y \<le> z" then show "x \<le> z"
```
```   301     by (cases rule: ereal3_cases[of x y z]) auto }
```
```   302 qed
```
```   303
```
```   304 end
```
```   305
```
```   306 instance ereal :: ordered_ab_semigroup_add
```
```   307 proof
```
```   308   fix a b c :: ereal assume "a \<le> b" then show "c + a \<le> c + b"
```
```   309     by (cases rule: ereal3_cases[of a b c]) auto
```
```   310 qed
```
```   311
```
```   312 lemma real_of_ereal_positive_mono:
```
```   313   fixes x y :: ereal shows "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
```
```   314   by (cases rule: ereal2_cases[of x y]) auto
```
```   315
```
```   316 lemma ereal_MInfty_lessI[intro, simp]:
```
```   317   fixes a :: ereal shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
```
```   318   by (cases a) auto
```
```   319
```
```   320 lemma ereal_less_PInfty[intro, simp]:
```
```   321   fixes a :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
```
```   322   by (cases a) auto
```
```   323
```
```   324 lemma ereal_less_ereal_Ex:
```
```   325   fixes a b :: ereal
```
```   326   shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
```
```   327   by (cases x) auto
```
```   328
```
```   329 lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))"
```
```   330 proof (cases x)
```
```   331   case (real r) then show ?thesis
```
```   332     using reals_Archimedean2[of r] by simp
```
```   333 qed simp_all
```
```   334
```
```   335 lemma ereal_add_mono:
```
```   336   fixes a b c d :: ereal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
```
```   337   using assms
```
```   338   apply (cases a)
```
```   339   apply (cases rule: ereal3_cases[of b c d], auto)
```
```   340   apply (cases rule: ereal3_cases[of b c d], auto)
```
```   341   done
```
```   342
```
```   343 lemma ereal_minus_le_minus[simp]:
```
```   344   fixes a b :: ereal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
```
```   345   by (cases rule: ereal2_cases[of a b]) auto
```
```   346
```
```   347 lemma ereal_minus_less_minus[simp]:
```
```   348   fixes a b :: ereal shows "- a < - b \<longleftrightarrow> b < a"
```
```   349   by (cases rule: ereal2_cases[of a b]) auto
```
```   350
```
```   351 lemma ereal_le_real_iff:
```
```   352   "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
```
```   353   by (cases y) auto
```
```   354
```
```   355 lemma real_le_ereal_iff:
```
```   356   "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
```
```   357   by (cases y) auto
```
```   358
```
```   359 lemma ereal_less_real_iff:
```
```   360   "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
```
```   361   by (cases y) auto
```
```   362
```
```   363 lemma real_less_ereal_iff:
```
```   364   "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
```
```   365   by (cases y) auto
```
```   366
```
```   367 lemma real_of_ereal_pos:
```
```   368   fixes x :: ereal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
```
```   369
```
```   370 lemmas real_of_ereal_ord_simps =
```
```   371   ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
```
```   372
```
```   373 lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
```
```   374   by (cases x) auto
```
```   375
```
```   376 lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
```
```   377   by (cases x) auto
```
```   378
```
```   379 lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
```
```   380   by (cases x) auto
```
```   381
```
```   382 lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> (x \<le> 0 \<or> x = \<infinity>)"
```
```   383   by (cases x) auto
```
```   384
```
```   385 lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
```
```   386   by (cases x) auto
```
```   387
```
```   388 lemma zero_less_real_of_ereal:
```
```   389   fixes x :: ereal shows "0 < real x \<longleftrightarrow> (0 < x \<and> x \<noteq> \<infinity>)"
```
```   390   by (cases x) auto
```
```   391
```
```   392 lemma ereal_0_le_uminus_iff[simp]:
```
```   393   fixes a :: ereal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
```
```   394   by (cases rule: ereal2_cases[of a]) auto
```
```   395
```
```   396 lemma ereal_uminus_le_0_iff[simp]:
```
```   397   fixes a :: ereal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a"
```
```   398   by (cases rule: ereal2_cases[of a]) auto
```
```   399
```
```   400 lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y"
```
```   401   using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto
```
```   402
```
```   403 lemma ereal_dense:
```
```   404   fixes x y :: ereal assumes "x < y"
```
```   405   shows "\<exists>z. x < z \<and> z < y"
```
```   406   using ereal_dense2[OF `x < y`] by blast
```
```   407
```
```   408 lemma ereal_add_strict_mono:
```
```   409   fixes a b c d :: ereal
```
```   410   assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
```
```   411   shows "a + c < b + d"
```
```   412   using assms by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
```
```   413
```
```   414 lemma ereal_less_add:
```
```   415   fixes a b c :: ereal shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
```
```   416   by (cases rule: ereal2_cases[of b c]) auto
```
```   417
```
```   418 lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" by auto
```
```   419
```
```   420 lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
```
```   421   by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
```
```   422
```
```   423 lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)"
```
```   424   by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)
```
```   425
```
```   426 lemmas ereal_uminus_reorder =
```
```   427   ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder
```
```   428
```
```   429 lemma ereal_bot:
```
```   430   fixes x :: ereal assumes "\<And>B. x \<le> ereal B" shows "x = - \<infinity>"
```
```   431 proof (cases x)
```
```   432   case (real r) with assms[of "r - 1"] show ?thesis by auto
```
```   433 next
```
```   434   case PInf with assms[of 0] show ?thesis by auto
```
```   435 next
```
```   436   case MInf then show ?thesis by simp
```
```   437 qed
```
```   438
```
```   439 lemma ereal_top:
```
```   440   fixes x :: ereal assumes "\<And>B. x \<ge> ereal B" shows "x = \<infinity>"
```
```   441 proof (cases x)
```
```   442   case (real r) with assms[of "r + 1"] show ?thesis by auto
```
```   443 next
```
```   444   case MInf with assms[of 0] show ?thesis by auto
```
```   445 next
```
```   446   case PInf then show ?thesis by simp
```
```   447 qed
```
```   448
```
```   449 lemma
```
```   450   shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
```
```   451     and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"
```
```   452   by (simp_all add: min_def max_def)
```
```   453
```
```   454 lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
```
```   455   by (auto simp: zero_ereal_def)
```
```   456
```
```   457 lemma
```
```   458   fixes f :: "nat \<Rightarrow> ereal"
```
```   459   shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
```
```   460   and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
```
```   461   unfolding decseq_def incseq_def by auto
```
```   462
```
```   463 lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
```
```   464   unfolding incseq_def by auto
```
```   465
```
```   466 lemma ereal_add_nonneg_nonneg:
```
```   467   fixes a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
```
```   468   using add_mono[of 0 a 0 b] by simp
```
```   469
```
```   470 lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
```
```   471   by auto
```
```   472
```
```   473 lemma incseq_setsumI:
```
```   474   fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
```
```   475   assumes "\<And>i. 0 \<le> f i"
```
```   476   shows "incseq (\<lambda>i. setsum f {..< i})"
```
```   477 proof (intro incseq_SucI)
```
```   478   fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
```
```   479     using assms by (rule add_left_mono)
```
```   480   then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
```
```   481     by auto
```
```   482 qed
```
```   483
```
```   484 lemma incseq_setsumI2:
```
```   485   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
```
```   486   assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
```
```   487   shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
```
```   488   using assms unfolding incseq_def by (auto intro: setsum_mono)
```
```   489
```
```   490 subsubsection "Multiplication"
```
```   491
```
```   492 instantiation ereal :: "{comm_monoid_mult, sgn}"
```
```   493 begin
```
```   494
```
```   495 definition "1 = ereal 1"
```
```   496
```
```   497 function sgn_ereal where
```
```   498   "sgn (ereal r) = ereal (sgn r)"
```
```   499 | "sgn (\<infinity>::ereal) = 1"
```
```   500 | "sgn (-\<infinity>::ereal) = -1"
```
```   501 by (auto intro: ereal_cases)
```
```   502 termination proof qed (rule wf_empty)
```
```   503
```
```   504 function times_ereal where
```
```   505 "ereal r * ereal p = ereal (r * p)" |
```
```   506 "ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
```
```   507 "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
```
```   508 "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
```
```   509 "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
```
```   510 "(\<infinity>::ereal) * \<infinity> = \<infinity>" |
```
```   511 "-(\<infinity>::ereal) * \<infinity> = -\<infinity>" |
```
```   512 "(\<infinity>::ereal) * -\<infinity> = -\<infinity>" |
```
```   513 "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
```
```   514 proof -
```
```   515   case (goal1 P x)
```
```   516   moreover then obtain a b where "x = (a, b)" by (cases x) auto
```
```   517   ultimately show P by (cases rule: ereal2_cases[of a b]) auto
```
```   518 qed simp_all
```
```   519 termination by (relation "{}") simp
```
```   520
```
```   521 instance
```
```   522 proof
```
```   523   fix a b c :: ereal show "1 * a = a"
```
```   524     by (cases a) (simp_all add: one_ereal_def)
```
```   525   show "a * b = b * a"
```
```   526     by (cases rule: ereal2_cases[of a b]) simp_all
```
```   527   show "a * b * c = a * (b * c)"
```
```   528     by (cases rule: ereal3_cases[of a b c])
```
```   529        (simp_all add: zero_ereal_def zero_less_mult_iff)
```
```   530 qed
```
```   531 end
```
```   532
```
```   533 lemma real_of_ereal_le_1:
```
```   534   fixes a :: ereal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
```
```   535   by (cases a) (auto simp: one_ereal_def)
```
```   536
```
```   537 lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
```
```   538   unfolding one_ereal_def by simp
```
```   539
```
```   540 lemma ereal_mult_zero[simp]:
```
```   541   fixes a :: ereal shows "a * 0 = 0"
```
```   542   by (cases a) (simp_all add: zero_ereal_def)
```
```   543
```
```   544 lemma ereal_zero_mult[simp]:
```
```   545   fixes a :: ereal shows "0 * a = 0"
```
```   546   by (cases a) (simp_all add: zero_ereal_def)
```
```   547
```
```   548 lemma ereal_m1_less_0[simp]:
```
```   549   "-(1::ereal) < 0"
```
```   550   by (simp add: zero_ereal_def one_ereal_def)
```
```   551
```
```   552 lemma ereal_zero_m1[simp]:
```
```   553   "1 \<noteq> (0::ereal)"
```
```   554   by (simp add: zero_ereal_def one_ereal_def)
```
```   555
```
```   556 lemma ereal_times_0[simp]:
```
```   557   fixes x :: ereal shows "0 * x = 0"
```
```   558   by (cases x) (auto simp: zero_ereal_def)
```
```   559
```
```   560 lemma ereal_times[simp]:
```
```   561   "1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
```
```   562   "1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1"
```
```   563   by (auto simp add: times_ereal_def one_ereal_def)
```
```   564
```
```   565 lemma ereal_plus_1[simp]:
```
```   566   "1 + ereal r = ereal (r + 1)" "ereal r + 1 = ereal (r + 1)"
```
```   567   "1 + -(\<infinity>::ereal) = -\<infinity>" "-(\<infinity>::ereal) + 1 = -\<infinity>"
```
```   568   unfolding one_ereal_def by auto
```
```   569
```
```   570 lemma ereal_zero_times[simp]:
```
```   571   fixes a b :: ereal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
```
```   572   by (cases rule: ereal2_cases[of a b]) auto
```
```   573
```
```   574 lemma ereal_mult_eq_PInfty[simp]:
```
```   575   shows "a * b = (\<infinity>::ereal) \<longleftrightarrow>
```
```   576     (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
```
```   577   by (cases rule: ereal2_cases[of a b]) auto
```
```   578
```
```   579 lemma ereal_mult_eq_MInfty[simp]:
```
```   580   shows "a * b = -(\<infinity>::ereal) \<longleftrightarrow>
```
```   581     (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
```
```   582   by (cases rule: ereal2_cases[of a b]) auto
```
```   583
```
```   584 lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
```
```   585   by (simp_all add: zero_ereal_def one_ereal_def)
```
```   586
```
```   587 lemma ereal_zero_one[simp]: "0 \<noteq> (1::ereal)"
```
```   588   by (simp_all add: zero_ereal_def one_ereal_def)
```
```   589
```
```   590 lemma ereal_mult_minus_left[simp]:
```
```   591   fixes a b :: ereal shows "-a * b = - (a * b)"
```
```   592   by (cases rule: ereal2_cases[of a b]) auto
```
```   593
```
```   594 lemma ereal_mult_minus_right[simp]:
```
```   595   fixes a b :: ereal shows "a * -b = - (a * b)"
```
```   596   by (cases rule: ereal2_cases[of a b]) auto
```
```   597
```
```   598 lemma ereal_mult_infty[simp]:
```
```   599   "a * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
```
```   600   by (cases a) auto
```
```   601
```
```   602 lemma ereal_infty_mult[simp]:
```
```   603   "(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
```
```   604   by (cases a) auto
```
```   605
```
```   606 lemma ereal_mult_strict_right_mono:
```
```   607   assumes "a < b" and "0 < c" "c < (\<infinity>::ereal)"
```
```   608   shows "a * c < b * c"
```
```   609   using assms
```
```   610   by (cases rule: ereal3_cases[of a b c])
```
```   611      (auto simp: zero_le_mult_iff)
```
```   612
```
```   613 lemma ereal_mult_strict_left_mono:
```
```   614   "\<lbrakk> a < b ; 0 < c ; c < (\<infinity>::ereal)\<rbrakk> \<Longrightarrow> c * a < c * b"
```
```   615   using ereal_mult_strict_right_mono by (simp add: mult_commute[of c])
```
```   616
```
```   617 lemma ereal_mult_right_mono:
```
```   618   fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
```
```   619   using assms
```
```   620   apply (cases "c = 0") apply simp
```
```   621   by (cases rule: ereal3_cases[of a b c])
```
```   622      (auto simp: zero_le_mult_iff)
```
```   623
```
```   624 lemma ereal_mult_left_mono:
```
```   625   fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
```
```   626   using ereal_mult_right_mono by (simp add: mult_commute[of c])
```
```   627
```
```   628 lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
```
```   629   by (simp add: one_ereal_def zero_ereal_def)
```
```   630
```
```   631 lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)"
```
```   632   by (cases rule: ereal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
```
```   633
```
```   634 lemma ereal_right_distrib:
```
```   635   fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
```
```   636   by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
```
```   637
```
```   638 lemma ereal_left_distrib:
```
```   639   fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
```
```   640   by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)
```
```   641
```
```   642 lemma ereal_mult_le_0_iff:
```
```   643   fixes a b :: ereal
```
```   644   shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
```
```   645   by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff)
```
```   646
```
```   647 lemma ereal_zero_le_0_iff:
```
```   648   fixes a b :: ereal
```
```   649   shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
```
```   650   by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
```
```   651
```
```   652 lemma ereal_mult_less_0_iff:
```
```   653   fixes a b :: ereal
```
```   654   shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
```
```   655   by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff)
```
```   656
```
```   657 lemma ereal_zero_less_0_iff:
```
```   658   fixes a b :: ereal
```
```   659   shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
```
```   660   by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
```
```   661
```
```   662 lemma ereal_distrib:
```
```   663   fixes a b c :: ereal
```
```   664   assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
```
```   665   shows "(a + b) * c = a * c + b * c"
```
```   666   using assms
```
```   667   by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
```
```   668
```
```   669 instance ereal :: numeral ..
```
```   670
```
```   671 lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
```
```   672   apply (induct w rule: num_induct)
```
```   673   apply (simp only: numeral_One one_ereal_def)
```
```   674   apply (simp only: numeral_inc ereal_plus_1)
```
```   675   done
```
```   676
```
```   677 lemma ereal_le_epsilon:
```
```   678   fixes x y :: ereal
```
```   679   assumes "ALL e. 0 < e --> x <= y + e"
```
```   680   shows "x <= y"
```
```   681 proof-
```
```   682 { assume a: "EX r. y = ereal r"
```
```   683   then obtain r where r_def: "y = ereal r" by auto
```
```   684   { assume "x=(-\<infinity>)" hence ?thesis by auto }
```
```   685   moreover
```
```   686   { assume "~(x=(-\<infinity>))"
```
```   687     then obtain p where p_def: "x = ereal p"
```
```   688     using a assms[rule_format, of 1] by (cases x) auto
```
```   689     { fix e have "0 < e --> p <= r + e"
```
```   690       using assms[rule_format, of "ereal e"] p_def r_def by auto }
```
```   691     hence "p <= r" apply (subst field_le_epsilon) by auto
```
```   692     hence ?thesis using r_def p_def by auto
```
```   693   } ultimately have ?thesis by blast
```
```   694 }
```
```   695 moreover
```
```   696 { assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
```
```   697     using assms[rule_format, of 1] by (cases x) auto
```
```   698 } ultimately show ?thesis by (cases y) auto
```
```   699 qed
```
```   700
```
```   701
```
```   702 lemma ereal_le_epsilon2:
```
```   703   fixes x y :: ereal
```
```   704   assumes "ALL e. 0 < e --> x <= y + ereal e"
```
```   705   shows "x <= y"
```
```   706 proof-
```
```   707 { fix e :: ereal assume "e>0"
```
```   708   { assume "e=\<infinity>" hence "x<=y+e" by auto }
```
```   709   moreover
```
```   710   { assume "e~=\<infinity>"
```
```   711     then obtain r where "e = ereal r" using `e>0` apply (cases e) by auto
```
```   712     hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
```
```   713   } ultimately have "x<=y+e" by blast
```
```   714 } then show ?thesis using ereal_le_epsilon by auto
```
```   715 qed
```
```   716
```
```   717 lemma ereal_le_real:
```
```   718   fixes x y :: ereal
```
```   719   assumes "ALL z. x <= ereal z --> y <= ereal z"
```
```   720   shows "y <= x"
```
```   721 by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
```
```   722
```
```   723 lemma ereal_le_ereal:
```
```   724   fixes x y :: ereal
```
```   725   assumes "\<And>B. B < x \<Longrightarrow> B <= y"
```
```   726   shows "x <= y"
```
```   727 by (metis assms ereal_dense leD linorder_le_less_linear)
```
```   728
```
```   729 lemma ereal_ge_ereal:
```
```   730   fixes x y :: ereal
```
```   731   assumes "ALL B. B>x --> B >= y"
```
```   732   shows "x >= y"
```
```   733 by (metis assms ereal_dense leD linorder_le_less_linear)
```
```   734
```
```   735 lemma setprod_ereal_0:
```
```   736   fixes f :: "'a \<Rightarrow> ereal"
```
```   737   shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
```
```   738 proof cases
```
```   739   assume "finite A"
```
```   740   then show ?thesis by (induct A) auto
```
```   741 qed auto
```
```   742
```
```   743 lemma setprod_ereal_pos:
```
```   744   fixes f :: "'a \<Rightarrow> ereal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"
```
```   745 proof cases
```
```   746   assume "finite I" from this pos show ?thesis by induct auto
```
```   747 qed simp
```
```   748
```
```   749 lemma setprod_PInf:
```
```   750   fixes f :: "'a \<Rightarrow> ereal"
```
```   751   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
```
```   752   shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
```
```   753 proof cases
```
```   754   assume "finite I" from this assms show ?thesis
```
```   755   proof (induct I)
```
```   756     case (insert i I)
```
```   757     then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_ereal_pos)
```
```   758     from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
```
```   759     also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
```
```   760       using setprod_ereal_pos[of I f] pos
```
```   761       by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
```
```   762     also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"
```
```   763       using insert by (auto simp: setprod_ereal_0)
```
```   764     finally show ?case .
```
```   765   qed simp
```
```   766 qed simp
```
```   767
```
```   768 lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)"
```
```   769 proof cases
```
```   770   assume "finite A" then show ?thesis
```
```   771     by induct (auto simp: one_ereal_def)
```
```   772 qed (simp add: one_ereal_def)
```
```   773
```
```   774 subsubsection {* Power *}
```
```   775
```
```   776 lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
```
```   777   by (induct n) (auto simp: one_ereal_def)
```
```   778
```
```   779 lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
```
```   780   by (induct n) (auto simp: one_ereal_def)
```
```   781
```
```   782 lemma ereal_power_uminus[simp]:
```
```   783   fixes x :: ereal
```
```   784   shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
```
```   785   by (induct n) (auto simp: one_ereal_def)
```
```   786
```
```   787 lemma ereal_power_numeral[simp]:
```
```   788   "(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
```
```   789   by (induct n) (auto simp: one_ereal_def)
```
```   790
```
```   791 lemma zero_le_power_ereal[simp]:
```
```   792   fixes a :: ereal assumes "0 \<le> a"
```
```   793   shows "0 \<le> a ^ n"
```
```   794   using assms by (induct n) (auto simp: ereal_zero_le_0_iff)
```
```   795
```
```   796 subsubsection {* Subtraction *}
```
```   797
```
```   798 lemma ereal_minus_minus_image[simp]:
```
```   799   fixes S :: "ereal set"
```
```   800   shows "uminus ` uminus ` S = S"
```
```   801   by (auto simp: image_iff)
```
```   802
```
```   803 lemma ereal_uminus_lessThan[simp]:
```
```   804   fixes a :: ereal shows "uminus ` {..<a} = {-a<..}"
```
```   805 proof -
```
```   806   {
```
```   807     fix x assume "-a < x"
```
```   808     then have "- x < - (- a)" by (simp del: ereal_uminus_uminus)
```
```   809     then have "- x < a" by simp
```
```   810   }
```
```   811   then show ?thesis by (auto intro!: image_eqI)
```
```   812 qed
```
```   813
```
```   814 lemma ereal_uminus_greaterThan[simp]:
```
```   815   "uminus ` {(a::ereal)<..} = {..<-a}"
```
```   816   by (metis ereal_uminus_lessThan ereal_uminus_uminus
```
```   817             ereal_minus_minus_image)
```
```   818
```
```   819 instantiation ereal :: minus
```
```   820 begin
```
```   821 definition "x - y = x + -(y::ereal)"
```
```   822 instance ..
```
```   823 end
```
```   824
```
```   825 lemma ereal_minus[simp]:
```
```   826   "ereal r - ereal p = ereal (r - p)"
```
```   827   "-\<infinity> - ereal r = -\<infinity>"
```
```   828   "ereal r - \<infinity> = -\<infinity>"
```
```   829   "(\<infinity>::ereal) - x = \<infinity>"
```
```   830   "-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
```
```   831   "x - -y = x + y"
```
```   832   "x - 0 = x"
```
```   833   "0 - x = -x"
```
```   834   by (simp_all add: minus_ereal_def)
```
```   835
```
```   836 lemma ereal_x_minus_x[simp]:
```
```   837   "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
```
```   838   by (cases x) simp_all
```
```   839
```
```   840 lemma ereal_eq_minus_iff:
```
```   841   fixes x y z :: ereal
```
```   842   shows "x = z - y \<longleftrightarrow>
```
```   843     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
```
```   844     (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
```
```   845     (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
```
```   846     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
```
```   847   by (cases rule: ereal3_cases[of x y z]) auto
```
```   848
```
```   849 lemma ereal_eq_minus:
```
```   850   fixes x y z :: ereal
```
```   851   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
```
```   852   by (auto simp: ereal_eq_minus_iff)
```
```   853
```
```   854 lemma ereal_less_minus_iff:
```
```   855   fixes x y z :: ereal
```
```   856   shows "x < z - y \<longleftrightarrow>
```
```   857     (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
```
```   858     (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
```
```   859     (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
```
```   860   by (cases rule: ereal3_cases[of x y z]) auto
```
```   861
```
```   862 lemma ereal_less_minus:
```
```   863   fixes x y z :: ereal
```
```   864   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
```
```   865   by (auto simp: ereal_less_minus_iff)
```
```   866
```
```   867 lemma ereal_le_minus_iff:
```
```   868   fixes x y z :: ereal
```
```   869   shows "x \<le> z - y \<longleftrightarrow>
```
```   870     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
```
```   871     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
```
```   872   by (cases rule: ereal3_cases[of x y z]) auto
```
```   873
```
```   874 lemma ereal_le_minus:
```
```   875   fixes x y z :: ereal
```
```   876   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
```
```   877   by (auto simp: ereal_le_minus_iff)
```
```   878
```
```   879 lemma ereal_minus_less_iff:
```
```   880   fixes x y z :: ereal
```
```   881   shows "x - y < z \<longleftrightarrow>
```
```   882     y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
```
```   883     (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
```
```   884   by (cases rule: ereal3_cases[of x y z]) auto
```
```   885
```
```   886 lemma ereal_minus_less:
```
```   887   fixes x y z :: ereal
```
```   888   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
```
```   889   by (auto simp: ereal_minus_less_iff)
```
```   890
```
```   891 lemma ereal_minus_le_iff:
```
```   892   fixes x y z :: ereal
```
```   893   shows "x - y \<le> z \<longleftrightarrow>
```
```   894     (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
```
```   895     (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
```
```   896     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
```
```   897   by (cases rule: ereal3_cases[of x y z]) auto
```
```   898
```
```   899 lemma ereal_minus_le:
```
```   900   fixes x y z :: ereal
```
```   901   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
```
```   902   by (auto simp: ereal_minus_le_iff)
```
```   903
```
```   904 lemma ereal_minus_eq_minus_iff:
```
```   905   fixes a b c :: ereal
```
```   906   shows "a - b = a - c \<longleftrightarrow>
```
```   907     b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
```
```   908   by (cases rule: ereal3_cases[of a b c]) auto
```
```   909
```
```   910 lemma ereal_add_le_add_iff:
```
```   911   fixes a b c :: ereal
```
```   912   shows "c + a \<le> c + b \<longleftrightarrow>
```
```   913     a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
```
```   914   by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
```
```   915
```
```   916 lemma ereal_mult_le_mult_iff:
```
```   917   fixes a b c :: ereal
```
```   918   shows "\<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)"
```
```   919   by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
```
```   920
```
```   921 lemma ereal_minus_mono:
```
```   922   fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C"
```
```   923   shows "A - C \<le> B - D"
```
```   924   using assms
```
```   925   by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all
```
```   926
```
```   927 lemma real_of_ereal_minus:
```
```   928   fixes a b :: ereal
```
```   929   shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
```
```   930   by (cases rule: ereal2_cases[of a b]) auto
```
```   931
```
```   932 lemma ereal_diff_positive:
```
```   933   fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
```
```   934   by (cases rule: ereal2_cases[of a b]) auto
```
```   935
```
```   936 lemma ereal_between:
```
```   937   fixes x e :: ereal
```
```   938   assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
```
```   939   shows "x - e < x" "x < x + e"
```
```   940 using assms apply (cases x, cases e) apply auto
```
```   941 using assms apply (cases x, cases e) apply auto
```
```   942 done
```
```   943
```
```   944 subsubsection {* Division *}
```
```   945
```
```   946 instantiation ereal :: inverse
```
```   947 begin
```
```   948
```
```   949 function inverse_ereal where
```
```   950 "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))" |
```
```   951 "inverse (\<infinity>::ereal) = 0" |
```
```   952 "inverse (-\<infinity>::ereal) = 0"
```
```   953   by (auto intro: ereal_cases)
```
```   954 termination by (relation "{}") simp
```
```   955
```
```   956 definition "x / y = x * inverse (y :: ereal)"
```
```   957
```
```   958 instance ..
```
```   959 end
```
```   960
```
```   961 lemma real_of_ereal_inverse[simp]:
```
```   962   fixes a :: ereal
```
```   963   shows "real (inverse a) = 1 / real a"
```
```   964   by (cases a) (auto simp: inverse_eq_divide)
```
```   965
```
```   966 lemma ereal_inverse[simp]:
```
```   967   "inverse (0::ereal) = \<infinity>"
```
```   968   "inverse (1::ereal) = 1"
```
```   969   by (simp_all add: one_ereal_def zero_ereal_def)
```
```   970
```
```   971 lemma ereal_divide[simp]:
```
```   972   "ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))"
```
```   973   unfolding divide_ereal_def by (auto simp: divide_real_def)
```
```   974
```
```   975 lemma ereal_divide_same[simp]:
```
```   976   fixes x :: ereal shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
```
```   977   by (cases x)
```
```   978      (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
```
```   979
```
```   980 lemma ereal_inv_inv[simp]:
```
```   981   fixes x :: ereal shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
```
```   982   by (cases x) auto
```
```   983
```
```   984 lemma ereal_inverse_minus[simp]:
```
```   985   fixes x :: ereal shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
```
```   986   by (cases x) simp_all
```
```   987
```
```   988 lemma ereal_uminus_divide[simp]:
```
```   989   fixes x y :: ereal shows "- x / y = - (x / y)"
```
```   990   unfolding divide_ereal_def by simp
```
```   991
```
```   992 lemma ereal_divide_Infty[simp]:
```
```   993   fixes x :: ereal shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
```
```   994   unfolding divide_ereal_def by simp_all
```
```   995
```
```   996 lemma ereal_divide_one[simp]:
```
```   997   "x / 1 = (x::ereal)"
```
```   998   unfolding divide_ereal_def by simp
```
```   999
```
```  1000 lemma ereal_divide_ereal[simp]:
```
```  1001   "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
```
```  1002   unfolding divide_ereal_def by simp
```
```  1003
```
```  1004 lemma zero_le_divide_ereal[simp]:
```
```  1005   fixes a :: ereal assumes "0 \<le> a" "0 \<le> b"
```
```  1006   shows "0 \<le> a / b"
```
```  1007   using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)
```
```  1008
```
```  1009 lemma ereal_le_divide_pos:
```
```  1010   fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
```
```  1011   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  1012
```
```  1013 lemma ereal_divide_le_pos:
```
```  1014   fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
```
```  1015   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  1016
```
```  1017 lemma ereal_le_divide_neg:
```
```  1018   fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
```
```  1019   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  1020
```
```  1021 lemma ereal_divide_le_neg:
```
```  1022   fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
```
```  1023   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  1024
```
```  1025 lemma ereal_inverse_antimono_strict:
```
```  1026   fixes x y :: ereal
```
```  1027   shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
```
```  1028   by (cases rule: ereal2_cases[of x y]) auto
```
```  1029
```
```  1030 lemma ereal_inverse_antimono:
```
```  1031   fixes x y :: ereal
```
```  1032   shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
```
```  1033   by (cases rule: ereal2_cases[of x y]) auto
```
```  1034
```
```  1035 lemma inverse_inverse_Pinfty_iff[simp]:
```
```  1036   fixes x :: ereal shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
```
```  1037   by (cases x) auto
```
```  1038
```
```  1039 lemma ereal_inverse_eq_0:
```
```  1040   fixes x :: ereal shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
```
```  1041   by (cases x) auto
```
```  1042
```
```  1043 lemma ereal_0_gt_inverse:
```
```  1044   fixes x :: ereal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
```
```  1045   by (cases x) auto
```
```  1046
```
```  1047 lemma ereal_mult_less_right:
```
```  1048   fixes a b c :: ereal
```
```  1049   assumes "b * a < c * a" "0 < a" "a < \<infinity>"
```
```  1050   shows "b < c"
```
```  1051   using assms
```
```  1052   by (cases rule: ereal3_cases[of a b c])
```
```  1053      (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
```
```  1054
```
```  1055 lemma ereal_power_divide:
```
```  1056   fixes x y :: ereal shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
```
```  1057   by (cases rule: ereal2_cases[of x y])
```
```  1058      (auto simp: one_ereal_def zero_ereal_def power_divide not_le
```
```  1059                  power_less_zero_eq zero_le_power_iff)
```
```  1060
```
```  1061 lemma ereal_le_mult_one_interval:
```
```  1062   fixes x y :: ereal
```
```  1063   assumes y: "y \<noteq> -\<infinity>"
```
```  1064   assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
```
```  1065   shows "x \<le> y"
```
```  1066 proof (cases x)
```
```  1067   case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_ereal_def)
```
```  1068 next
```
```  1069   case (real r) note r = this
```
```  1070   show "x \<le> y"
```
```  1071   proof (cases y)
```
```  1072     case (real p) note p = this
```
```  1073     have "r \<le> p"
```
```  1074     proof (rule field_le_mult_one_interval)
```
```  1075       fix z :: real assume "0 < z" and "z < 1"
```
```  1076       with z[of "ereal z"]
```
```  1077       show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_ereal_def)
```
```  1078     qed
```
```  1079     then show "x \<le> y" using p r by simp
```
```  1080   qed (insert y, simp_all)
```
```  1081 qed simp
```
```  1082
```
```  1083 lemma ereal_divide_right_mono[simp]:
```
```  1084   fixes x y z :: ereal
```
```  1085   assumes "x \<le> y" "0 < z" shows "x / z \<le> y / z"
```
```  1086 using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)
```
```  1087
```
```  1088 lemma ereal_divide_left_mono[simp]:
```
```  1089   fixes x y z :: ereal
```
```  1090   assumes "y \<le> x" "0 < z" "0 < x * y"
```
```  1091   shows "z / x \<le> z / y"
```
```  1092 using assms by (cases x y z rule: ereal3_cases)
```
```  1093   (auto intro: divide_left_mono simp: field_simps sign_simps split: split_if_asm)
```
```  1094
```
```  1095 lemma ereal_divide_zero_left[simp]:
```
```  1096   fixes a :: ereal
```
```  1097   shows "0 / a = 0"
```
```  1098   by (cases a) (auto simp: zero_ereal_def)
```
```  1099
```
```  1100 lemma ereal_times_divide_eq_left[simp]:
```
```  1101   fixes a b c :: ereal
```
```  1102   shows "b / c * a = b * a / c"
```
```  1103   by (cases a b c rule: ereal3_cases) (auto simp: field_simps sign_simps)
```
```  1104
```
```  1105 subsection "Complete lattice"
```
```  1106
```
```  1107 instantiation ereal :: lattice
```
```  1108 begin
```
```  1109 definition [simp]: "sup x y = (max x y :: ereal)"
```
```  1110 definition [simp]: "inf x y = (min x y :: ereal)"
```
```  1111 instance by default simp_all
```
```  1112 end
```
```  1113
```
```  1114 instantiation ereal :: complete_lattice
```
```  1115 begin
```
```  1116
```
```  1117 definition "bot = (-\<infinity>::ereal)"
```
```  1118 definition "top = (\<infinity>::ereal)"
```
```  1119
```
```  1120 definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: ereal)"
```
```  1121 definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: ereal)"
```
```  1122
```
```  1123 lemma ereal_complete_Sup:
```
```  1124   fixes S :: "ereal set" assumes "S \<noteq> {}"
```
```  1125   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
```
```  1126 proof cases
```
```  1127   assume "\<exists>x. \<forall>a\<in>S. a \<le> ereal x"
```
```  1128   then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" by auto
```
```  1129   then have "\<infinity> \<notin> S" by force
```
```  1130   show ?thesis
```
```  1131   proof cases
```
```  1132     assume "S = {-\<infinity>}"
```
```  1133     then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
```
```  1134   next
```
```  1135     assume "S \<noteq> {-\<infinity>}"
```
```  1136     with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
```
```  1137     with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
```
```  1138       by (auto simp: real_of_ereal_ord_simps)
```
```  1139     with complete_real[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
```
```  1140     obtain s where s:
```
```  1141        "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
```
```  1142        by auto
```
```  1143     show ?thesis
```
```  1144     proof (safe intro!: exI[of _ "ereal s"])
```
```  1145       fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> ereal s"
```
```  1146       proof (cases z)
```
```  1147         case (real r)
```
```  1148         then show ?thesis
```
```  1149           using s(1)[rule_format, of z] `z \<in> S` `z = ereal r` by auto
```
```  1150       qed auto
```
```  1151     next
```
```  1152       fix z assume *: "\<forall>y\<in>S. y \<le> z"
```
```  1153       with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "ereal s \<le> z"
```
```  1154       proof (cases z)
```
```  1155         case (real u)
```
```  1156         with * have "s \<le> u"
```
```  1157           by (intro s(2)[of u]) (auto simp: real_of_ereal_ord_simps)
```
```  1158         then show ?thesis using real by simp
```
```  1159       qed auto
```
```  1160     qed
```
```  1161   qed
```
```  1162 next
```
```  1163   assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> ereal x)"
```
```  1164   show ?thesis
```
```  1165   proof (safe intro!: exI[of _ \<infinity>])
```
```  1166     fix y assume **: "\<forall>z\<in>S. z \<le> y"
```
```  1167     with * show "\<infinity> \<le> y"
```
```  1168     proof (cases y)
```
```  1169       case MInf with * ** show ?thesis by (force simp: not_le)
```
```  1170     qed auto
```
```  1171   qed simp
```
```  1172 qed
```
```  1173
```
```  1174 lemma ereal_complete_Inf:
```
```  1175   fixes S :: "ereal set" assumes "S ~= {}"
```
```  1176   shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
```
```  1177 proof-
```
```  1178 def S1 == "uminus ` S"
```
```  1179 hence "S1 ~= {}" using assms by auto
```
```  1180 then obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
```
```  1181    using ereal_complete_Sup[of S1] by auto
```
```  1182 { fix z assume "ALL y:S. z <= y"
```
```  1183   hence "ALL y:S1. y <= -z" unfolding S1_def by auto
```
```  1184   hence "x <= -z" using x_def by auto
```
```  1185   hence "z <= -x"
```
```  1186     apply (subst ereal_uminus_uminus[symmetric])
```
```  1187     unfolding ereal_minus_le_minus . }
```
```  1188 moreover have "(ALL y:S. -x <= y)"
```
```  1189    using x_def unfolding S1_def
```
```  1190    apply simp
```
```  1191    apply (subst (3) ereal_uminus_uminus[symmetric])
```
```  1192    unfolding ereal_minus_le_minus by simp
```
```  1193 ultimately show ?thesis by auto
```
```  1194 qed
```
```  1195
```
```  1196 lemma ereal_complete_uminus_eq:
```
```  1197   fixes S :: "ereal set"
```
```  1198   shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
```
```  1199      \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
```
```  1200   by simp (metis ereal_minus_le_minus ereal_uminus_uminus)
```
```  1201
```
```  1202 lemma ereal_Sup_uminus_image_eq:
```
```  1203   fixes S :: "ereal set"
```
```  1204   shows "Sup (uminus ` S) = - Inf S"
```
```  1205 proof cases
```
```  1206   assume "S = {}"
```
```  1207   moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::ereal)"
```
```  1208     by (rule the_equality) (auto intro!: ereal_bot)
```
```  1209   moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::ereal)"
```
```  1210     by (rule some_equality) (auto intro!: ereal_top)
```
```  1211   ultimately show ?thesis unfolding Inf_ereal_def Sup_ereal_def
```
```  1212     Least_def Greatest_def GreatestM_def by simp
```
```  1213 next
```
```  1214   assume "S \<noteq> {}"
```
```  1215   with ereal_complete_Sup[of "uminus`S"]
```
```  1216   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)"
```
```  1217     unfolding ereal_complete_uminus_eq by auto
```
```  1218   show "Sup (uminus ` S) = - Inf S"
```
```  1219     unfolding Inf_ereal_def Greatest_def GreatestM_def
```
```  1220   proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
```
```  1221     show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
```
```  1222       using x .
```
```  1223     fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
```
```  1224     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)"
```
```  1225       unfolding ereal_complete_uminus_eq by simp
```
```  1226     then show "Sup (uminus ` S) = -x'"
```
```  1227       unfolding Sup_ereal_def ereal_uminus_eq_iff
```
```  1228       by (intro Least_equality) auto
```
```  1229   qed
```
```  1230 qed
```
```  1231
```
```  1232 instance
```
```  1233 proof
```
```  1234   { fix x :: ereal and A
```
```  1235     show "bot <= x" by (cases x) (simp_all add: bot_ereal_def)
```
```  1236     show "x <= top" by (simp add: top_ereal_def) }
```
```  1237
```
```  1238   { fix x :: ereal and A assume "x : A"
```
```  1239     with ereal_complete_Sup[of A]
```
```  1240     obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
```
```  1241     hence "x <= s" using `x : A` by auto
```
```  1242     also have "... = Sup A" using s unfolding Sup_ereal_def
```
```  1243       by (auto intro!: Least_equality[symmetric])
```
```  1244     finally show "x <= Sup A" . }
```
```  1245   note le_Sup = this
```
```  1246
```
```  1247   { fix x :: ereal and A assume *: "!!z. (z : A ==> z <= x)"
```
```  1248     show "Sup A <= x"
```
```  1249     proof (cases "A = {}")
```
```  1250       case True
```
```  1251       hence "Sup A = -\<infinity>" unfolding Sup_ereal_def
```
```  1252         by (auto intro!: Least_equality)
```
```  1253       thus "Sup A <= x" by simp
```
```  1254     next
```
```  1255       case False
```
```  1256       with ereal_complete_Sup[of A]
```
```  1257       obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
```
```  1258       hence "Sup A = s"
```
```  1259         unfolding Sup_ereal_def by (auto intro!: Least_equality)
```
```  1260       also have "s <= x" using * s by auto
```
```  1261       finally show "Sup A <= x" .
```
```  1262     qed }
```
```  1263   note Sup_le = this
```
```  1264
```
```  1265   { fix x :: ereal and A assume "x \<in> A"
```
```  1266     with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
```
```  1267       unfolding ereal_Sup_uminus_image_eq by simp }
```
```  1268
```
```  1269   { fix x :: ereal and A assume *: "!!z. (z : A ==> x <= z)"
```
```  1270     with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
```
```  1271       unfolding ereal_Sup_uminus_image_eq by force }
```
```  1272 qed
```
```  1273
```
```  1274 end
```
```  1275
```
```  1276 instance ereal :: complete_linorder ..
```
```  1277
```
```  1278 lemma ereal_SUPR_uminus:
```
```  1279   fixes f :: "'a => ereal"
```
```  1280   shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
```
```  1281   unfolding SUP_def INF_def
```
```  1282   using ereal_Sup_uminus_image_eq[of "f`R"]
```
```  1283   by (simp add: image_image)
```
```  1284
```
```  1285 lemma ereal_INFI_uminus:
```
```  1286   fixes f :: "'a => ereal"
```
```  1287   shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
```
```  1288   using ereal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
```
```  1289
```
```  1290 lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::ereal set)"
```
```  1291   using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
```
```  1292
```
```  1293 lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)"
```
```  1294   by (auto intro!: inj_onI)
```
```  1295
```
```  1296 lemma ereal_image_uminus_shift:
```
```  1297   fixes X Y :: "ereal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
```
```  1298 proof
```
```  1299   assume "uminus ` X = Y"
```
```  1300   then have "uminus ` uminus ` X = uminus ` Y"
```
```  1301     by (simp add: inj_image_eq_iff)
```
```  1302   then show "X = uminus ` Y" by (simp add: image_image)
```
```  1303 qed (simp add: image_image)
```
```  1304
```
```  1305 lemma Inf_ereal_iff:
```
```  1306   fixes z :: ereal
```
```  1307   shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
```
```  1308   by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
```
```  1309             order_less_le_trans)
```
```  1310
```
```  1311 lemma Sup_eq_MInfty:
```
```  1312   fixes S :: "ereal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
```
```  1313 proof
```
```  1314   assume a: "Sup S = -\<infinity>"
```
```  1315   with complete_lattice_class.Sup_upper[of _ S]
```
```  1316   show "S={} \<or> S={-\<infinity>}" by auto
```
```  1317 next
```
```  1318   assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
```
```  1319     unfolding Sup_ereal_def by (auto intro!: Least_equality)
```
```  1320 qed
```
```  1321
```
```  1322 lemma Inf_eq_PInfty:
```
```  1323   fixes S :: "ereal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
```
```  1324   using Sup_eq_MInfty[of "uminus`S"]
```
```  1325   unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp
```
```  1326
```
```  1327 lemma Inf_eq_MInfty:
```
```  1328   fixes S :: "ereal set" shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
```
```  1329   unfolding Inf_ereal_def
```
```  1330   by (auto intro!: Greatest_equality)
```
```  1331
```
```  1332 lemma Sup_eq_PInfty:
```
```  1333   fixes S :: "ereal set" shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
```
```  1334   unfolding Sup_ereal_def
```
```  1335   by (auto intro!: Least_equality)
```
```  1336
```
```  1337 lemma ereal_SUPI:
```
```  1338   fixes x :: ereal
```
```  1339   assumes "!!i. i : A ==> f i <= x"
```
```  1340   assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
```
```  1341   shows "(SUP i:A. f i) = x"
```
```  1342   unfolding SUP_def Sup_ereal_def
```
```  1343   using assms by (auto intro!: Least_equality)
```
```  1344
```
```  1345 lemma ereal_INFI:
```
```  1346   fixes x :: ereal
```
```  1347   assumes "!!i. i : A ==> f i >= x"
```
```  1348   assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
```
```  1349   shows "(INF i:A. f i) = x"
```
```  1350   unfolding INF_def Inf_ereal_def
```
```  1351   using assms by (auto intro!: Greatest_equality)
```
```  1352
```
```  1353 lemma Sup_ereal_close:
```
```  1354   fixes e :: ereal
```
```  1355   assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
```
```  1356   shows "\<exists>x\<in>S. Sup S - e < x"
```
```  1357   using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
```
```  1358
```
```  1359 lemma Inf_ereal_close:
```
```  1360   fixes e :: ereal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
```
```  1361   shows "\<exists>x\<in>X. x < Inf X + e"
```
```  1362 proof (rule Inf_less_iff[THEN iffD1])
```
```  1363   show "Inf X < Inf X + e" using assms
```
```  1364     by (cases e) auto
```
```  1365 qed
```
```  1366
```
```  1367 lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>"
```
```  1368 proof -
```
```  1369   { fix x ::ereal assume "x \<noteq> \<infinity>"
```
```  1370     then have "\<exists>k::nat. x < ereal (real k)"
```
```  1371     proof (cases x)
```
```  1372       case MInf then show ?thesis by (intro exI[of _ 0]) auto
```
```  1373     next
```
```  1374       case (real r)
```
```  1375       moreover obtain k :: nat where "r < real k"
```
```  1376         using ex_less_of_nat by (auto simp: real_eq_of_nat)
```
```  1377       ultimately show ?thesis by auto
```
```  1378     qed simp }
```
```  1379   then show ?thesis
```
```  1380     using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. ereal (real n)"]
```
```  1381     by (auto simp: top_ereal_def)
```
```  1382 qed
```
```  1383
```
```  1384 lemma ereal_le_Sup:
```
```  1385   fixes x :: ereal
```
```  1386   shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
```
```  1387 (is "?lhs <-> ?rhs")
```
```  1388 proof-
```
```  1389 { assume "?rhs"
```
```  1390   { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
```
```  1391     then obtain y where y_def: "(SUP i:A. f i)<y & y<x" using ereal_dense by auto
```
```  1392     then obtain i where "i : A & y <= f i" using `?rhs` by auto
```
```  1393     hence "y <= (SUP i:A. f i)" using SUP_upper[of i A f] by auto
```
```  1394     hence False using y_def by auto
```
```  1395   } hence "?lhs" by auto
```
```  1396 }
```
```  1397 moreover
```
```  1398 { assume "?lhs" hence "?rhs"
```
```  1399   by (metis less_SUP_iff order_less_imp_le order_less_le_trans)
```
```  1400 } ultimately show ?thesis by auto
```
```  1401 qed
```
```  1402
```
```  1403 lemma ereal_Inf_le:
```
```  1404   fixes x :: ereal
```
```  1405   shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
```
```  1406 (is "?lhs <-> ?rhs")
```
```  1407 proof-
```
```  1408 { assume "?rhs"
```
```  1409   { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
```
```  1410     then obtain y where y_def: "x<y & y<(INF i:A. f i)" using ereal_dense by auto
```
```  1411     then obtain i where "i : A & f i <= y" using `?rhs` by auto
```
```  1412     hence "(INF i:A. f i) <= y" using INF_lower[of i A f] by auto
```
```  1413     hence False using y_def by auto
```
```  1414   } hence "?lhs" by auto
```
```  1415 }
```
```  1416 moreover
```
```  1417 { assume "?lhs" hence "?rhs"
```
```  1418   by (metis INF_less_iff order_le_less order_less_le_trans)
```
```  1419 } ultimately show ?thesis by auto
```
```  1420 qed
```
```  1421
```
```  1422 lemma Inf_less:
```
```  1423   fixes x :: ereal
```
```  1424   assumes "(INF i:A. f i) < x"
```
```  1425   shows "EX i. i : A & f i <= x"
```
```  1426 proof(rule ccontr)
```
```  1427   assume "~ (EX i. i : A & f i <= x)"
```
```  1428   hence "ALL i:A. f i > x" by auto
```
```  1429   hence "(INF i:A. f i) >= x" apply (subst INF_greatest) by auto
```
```  1430   thus False using assms by auto
```
```  1431 qed
```
```  1432
```
```  1433 lemma same_INF:
```
```  1434   assumes "ALL e:A. f e = g e"
```
```  1435   shows "(INF e:A. f e) = (INF e:A. g e)"
```
```  1436 proof-
```
```  1437 have "f ` A = g ` A" unfolding image_def using assms by auto
```
```  1438 thus ?thesis unfolding INF_def by auto
```
```  1439 qed
```
```  1440
```
```  1441 lemma same_SUP:
```
```  1442   assumes "ALL e:A. f e = g e"
```
```  1443   shows "(SUP e:A. f e) = (SUP e:A. g e)"
```
```  1444 proof-
```
```  1445 have "f ` A = g ` A" unfolding image_def using assms by auto
```
```  1446 thus ?thesis unfolding SUP_def by auto
```
```  1447 qed
```
```  1448
```
```  1449 lemma SUPR_eq:
```
```  1450   assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
```
```  1451   assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
```
```  1452   shows "(SUP i:A. f i) = (SUP j:B. g j)"
```
```  1453 proof (intro antisym)
```
```  1454   show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
```
```  1455     using assms by (metis SUP_least SUP_upper2)
```
```  1456   show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
```
```  1457     using assms by (metis SUP_least SUP_upper2)
```
```  1458 qed
```
```  1459
```
```  1460 lemma SUP_ereal_le_addI:
```
```  1461   fixes f :: "'i \<Rightarrow> ereal"
```
```  1462   assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
```
```  1463   shows "SUPR UNIV f + y \<le> z"
```
```  1464 proof (cases y)
```
```  1465   case (real r)
```
```  1466   then have "\<And>i. f i \<le> z - y" using assms by (simp add: ereal_le_minus_iff)
```
```  1467   then have "SUPR UNIV f \<le> z - y" by (rule SUP_least)
```
```  1468   then show ?thesis using real by (simp add: ereal_le_minus_iff)
```
```  1469 qed (insert assms, auto)
```
```  1470
```
```  1471 lemma SUPR_ereal_add:
```
```  1472   fixes f g :: "nat \<Rightarrow> ereal"
```
```  1473   assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
```
```  1474   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
```
```  1475 proof (rule ereal_SUPI)
```
```  1476   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
```
```  1477   have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
```
```  1478     unfolding SUP_def Sup_eq_MInfty by (auto dest: image_eqD)
```
```  1479   { fix j
```
```  1480     { fix i
```
```  1481       have "f i + g j \<le> f i + g (max i j)"
```
```  1482         using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
```
```  1483       also have "\<dots> \<le> f (max i j) + g (max i j)"
```
```  1484         using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
```
```  1485       also have "\<dots> \<le> y" using * by auto
```
```  1486       finally have "f i + g j \<le> y" . }
```
```  1487     then have "SUPR UNIV f + g j \<le> y"
```
```  1488       using assms(4)[of j] by (intro SUP_ereal_le_addI) auto
```
```  1489     then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
```
```  1490   then have "SUPR UNIV g + SUPR UNIV f \<le> y"
```
```  1491     using f by (rule SUP_ereal_le_addI)
```
```  1492   then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
```
```  1493 qed (auto intro!: add_mono SUP_upper)
```
```  1494
```
```  1495 lemma SUPR_ereal_add_pos:
```
```  1496   fixes f g :: "nat \<Rightarrow> ereal"
```
```  1497   assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
```
```  1498   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
```
```  1499 proof (intro SUPR_ereal_add inc)
```
```  1500   fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
```
```  1501 qed
```
```  1502
```
```  1503 lemma SUPR_ereal_setsum:
```
```  1504   fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
```
```  1505   assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
```
```  1506   shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
```
```  1507 proof cases
```
```  1508   assume "finite A" then show ?thesis using assms
```
```  1509     by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_ereal_add_pos)
```
```  1510 qed simp
```
```  1511
```
```  1512 lemma SUPR_ereal_cmult:
```
```  1513   fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
```
```  1514   shows "(SUP i. c * f i) = c * SUPR UNIV f"
```
```  1515 proof (rule ereal_SUPI)
```
```  1516   fix i have "f i \<le> SUPR UNIV f" by (rule SUP_upper) auto
```
```  1517   then show "c * f i \<le> c * SUPR UNIV f"
```
```  1518     using `0 \<le> c` by (rule ereal_mult_left_mono)
```
```  1519 next
```
```  1520   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
```
```  1521   show "c * SUPR UNIV f \<le> y"
```
```  1522   proof cases
```
```  1523     assume c: "0 < c \<and> c \<noteq> \<infinity>"
```
```  1524     with * have "SUPR UNIV f \<le> y / c"
```
```  1525       by (intro SUP_least) (auto simp: ereal_le_divide_pos)
```
```  1526     with c show ?thesis
```
```  1527       by (auto simp: ereal_le_divide_pos)
```
```  1528   next
```
```  1529     { assume "c = \<infinity>" have ?thesis
```
```  1530       proof cases
```
```  1531         assume "\<forall>i. f i = 0"
```
```  1532         moreover then have "range f = {0}" by auto
```
```  1533         ultimately show "c * SUPR UNIV f \<le> y" using *
```
```  1534           by (auto simp: SUP_def min_max.sup_absorb1)
```
```  1535       next
```
```  1536         assume "\<not> (\<forall>i. f i = 0)"
```
```  1537         then obtain i where "f i \<noteq> 0" by auto
```
```  1538         with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
```
```  1539       qed }
```
```  1540     moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
```
```  1541     ultimately show ?thesis using * `0 \<le> c` by auto
```
```  1542   qed
```
```  1543 qed
```
```  1544
```
```  1545 lemma SUP_PInfty:
```
```  1546   fixes f :: "'a \<Rightarrow> ereal"
```
```  1547   assumes "\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i"
```
```  1548   shows "(SUP i:A. f i) = \<infinity>"
```
```  1549   unfolding SUP_def Sup_eq_top_iff[where 'a=ereal, unfolded top_ereal_def]
```
```  1550   apply simp
```
```  1551 proof safe
```
```  1552   fix x :: ereal assume "x \<noteq> \<infinity>"
```
```  1553   show "\<exists>i\<in>A. x < f i"
```
```  1554   proof (cases x)
```
```  1555     case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
```
```  1556   next
```
```  1557     case MInf with assms[of "0"] show ?thesis by force
```
```  1558   next
```
```  1559     case (real r)
```
```  1560     with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)" by auto
```
```  1561     moreover from assms[of n] guess i ..
```
```  1562     ultimately show ?thesis
```
```  1563       by (auto intro!: bexI[of _ i])
```
```  1564   qed
```
```  1565 qed
```
```  1566
```
```  1567 lemma Sup_countable_SUPR:
```
```  1568   assumes "A \<noteq> {}"
```
```  1569   shows "\<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
```
```  1570 proof (cases "Sup A")
```
```  1571   case (real r)
```
```  1572   have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
```
```  1573   proof
```
```  1574     fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x"
```
```  1575       using assms real by (intro Sup_ereal_close) (auto simp: one_ereal_def)
```
```  1576     then guess x ..
```
```  1577     then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
```
```  1578       by (auto intro!: exI[of _ x] simp: ereal_minus_less_iff)
```
```  1579   qed
```
```  1580   from choice[OF this] guess f .. note f = this
```
```  1581   have "SUPR UNIV f = Sup A"
```
```  1582   proof (rule ereal_SUPI)
```
```  1583     fix i show "f i \<le> Sup A" using f
```
```  1584       by (auto intro!: complete_lattice_class.Sup_upper)
```
```  1585   next
```
```  1586     fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
```
```  1587     show "Sup A \<le> y"
```
```  1588     proof (rule ereal_le_epsilon, intro allI impI)
```
```  1589       fix e :: ereal assume "0 < e"
```
```  1590       show "Sup A \<le> y + e"
```
```  1591       proof (cases e)
```
```  1592         case (real r)
```
```  1593         hence "0 < r" using `0 < e` by auto
```
```  1594         then obtain n ::nat where *: "1 / real n < r" "0 < n"
```
```  1595           using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
```
```  1596         have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n]
```
```  1597           by auto
```
```  1598         also have "1 / ereal (real n) \<le> e" using real * by (auto simp: one_ereal_def )
```
```  1599         with bound have "f n + 1 / ereal (real n) \<le> y + e" by (rule add_mono) simp
```
```  1600         finally show "Sup A \<le> y + e" .
```
```  1601       qed (insert `0 < e`, auto)
```
```  1602     qed
```
```  1603   qed
```
```  1604   with f show ?thesis by (auto intro!: exI[of _ f])
```
```  1605 next
```
```  1606   case PInf
```
```  1607   from `A \<noteq> {}` obtain x where "x \<in> A" by auto
```
```  1608   show ?thesis
```
```  1609   proof cases
```
```  1610     assume "\<infinity> \<in> A"
```
```  1611     moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
```
```  1612     ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
```
```  1613   next
```
```  1614     assume "\<infinity> \<notin> A"
```
```  1615     have "\<exists>x\<in>A. 0 \<le> x"
```
```  1616       by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least ereal_infty_less_eq2 linorder_linear)
```
```  1617     then obtain x where "x \<in> A" "0 \<le> x" by auto
```
```  1618     have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + ereal (real n) \<le> f"
```
```  1619     proof (rule ccontr)
```
```  1620       assume "\<not> ?thesis"
```
```  1621       then have "\<exists>n::nat. Sup A \<le> x + ereal (real n)"
```
```  1622         by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
```
```  1623       then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
```
```  1624         by(cases x) auto
```
```  1625     qed
```
```  1626     from choice[OF this] guess f .. note f = this
```
```  1627     have "SUPR UNIV f = \<infinity>"
```
```  1628     proof (rule SUP_PInfty)
```
```  1629       fix n :: nat show "\<exists>i\<in>UNIV. ereal (real n) \<le> f i"
```
```  1630         using f[THEN spec, of n] `0 \<le> x`
```
```  1631         by (cases rule: ereal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
```
```  1632     qed
```
```  1633     then show ?thesis using f PInf by (auto intro!: exI[of _ f])
```
```  1634   qed
```
```  1635 next
```
```  1636   case MInf
```
```  1637   with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
```
```  1638   then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
```
```  1639 qed
```
```  1640
```
```  1641 lemma SUPR_countable_SUPR:
```
```  1642   "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
```
```  1643   using Sup_countable_SUPR[of "g`A"] by (auto simp: SUP_def)
```
```  1644
```
```  1645 lemma Sup_ereal_cadd:
```
```  1646   fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1647   shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
```
```  1648 proof (rule antisym)
```
```  1649   have *: "\<And>a::ereal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
```
```  1650     by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
```
```  1651   then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
```
```  1652   show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
```
```  1653   proof (cases a)
```
```  1654     case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant min_max.sup_absorb1)
```
```  1655   next
```
```  1656     case (real r)
```
```  1657     then have **: "op + (- a) ` op + a ` A = A"
```
```  1658       by (auto simp: image_iff ac_simps zero_ereal_def[symmetric])
```
```  1659     from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
```
```  1660       by (cases rule: ereal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
```
```  1661   qed (insert `a \<noteq> -\<infinity>`, auto)
```
```  1662 qed
```
```  1663
```
```  1664 lemma Sup_ereal_cminus:
```
```  1665   fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1666   shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
```
```  1667   using Sup_ereal_cadd[of "uminus ` A" a] assms
```
```  1668   by (simp add: comp_def image_image minus_ereal_def
```
```  1669                  ereal_Sup_uminus_image_eq)
```
```  1670
```
```  1671 lemma SUPR_ereal_cminus:
```
```  1672   fixes f :: "'i \<Rightarrow> ereal"
```
```  1673   fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
```
```  1674   shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
```
```  1675   using Sup_ereal_cminus[of "f`A" a] assms
```
```  1676   unfolding SUP_def INF_def image_image by auto
```
```  1677
```
```  1678 lemma Inf_ereal_cminus:
```
```  1679   fixes A :: "ereal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
```
```  1680   shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
```
```  1681 proof -
```
```  1682   { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto }
```
```  1683   moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
```
```  1684     by (auto simp: image_image)
```
```  1685   ultimately show ?thesis
```
```  1686     using Sup_ereal_cminus[of "uminus ` A" "-a"] assms
```
```  1687     by (auto simp add: ereal_Sup_uminus_image_eq ereal_Inf_uminus_image_eq)
```
```  1688 qed
```
```  1689
```
```  1690 lemma INFI_ereal_cminus:
```
```  1691   fixes a :: ereal assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
```
```  1692   shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
```
```  1693   using Inf_ereal_cminus[of "f`A" a] assms
```
```  1694   unfolding SUP_def INF_def image_image
```
```  1695   by auto
```
```  1696
```
```  1697 lemma uminus_ereal_add_uminus_uminus:
```
```  1698   fixes a b :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
```
```  1699   by (cases rule: ereal2_cases[of a b]) auto
```
```  1700
```
```  1701 lemma INFI_ereal_add:
```
```  1702   fixes f :: "nat \<Rightarrow> ereal"
```
```  1703   assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
```
```  1704   shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g"
```
```  1705 proof -
```
```  1706   have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
```
```  1707     using assms unfolding INF_less_iff by auto
```
```  1708   { fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
```
```  1709       by (rule uminus_ereal_add_uminus_uminus) }
```
```  1710   then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
```
```  1711     by simp
```
```  1712   also have "\<dots> = INFI UNIV f + INFI UNIV g"
```
```  1713     unfolding ereal_INFI_uminus
```
```  1714     using assms INF_less
```
```  1715     by (subst SUPR_ereal_add)
```
```  1716        (auto simp: ereal_SUPR_uminus intro!: uminus_ereal_add_uminus_uminus)
```
```  1717   finally show ?thesis .
```
```  1718 qed
```
```  1719
```
```  1720 subsection "Relation to @{typ enat}"
```
```  1721
```
```  1722 definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
```
```  1723
```
```  1724 declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]]
```
```  1725 declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]]
```
```  1726
```
```  1727 lemma ereal_of_enat_simps[simp]:
```
```  1728   "ereal_of_enat (enat n) = ereal n"
```
```  1729   "ereal_of_enat \<infinity> = \<infinity>"
```
```  1730   by (simp_all add: ereal_of_enat_def)
```
```  1731
```
```  1732 lemma ereal_of_enat_le_iff[simp]:
```
```  1733   "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
```
```  1734 by (cases m n rule: enat2_cases) auto
```
```  1735
```
```  1736 lemma numeral_le_ereal_of_enat_iff[simp]:
```
```  1737   shows "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
```
```  1738 by (cases n) (auto dest: natceiling_le intro: natceiling_le_eq[THEN iffD1])
```
```  1739
```
```  1740 lemma ereal_of_enat_ge_zero_cancel_iff[simp]:
```
```  1741   "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
```
```  1742 by (cases n) (auto simp: enat_0[symmetric])
```
```  1743
```
```  1744 lemma ereal_of_enat_gt_zero_cancel_iff[simp]:
```
```  1745   "0 < ereal_of_enat n \<longleftrightarrow> 0 < n"
```
```  1746 by (cases n) (auto simp: enat_0[symmetric])
```
```  1747
```
```  1748 lemma ereal_of_enat_zero[simp]:
```
```  1749   "ereal_of_enat 0 = 0"
```
```  1750 by (auto simp: enat_0[symmetric])
```
```  1751
```
```  1752 lemma ereal_of_enat_add:
```
```  1753   "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
```
```  1754 by (cases m n rule: enat2_cases) auto
```
```  1755
```
```  1756 lemma ereal_of_enat_sub:
```
```  1757   assumes "n \<le> m" shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
```
```  1758 using assms by (cases m n rule: enat2_cases) auto
```
```  1759
```
```  1760 lemma ereal_of_enat_mult:
```
```  1761   "ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
```
```  1762 by (cases m n rule: enat2_cases) auto
```
```  1763
```
```  1764 lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult
```
```  1765 lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric]
```
```  1766
```
```  1767
```
```  1768 subsection "Limits on @{typ ereal}"
```
```  1769
```
```  1770 subsubsection "Topological space"
```
```  1771
```
```  1772 instantiation ereal :: topological_space
```
```  1773 begin
```
```  1774
```
```  1775 definition "open A \<longleftrightarrow> open (ereal -` A)
```
```  1776        \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A))
```
```  1777        \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
```
```  1778
```
```  1779 lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
```
```  1780   unfolding open_ereal_def by auto
```
```  1781
```
```  1782 lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)"
```
```  1783   unfolding open_ereal_def by auto
```
```  1784
```
```  1785 lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{ereal x<..} \<subseteq> A"
```
```  1786   using open_PInfty[OF assms] by auto
```
```  1787
```
```  1788 lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<ereal x} \<subseteq> A"
```
```  1789   using open_MInfty[OF assms] by auto
```
```  1790
```
```  1791 lemma ereal_openE: assumes "open A" obtains x y where
```
```  1792   "open (ereal -` A)"
```
```  1793   "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A"
```
```  1794   "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
```
```  1795   using assms open_ereal_def by auto
```
```  1796
```
```  1797 instance
```
```  1798 proof
```
```  1799   let ?U = "UNIV::ereal set"
```
```  1800   show "open ?U" unfolding open_ereal_def
```
```  1801     by (auto intro!: exI[of _ 0])
```
```  1802 next
```
```  1803   fix S T::"ereal set" assume "open S" and "open T"
```
```  1804   from `open S`[THEN ereal_openE] guess xS yS .
```
```  1805   moreover from `open T`[THEN ereal_openE] guess xT yT .
```
```  1806   ultimately have
```
```  1807     "open (ereal -` (S \<inter> T))"
```
```  1808     "\<infinity> \<in> S \<inter> T \<Longrightarrow> {ereal (max xS xT) <..} \<subseteq> S \<inter> T"
```
```  1809     "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< ereal (min yS yT)} \<subseteq> S \<inter> T"
```
```  1810     by auto
```
```  1811   then show "open (S Int T)" unfolding open_ereal_def by blast
```
```  1812 next
```
```  1813   fix K :: "ereal set set" assume "\<forall>S\<in>K. open S"
```
```  1814   then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (ereal -` S) \<and>
```
```  1815     (\<infinity> \<in> S \<longrightarrow> {ereal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< ereal y} \<subseteq> S)"
```
```  1816     by (auto simp: open_ereal_def)
```
```  1817   then show "open (Union K)" unfolding open_ereal_def
```
```  1818   proof (intro conjI impI)
```
```  1819     show "open (ereal -` \<Union>K)"
```
```  1820       using *[THEN choice] by (auto simp: vimage_Union)
```
```  1821   qed ((metis UnionE Union_upper subset_trans *)+)
```
```  1822 qed
```
```  1823 end
```
```  1824
```
```  1825 lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)"
```
```  1826   by (auto simp: inj_vimage_image_eq open_ereal_def)
```
```  1827
```
```  1828 lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)"
```
```  1829   unfolding open_ereal_def by auto
```
```  1830
```
```  1831 lemma open_ereal_lessThan[intro, simp]: "open {..< a :: ereal}"
```
```  1832 proof -
```
```  1833   have "\<And>x. ereal -` {..<ereal x} = {..< x}"
```
```  1834     "ereal -` {..< \<infinity>} = UNIV" "ereal -` {..< -\<infinity>} = {}" by auto
```
```  1835   then show ?thesis by (cases a) (auto simp: open_ereal_def)
```
```  1836 qed
```
```  1837
```
```  1838 lemma open_ereal_greaterThan[intro, simp]:
```
```  1839   "open {a :: ereal <..}"
```
```  1840 proof -
```
```  1841   have "\<And>x. ereal -` {ereal x<..} = {x<..}"
```
```  1842     "ereal -` {\<infinity><..} = {}" "ereal -` {-\<infinity><..} = UNIV" by auto
```
```  1843   then show ?thesis by (cases a) (auto simp: open_ereal_def)
```
```  1844 qed
```
```  1845
```
```  1846 lemma ereal_open_greaterThanLessThan[intro, simp]: "open {a::ereal <..< b}"
```
```  1847   unfolding greaterThanLessThan_def by auto
```
```  1848
```
```  1849 lemma closed_ereal_atLeast[simp, intro]: "closed {a :: ereal ..}"
```
```  1850 proof -
```
```  1851   have "- {a ..} = {..< a}" by auto
```
```  1852   then show "closed {a ..}"
```
```  1853     unfolding closed_def using open_ereal_lessThan by auto
```
```  1854 qed
```
```  1855
```
```  1856 lemma closed_ereal_atMost[simp, intro]: "closed {.. b :: ereal}"
```
```  1857 proof -
```
```  1858   have "- {.. b} = {b <..}" by auto
```
```  1859   then show "closed {.. b}"
```
```  1860     unfolding closed_def using open_ereal_greaterThan by auto
```
```  1861 qed
```
```  1862
```
```  1863 lemma closed_ereal_atLeastAtMost[simp, intro]:
```
```  1864   shows "closed {a :: ereal .. b}"
```
```  1865   unfolding atLeastAtMost_def by auto
```
```  1866
```
```  1867 lemma closed_ereal_singleton:
```
```  1868   "closed {a :: ereal}"
```
```  1869 by (metis atLeastAtMost_singleton closed_ereal_atLeastAtMost)
```
```  1870
```
```  1871 lemma ereal_open_cont_interval:
```
```  1872   fixes S :: "ereal set"
```
```  1873   assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  1874   obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
```
```  1875 proof-
```
```  1876   from `open S` have "open (ereal -` S)" by (rule ereal_openE)
```
```  1877   then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
```
```  1878     using assms unfolding open_dist by force
```
```  1879   show thesis
```
```  1880   proof (intro that subsetI)
```
```  1881     show "0 < ereal e" using `0 < e` by auto
```
```  1882     fix y assume "y \<in> {x - ereal e<..<x + ereal e}"
```
```  1883     with assms obtain t where "y = ereal t" "dist t (real x) < e"
```
```  1884       apply (cases y) by (auto simp: dist_real_def)
```
```  1885     then show "y \<in> S" using e[of t] by auto
```
```  1886   qed
```
```  1887 qed
```
```  1888
```
```  1889 lemma ereal_open_cont_interval2:
```
```  1890   fixes S :: "ereal set"
```
```  1891   assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  1892   obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
```
```  1893 proof-
```
```  1894   guess e using ereal_open_cont_interval[OF assms] .
```
```  1895   with that[of "x-e" "x+e"] ereal_between[OF x, of e]
```
```  1896   show thesis by auto
```
```  1897 qed
```
```  1898
```
```  1899 instance ereal :: t2_space
```
```  1900 proof
```
```  1901   fix x y :: ereal assume "x ~= y"
```
```  1902   let "?P x (y::ereal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
```
```  1903
```
```  1904   { fix x y :: ereal assume "x < y"
```
```  1905     from ereal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
```
```  1906     have "?P x y"
```
```  1907       apply (rule exI[of _ "{..<z}"])
```
```  1908       apply (rule exI[of _ "{z<..}"])
```
```  1909       using z by auto }
```
```  1910   note * = this
```
```  1911
```
```  1912   from `x ~= y`
```
```  1913   show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
```
```  1914   proof (cases rule: linorder_cases)
```
```  1915     assume "x = y" with `x ~= y` show ?thesis by simp
```
```  1916   next assume "x < y" from *[OF this] show ?thesis by auto
```
```  1917   next assume "y < x" from *[OF this] show ?thesis by auto
```
```  1918   qed
```
```  1919 qed
```
```  1920
```
```  1921 subsubsection {* Convergent sequences *}
```
```  1922
```
```  1923 lemma lim_ereal[simp]:
```
```  1924   "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
```
```  1925 proof (intro iffI topological_tendstoI)
```
```  1926   fix S assume "?l" "open S" "x \<in> S"
```
```  1927   then show "eventually (\<lambda>x. f x \<in> S) net"
```
```  1928     using `?l`[THEN topological_tendstoD, OF open_ereal, OF `open S`]
```
```  1929     by (simp add: inj_image_mem_iff)
```
```  1930 next
```
```  1931   fix S assume "?r" "open S" "ereal x \<in> S"
```
```  1932   show "eventually (\<lambda>x. ereal (f x) \<in> S) net"
```
```  1933     using `?r`[THEN topological_tendstoD, OF open_ereal_vimage, OF `open S`]
```
```  1934     using `ereal x \<in> S` by auto
```
```  1935 qed
```
```  1936
```
```  1937 lemma lim_real_of_ereal[simp]:
```
```  1938   assumes lim: "(f ---> ereal x) net"
```
```  1939   shows "((\<lambda>x. real (f x)) ---> x) net"
```
```  1940 proof (intro topological_tendstoI)
```
```  1941   fix S assume "open S" "x \<in> S"
```
```  1942   then have S: "open S" "ereal x \<in> ereal ` S"
```
```  1943     by (simp_all add: inj_image_mem_iff)
```
```  1944   have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S" by auto
```
```  1945   from this lim[THEN topological_tendstoD, OF open_ereal, OF S]
```
```  1946   show "eventually (\<lambda>x. real (f x) \<in> S) net"
```
```  1947     by (rule eventually_mono)
```
```  1948 qed
```
```  1949
```
```  1950 lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= ereal B)" (is "?l = ?r")
```
```  1951 proof
```
```  1952   assume ?r
```
```  1953   show ?l
```
```  1954     apply(rule topological_tendstoI)
```
```  1955     unfolding eventually_sequentially
```
```  1956   proof-
```
```  1957     fix S :: "ereal set" assume "open S" "\<infinity> : S"
```
```  1958     from open_PInfty[OF this] guess B .. note B=this
```
```  1959     from `?r`[rule_format,of "B+1"] guess N .. note N=this
```
```  1960     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
```
```  1961     proof safe case goal1
```
```  1962       have "ereal B < ereal (B + 1)" by auto
```
```  1963       also have "... <= f n" using goal1 N by auto
```
```  1964       finally show ?case using B by fastforce
```
```  1965     qed
```
```  1966   qed
```
```  1967 next
```
```  1968   assume ?l
```
```  1969   show ?r
```
```  1970   proof fix B::real have "open {ereal B<..}" "\<infinity> : {ereal B<..}" by auto
```
```  1971     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
```
```  1972     guess N .. note N=this
```
```  1973     show "EX N. ALL n>=N. ereal B <= f n" apply(rule_tac x=N in exI) using N by auto
```
```  1974   qed
```
```  1975 qed
```
```  1976
```
```  1977
```
```  1978 lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= ereal B)" (is "?l = ?r")
```
```  1979 proof
```
```  1980   assume ?r
```
```  1981   show ?l
```
```  1982     apply(rule topological_tendstoI)
```
```  1983     unfolding eventually_sequentially
```
```  1984   proof-
```
```  1985     fix S :: "ereal set"
```
```  1986     assume "open S" "(-\<infinity>) : S"
```
```  1987     from open_MInfty[OF this] guess B .. note B=this
```
```  1988     from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
```
```  1989     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
```
```  1990     proof safe case goal1
```
```  1991       have "ereal (B - 1) >= f n" using goal1 N by auto
```
```  1992       also have "... < ereal B" by auto
```
```  1993       finally show ?case using B by fastforce
```
```  1994     qed
```
```  1995   qed
```
```  1996 next assume ?l show ?r
```
```  1997   proof fix B::real have "open {..<ereal B}" "(-\<infinity>) : {..<ereal B}" by auto
```
```  1998     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
```
```  1999     guess N .. note N=this
```
```  2000     show "EX N. ALL n>=N. ereal B >= f n" apply(rule_tac x=N in exI) using N by auto
```
```  2001   qed
```
```  2002 qed
```
```  2003
```
```  2004
```
```  2005 lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= ereal B" shows "l ~= \<infinity>"
```
```  2006 proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
```
```  2007   from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
```
```  2008   guess N .. note N=this[rule_format,OF le_refl]
```
```  2009   hence "ereal ?B <= ereal B" using assms(2)[of N] by(rule order_trans)
```
```  2010   hence "ereal ?B < ereal ?B" apply (rule le_less_trans) by auto
```
```  2011   thus False by auto
```
```  2012 qed
```
```  2013
```
```  2014
```
```  2015 lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= ereal B" shows "l ~= (-\<infinity>)"
```
```  2016 proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
```
```  2017   from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
```
```  2018   guess N .. note N=this[rule_format,OF le_refl]
```
```  2019   hence "ereal B <= ereal ?B" using assms(2)[of N] order_trans[of "ereal B" "f N" "ereal(B - 1)"] by blast
```
```  2020   thus False by auto
```
```  2021 qed
```
```  2022
```
```  2023
```
```  2024 lemma tendsto_explicit:
```
```  2025   "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
```
```  2026   unfolding tendsto_def eventually_sequentially by auto
```
```  2027
```
```  2028
```
```  2029 lemma tendsto_obtains_N:
```
```  2030   assumes "f ----> f0"
```
```  2031   assumes "open S" "f0 : S"
```
```  2032   obtains N where "ALL n>=N. f n : S"
```
```  2033   using tendsto_explicit[of f f0] assms by auto
```
```  2034
```
```  2035
```
```  2036 lemma tail_same_limit:
```
```  2037   fixes X Y N
```
```  2038   assumes "X ----> L" "ALL n>=N. X n = Y n"
```
```  2039   shows "Y ----> L"
```
```  2040 proof-
```
```  2041 { fix S assume "open S" and "L:S"
```
```  2042   then obtain N1 where "ALL n>=N1. X n : S"
```
```  2043      using assms unfolding tendsto_def eventually_sequentially by auto
```
```  2044   hence "ALL n>=max N N1. Y n : S" using assms by auto
```
```  2045   hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
```
```  2046 }
```
```  2047 thus ?thesis using tendsto_explicit by auto
```
```  2048 qed
```
```  2049
```
```  2050
```
```  2051 lemma Lim_bounded_PInfty2:
```
```  2052 assumes lim:"f ----> l" and "ALL n>=N. f n <= ereal B"
```
```  2053 shows "l ~= \<infinity>"
```
```  2054 proof-
```
```  2055   def g == "(%n. if n>=N then f n else ereal B)"
```
```  2056   hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
```
```  2057   moreover have "!!n. g n <= ereal B" using g_def assms by auto
```
```  2058   ultimately show ?thesis using  Lim_bounded_PInfty by auto
```
```  2059 qed
```
```  2060
```
```  2061 lemma Lim_bounded_ereal:
```
```  2062   assumes lim:"f ----> (l :: ereal)"
```
```  2063   and "ALL n>=M. f n <= C"
```
```  2064   shows "l<=C"
```
```  2065 proof-
```
```  2066 { assume "l=(-\<infinity>)" hence ?thesis by auto }
```
```  2067 moreover
```
```  2068 { assume "~(l=(-\<infinity>))"
```
```  2069   { assume "C=\<infinity>" hence ?thesis by auto }
```
```  2070   moreover
```
```  2071   { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
```
```  2072     hence "l=(-\<infinity>)" using assms
```
```  2073        tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
```
```  2074     hence ?thesis by auto }
```
```  2075   moreover
```
```  2076   { assume "EX B. C = ereal B"
```
```  2077     then obtain B where B_def: "C=ereal B" by auto
```
```  2078     hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
```
```  2079     then obtain m where m_def: "ereal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
```
```  2080     then obtain N where N_def: "ALL n>=N. f n : {ereal(m - 1) <..< ereal(m+1)}"
```
```  2081        apply (subst tendsto_obtains_N[of f l "{ereal(m - 1) <..< ereal(m+1)}"]) using assms by auto
```
```  2082     { fix n assume "n>=N"
```
```  2083       hence "EX r. ereal r = f n" using N_def by (cases "f n") auto
```
```  2084     } then obtain g where g_def: "ALL n>=N. ereal (g n) = f n" by metis
```
```  2085     hence "(%n. ereal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
```
```  2086     hence *: "(%n. g n) ----> m" using m_def by auto
```
```  2087     { fix n assume "n>=max N M"
```
```  2088       hence "ereal (g n) <= ereal B" using assms g_def B_def by auto
```
```  2089       hence "g n <= B" by auto
```
```  2090     } hence "EX N. ALL n>=N. g n <= B" by blast
```
```  2091     hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
```
```  2092     hence ?thesis using m_def B_def by auto
```
```  2093   } ultimately have ?thesis by (cases C) auto
```
```  2094 } ultimately show ?thesis by blast
```
```  2095 qed
```
```  2096
```
```  2097 lemma real_of_ereal_mult[simp]:
```
```  2098   fixes a b :: ereal shows "real (a * b) = real a * real b"
```
```  2099   by (cases rule: ereal2_cases[of a b]) auto
```
```  2100
```
```  2101 lemma real_of_ereal_eq_0:
```
```  2102   fixes x :: ereal shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
```
```  2103   by (cases x) auto
```
```  2104
```
```  2105 lemma tendsto_ereal_realD:
```
```  2106   fixes f :: "'a \<Rightarrow> ereal"
```
```  2107   assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
```
```  2108   shows "(f ---> x) net"
```
```  2109 proof (intro topological_tendstoI)
```
```  2110   fix S assume S: "open S" "x \<in> S"
```
```  2111   with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
```
```  2112   from tendsto[THEN topological_tendstoD, OF this]
```
```  2113   show "eventually (\<lambda>x. f x \<in> S) net"
```
```  2114     by (rule eventually_rev_mp) (auto simp: ereal_real)
```
```  2115 qed
```
```  2116
```
```  2117 lemma tendsto_ereal_realI:
```
```  2118   fixes f :: "'a \<Rightarrow> ereal"
```
```  2119   assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
```
```  2120   shows "((\<lambda>x. ereal (real (f x))) ---> x) net"
```
```  2121 proof (intro topological_tendstoI)
```
```  2122   fix S assume "open S" "x \<in> S"
```
```  2123   with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
```
```  2124   from tendsto[THEN topological_tendstoD, OF this]
```
```  2125   show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net"
```
```  2126     by (elim eventually_elim1) (auto simp: ereal_real)
```
```  2127 qed
```
```  2128
```
```  2129 lemma ereal_mult_cancel_left:
```
```  2130   fixes a b c :: ereal shows "a * b = a * c \<longleftrightarrow>
```
```  2131     ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
```
```  2132   by (cases rule: ereal3_cases[of a b c])
```
```  2133      (simp_all add: zero_less_mult_iff)
```
```  2134
```
```  2135 lemma ereal_inj_affinity:
```
```  2136   fixes m t :: ereal
```
```  2137   assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
```
```  2138   shows "inj_on (\<lambda>x. m * x + t) A"
```
```  2139   using assms
```
```  2140   by (cases rule: ereal2_cases[of m t])
```
```  2141      (auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left)
```
```  2142
```
```  2143 lemma ereal_PInfty_eq_plus[simp]:
```
```  2144   fixes a b :: ereal
```
```  2145   shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
```
```  2146   by (cases rule: ereal2_cases[of a b]) auto
```
```  2147
```
```  2148 lemma ereal_MInfty_eq_plus[simp]:
```
```  2149   fixes a b :: ereal
```
```  2150   shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
```
```  2151   by (cases rule: ereal2_cases[of a b]) auto
```
```  2152
```
```  2153 lemma ereal_less_divide_pos:
```
```  2154   fixes x y :: ereal
```
```  2155   shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
```
```  2156   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  2157
```
```  2158 lemma ereal_divide_less_pos:
```
```  2159   fixes x y z :: ereal
```
```  2160   shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
```
```  2161   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
```
```  2162
```
```  2163 lemma ereal_divide_eq:
```
```  2164   fixes a b c :: ereal
```
```  2165   shows "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
```
```  2166   by (cases rule: ereal3_cases[of a b c])
```
```  2167      (simp_all add: field_simps)
```
```  2168
```
```  2169 lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>"
```
```  2170   by (cases a) auto
```
```  2171
```
```  2172 lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
```
```  2173   by (cases x) auto
```
```  2174
```
```  2175 lemma ereal_LimI_finite:
```
```  2176   fixes x :: ereal
```
```  2177   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  2178   assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
```
```  2179   shows "u ----> x"
```
```  2180 proof (rule topological_tendstoI, unfold eventually_sequentially)
```
```  2181   obtain rx where rx_def: "x=ereal rx" using assms by (cases x) auto
```
```  2182   fix S assume "open S" "x : S"
```
```  2183   then have "open (ereal -` S)" unfolding open_ereal_def by auto
```
```  2184   with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> ereal y \<in> S"
```
```  2185     unfolding open_real_def rx_def by auto
```
```  2186   then obtain n where
```
```  2187     upper: "!!N. n <= N ==> u N < x + ereal r" and
```
```  2188     lower: "!!N. n <= N ==> x < u N + ereal r" using assms(2)[of "ereal r"] by auto
```
```  2189   show "EX N. ALL n>=N. u n : S"
```
```  2190   proof (safe intro!: exI[of _ n])
```
```  2191     fix N assume "n <= N"
```
```  2192     from upper[OF this] lower[OF this] assms `0 < r`
```
```  2193     have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
```
```  2194     then obtain ra where ra_def: "(u N) = ereal ra" by (cases "u N") auto
```
```  2195     hence "rx < ra + r" and "ra < rx + r"
```
```  2196        using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
```
```  2197     hence "dist (real (u N)) rx < r"
```
```  2198       using rx_def ra_def
```
```  2199       by (auto simp: dist_real_def abs_diff_less_iff field_simps)
```
```  2200     from dist[OF this] show "u N : S" using `u N  ~: {\<infinity>, -\<infinity>}`
```
```  2201       by (auto simp: ereal_real split: split_if_asm)
```
```  2202   qed
```
```  2203 qed
```
```  2204
```
```  2205 lemma ereal_LimI_finite_iff:
```
```  2206   fixes x :: ereal
```
```  2207   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
```
```  2208   shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
```
```  2209   (is "?lhs <-> ?rhs")
```
```  2210 proof
```
```  2211   assume lim: "u ----> x"
```
```  2212   { fix r assume "(r::ereal)>0"
```
```  2213     then obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
```
```  2214        apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
```
```  2215        using lim ereal_between[of x r] assms `r>0` by auto
```
```  2216     hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
```
```  2217       using ereal_minus_less[of r x] by (cases r) auto
```
```  2218   } then show "?rhs" by auto
```
```  2219 next
```
```  2220   assume ?rhs then show "u ----> x"
```
```  2221     using ereal_LimI_finite[of x] assms by auto
```
```  2222 qed
```
```  2223
```
```  2224
```
```  2225 subsubsection {* @{text Liminf} and @{text Limsup} *}
```
```  2226
```
```  2227 definition
```
```  2228   "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
```
```  2229
```
```  2230 definition
```
```  2231   "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
```
```  2232
```
```  2233 lemma Liminf_Sup:
```
```  2234   fixes f :: "'a => 'b::complete_linorder"
```
```  2235   shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
```
```  2236   by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
```
```  2237
```
```  2238 lemma Limsup_Inf:
```
```  2239   fixes f :: "'a => 'b::complete_linorder"
```
```  2240   shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
```
```  2241   by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
```
```  2242
```
```  2243 lemma ereal_SupI:
```
```  2244   fixes x :: ereal
```
```  2245   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
```
```  2246   assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
```
```  2247   shows "Sup A = x"
```
```  2248   unfolding Sup_ereal_def
```
```  2249   using assms by (auto intro!: Least_equality)
```
```  2250
```
```  2251 lemma ereal_InfI:
```
```  2252   fixes x :: ereal
```
```  2253   assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
```
```  2254   assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
```
```  2255   shows "Inf A = x"
```
```  2256   unfolding Inf_ereal_def
```
```  2257   using assms by (auto intro!: Greatest_equality)
```
```  2258
```
```  2259 lemma Limsup_const:
```
```  2260   fixes c :: "'a::complete_linorder"
```
```  2261   assumes ntriv: "\<not> trivial_limit net"
```
```  2262   shows "Limsup net (\<lambda>x. c) = c"
```
```  2263   unfolding Limsup_Inf
```
```  2264 proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
```
```  2265   fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
```
```  2266   show "c \<le> x"
```
```  2267   proof (rule ccontr)
```
```  2268     assume "\<not> c \<le> x" then have "x < c" by auto
```
```  2269     then show False using ntriv * by (auto simp: trivial_limit_def)
```
```  2270   qed
```
```  2271 qed auto
```
```  2272
```
```  2273 lemma Liminf_const:
```
```  2274   fixes c :: "'a::complete_linorder"
```
```  2275   assumes ntriv: "\<not> trivial_limit net"
```
```  2276   shows "Liminf net (\<lambda>x. c) = c"
```
```  2277   unfolding Liminf_Sup
```
```  2278 proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
```
```  2279   fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
```
```  2280   show "x \<le> c"
```
```  2281   proof (rule ccontr)
```
```  2282     assume "\<not> x \<le> c" then have "c < x" by auto
```
```  2283     then show False using ntriv * by (auto simp: trivial_limit_def)
```
```  2284   qed
```
```  2285 qed auto
```
```  2286
```
```  2287 definition (in order) mono_set:
```
```  2288   "mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
```
```  2289
```
```  2290 lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
```
```  2291 lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
```
```  2292 lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
```
```  2293 lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
```
```  2294
```
```  2295 lemma (in complete_linorder) mono_set_iff:
```
```  2296   fixes S :: "'a set"
```
```  2297   defines "a \<equiv> Inf S"
```
```  2298   shows "mono_set S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
```
```  2299 proof
```
```  2300   assume "mono_set S"
```
```  2301   then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
```
```  2302   show ?c
```
```  2303   proof cases
```
```  2304     assume "a \<in> S"
```
```  2305     show ?c
```
```  2306       using mono[OF _ `a \<in> S`]
```
```  2307       by (auto intro: Inf_lower simp: a_def)
```
```  2308   next
```
```  2309     assume "a \<notin> S"
```
```  2310     have "S = {a <..}"
```
```  2311     proof safe
```
```  2312       fix x assume "x \<in> S"
```
```  2313       then have "a \<le> x" unfolding a_def by (rule Inf_lower)
```
```  2314       then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
```
```  2315     next
```
```  2316       fix x assume "a < x"
```
```  2317       then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
```
```  2318       with mono[of y x] show "x \<in> S" by auto
```
```  2319     qed
```
```  2320     then show ?c ..
```
```  2321   qed
```
```  2322 qed auto
```
```  2323
```
```  2324 lemma lim_imp_Liminf:
```
```  2325   fixes f :: "'a \<Rightarrow> ereal"
```
```  2326   assumes ntriv: "\<not> trivial_limit net"
```
```  2327   assumes lim: "(f ---> f0) net"
```
```  2328   shows "Liminf net f = f0"
```
```  2329   unfolding Liminf_Sup
```
```  2330 proof (safe intro!: ereal_SupI)
```
```  2331   fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
```
```  2332   show "y \<le> f0"
```
```  2333   proof (rule ereal_le_ereal)
```
```  2334     fix B assume "B < y"
```
```  2335     { assume "f0 < B"
```
```  2336       then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
```
```  2337          using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
```
```  2338          by (auto intro: eventually_conj)
```
```  2339       also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
```
```  2340       finally have False using ntriv[unfolded trivial_limit_def] by auto
```
```  2341     } then show "B \<le> f0" by (metis linorder_le_less_linear)
```
```  2342   qed
```
```  2343 next
```
```  2344   fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
```
```  2345   show "f0 \<le> y"
```
```  2346   proof (safe intro!: *[rule_format])
```
```  2347     fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
```
```  2348       using lim[THEN topological_tendstoD, of "{y <..}"] by auto
```
```  2349   qed
```
```  2350 qed
```
```  2351
```
```  2352 lemma ereal_Liminf_le_Limsup:
```
```  2353   fixes f :: "'a \<Rightarrow> ereal"
```
```  2354   assumes ntriv: "\<not> trivial_limit net"
```
```  2355   shows "Liminf net f \<le> Limsup net f"
```
```  2356   unfolding Limsup_Inf Liminf_Sup
```
```  2357 proof (safe intro!: complete_lattice_class.Inf_greatest  complete_lattice_class.Sup_least)
```
```  2358   fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
```
```  2359   show "u \<le> v"
```
```  2360   proof (rule ccontr)
```
```  2361     assume "\<not> u \<le> v"
```
```  2362     then obtain t where "t < u" "v < t"
```
```  2363       using ereal_dense[of v u] by (auto simp: not_le)
```
```  2364     then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
```
```  2365       using * by (auto intro: eventually_conj)
```
```  2366     also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
```
```  2367     finally show False using ntriv by (auto simp: trivial_limit_def)
```
```  2368   qed
```
```  2369 qed
```
```  2370
```
```  2371 lemma Liminf_mono:
```
```  2372   fixes f g :: "'a => ereal"
```
```  2373   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
```
```  2374   shows "Liminf net f \<le> Liminf net g"
```
```  2375   unfolding Liminf_Sup
```
```  2376 proof (safe intro!: Sup_mono bexI)
```
```  2377   fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
```
```  2378   then have "eventually (\<lambda>x. y < f x) net" by auto
```
```  2379   then show "eventually (\<lambda>x. y < g x) net"
```
```  2380     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
```
```  2381 qed simp
```
```  2382
```
```  2383 lemma Liminf_eq:
```
```  2384   fixes f g :: "'a \<Rightarrow> ereal"
```
```  2385   assumes "eventually (\<lambda>x. f x = g x) net"
```
```  2386   shows "Liminf net f = Liminf net g"
```
```  2387   by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
```
```  2388
```
```  2389 lemma Liminf_mono_all:
```
```  2390   fixes f g :: "'a \<Rightarrow> ereal"
```
```  2391   assumes "\<And>x. f x \<le> g x"
```
```  2392   shows "Liminf net f \<le> Liminf net g"
```
```  2393   using assms by (intro Liminf_mono always_eventually) auto
```
```  2394
```
```  2395 lemma Limsup_mono:
```
```  2396   fixes f g :: "'a \<Rightarrow> ereal"
```
```  2397   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
```
```  2398   shows "Limsup net f \<le> Limsup net g"
```
```  2399   unfolding Limsup_Inf
```
```  2400 proof (safe intro!: Inf_mono bexI)
```
```  2401   fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
```
```  2402   then have "eventually (\<lambda>x. g x < y) net" by auto
```
```  2403   then show "eventually (\<lambda>x. f x < y) net"
```
```  2404     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
```
```  2405 qed simp
```
```  2406
```
```  2407 lemma Limsup_mono_all:
```
```  2408   fixes f g :: "'a \<Rightarrow> ereal"
```
```  2409   assumes "\<And>x. f x \<le> g x"
```
```  2410   shows "Limsup net f \<le> Limsup net g"
```
```  2411   using assms by (intro Limsup_mono always_eventually) auto
```
```  2412
```
```  2413 lemma Limsup_eq:
```
```  2414   fixes f g :: "'a \<Rightarrow> ereal"
```
```  2415   assumes "eventually (\<lambda>x. f x = g x) net"
```
```  2416   shows "Limsup net f = Limsup net g"
```
```  2417   by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
```
```  2418
```
```  2419 abbreviation "liminf \<equiv> Liminf sequentially"
```
```  2420
```
```  2421 abbreviation "limsup \<equiv> Limsup sequentially"
```
```  2422
```
```  2423 lemma liminf_SUPR_INFI:
```
```  2424   fixes f :: "nat \<Rightarrow> ereal"
```
```  2425   shows "liminf f = (SUP n. INF m:{n..}. f m)"
```
```  2426   unfolding Liminf_Sup eventually_sequentially
```
```  2427 proof (safe intro!: antisym complete_lattice_class.Sup_least)
```
```  2428   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)"
```
```  2429   proof (rule ereal_le_ereal)
```
```  2430     fix y assume "y < x"
```
```  2431     with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
```
```  2432     then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
```
```  2433     also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro SUP_upper) auto
```
```  2434     finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
```
```  2435   qed
```
```  2436 next
```
```  2437   show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
```
```  2438   proof (unfold SUP_def, safe intro!: Sup_mono bexI)
```
```  2439     fix y n assume "y < INFI {n..} f"
```
```  2440     from less_INF_D[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
```
```  2441   qed (rule order_refl)
```
```  2442 qed
```
```  2443
```
```  2444 lemma tail_same_limsup:
```
```  2445   fixes X Y :: "nat => ereal"
```
```  2446   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
```
```  2447   shows "limsup X = limsup Y"
```
```  2448   using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
```
```  2449
```
```  2450 lemma tail_same_liminf:
```
```  2451   fixes X Y :: "nat => ereal"
```
```  2452   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
```
```  2453   shows "liminf X = liminf Y"
```
```  2454   using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
```
```  2455
```
```  2456 lemma liminf_mono:
```
```  2457   fixes X Y :: "nat \<Rightarrow> ereal"
```
```  2458   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
```
```  2459   shows "liminf X \<le> liminf Y"
```
```  2460   using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
```
```  2461
```
```  2462 lemma limsup_mono:
```
```  2463   fixes X Y :: "nat => ereal"
```
```  2464   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
```
```  2465   shows "limsup X \<le> limsup Y"
```
```  2466   using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
```
```  2467
```
```  2468 lemma
```
```  2469   fixes X :: "nat \<Rightarrow> ereal"
```
```  2470   shows ereal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
```
```  2471     and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
```
```  2472   unfolding incseq_def decseq_def by auto
```
```  2473
```
```  2474 lemma liminf_bounded:
```
```  2475   fixes X Y :: "nat \<Rightarrow> ereal"
```
```  2476   assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
```
```  2477   shows "C \<le> liminf X"
```
```  2478   using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
```
```  2479
```
```  2480 lemma limsup_bounded:
```
```  2481   fixes X Y :: "nat => ereal"
```
```  2482   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
```
```  2483   shows "limsup X \<le> C"
```
```  2484   using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
```
```  2485
```
```  2486 lemma liminf_bounded_iff:
```
```  2487   fixes x :: "nat \<Rightarrow> ereal"
```
```  2488   shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
```
```  2489 proof safe
```
```  2490   fix B assume "B < C" "C \<le> liminf x"
```
```  2491   then have "B < liminf x" by auto
```
```  2492   then obtain N where "B < (INF m:{N..}. x m)"
```
```  2493     unfolding liminf_SUPR_INFI SUP_def less_Sup_iff by auto
```
```  2494   from less_INF_D[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
```
```  2495 next
```
```  2496   assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
```
```  2497   { fix B assume "B<C"
```
```  2498     then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
```
```  2499     hence "B \<le> (INF m:{N..}. x m)" by (intro INF_greatest) auto
```
```  2500     also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro SUP_upper) simp
```
```  2501     finally have "B \<le> liminf x" .
```
```  2502   } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
```
```  2503 qed
```
```  2504
```
```  2505 lemma liminf_subseq_mono:
```
```  2506   fixes X :: "nat \<Rightarrow> ereal"
```
```  2507   assumes "subseq r"
```
```  2508   shows "liminf X \<le> liminf (X \<circ> r) "
```
```  2509 proof-
```
```  2510   have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
```
```  2511   proof (safe intro!: INF_mono)
```
```  2512     fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
```
```  2513       using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
```
```  2514   qed
```
```  2515   then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
```
```  2516 qed
```
```  2517
```
```  2518 lemma ereal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "ereal (real x) = x"
```
```  2519   using assms by auto
```
```  2520
```
```  2521 lemma ereal_le_ereal_bounded:
```
```  2522   fixes x y z :: ereal
```
```  2523   assumes "z \<le> y"
```
```  2524   assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
```
```  2525   shows "x \<le> y"
```
```  2526 proof (rule ereal_le_ereal)
```
```  2527   fix B assume "B < x"
```
```  2528   show "B \<le> y"
```
```  2529   proof cases
```
```  2530     assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
```
```  2531   next
```
```  2532     assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
```
```  2533   qed
```
```  2534 qed
```
```  2535
```
```  2536 lemma fixes x y :: ereal
```
```  2537   shows Sup_atMost[simp]: "Sup {.. y} = y"
```
```  2538     and Sup_lessThan[simp]: "Sup {..< y} = y"
```
```  2539     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
```
```  2540     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
```
```  2541     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
```
```  2542   by (auto simp: Sup_ereal_def intro!: Least_equality
```
```  2543            intro: ereal_le_ereal ereal_le_ereal_bounded[of x])
```
```  2544
```
```  2545 lemma Sup_greaterThanlessThan[simp]:
```
```  2546   fixes x y :: ereal assumes "x < y" shows "Sup { x <..< y} = y"
```
```  2547   unfolding Sup_ereal_def
```
```  2548 proof (intro Least_equality ereal_le_ereal_bounded[of _ _ y])
```
```  2549   fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
```
```  2550   from ereal_dense[OF `x < y`] guess w .. note w = this
```
```  2551   with z[THEN bspec, of w] show "x \<le> z" by auto
```
```  2552 qed auto
```
```  2553
```
```  2554 lemma real_ereal_id: "real o ereal = id"
```
```  2555 proof-
```
```  2556   { fix x have "(real o ereal) x = id x" by auto }
```
```  2557   then show ?thesis using ext by blast
```
```  2558 qed
```
```  2559
```
```  2560 lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})"
```
```  2561 by (metis range_ereal open_ereal open_UNIV)
```
```  2562
```
```  2563 lemma ereal_le_distrib:
```
```  2564   fixes a b c :: ereal shows "c * (a + b) \<le> c * a + c * b"
```
```  2565   by (cases rule: ereal3_cases[of a b c])
```
```  2566      (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
```
```  2567
```
```  2568 lemma ereal_pos_distrib:
```
```  2569   fixes a b c :: ereal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
```
```  2570   using assms by (cases rule: ereal3_cases[of a b c])
```
```  2571                  (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
```
```  2572
```
```  2573 lemma ereal_pos_le_distrib:
```
```  2574 fixes a b c :: ereal
```
```  2575 assumes "c>=0"
```
```  2576 shows "c * (a + b) <= c * a + c * b"
```
```  2577   using assms by (cases rule: ereal3_cases[of a b c])
```
```  2578                  (auto simp add: field_simps)
```
```  2579
```
```  2580 lemma ereal_max_mono:
```
```  2581   "[| (a::ereal) <= b; c <= d |] ==> max a c <= max b d"
```
```  2582   by (metis sup_ereal_def sup_mono)
```
```  2583
```
```  2584
```
```  2585 lemma ereal_max_least:
```
```  2586   "[| (a::ereal) <= x; c <= x |] ==> max a c <= x"
```
```  2587   by (metis sup_ereal_def sup_least)
```
```  2588
```
```  2589 subsubsection {* Tests for code generator *}
```
```  2590
```
```  2591 (* A small list of simple arithmetic expressions *)
```
```  2592
```
```  2593 value [code] "- \<infinity> :: ereal"
```
```  2594 value [code] "\<bar>-\<infinity>\<bar> :: ereal"
```
```  2595 value [code] "4 + 5 / 4 - ereal 2 :: ereal"
```
```  2596 value [code] "ereal 3 < \<infinity>"
```
```  2597 value [code] "real (\<infinity>::ereal) = 0"
```
```  2598
```
```  2599 end
```