src/HOL/Library/Extended_Real.thy
 author kuncar Fri Dec 09 18:07:04 2011 +0100 (2011-12-09) changeset 45802 b16f976db515 parent 45769 2d5b1af2426a child 45934 9321cd2572fe permissions -rw-r--r--
Quotient_Info stores only relation maps
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 *)
8 header {* Extended real number line *}
10 theory Extended_Real
11 imports Complex_Main Extended_Nat
12 begin
14 text {*
16 For more lemmas about the extended real numbers go to
17   @{text "src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"}
19 *}
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
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)
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)
39 subsection {* Definition and basic properties *}
41 datatype ereal = ereal real | PInfty | MInfty
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
52 instantiation ereal :: infinity
53 begin
54   definition "(\<infinity>::ereal) = PInfty"
55   instance ..
56 end
58 definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
60 declare [[coercion "ereal :: real \<Rightarrow> ereal"]]
61 declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]]
62 declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]]
64 lemma ereal_uminus_uminus[simp]:
65   fixes a :: ereal shows "- (- a) = a"
66   by (cases a) simp_all
68 lemma
69   shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>"
70     and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>"
71     and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)"
72     and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r"
73     and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r"
74     and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y"
75     and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z"
78 declare
79   PInfty_eq_infinity[code_post]
80   MInfty_eq_minfinity[code_post]
82 lemma [code_unfold]:
83   "\<infinity> = PInfty"
84   "-PInfty = MInfty"
85   by simp_all
87 lemma inj_ereal[simp]: "inj_on ereal A"
88   unfolding inj_on_def by auto
90 lemma ereal_cases[case_names real PInf MInf, cases type: ereal]:
91   assumes "\<And>r. x = ereal r \<Longrightarrow> P"
92   assumes "x = \<infinity> \<Longrightarrow> P"
93   assumes "x = -\<infinity> \<Longrightarrow> P"
94   shows P
95   using assms by (cases x) auto
97 lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
98 lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]
100 lemma ereal_uminus_eq_iff[simp]:
101   fixes a b :: ereal shows "-a = -b \<longleftrightarrow> a = b"
102   by (cases rule: ereal2_cases[of a b]) simp_all
104 function of_ereal :: "ereal \<Rightarrow> real" where
105 "of_ereal (ereal r) = r" |
106 "of_ereal \<infinity> = 0" |
107 "of_ereal (-\<infinity>) = 0"
108   by (auto intro: ereal_cases)
109 termination proof qed (rule wf_empty)
112   real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal"
114 lemma real_of_ereal[simp]:
115     "real (- x :: ereal) = - (real x)"
116     "real (ereal r) = r"
117     "real (\<infinity>::ereal) = 0"
118   by (cases x) (simp_all add: real_of_ereal_def)
120 lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}"
121 proof safe
122   fix x assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
123   then show "x = -\<infinity>" by (cases x) auto
124 qed auto
126 lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
127 proof safe
128   fix x :: ereal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
129 qed auto
131 instantiation ereal :: number
132 begin
133 definition [simp]: "number_of x = ereal (number_of x)"
134 instance proof qed
135 end
137 instantiation ereal :: abs
138 begin
139   function abs_ereal where
140     "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
141   | "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
142   | "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
143   by (auto intro: ereal_cases)
144   termination proof qed (rule wf_empty)
145   instance ..
146 end
148 lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
149   by (cases x) auto
151 lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> \<noteq> \<infinity> ; \<And>r. x = ereal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
152   by (cases x) auto
154 lemma abs_ereal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::ereal\<bar>"
155   by (cases x) auto
160 begin
162 definition "0 = ereal 0"
164 function plus_ereal where
165 "ereal r + ereal p = ereal (r + p)" |
166 "\<infinity> + a = (\<infinity>::ereal)" |
167 "a + \<infinity> = (\<infinity>::ereal)" |
168 "ereal r + -\<infinity> = - \<infinity>" |
169 "-\<infinity> + ereal p = -(\<infinity>::ereal)" |
170 "-\<infinity> + -\<infinity> = -(\<infinity>::ereal)"
171 proof -
172   case (goal1 P x)
173   moreover then obtain a b where "x = (a, b)" by (cases x) auto
174   ultimately show P
175    by (cases rule: ereal2_cases[of a b]) auto
176 qed auto
177 termination proof qed (rule wf_empty)
179 lemma Infty_neq_0[simp]:
180   "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
181   "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
184 lemma ereal_eq_0[simp]:
185   "ereal r = 0 \<longleftrightarrow> r = 0"
186   "0 = ereal r \<longleftrightarrow> r = 0"
187   unfolding zero_ereal_def by simp_all
189 instance
190 proof
191   fix a :: ereal show "0 + a = a"
192     by (cases a) (simp_all add: zero_ereal_def)
193   fix b :: ereal show "a + b = b + a"
194     by (cases rule: ereal2_cases[of a b]) simp_all
195   fix c :: ereal show "a + b + c = a + (b + c)"
196     by (cases rule: ereal3_cases[of a b c]) simp_all
197 qed
198 end
200 lemma real_of_ereal_0[simp]: "real (0::ereal) = 0"
201   unfolding real_of_ereal_def zero_ereal_def by simp
203 lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)"
204   unfolding zero_ereal_def abs_ereal.simps by simp
206 lemma ereal_uminus_zero[simp]:
207   "- 0 = (0::ereal)"
210 lemma ereal_uminus_zero_iff[simp]:
211   fixes a :: ereal shows "-a = 0 \<longleftrightarrow> a = 0"
212   by (cases a) simp_all
214 lemma ereal_plus_eq_PInfty[simp]:
215   fixes a b :: ereal shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
216   by (cases rule: ereal2_cases[of a b]) auto
218 lemma ereal_plus_eq_MInfty[simp]:
219   fixes a b :: ereal shows "a + b = -\<infinity> \<longleftrightarrow>
220     (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
221   by (cases rule: ereal2_cases[of a b]) auto
224   fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
225   shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
226   using assms by (cases rule: ereal3_cases[of a b c]) auto
229   fixes a b :: ereal assumes "a \<noteq> -\<infinity>"
230   shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
231   using assms by (cases rule: ereal3_cases[of a b c]) auto
233 lemma ereal_real:
234   "ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
235   by (cases x) simp_all
238   fixes a b :: ereal
239   shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)"
240   by (cases rule: ereal2_cases[of a b]) auto
242 subsubsection "Linear order on @{typ ereal}"
244 instantiation ereal :: linorder
245 begin
247 function less_ereal where
248 "   ereal x < ereal y     \<longleftrightarrow> x < y" |
249 "(\<infinity>::ereal) < a           \<longleftrightarrow> False" |
250 "         a < -(\<infinity>::ereal) \<longleftrightarrow> False" |
251 "ereal x    < \<infinity>           \<longleftrightarrow> True" |
252 "        -\<infinity> < ereal r     \<longleftrightarrow> True" |
253 "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True"
254 proof -
255   case (goal1 P x)
256   moreover then obtain a b where "x = (a,b)" by (cases x) auto
257   ultimately show P by (cases rule: ereal2_cases[of a b]) auto
258 qed simp_all
259 termination by (relation "{}") simp
261 definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"
263 lemma ereal_infty_less[simp]:
264   fixes x :: ereal
265   shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
266     "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
267   by (cases x, simp_all) (cases x, simp_all)
269 lemma ereal_infty_less_eq[simp]:
270   fixes x :: ereal
271   shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
272   "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
273   by (auto simp add: less_eq_ereal_def)
275 lemma ereal_less[simp]:
276   "ereal r < 0 \<longleftrightarrow> (r < 0)"
277   "0 < ereal r \<longleftrightarrow> (0 < r)"
278   "0 < (\<infinity>::ereal)"
279   "-(\<infinity>::ereal) < 0"
282 lemma ereal_less_eq[simp]:
283   "x \<le> (\<infinity>::ereal)"
284   "-(\<infinity>::ereal) \<le> x"
285   "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
286   "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
287   "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
288   by (auto simp add: less_eq_ereal_def zero_ereal_def)
290 lemma ereal_infty_less_eq2:
291   "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
292   "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
293   by simp_all
295 instance
296 proof
297   fix x :: ereal show "x \<le> x"
298     by (cases x) simp_all
299   fix y :: ereal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
300     by (cases rule: ereal2_cases[of x y]) auto
301   show "x \<le> y \<or> y \<le> x "
302     by (cases rule: ereal2_cases[of x y]) auto
303   { assume "x \<le> y" "y \<le> x" then show "x = y"
304     by (cases rule: ereal2_cases[of x y]) auto }
305   { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
306     by (cases rule: ereal3_cases[of x y z]) auto }
307 qed
308 end
311 proof
312   fix a b c :: ereal assume "a \<le> b" then show "c + a \<le> c + b"
313     by (cases rule: ereal3_cases[of a b c]) auto
314 qed
316 lemma real_of_ereal_positive_mono:
317   fixes x y :: ereal shows "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
318   by (cases rule: ereal2_cases[of x y]) auto
320 lemma ereal_MInfty_lessI[intro, simp]:
321   fixes a :: ereal shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
322   by (cases a) auto
324 lemma ereal_less_PInfty[intro, simp]:
325   fixes a :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
326   by (cases a) auto
328 lemma ereal_less_ereal_Ex:
329   fixes a b :: ereal
330   shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
331   by (cases x) auto
333 lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))"
334 proof (cases x)
335   case (real r) then show ?thesis
336     using reals_Archimedean2[of r] by simp
337 qed simp_all
340   fixes a b c d :: ereal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
341   using assms
342   apply (cases a)
343   apply (cases rule: ereal3_cases[of b c d], auto)
344   apply (cases rule: ereal3_cases[of b c d], auto)
345   done
347 lemma ereal_minus_le_minus[simp]:
348   fixes a b :: ereal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
349   by (cases rule: ereal2_cases[of a b]) auto
351 lemma ereal_minus_less_minus[simp]:
352   fixes a b :: ereal shows "- a < - b \<longleftrightarrow> b < a"
353   by (cases rule: ereal2_cases[of a b]) auto
355 lemma ereal_le_real_iff:
356   "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
357   by (cases y) auto
359 lemma real_le_ereal_iff:
360   "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
361   by (cases y) auto
363 lemma ereal_less_real_iff:
364   "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
365   by (cases y) auto
367 lemma real_less_ereal_iff:
368   "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
369   by (cases y) auto
371 lemma real_of_ereal_pos:
372   fixes x :: ereal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
374 lemmas real_of_ereal_ord_simps =
375   ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff
377 lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
378   by (cases x) auto
380 lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
381   by (cases x) auto
383 lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
384   by (cases x) auto
386 lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> (x \<le> 0 \<or> x = \<infinity>)"
387   by (cases x) auto
389 lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>"
390   by (cases x) auto
392 lemma zero_less_real_of_ereal:
393   fixes x :: ereal shows "0 < real x \<longleftrightarrow> (0 < x \<and> x \<noteq> \<infinity>)"
394   by (cases x) auto
396 lemma ereal_0_le_uminus_iff[simp]:
397   fixes a :: ereal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
398   by (cases rule: ereal2_cases[of a]) auto
400 lemma ereal_uminus_le_0_iff[simp]:
401   fixes a :: ereal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a"
402   by (cases rule: ereal2_cases[of a]) auto
404 lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y"
405   using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto
407 lemma ereal_dense:
408   fixes x y :: ereal assumes "x < y"
409   shows "\<exists>z. x < z \<and> z < y"
410   using ereal_dense2[OF `x < y`] by blast
413   fixes a b c d :: ereal
414   assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
415   shows "a + c < b + d"
416   using assms by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto
419   fixes a b c :: ereal shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
420   by (cases rule: ereal2_cases[of b c]) auto
422 lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" by auto
424 lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)"
425   by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)
427 lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)"
428   by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)
430 lemmas ereal_uminus_reorder =
431   ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder
433 lemma ereal_bot:
434   fixes x :: ereal assumes "\<And>B. x \<le> ereal B" shows "x = - \<infinity>"
435 proof (cases x)
436   case (real r) with assms[of "r - 1"] show ?thesis by auto
437 next case PInf with assms[of 0] show ?thesis by auto
438 next case MInf then show ?thesis by simp
439 qed
441 lemma ereal_top:
442   fixes x :: ereal assumes "\<And>B. x \<ge> ereal B" shows "x = \<infinity>"
443 proof (cases x)
444   case (real r) with assms[of "r + 1"] show ?thesis by auto
445 next case MInf with assms[of 0] show ?thesis by auto
446 next case PInf then show ?thesis by simp
447 qed
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)
454 lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
455   by (auto simp: zero_ereal_def)
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
463 lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
464   unfolding incseq_def by auto
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
470 lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
471   by auto
473 lemma incseq_setsumI:
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
484 lemma incseq_setsumI2:
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)
490 subsubsection "Multiplication"
492 instantiation ereal :: "{comm_monoid_mult, sgn}"
493 begin
495 definition "1 = ereal 1"
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)
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
521 instance
522 proof
523   fix a :: ereal show "1 * a = a"
524     by (cases a) (simp_all add: one_ereal_def)
525   fix b :: ereal show "a * b = b * a"
526     by (cases rule: ereal2_cases[of a b]) simp_all
527   fix c :: ereal show "a * b * c = a * (b * c)"
528     by (cases rule: ereal3_cases[of a b c])
530 qed
531 end
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)
537 lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)"
538   unfolding one_ereal_def by simp
540 lemma ereal_mult_zero[simp]:
541   fixes a :: ereal shows "a * 0 = 0"
542   by (cases a) (simp_all add: zero_ereal_def)
544 lemma ereal_zero_mult[simp]:
545   fixes a :: ereal shows "0 * a = 0"
546   by (cases a) (simp_all add: zero_ereal_def)
548 lemma ereal_m1_less_0[simp]:
549   "-(1::ereal) < 0"
550   by (simp add: zero_ereal_def one_ereal_def)
552 lemma ereal_zero_m1[simp]:
553   "1 \<noteq> (0::ereal)"
554   by (simp add: zero_ereal_def one_ereal_def)
556 lemma ereal_times_0[simp]:
557   fixes x :: ereal shows "0 * x = 0"
558   by (cases x) (auto simp: zero_ereal_def)
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)
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
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
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
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
584 lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
585   by (simp_all add: zero_ereal_def one_ereal_def)
587 lemma ereal_zero_one[simp]: "0 \<noteq> (1::ereal)"
588   by (simp_all add: zero_ereal_def one_ereal_def)
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
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
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
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
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)
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])
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)
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])
628 lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)"
629   by (simp add: one_ereal_def zero_ereal_def)
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)
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)
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)
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)
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)
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)
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)
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)
669 lemma ereal_le_epsilon:
670   fixes x y :: ereal
671   assumes "ALL e. 0 < e --> x <= y + e"
672   shows "x <= y"
673 proof-
674 { assume a: "EX r. y = ereal r"
675   from this obtain r where r_def: "y = ereal r" by auto
676   { assume "x=(-\<infinity>)" hence ?thesis by auto }
677   moreover
678   { assume "~(x=(-\<infinity>))"
679     from this obtain p where p_def: "x = ereal p"
680     using a assms[rule_format, of 1] by (cases x) auto
681     { fix e have "0 < e --> p <= r + e"
682       using assms[rule_format, of "ereal e"] p_def r_def by auto }
683     hence "p <= r" apply (subst field_le_epsilon) by auto
684     hence ?thesis using r_def p_def by auto
685   } ultimately have ?thesis by blast
686 }
687 moreover
688 { assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
689     using assms[rule_format, of 1] by (cases x) auto
690 } ultimately show ?thesis by (cases y) auto
691 qed
694 lemma ereal_le_epsilon2:
695   fixes x y :: ereal
696   assumes "ALL e. 0 < e --> x <= y + ereal e"
697   shows "x <= y"
698 proof-
699 { fix e :: ereal assume "e>0"
700   { assume "e=\<infinity>" hence "x<=y+e" by auto }
701   moreover
702   { assume "e~=\<infinity>"
703     from this obtain r where "e = ereal r" using `e>0` apply (cases e) by auto
704     hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
705   } ultimately have "x<=y+e" by blast
706 } from this show ?thesis using ereal_le_epsilon by auto
707 qed
709 lemma ereal_le_real:
710   fixes x y :: ereal
711   assumes "ALL z. x <= ereal z --> y <= ereal z"
712   shows "y <= x"
713 by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)
715 lemma ereal_le_ereal:
716   fixes x y :: ereal
717   assumes "\<And>B. B < x \<Longrightarrow> B <= y"
718   shows "x <= y"
719 by (metis assms ereal_dense leD linorder_le_less_linear)
721 lemma ereal_ge_ereal:
722   fixes x y :: ereal
723   assumes "ALL B. B>x --> B >= y"
724   shows "x >= y"
725 by (metis assms ereal_dense leD linorder_le_less_linear)
727 lemma setprod_ereal_0:
728   fixes f :: "'a \<Rightarrow> ereal"
729   shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
730 proof cases
731   assume "finite A"
732   then show ?thesis by (induct A) auto
733 qed auto
735 lemma setprod_ereal_pos:
736   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)"
737 proof cases
738   assume "finite I" from this pos show ?thesis by induct auto
739 qed simp
741 lemma setprod_PInf:
742   fixes f :: "'a \<Rightarrow> ereal"
743   assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
744   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)"
745 proof cases
746   assume "finite I" from this assms show ?thesis
747   proof (induct I)
748     case (insert i I)
749     then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_ereal_pos)
750     from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
751     also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
752       using setprod_ereal_pos[of I f] pos
753       by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto
754     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)"
755       using insert by (auto simp: setprod_ereal_0)
756     finally show ?case .
757   qed simp
758 qed simp
760 lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)"
761 proof cases
762   assume "finite A" then show ?thesis
763     by induct (auto simp: one_ereal_def)
766 subsubsection {* Power *}
768 lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
769   by (induct n) (auto simp: one_ereal_def)
771 lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
772   by (induct n) (auto simp: one_ereal_def)
774 lemma ereal_power_uminus[simp]:
775   fixes x :: ereal
776   shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
777   by (induct n) (auto simp: one_ereal_def)
779 lemma ereal_power_number_of[simp]:
780   "(number_of num :: ereal) ^ n = ereal (number_of num ^ n)"
781   by (induct n) (auto simp: one_ereal_def)
783 lemma zero_le_power_ereal[simp]:
784   fixes a :: ereal assumes "0 \<le> a"
785   shows "0 \<le> a ^ n"
786   using assms by (induct n) (auto simp: ereal_zero_le_0_iff)
788 subsubsection {* Subtraction *}
790 lemma ereal_minus_minus_image[simp]:
791   fixes S :: "ereal set"
792   shows "uminus ` uminus ` S = S"
793   by (auto simp: image_iff)
795 lemma ereal_uminus_lessThan[simp]:
796   fixes a :: ereal shows "uminus ` {..<a} = {-a<..}"
797 proof (safe intro!: image_eqI)
798   fix x assume "-a < x"
799   then have "- x < - (- a)" by (simp del: ereal_uminus_uminus)
800   then show "- x < a" by simp
801 qed auto
803 lemma ereal_uminus_greaterThan[simp]:
804   "uminus ` {(a::ereal)<..} = {..<-a}"
805   by (metis ereal_uminus_lessThan ereal_uminus_uminus
806             ereal_minus_minus_image)
808 instantiation ereal :: minus
809 begin
810 definition "x - y = x + -(y::ereal)"
811 instance ..
812 end
814 lemma ereal_minus[simp]:
815   "ereal r - ereal p = ereal (r - p)"
816   "-\<infinity> - ereal r = -\<infinity>"
817   "ereal r - \<infinity> = -\<infinity>"
818   "(\<infinity>::ereal) - x = \<infinity>"
819   "-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
820   "x - -y = x + y"
821   "x - 0 = x"
822   "0 - x = -x"
825 lemma ereal_x_minus_x[simp]:
826   "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
827   by (cases x) simp_all
829 lemma ereal_eq_minus_iff:
830   fixes x y z :: ereal
831   shows "x = z - y \<longleftrightarrow>
832     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
833     (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
834     (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
835     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
836   by (cases rule: ereal3_cases[of x y z]) auto
838 lemma ereal_eq_minus:
839   fixes x y z :: ereal
840   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
841   by (auto simp: ereal_eq_minus_iff)
843 lemma ereal_less_minus_iff:
844   fixes x y z :: ereal
845   shows "x < z - y \<longleftrightarrow>
846     (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
847     (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
848     (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
849   by (cases rule: ereal3_cases[of x y z]) auto
851 lemma ereal_less_minus:
852   fixes x y z :: ereal
853   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
854   by (auto simp: ereal_less_minus_iff)
856 lemma ereal_le_minus_iff:
857   fixes x y z :: ereal
858   shows "x \<le> z - y \<longleftrightarrow>
859     (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
860     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
861   by (cases rule: ereal3_cases[of x y z]) auto
863 lemma ereal_le_minus:
864   fixes x y z :: ereal
865   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
866   by (auto simp: ereal_le_minus_iff)
868 lemma ereal_minus_less_iff:
869   fixes x y z :: ereal
870   shows "x - y < z \<longleftrightarrow>
871     y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
872     (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
873   by (cases rule: ereal3_cases[of x y z]) auto
875 lemma ereal_minus_less:
876   fixes x y z :: ereal
877   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
878   by (auto simp: ereal_minus_less_iff)
880 lemma ereal_minus_le_iff:
881   fixes x y z :: ereal
882   shows "x - y \<le> z \<longleftrightarrow>
883     (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
884     (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
885     (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
886   by (cases rule: ereal3_cases[of x y z]) auto
888 lemma ereal_minus_le:
889   fixes x y z :: ereal
890   shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
891   by (auto simp: ereal_minus_le_iff)
893 lemma ereal_minus_eq_minus_iff:
894   fixes a b c :: ereal
895   shows "a - b = a - c \<longleftrightarrow>
896     b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
897   by (cases rule: ereal3_cases[of a b c]) auto
900   fixes a b c :: ereal
901   shows "c + a \<le> c + b \<longleftrightarrow>
902     a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
903   by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)
905 lemma ereal_mult_le_mult_iff:
906   fixes a b c :: ereal
907   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)"
908   by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
910 lemma ereal_minus_mono:
911   fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C"
912   shows "A - C \<le> B - D"
913   using assms
914   by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all
916 lemma real_of_ereal_minus:
917   fixes a b :: ereal
918   shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
919   by (cases rule: ereal2_cases[of a b]) auto
921 lemma ereal_diff_positive:
922   fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
923   by (cases rule: ereal2_cases[of a b]) auto
925 lemma ereal_between:
926   fixes x e :: ereal
927   assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
928   shows "x - e < x" "x < x + e"
929 using assms apply (cases x, cases e) apply auto
930 using assms by (cases x, cases e) auto
932 subsubsection {* Division *}
934 instantiation ereal :: inverse
935 begin
937 function inverse_ereal where
938 "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))" |
939 "inverse (\<infinity>::ereal) = 0" |
940 "inverse (-\<infinity>::ereal) = 0"
941   by (auto intro: ereal_cases)
942 termination by (relation "{}") simp
944 definition "x / y = x * inverse (y :: ereal)"
946 instance proof qed
947 end
949 lemma real_of_ereal_inverse[simp]:
950   fixes a :: ereal
951   shows "real (inverse a) = 1 / real a"
952   by (cases a) (auto simp: inverse_eq_divide)
954 lemma ereal_inverse[simp]:
955   "inverse (0::ereal) = \<infinity>"
956   "inverse (1::ereal) = 1"
957   by (simp_all add: one_ereal_def zero_ereal_def)
959 lemma ereal_divide[simp]:
960   "ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))"
961   unfolding divide_ereal_def by (auto simp: divide_real_def)
963 lemma ereal_divide_same[simp]:
964   fixes x :: ereal shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
965   by (cases x)
966      (simp_all add: divide_real_def divide_ereal_def one_ereal_def)
968 lemma ereal_inv_inv[simp]:
969   fixes x :: ereal shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
970   by (cases x) auto
972 lemma ereal_inverse_minus[simp]:
973   fixes x :: ereal shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
974   by (cases x) simp_all
976 lemma ereal_uminus_divide[simp]:
977   fixes x y :: ereal shows "- x / y = - (x / y)"
978   unfolding divide_ereal_def by simp
980 lemma ereal_divide_Infty[simp]:
981   fixes x :: ereal shows "x / \<infinity> = 0" "x / -\<infinity> = 0"
982   unfolding divide_ereal_def by simp_all
984 lemma ereal_divide_one[simp]:
985   "x / 1 = (x::ereal)"
986   unfolding divide_ereal_def by simp
988 lemma ereal_divide_ereal[simp]:
989   "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
990   unfolding divide_ereal_def by simp
992 lemma zero_le_divide_ereal[simp]:
993   fixes a :: ereal assumes "0 \<le> a" "0 \<le> b"
994   shows "0 \<le> a / b"
995   using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)
997 lemma ereal_le_divide_pos:
998   fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
999   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
1001 lemma ereal_divide_le_pos:
1002   fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
1003   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
1005 lemma ereal_le_divide_neg:
1006   fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
1007   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
1009 lemma ereal_divide_le_neg:
1010   fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
1011   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
1013 lemma ereal_inverse_antimono_strict:
1014   fixes x y :: ereal
1015   shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
1016   by (cases rule: ereal2_cases[of x y]) auto
1018 lemma ereal_inverse_antimono:
1019   fixes x y :: ereal
1020   shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
1021   by (cases rule: ereal2_cases[of x y]) auto
1023 lemma inverse_inverse_Pinfty_iff[simp]:
1024   fixes x :: ereal shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
1025   by (cases x) auto
1027 lemma ereal_inverse_eq_0:
1028   fixes x :: ereal shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
1029   by (cases x) auto
1031 lemma ereal_0_gt_inverse:
1032   fixes x :: ereal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
1033   by (cases x) auto
1035 lemma ereal_mult_less_right:
1036   fixes a b c :: ereal
1037   assumes "b * a < c * a" "0 < a" "a < \<infinity>"
1038   shows "b < c"
1039   using assms
1040   by (cases rule: ereal3_cases[of a b c])
1041      (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
1043 lemma ereal_power_divide:
1044   fixes x y :: ereal shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n"
1045   by (cases rule: ereal2_cases[of x y])
1046      (auto simp: one_ereal_def zero_ereal_def power_divide not_le
1047                  power_less_zero_eq zero_le_power_iff)
1049 lemma ereal_le_mult_one_interval:
1050   fixes x y :: ereal
1051   assumes y: "y \<noteq> -\<infinity>"
1052   assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
1053   shows "x \<le> y"
1054 proof (cases x)
1055   case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_ereal_def)
1056 next
1057   case (real r) note r = this
1058   show "x \<le> y"
1059   proof (cases y)
1060     case (real p) note p = this
1061     have "r \<le> p"
1062     proof (rule field_le_mult_one_interval)
1063       fix z :: real assume "0 < z" and "z < 1"
1064       with z[of "ereal z"]
1065       show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_ereal_def)
1066     qed
1067     then show "x \<le> y" using p r by simp
1068   qed (insert y, simp_all)
1069 qed simp
1071 subsection "Complete lattice"
1073 instantiation ereal :: lattice
1074 begin
1075 definition [simp]: "sup x y = (max x y :: ereal)"
1076 definition [simp]: "inf x y = (min x y :: ereal)"
1077 instance proof qed simp_all
1078 end
1080 instantiation ereal :: complete_lattice
1081 begin
1083 definition "bot = (-\<infinity>::ereal)"
1084 definition "top = (\<infinity>::ereal)"
1086 definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: ereal)"
1087 definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: ereal)"
1089 lemma ereal_complete_Sup:
1090   fixes S :: "ereal set" assumes "S \<noteq> {}"
1091   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
1092 proof cases
1093   assume "\<exists>x. \<forall>a\<in>S. a \<le> ereal x"
1094   then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" by auto
1095   then have "\<infinity> \<notin> S" by force
1096   show ?thesis
1097   proof cases
1098     assume "S = {-\<infinity>}"
1099     then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
1100   next
1101     assume "S \<noteq> {-\<infinity>}"
1102     with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
1103     with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
1104       by (auto simp: real_of_ereal_ord_simps)
1105     with complete_real[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
1106     obtain s where s:
1107        "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
1108        by auto
1109     show ?thesis
1110     proof (safe intro!: exI[of _ "ereal s"])
1111       fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> ereal s"
1112       proof (cases z)
1113         case (real r)
1114         then show ?thesis
1115           using s(1)[rule_format, of z] `z \<in> S` `z = ereal r` by auto
1116       qed auto
1117     next
1118       fix z assume *: "\<forall>y\<in>S. y \<le> z"
1119       with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "ereal s \<le> z"
1120       proof (cases z)
1121         case (real u)
1122         with * have "s \<le> u"
1123           by (intro s(2)[of u]) (auto simp: real_of_ereal_ord_simps)
1124         then show ?thesis using real by simp
1125       qed auto
1126     qed
1127   qed
1128 next
1129   assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> ereal x)"
1130   show ?thesis
1131   proof (safe intro!: exI[of _ \<infinity>])
1132     fix y assume **: "\<forall>z\<in>S. z \<le> y"
1133     with * show "\<infinity> \<le> y"
1134     proof (cases y)
1135       case MInf with * ** show ?thesis by (force simp: not_le)
1136     qed auto
1137   qed simp
1138 qed
1140 lemma ereal_complete_Inf:
1141   fixes S :: "ereal set" assumes "S ~= {}"
1142   shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
1143 proof-
1144 def S1 == "uminus ` S"
1145 hence "S1 ~= {}" using assms by auto
1146 from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
1147    using ereal_complete_Sup[of S1] by auto
1148 { fix z assume "ALL y:S. z <= y"
1149   hence "ALL y:S1. y <= -z" unfolding S1_def by auto
1150   hence "x <= -z" using x_def by auto
1151   hence "z <= -x"
1152     apply (subst ereal_uminus_uminus[symmetric])
1153     unfolding ereal_minus_le_minus . }
1154 moreover have "(ALL y:S. -x <= y)"
1155    using x_def unfolding S1_def
1156    apply simp
1157    apply (subst (3) ereal_uminus_uminus[symmetric])
1158    unfolding ereal_minus_le_minus by simp
1159 ultimately show ?thesis by auto
1160 qed
1162 lemma ereal_complete_uminus_eq:
1163   fixes S :: "ereal set"
1164   shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
1165      \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
1166   by simp (metis ereal_minus_le_minus ereal_uminus_uminus)
1168 lemma ereal_Sup_uminus_image_eq:
1169   fixes S :: "ereal set"
1170   shows "Sup (uminus ` S) = - Inf S"
1171 proof cases
1172   assume "S = {}"
1173   moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::ereal)"
1174     by (rule the_equality) (auto intro!: ereal_bot)
1175   moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::ereal)"
1176     by (rule some_equality) (auto intro!: ereal_top)
1177   ultimately show ?thesis unfolding Inf_ereal_def Sup_ereal_def
1178     Least_def Greatest_def GreatestM_def by simp
1179 next
1180   assume "S \<noteq> {}"
1181   with ereal_complete_Sup[of "uminus`S"]
1182   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)"
1183     unfolding ereal_complete_uminus_eq by auto
1184   show "Sup (uminus ` S) = - Inf S"
1185     unfolding Inf_ereal_def Greatest_def GreatestM_def
1186   proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
1187     show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
1188       using x .
1189     fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
1190     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)"
1191       unfolding ereal_complete_uminus_eq by simp
1192     then show "Sup (uminus ` S) = -x'"
1193       unfolding Sup_ereal_def ereal_uminus_eq_iff
1194       by (intro Least_equality) auto
1195   qed
1196 qed
1198 instance
1199 proof
1200   { fix x :: ereal and A
1201     show "bot <= x" by (cases x) (simp_all add: bot_ereal_def)
1202     show "x <= top" by (simp add: top_ereal_def) }
1204   { fix x :: ereal and A assume "x : A"
1205     with ereal_complete_Sup[of A]
1206     obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
1207     hence "x <= s" using `x : A` by auto
1208     also have "... = Sup A" using s unfolding Sup_ereal_def
1209       by (auto intro!: Least_equality[symmetric])
1210     finally show "x <= Sup A" . }
1211   note le_Sup = this
1213   { fix x :: ereal and A assume *: "!!z. (z : A ==> z <= x)"
1214     show "Sup A <= x"
1215     proof (cases "A = {}")
1216       case True
1217       hence "Sup A = -\<infinity>" unfolding Sup_ereal_def
1218         by (auto intro!: Least_equality)
1219       thus "Sup A <= x" by simp
1220     next
1221       case False
1222       with ereal_complete_Sup[of A]
1223       obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
1224       hence "Sup A = s"
1225         unfolding Sup_ereal_def by (auto intro!: Least_equality)
1226       also have "s <= x" using * s by auto
1227       finally show "Sup A <= x" .
1228     qed }
1229   note Sup_le = this
1231   { fix x :: ereal and A assume "x \<in> A"
1232     with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
1233       unfolding ereal_Sup_uminus_image_eq by simp }
1235   { fix x :: ereal and A assume *: "!!z. (z : A ==> x <= z)"
1236     with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
1237       unfolding ereal_Sup_uminus_image_eq by force }
1238 qed
1240 end
1242 instance ereal :: complete_linorder ..
1244 lemma ereal_SUPR_uminus:
1245   fixes f :: "'a => ereal"
1246   shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
1247   unfolding SUP_def INF_def
1248   using ereal_Sup_uminus_image_eq[of "f`R"]
1251 lemma ereal_INFI_uminus:
1252   fixes f :: "'a => ereal"
1253   shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
1254   using ereal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
1256 lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::ereal set)"
1257   using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
1259 lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)"
1260   by (auto intro!: inj_onI)
1262 lemma ereal_image_uminus_shift:
1263   fixes X Y :: "ereal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
1264 proof
1265   assume "uminus ` X = Y"
1266   then have "uminus ` uminus ` X = uminus ` Y"
1268   then show "X = uminus ` Y" by (simp add: image_image)
1271 lemma Inf_ereal_iff:
1272   fixes z :: ereal
1273   shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
1274   by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
1275             order_less_le_trans)
1277 lemma Sup_eq_MInfty:
1278   fixes S :: "ereal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
1279 proof
1280   assume a: "Sup S = -\<infinity>"
1281   with complete_lattice_class.Sup_upper[of _ S]
1282   show "S={} \<or> S={-\<infinity>}" by auto
1283 next
1284   assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
1285     unfolding Sup_ereal_def by (auto intro!: Least_equality)
1286 qed
1288 lemma Inf_eq_PInfty:
1289   fixes S :: "ereal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
1290   using Sup_eq_MInfty[of "uminus`S"]
1291   unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp
1293 lemma Inf_eq_MInfty:
1294   fixes S :: "ereal set" shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
1295   unfolding Inf_ereal_def
1296   by (auto intro!: Greatest_equality)
1298 lemma Sup_eq_PInfty:
1299   fixes S :: "ereal set" shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
1300   unfolding Sup_ereal_def
1301   by (auto intro!: Least_equality)
1303 lemma ereal_SUPI:
1304   fixes x :: ereal
1305   assumes "!!i. i : A ==> f i <= x"
1306   assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
1307   shows "(SUP i:A. f i) = x"
1308   unfolding SUP_def Sup_ereal_def
1309   using assms by (auto intro!: Least_equality)
1311 lemma ereal_INFI:
1312   fixes x :: ereal
1313   assumes "!!i. i : A ==> f i >= x"
1314   assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
1315   shows "(INF i:A. f i) = x"
1316   unfolding INF_def Inf_ereal_def
1317   using assms by (auto intro!: Greatest_equality)
1319 lemma Sup_ereal_close:
1320   fixes e :: ereal
1321   assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
1322   shows "\<exists>x\<in>S. Sup S - e < x"
1323   using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
1325 lemma Inf_ereal_close:
1326   fixes e :: ereal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
1327   shows "\<exists>x\<in>X. x < Inf X + e"
1328 proof (rule Inf_less_iff[THEN iffD1])
1329   show "Inf X < Inf X + e" using assms
1330     by (cases e) auto
1331 qed
1333 lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>"
1334 proof -
1335   { fix x ::ereal assume "x \<noteq> \<infinity>"
1336     then have "\<exists>k::nat. x < ereal (real k)"
1337     proof (cases x)
1338       case MInf then show ?thesis by (intro exI[of _ 0]) auto
1339     next
1340       case (real r)
1341       moreover obtain k :: nat where "r < real k"
1342         using ex_less_of_nat by (auto simp: real_eq_of_nat)
1343       ultimately show ?thesis by auto
1344     qed simp }
1345   then show ?thesis
1346     using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. ereal (real n)"]
1347     by (auto simp: top_ereal_def)
1348 qed
1350 lemma ereal_le_Sup:
1351   fixes x :: ereal
1352   shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
1353 (is "?lhs <-> ?rhs")
1354 proof-
1355 { assume "?rhs"
1356   { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
1357     from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using ereal_dense by auto
1358     from this obtain i where "i : A & y <= f i" using `?rhs` by auto
1359     hence "y <= (SUP i:A. f i)" using SUP_upper[of i A f] by auto
1360     hence False using y_def by auto
1361   } hence "?lhs" by auto
1362 }
1363 moreover
1364 { assume "?lhs" hence "?rhs"
1365   by (metis less_SUP_iff order_less_imp_le order_less_le_trans)
1366 } ultimately show ?thesis by auto
1367 qed
1369 lemma ereal_Inf_le:
1370   fixes x :: ereal
1371   shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
1372 (is "?lhs <-> ?rhs")
1373 proof-
1374 { assume "?rhs"
1375   { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
1376     from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using ereal_dense by auto
1377     from this obtain i where "i : A & f i <= y" using `?rhs` by auto
1378     hence "(INF i:A. f i) <= y" using INF_lower[of i A f] by auto
1379     hence False using y_def by auto
1380   } hence "?lhs" by auto
1381 }
1382 moreover
1383 { assume "?lhs" hence "?rhs"
1384   by (metis INF_less_iff order_le_less order_less_le_trans)
1385 } ultimately show ?thesis by auto
1386 qed
1388 lemma Inf_less:
1389   fixes x :: ereal
1390   assumes "(INF i:A. f i) < x"
1391   shows "EX i. i : A & f i <= x"
1392 proof(rule ccontr)
1393   assume "~ (EX i. i : A & f i <= x)"
1394   hence "ALL i:A. f i > x" by auto
1395   hence "(INF i:A. f i) >= x" apply (subst INF_greatest) by auto
1396   thus False using assms by auto
1397 qed
1399 lemma same_INF:
1400   assumes "ALL e:A. f e = g e"
1401   shows "(INF e:A. f e) = (INF e:A. g e)"
1402 proof-
1403 have "f ` A = g ` A" unfolding image_def using assms by auto
1404 thus ?thesis unfolding INF_def by auto
1405 qed
1407 lemma same_SUP:
1408   assumes "ALL e:A. f e = g e"
1409   shows "(SUP e:A. f e) = (SUP e:A. g e)"
1410 proof-
1411 have "f ` A = g ` A" unfolding image_def using assms by auto
1412 thus ?thesis unfolding SUP_def by auto
1413 qed
1415 lemma SUPR_eq:
1416   assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
1417   assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
1418   shows "(SUP i:A. f i) = (SUP j:B. g j)"
1419 proof (intro antisym)
1420   show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
1421     using assms by (metis SUP_least SUP_upper2)
1422   show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
1423     using assms by (metis SUP_least SUP_upper2)
1424 qed
1427   fixes f :: "'i \<Rightarrow> ereal"
1428   assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
1429   shows "SUPR UNIV f + y \<le> z"
1430 proof (cases y)
1431   case (real r)
1432   then have "\<And>i. f i \<le> z - y" using assms by (simp add: ereal_le_minus_iff)
1433   then have "SUPR UNIV f \<le> z - y" by (rule SUP_least)
1434   then show ?thesis using real by (simp add: ereal_le_minus_iff)
1435 qed (insert assms, auto)
1438   fixes f g :: "nat \<Rightarrow> ereal"
1439   assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
1440   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
1441 proof (rule ereal_SUPI)
1442   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
1443   have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
1444     unfolding SUP_def Sup_eq_MInfty by (auto dest: image_eqD)
1445   { fix j
1446     { fix i
1447       have "f i + g j \<le> f i + g (max i j)"
1448         using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
1449       also have "\<dots> \<le> f (max i j) + g (max i j)"
1450         using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
1451       also have "\<dots> \<le> y" using * by auto
1452       finally have "f i + g j \<le> y" . }
1453     then have "SUPR UNIV f + g j \<le> y"
1454       using assms(4)[of j] by (intro SUP_ereal_le_addI) auto
1455     then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
1456   then have "SUPR UNIV g + SUPR UNIV f \<le> y"
1457     using f by (rule SUP_ereal_le_addI)
1458   then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
1459 qed (auto intro!: add_mono SUP_upper)
1462   fixes f g :: "nat \<Rightarrow> ereal"
1463   assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
1464   shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
1466   fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
1467 qed
1469 lemma SUPR_ereal_setsum:
1470   fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
1471   assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
1472   shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
1473 proof cases
1474   assume "finite A" then show ?thesis using assms
1475     by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_ereal_add_pos)
1476 qed simp
1478 lemma SUPR_ereal_cmult:
1479   fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
1480   shows "(SUP i. c * f i) = c * SUPR UNIV f"
1481 proof (rule ereal_SUPI)
1482   fix i have "f i \<le> SUPR UNIV f" by (rule SUP_upper) auto
1483   then show "c * f i \<le> c * SUPR UNIV f"
1484     using `0 \<le> c` by (rule ereal_mult_left_mono)
1485 next
1486   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
1487   show "c * SUPR UNIV f \<le> y"
1488   proof cases
1489     assume c: "0 < c \<and> c \<noteq> \<infinity>"
1490     with * have "SUPR UNIV f \<le> y / c"
1491       by (intro SUP_least) (auto simp: ereal_le_divide_pos)
1492     with c show ?thesis
1493       by (auto simp: ereal_le_divide_pos)
1494   next
1495     { assume "c = \<infinity>" have ?thesis
1496       proof cases
1497         assume "\<forall>i. f i = 0"
1498         moreover then have "range f = {0}" by auto
1499         ultimately show "c * SUPR UNIV f \<le> y" using *
1500           by (auto simp: SUP_def min_max.sup_absorb1)
1501       next
1502         assume "\<not> (\<forall>i. f i = 0)"
1503         then obtain i where "f i \<noteq> 0" by auto
1504         with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
1505       qed }
1506     moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
1507     ultimately show ?thesis using * `0 \<le> c` by auto
1508   qed
1509 qed
1511 lemma SUP_PInfty:
1512   fixes f :: "'a \<Rightarrow> ereal"
1513   assumes "\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i"
1514   shows "(SUP i:A. f i) = \<infinity>"
1515   unfolding SUP_def Sup_eq_top_iff[where 'a=ereal, unfolded top_ereal_def]
1516   apply simp
1517 proof safe
1518   fix x :: ereal assume "x \<noteq> \<infinity>"
1519   show "\<exists>i\<in>A. x < f i"
1520   proof (cases x)
1521     case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
1522   next
1523     case MInf with assms[of "0"] show ?thesis by force
1524   next
1525     case (real r)
1526     with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)" by auto
1527     moreover from assms[of n] guess i ..
1528     ultimately show ?thesis
1529       by (auto intro!: bexI[of _ i])
1530   qed
1531 qed
1533 lemma Sup_countable_SUPR:
1534   assumes "A \<noteq> {}"
1535   shows "\<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
1536 proof (cases "Sup A")
1537   case (real r)
1538   have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
1539   proof
1540     fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x"
1541       using assms real by (intro Sup_ereal_close) (auto simp: one_ereal_def)
1542     then guess x ..
1543     then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)"
1544       by (auto intro!: exI[of _ x] simp: ereal_minus_less_iff)
1545   qed
1546   from choice[OF this] guess f .. note f = this
1547   have "SUPR UNIV f = Sup A"
1548   proof (rule ereal_SUPI)
1549     fix i show "f i \<le> Sup A" using f
1550       by (auto intro!: complete_lattice_class.Sup_upper)
1551   next
1552     fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
1553     show "Sup A \<le> y"
1554     proof (rule ereal_le_epsilon, intro allI impI)
1555       fix e :: ereal assume "0 < e"
1556       show "Sup A \<le> y + e"
1557       proof (cases e)
1558         case (real r)
1559         hence "0 < r" using `0 < e` by auto
1560         then obtain n ::nat where *: "1 / real n < r" "0 < n"
1561           using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
1562         have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n]
1563           by auto
1564         also have "1 / ereal (real n) \<le> e" using real * by (auto simp: one_ereal_def )
1565         with bound have "f n + 1 / ereal (real n) \<le> y + e" by (rule add_mono) simp
1566         finally show "Sup A \<le> y + e" .
1567       qed (insert `0 < e`, auto)
1568     qed
1569   qed
1570   with f show ?thesis by (auto intro!: exI[of _ f])
1571 next
1572   case PInf
1573   from `A \<noteq> {}` obtain x where "x \<in> A" by auto
1574   show ?thesis
1575   proof cases
1576     assume "\<infinity> \<in> A"
1577     moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
1578     ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
1579   next
1580     assume "\<infinity> \<notin> A"
1581     have "\<exists>x\<in>A. 0 \<le> x"
1582       by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least ereal_infty_less_eq2 linorder_linear)
1583     then obtain x where "x \<in> A" "0 \<le> x" by auto
1584     have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + ereal (real n) \<le> f"
1585     proof (rule ccontr)
1586       assume "\<not> ?thesis"
1587       then have "\<exists>n::nat. Sup A \<le> x + ereal (real n)"
1588         by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
1589       then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
1590         by(cases x) auto
1591     qed
1592     from choice[OF this] guess f .. note f = this
1593     have "SUPR UNIV f = \<infinity>"
1594     proof (rule SUP_PInfty)
1595       fix n :: nat show "\<exists>i\<in>UNIV. ereal (real n) \<le> f i"
1596         using f[THEN spec, of n] `0 \<le> x`
1597         by (cases rule: ereal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
1598     qed
1599     then show ?thesis using f PInf by (auto intro!: exI[of _ f])
1600   qed
1601 next
1602   case MInf
1603   with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
1604   then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
1605 qed
1607 lemma SUPR_countable_SUPR:
1608   "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
1609   using Sup_countable_SUPR[of "g`A"] by (auto simp: SUP_def)
1612   fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1613   shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
1614 proof (rule antisym)
1615   have *: "\<And>a::ereal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
1616     by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
1617   then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
1618   show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
1619   proof (cases a)
1620     case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant min_max.sup_absorb1)
1621   next
1622     case (real r)
1623     then have **: "op + (- a) ` op + a ` A = A"
1624       by (auto simp: image_iff ac_simps zero_ereal_def[symmetric])
1625     from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
1626       by (cases rule: ereal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
1627   qed (insert `a \<noteq> -\<infinity>`, auto)
1628 qed
1630 lemma Sup_ereal_cminus:
1631   fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1632   shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
1633   using Sup_ereal_cadd[of "uminus ` A" a] assms
1634   by (simp add: comp_def image_image minus_ereal_def
1635                  ereal_Sup_uminus_image_eq)
1637 lemma SUPR_ereal_cminus:
1638   fixes f :: "'i \<Rightarrow> ereal"
1639   fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1640   shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
1641   using Sup_ereal_cminus[of "f`A" a] assms
1642   unfolding SUP_def INF_def image_image by auto
1644 lemma Inf_ereal_cminus:
1645   fixes A :: "ereal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
1646   shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
1647 proof -
1648   { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto }
1649   moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
1650     by (auto simp: image_image)
1651   ultimately show ?thesis
1652     using Sup_ereal_cminus[of "uminus ` A" "-a"] assms
1653     by (auto simp add: ereal_Sup_uminus_image_eq ereal_Inf_uminus_image_eq)
1654 qed
1656 lemma INFI_ereal_cminus:
1657   fixes a :: ereal assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
1658   shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
1659   using Inf_ereal_cminus[of "f`A" a] assms
1660   unfolding SUP_def INF_def image_image
1661   by auto
1664   fixes a b :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
1665   by (cases rule: ereal2_cases[of a b]) auto
1668   fixes f :: "nat \<Rightarrow> ereal"
1669   assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
1670   shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g"
1671 proof -
1672   have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
1673     using assms unfolding INF_less_iff by auto
1674   { fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
1676   then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
1677     by simp
1678   also have "\<dots> = INFI UNIV f + INFI UNIV g"
1679     unfolding ereal_INFI_uminus
1680     using assms INF_less
1682        (auto simp: ereal_SUPR_uminus intro!: uminus_ereal_add_uminus_uminus)
1683   finally show ?thesis .
1684 qed
1686 subsection "Limits on @{typ ereal}"
1688 subsubsection "Topological space"
1690 instantiation ereal :: topological_space
1691 begin
1693 definition "open A \<longleftrightarrow> open (ereal -` A)
1694        \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A))
1695        \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))"
1697 lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
1698   unfolding open_ereal_def by auto
1700 lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)"
1701   unfolding open_ereal_def by auto
1703 lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{ereal x<..} \<subseteq> A"
1704   using open_PInfty[OF assms] by auto
1706 lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<ereal x} \<subseteq> A"
1707   using open_MInfty[OF assms] by auto
1709 lemma ereal_openE: assumes "open A" obtains x y where
1710   "open (ereal -` A)"
1711   "\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A"
1712   "-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A"
1713   using assms open_ereal_def by auto
1715 instance
1716 proof
1717   let ?U = "UNIV::ereal set"
1718   show "open ?U" unfolding open_ereal_def
1719     by (auto intro!: exI[of _ 0])
1720 next
1721   fix S T::"ereal set" assume "open S" and "open T"
1722   from `open S`[THEN ereal_openE] guess xS yS .
1723   moreover from `open T`[THEN ereal_openE] guess xT yT .
1724   ultimately have
1725     "open (ereal -` (S \<inter> T))"
1726     "\<infinity> \<in> S \<inter> T \<Longrightarrow> {ereal (max xS xT) <..} \<subseteq> S \<inter> T"
1727     "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< ereal (min yS yT)} \<subseteq> S \<inter> T"
1728     by auto
1729   then show "open (S Int T)" unfolding open_ereal_def by blast
1730 next
1731   fix K :: "ereal set set" assume "\<forall>S\<in>K. open S"
1732   then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (ereal -` S) \<and>
1733     (\<infinity> \<in> S \<longrightarrow> {ereal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< ereal y} \<subseteq> S)"
1734     by (auto simp: open_ereal_def)
1735   then show "open (Union K)" unfolding open_ereal_def
1736   proof (intro conjI impI)
1737     show "open (ereal -` \<Union>K)"
1738       using *[THEN choice] by (auto simp: vimage_Union)
1739   qed ((metis UnionE Union_upper subset_trans *)+)
1740 qed
1741 end
1743 lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)"
1744   by (auto simp: inj_vimage_image_eq open_ereal_def)
1746 lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)"
1747   unfolding open_ereal_def by auto
1749 lemma open_ereal_lessThan[intro, simp]: "open {..< a :: ereal}"
1750 proof -
1751   have "\<And>x. ereal -` {..<ereal x} = {..< x}"
1752     "ereal -` {..< \<infinity>} = UNIV" "ereal -` {..< -\<infinity>} = {}" by auto
1753   then show ?thesis by (cases a) (auto simp: open_ereal_def)
1754 qed
1756 lemma open_ereal_greaterThan[intro, simp]:
1757   "open {a :: ereal <..}"
1758 proof -
1759   have "\<And>x. ereal -` {ereal x<..} = {x<..}"
1760     "ereal -` {\<infinity><..} = {}" "ereal -` {-\<infinity><..} = UNIV" by auto
1761   then show ?thesis by (cases a) (auto simp: open_ereal_def)
1762 qed
1764 lemma ereal_open_greaterThanLessThan[intro, simp]: "open {a::ereal <..< b}"
1765   unfolding greaterThanLessThan_def by auto
1767 lemma closed_ereal_atLeast[simp, intro]: "closed {a :: ereal ..}"
1768 proof -
1769   have "- {a ..} = {..< a}" by auto
1770   then show "closed {a ..}"
1771     unfolding closed_def using open_ereal_lessThan by auto
1772 qed
1774 lemma closed_ereal_atMost[simp, intro]: "closed {.. b :: ereal}"
1775 proof -
1776   have "- {.. b} = {b <..}" by auto
1777   then show "closed {.. b}"
1778     unfolding closed_def using open_ereal_greaterThan by auto
1779 qed
1781 lemma closed_ereal_atLeastAtMost[simp, intro]:
1782   shows "closed {a :: ereal .. b}"
1783   unfolding atLeastAtMost_def by auto
1785 lemma closed_ereal_singleton:
1786   "closed {a :: ereal}"
1787 by (metis atLeastAtMost_singleton closed_ereal_atLeastAtMost)
1789 lemma ereal_open_cont_interval:
1790   fixes S :: "ereal set"
1791   assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
1792   obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
1793 proof-
1794   from `open S` have "open (ereal -` S)" by (rule ereal_openE)
1795   then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S"
1796     using assms unfolding open_dist by force
1797   show thesis
1798   proof (intro that subsetI)
1799     show "0 < ereal e" using `0 < e` by auto
1800     fix y assume "y \<in> {x - ereal e<..<x + ereal e}"
1801     with assms obtain t where "y = ereal t" "dist t (real x) < e"
1802       apply (cases y) by (auto simp: dist_real_def)
1803     then show "y \<in> S" using e[of t] by auto
1804   qed
1805 qed
1807 lemma ereal_open_cont_interval2:
1808   fixes S :: "ereal set"
1809   assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
1810   obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
1811 proof-
1812   guess e using ereal_open_cont_interval[OF assms] .
1813   with that[of "x-e" "x+e"] ereal_between[OF x, of e]
1814   show thesis by auto
1815 qed
1817 instance ereal :: t2_space
1818 proof
1819   fix x y :: ereal assume "x ~= y"
1820   let "?P x (y::ereal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
1822   { fix x y :: ereal assume "x < y"
1823     from ereal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
1824     have "?P x y"
1825       apply (rule exI[of _ "{..<z}"])
1826       apply (rule exI[of _ "{z<..}"])
1827       using z by auto }
1828   note * = this
1830   from `x ~= y`
1831   show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
1832   proof (cases rule: linorder_cases)
1833     assume "x = y" with `x ~= y` show ?thesis by simp
1834   next assume "x < y" from *[OF this] show ?thesis by auto
1835   next assume "y < x" from *[OF this] show ?thesis by auto
1836   qed
1837 qed
1839 subsubsection {* Convergent sequences *}
1841 lemma lim_ereal[simp]:
1842   "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
1843 proof (intro iffI topological_tendstoI)
1844   fix S assume "?l" "open S" "x \<in> S"
1845   then show "eventually (\<lambda>x. f x \<in> S) net"
1846     using `?l`[THEN topological_tendstoD, OF open_ereal, OF `open S`]
1848 next
1849   fix S assume "?r" "open S" "ereal x \<in> S"
1850   show "eventually (\<lambda>x. ereal (f x) \<in> S) net"
1851     using `?r`[THEN topological_tendstoD, OF open_ereal_vimage, OF `open S`]
1852     using `ereal x \<in> S` by auto
1853 qed
1855 lemma lim_real_of_ereal[simp]:
1856   assumes lim: "(f ---> ereal x) net"
1857   shows "((\<lambda>x. real (f x)) ---> x) net"
1858 proof (intro topological_tendstoI)
1859   fix S assume "open S" "x \<in> S"
1860   then have S: "open S" "ereal x \<in> ereal ` S"
1862   have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S" by auto
1863   from this lim[THEN topological_tendstoD, OF open_ereal, OF S]
1864   show "eventually (\<lambda>x. real (f x) \<in> S) net"
1865     by (rule eventually_mono)
1866 qed
1868 lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= ereal B)" (is "?l = ?r")
1869 proof
1870   assume ?r
1871   show ?l
1872     apply(rule topological_tendstoI)
1873     unfolding eventually_sequentially
1874   proof-
1875     fix S :: "ereal set" assume "open S" "\<infinity> : S"
1876     from open_PInfty[OF this] guess B .. note B=this
1877     from `?r`[rule_format,of "B+1"] guess N .. note N=this
1878     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
1879     proof safe case goal1
1880       have "ereal B < ereal (B + 1)" by auto
1881       also have "... <= f n" using goal1 N by auto
1882       finally show ?case using B by fastforce
1883     qed
1884   qed
1885 next
1886   assume ?l
1887   show ?r
1888   proof fix B::real have "open {ereal B<..}" "\<infinity> : {ereal B<..}" by auto
1889     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
1890     guess N .. note N=this
1891     show "EX N. ALL n>=N. ereal B <= f n" apply(rule_tac x=N in exI) using N by auto
1892   qed
1893 qed
1896 lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= ereal B)" (is "?l = ?r")
1897 proof
1898   assume ?r
1899   show ?l
1900     apply(rule topological_tendstoI)
1901     unfolding eventually_sequentially
1902   proof-
1903     fix S :: "ereal set"
1904     assume "open S" "(-\<infinity>) : S"
1905     from open_MInfty[OF this] guess B .. note B=this
1906     from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
1907     show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
1908     proof safe case goal1
1909       have "ereal (B - 1) >= f n" using goal1 N by auto
1910       also have "... < ereal B" by auto
1911       finally show ?case using B by fastforce
1912     qed
1913   qed
1914 next assume ?l show ?r
1915   proof fix B::real have "open {..<ereal B}" "(-\<infinity>) : {..<ereal B}" by auto
1916     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
1917     guess N .. note N=this
1918     show "EX N. ALL n>=N. ereal B >= f n" apply(rule_tac x=N in exI) using N by auto
1919   qed
1920 qed
1923 lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= ereal B" shows "l ~= \<infinity>"
1924 proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
1925   from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
1926   guess N .. note N=this[rule_format,OF le_refl]
1927   hence "ereal ?B <= ereal B" using assms(2)[of N] by(rule order_trans)
1928   hence "ereal ?B < ereal ?B" apply (rule le_less_trans) by auto
1929   thus False by auto
1930 qed
1933 lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= ereal B" shows "l ~= (-\<infinity>)"
1934 proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
1935   from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
1936   guess N .. note N=this[rule_format,OF le_refl]
1937   hence "ereal B <= ereal ?B" using assms(2)[of N] order_trans[of "ereal B" "f N" "ereal(B - 1)"] by blast
1938   thus False by auto
1939 qed
1942 lemma tendsto_explicit:
1943   "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
1944   unfolding tendsto_def eventually_sequentially by auto
1947 lemma tendsto_obtains_N:
1948   assumes "f ----> f0"
1949   assumes "open S" "f0 : S"
1950   obtains N where "ALL n>=N. f n : S"
1951   using tendsto_explicit[of f f0] assms by auto
1954 lemma tail_same_limit:
1955   fixes X Y N
1956   assumes "X ----> L" "ALL n>=N. X n = Y n"
1957   shows "Y ----> L"
1958 proof-
1959 { fix S assume "open S" and "L:S"
1960   from this obtain N1 where "ALL n>=N1. X n : S"
1961      using assms unfolding tendsto_def eventually_sequentially by auto
1962   hence "ALL n>=max N N1. Y n : S" using assms by auto
1963   hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
1964 }
1965 thus ?thesis using tendsto_explicit by auto
1966 qed
1969 lemma Lim_bounded_PInfty2:
1970 assumes lim:"f ----> l" and "ALL n>=N. f n <= ereal B"
1971 shows "l ~= \<infinity>"
1972 proof-
1973   def g == "(%n. if n>=N then f n else ereal B)"
1974   hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
1975   moreover have "!!n. g n <= ereal B" using g_def assms by auto
1976   ultimately show ?thesis using  Lim_bounded_PInfty by auto
1977 qed
1979 lemma Lim_bounded_ereal:
1980   assumes lim:"f ----> (l :: ereal)"
1981   and "ALL n>=M. f n <= C"
1982   shows "l<=C"
1983 proof-
1984 { assume "l=(-\<infinity>)" hence ?thesis by auto }
1985 moreover
1986 { assume "~(l=(-\<infinity>))"
1987   { assume "C=\<infinity>" hence ?thesis by auto }
1988   moreover
1989   { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
1990     hence "l=(-\<infinity>)" using assms
1991        tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
1992     hence ?thesis by auto }
1993   moreover
1994   { assume "EX B. C = ereal B"
1995     from this obtain B where B_def: "C=ereal B" by auto
1996     hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
1997     from this obtain m where m_def: "ereal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
1998     from this obtain N where N_def: "ALL n>=N. f n : {ereal(m - 1) <..< ereal(m+1)}"
1999        apply (subst tendsto_obtains_N[of f l "{ereal(m - 1) <..< ereal(m+1)}"]) using assms by auto
2000     { fix n assume "n>=N"
2001       hence "EX r. ereal r = f n" using N_def by (cases "f n") auto
2002     } from this obtain g where g_def: "ALL n>=N. ereal (g n) = f n" by metis
2003     hence "(%n. ereal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
2004     hence *: "(%n. g n) ----> m" using m_def by auto
2005     { fix n assume "n>=max N M"
2006       hence "ereal (g n) <= ereal B" using assms g_def B_def by auto
2007       hence "g n <= B" by auto
2008     } hence "EX N. ALL n>=N. g n <= B" by blast
2009     hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
2010     hence ?thesis using m_def B_def by auto
2011   } ultimately have ?thesis by (cases C) auto
2012 } ultimately show ?thesis by blast
2013 qed
2015 lemma real_of_ereal_mult[simp]:
2016   fixes a b :: ereal shows "real (a * b) = real a * real b"
2017   by (cases rule: ereal2_cases[of a b]) auto
2019 lemma real_of_ereal_eq_0:
2020   fixes x :: ereal shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
2021   by (cases x) auto
2023 lemma tendsto_ereal_realD:
2024   fixes f :: "'a \<Rightarrow> ereal"
2025   assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net"
2026   shows "(f ---> x) net"
2027 proof (intro topological_tendstoI)
2028   fix S assume S: "open S" "x \<in> S"
2029   with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
2030   from tendsto[THEN topological_tendstoD, OF this]
2031   show "eventually (\<lambda>x. f x \<in> S) net"
2032     by (rule eventually_rev_mp) (auto simp: ereal_real)
2033 qed
2035 lemma tendsto_ereal_realI:
2036   fixes f :: "'a \<Rightarrow> ereal"
2037   assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
2038   shows "((\<lambda>x. ereal (real (f x))) ---> x) net"
2039 proof (intro topological_tendstoI)
2040   fix S assume "open S" "x \<in> S"
2041   with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
2042   from tendsto[THEN topological_tendstoD, OF this]
2043   show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net"
2044     by (elim eventually_elim1) (auto simp: ereal_real)
2045 qed
2047 lemma ereal_mult_cancel_left:
2048   fixes a b c :: ereal shows "a * b = a * c \<longleftrightarrow>
2049     ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
2050   by (cases rule: ereal3_cases[of a b c])
2053 lemma ereal_inj_affinity:
2054   fixes m t :: ereal
2055   assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
2056   shows "inj_on (\<lambda>x. m * x + t) A"
2057   using assms
2058   by (cases rule: ereal2_cases[of m t])
2059      (auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left)
2061 lemma ereal_PInfty_eq_plus[simp]:
2062   fixes a b :: ereal
2063   shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
2064   by (cases rule: ereal2_cases[of a b]) auto
2066 lemma ereal_MInfty_eq_plus[simp]:
2067   fixes a b :: ereal
2068   shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
2069   by (cases rule: ereal2_cases[of a b]) auto
2071 lemma ereal_less_divide_pos:
2072   fixes x y :: ereal
2073   shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
2074   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
2076 lemma ereal_divide_less_pos:
2077   fixes x y z :: ereal
2078   shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
2079   by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)
2081 lemma ereal_divide_eq:
2082   fixes a b c :: ereal
2083   shows "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
2084   by (cases rule: ereal3_cases[of a b c])
2087 lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>"
2088   by (cases a) auto
2090 lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
2091   by (cases x) auto
2093 lemma ereal_LimI_finite:
2094   fixes x :: ereal
2095   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
2096   assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
2097   shows "u ----> x"
2098 proof (rule topological_tendstoI, unfold eventually_sequentially)
2099   obtain rx where rx_def: "x=ereal rx" using assms by (cases x) auto
2100   fix S assume "open S" "x : S"
2101   then have "open (ereal -` S)" unfolding open_ereal_def by auto
2102   with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> ereal y \<in> S"
2103     unfolding open_real_def rx_def by auto
2104   then obtain n where
2105     upper: "!!N. n <= N ==> u N < x + ereal r" and
2106     lower: "!!N. n <= N ==> x < u N + ereal r" using assms(2)[of "ereal r"] by auto
2107   show "EX N. ALL n>=N. u n : S"
2108   proof (safe intro!: exI[of _ n])
2109     fix N assume "n <= N"
2110     from upper[OF this] lower[OF this] assms `0 < r`
2111     have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
2112     from this obtain ra where ra_def: "(u N) = ereal ra" by (cases "u N") auto
2113     hence "rx < ra + r" and "ra < rx + r"
2114        using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
2115     hence "dist (real (u N)) rx < r"
2116       using rx_def ra_def
2117       by (auto simp: dist_real_def abs_diff_less_iff field_simps)
2118     from dist[OF this] show "u N : S" using `u N  ~: {\<infinity>, -\<infinity>}`
2119       by (auto simp: ereal_real split: split_if_asm)
2120   qed
2121 qed
2123 lemma ereal_LimI_finite_iff:
2124   fixes x :: ereal
2125   assumes "\<bar>x\<bar> \<noteq> \<infinity>"
2126   shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
2127   (is "?lhs <-> ?rhs")
2128 proof
2129   assume lim: "u ----> x"
2130   { fix r assume "(r::ereal)>0"
2131     from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
2132        apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
2133        using lim ereal_between[of x r] assms `r>0` by auto
2134     hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
2135       using ereal_minus_less[of r x] by (cases r) auto
2136   } then show "?rhs" by auto
2137 next
2138   assume ?rhs then show "u ----> x"
2139     using ereal_LimI_finite[of x] assms by auto
2140 qed
2143 subsubsection {* @{text Liminf} and @{text Limsup} *}
2145 definition
2146   "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
2148 definition
2149   "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
2151 lemma Liminf_Sup:
2152   fixes f :: "'a => 'b::complete_linorder"
2153   shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
2154   by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
2156 lemma Limsup_Inf:
2157   fixes f :: "'a => 'b::complete_linorder"
2158   shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
2159   by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
2161 lemma ereal_SupI:
2162   fixes x :: ereal
2163   assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
2164   assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
2165   shows "Sup A = x"
2166   unfolding Sup_ereal_def
2167   using assms by (auto intro!: Least_equality)
2169 lemma ereal_InfI:
2170   fixes x :: ereal
2171   assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
2172   assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
2173   shows "Inf A = x"
2174   unfolding Inf_ereal_def
2175   using assms by (auto intro!: Greatest_equality)
2177 lemma Limsup_const:
2178   fixes c :: "'a::complete_linorder"
2179   assumes ntriv: "\<not> trivial_limit net"
2180   shows "Limsup net (\<lambda>x. c) = c"
2181   unfolding Limsup_Inf
2182 proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
2183   fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
2184   show "c \<le> x"
2185   proof (rule ccontr)
2186     assume "\<not> c \<le> x" then have "x < c" by auto
2187     then show False using ntriv * by (auto simp: trivial_limit_def)
2188   qed
2189 qed auto
2191 lemma Liminf_const:
2192   fixes c :: "'a::complete_linorder"
2193   assumes ntriv: "\<not> trivial_limit net"
2194   shows "Liminf net (\<lambda>x. c) = c"
2195   unfolding Liminf_Sup
2196 proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
2197   fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
2198   show "x \<le> c"
2199   proof (rule ccontr)
2200     assume "\<not> x \<le> c" then have "c < x" by auto
2201     then show False using ntriv * by (auto simp: trivial_limit_def)
2202   qed
2203 qed auto
2205 definition (in order) mono_set:
2206   "mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
2208 lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
2209 lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
2210 lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
2211 lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
2213 lemma (in complete_linorder) mono_set_iff:
2214   fixes S :: "'a set"
2215   defines "a \<equiv> Inf S"
2216   shows "mono_set S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
2217 proof
2218   assume "mono_set S"
2219   then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
2220   show ?c
2221   proof cases
2222     assume "a \<in> S"
2223     show ?c
2224       using mono[OF _ `a \<in> S`]
2225       by (auto intro: Inf_lower simp: a_def)
2226   next
2227     assume "a \<notin> S"
2228     have "S = {a <..}"
2229     proof safe
2230       fix x assume "x \<in> S"
2231       then have "a \<le> x" unfolding a_def by (rule Inf_lower)
2232       then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
2233     next
2234       fix x assume "a < x"
2235       then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
2236       with mono[of y x] show "x \<in> S" by auto
2237     qed
2238     then show ?c ..
2239   qed
2240 qed auto
2242 lemma lim_imp_Liminf:
2243   fixes f :: "'a \<Rightarrow> ereal"
2244   assumes ntriv: "\<not> trivial_limit net"
2245   assumes lim: "(f ---> f0) net"
2246   shows "Liminf net f = f0"
2247   unfolding Liminf_Sup
2248 proof (safe intro!: ereal_SupI)
2249   fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
2250   show "y \<le> f0"
2251   proof (rule ereal_le_ereal)
2252     fix B assume "B < y"
2253     { assume "f0 < B"
2254       then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
2255          using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
2256          by (auto intro: eventually_conj)
2257       also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
2258       finally have False using ntriv[unfolded trivial_limit_def] by auto
2259     } then show "B \<le> f0" by (metis linorder_le_less_linear)
2260   qed
2261 next
2262   fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
2263   show "f0 \<le> y"
2264   proof (safe intro!: *[rule_format])
2265     fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
2266       using lim[THEN topological_tendstoD, of "{y <..}"] by auto
2267   qed
2268 qed
2270 lemma ereal_Liminf_le_Limsup:
2271   fixes f :: "'a \<Rightarrow> ereal"
2272   assumes ntriv: "\<not> trivial_limit net"
2273   shows "Liminf net f \<le> Limsup net f"
2274   unfolding Limsup_Inf Liminf_Sup
2275 proof (safe intro!: complete_lattice_class.Inf_greatest  complete_lattice_class.Sup_least)
2276   fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
2277   show "u \<le> v"
2278   proof (rule ccontr)
2279     assume "\<not> u \<le> v"
2280     then obtain t where "t < u" "v < t"
2281       using ereal_dense[of v u] by (auto simp: not_le)
2282     then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
2283       using * by (auto intro: eventually_conj)
2284     also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
2285     finally show False using ntriv by (auto simp: trivial_limit_def)
2286   qed
2287 qed
2289 lemma Liminf_mono:
2290   fixes f g :: "'a => ereal"
2291   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
2292   shows "Liminf net f \<le> Liminf net g"
2293   unfolding Liminf_Sup
2294 proof (safe intro!: Sup_mono bexI)
2295   fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
2296   then have "eventually (\<lambda>x. y < f x) net" by auto
2297   then show "eventually (\<lambda>x. y < g x) net"
2298     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
2299 qed simp
2301 lemma Liminf_eq:
2302   fixes f g :: "'a \<Rightarrow> ereal"
2303   assumes "eventually (\<lambda>x. f x = g x) net"
2304   shows "Liminf net f = Liminf net g"
2305   by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
2307 lemma Liminf_mono_all:
2308   fixes f g :: "'a \<Rightarrow> ereal"
2309   assumes "\<And>x. f x \<le> g x"
2310   shows "Liminf net f \<le> Liminf net g"
2311   using assms by (intro Liminf_mono always_eventually) auto
2313 lemma Limsup_mono:
2314   fixes f g :: "'a \<Rightarrow> ereal"
2315   assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
2316   shows "Limsup net f \<le> Limsup net g"
2317   unfolding Limsup_Inf
2318 proof (safe intro!: Inf_mono bexI)
2319   fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
2320   then have "eventually (\<lambda>x. g x < y) net" by auto
2321   then show "eventually (\<lambda>x. f x < y) net"
2322     by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
2323 qed simp
2325 lemma Limsup_mono_all:
2326   fixes f g :: "'a \<Rightarrow> ereal"
2327   assumes "\<And>x. f x \<le> g x"
2328   shows "Limsup net f \<le> Limsup net g"
2329   using assms by (intro Limsup_mono always_eventually) auto
2331 lemma Limsup_eq:
2332   fixes f g :: "'a \<Rightarrow> ereal"
2333   assumes "eventually (\<lambda>x. f x = g x) net"
2334   shows "Limsup net f = Limsup net g"
2335   by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
2337 abbreviation "liminf \<equiv> Liminf sequentially"
2339 abbreviation "limsup \<equiv> Limsup sequentially"
2341 lemma liminf_SUPR_INFI:
2342   fixes f :: "nat \<Rightarrow> ereal"
2343   shows "liminf f = (SUP n. INF m:{n..}. f m)"
2344   unfolding Liminf_Sup eventually_sequentially
2345 proof (safe intro!: antisym complete_lattice_class.Sup_least)
2346   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)"
2347   proof (rule ereal_le_ereal)
2348     fix y assume "y < x"
2349     with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
2350     then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
2351     also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro SUP_upper) auto
2352     finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
2353   qed
2354 next
2355   show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
2356   proof (unfold SUP_def, safe intro!: Sup_mono bexI)
2357     fix y n assume "y < INFI {n..} f"
2358     from less_INF_D[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
2359   qed (rule order_refl)
2360 qed
2362 lemma tail_same_limsup:
2363   fixes X Y :: "nat => ereal"
2364   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
2365   shows "limsup X = limsup Y"
2366   using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
2368 lemma tail_same_liminf:
2369   fixes X Y :: "nat => ereal"
2370   assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
2371   shows "liminf X = liminf Y"
2372   using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
2374 lemma liminf_mono:
2375   fixes X Y :: "nat \<Rightarrow> ereal"
2376   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
2377   shows "liminf X \<le> liminf Y"
2378   using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
2380 lemma limsup_mono:
2381   fixes X Y :: "nat => ereal"
2382   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
2383   shows "limsup X \<le> limsup Y"
2384   using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
2386 lemma
2387   fixes X :: "nat \<Rightarrow> ereal"
2388   shows ereal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
2389     and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
2390   unfolding incseq_def decseq_def by auto
2392 lemma liminf_bounded:
2393   fixes X Y :: "nat \<Rightarrow> ereal"
2394   assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
2395   shows "C \<le> liminf X"
2396   using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
2398 lemma limsup_bounded:
2399   fixes X Y :: "nat => ereal"
2400   assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
2401   shows "limsup X \<le> C"
2402   using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
2404 lemma liminf_bounded_iff:
2405   fixes x :: "nat \<Rightarrow> ereal"
2406   shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
2407 proof safe
2408   fix B assume "B < C" "C \<le> liminf x"
2409   then have "B < liminf x" by auto
2410   then obtain N where "B < (INF m:{N..}. x m)"
2411     unfolding liminf_SUPR_INFI SUP_def less_Sup_iff by auto
2412   from less_INF_D[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
2413 next
2414   assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
2415   { fix B assume "B<C"
2416     then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
2417     hence "B \<le> (INF m:{N..}. x m)" by (intro INF_greatest) auto
2418     also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro SUP_upper) simp
2419     finally have "B \<le> liminf x" .
2420   } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
2421 qed
2423 lemma liminf_subseq_mono:
2424   fixes X :: "nat \<Rightarrow> ereal"
2425   assumes "subseq r"
2426   shows "liminf X \<le> liminf (X \<circ> r) "
2427 proof-
2428   have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
2429   proof (safe intro!: INF_mono)
2430     fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
2431       using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
2432   qed
2433   then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
2434 qed
2436 lemma ereal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "ereal (real x) = x"
2437   using assms by auto
2439 lemma ereal_le_ereal_bounded:
2440   fixes x y z :: ereal
2441   assumes "z \<le> y"
2442   assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
2443   shows "x \<le> y"
2444 proof (rule ereal_le_ereal)
2445   fix B assume "B < x"
2446   show "B \<le> y"
2447   proof cases
2448     assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
2449   next
2450     assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
2451   qed
2452 qed
2454 lemma fixes x y :: ereal
2455   shows Sup_atMost[simp]: "Sup {.. y} = y"
2456     and Sup_lessThan[simp]: "Sup {..< y} = y"
2457     and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
2458     and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
2459     and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
2460   by (auto simp: Sup_ereal_def intro!: Least_equality
2461            intro: ereal_le_ereal ereal_le_ereal_bounded[of x])
2463 lemma Sup_greaterThanlessThan[simp]:
2464   fixes x y :: ereal assumes "x < y" shows "Sup { x <..< y} = y"
2465   unfolding Sup_ereal_def
2466 proof (intro Least_equality ereal_le_ereal_bounded[of _ _ y])
2467   fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
2468   from ereal_dense[OF `x < y`] guess w .. note w = this
2469   with z[THEN bspec, of w] show "x \<le> z" by auto
2470 qed auto
2472 lemma real_ereal_id: "real o ereal = id"
2473 proof-
2474 { fix x have "(real o ereal) x = id x" by auto }
2475 from this show ?thesis using ext by blast
2476 qed
2478 lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})"
2479 by (metis range_ereal open_ereal open_UNIV)
2481 lemma ereal_le_distrib:
2482   fixes a b c :: ereal shows "c * (a + b) \<le> c * a + c * b"
2483   by (cases rule: ereal3_cases[of a b c])
2484      (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
2486 lemma ereal_pos_distrib:
2487   fixes a b c :: ereal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
2488   using assms by (cases rule: ereal3_cases[of a b c])
2489                  (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
2491 lemma ereal_pos_le_distrib:
2492 fixes a b c :: ereal
2493 assumes "c>=0"
2494 shows "c * (a + b) <= c * a + c * b"
2495   using assms by (cases rule: ereal3_cases[of a b c])
2498 lemma ereal_max_mono:
2499   "[| (a::ereal) <= b; c <= d |] ==> max a c <= max b d"
2500   by (metis sup_ereal_def sup_mono)
2503 lemma ereal_max_least:
2504   "[| (a::ereal) <= x; c <= x |] ==> max a c <= x"
2505   by (metis sup_ereal_def sup_least)
2507 subsubsection {* Tests for code generator *}
2509 (* A small list of simple arithmetic expressions *)
2511 value [code] "- \<infinity> :: ereal"
2512 value [code] "\<bar>-\<infinity>\<bar> :: ereal"
2513 value [code] "4 + 5 / 4 - ereal 2 :: ereal"
2514 value [code] "ereal 3 < \<infinity>"
2515 value [code] "real (\<infinity>::ereal) = 0"
2517 end