1 (* Title: HOL/Library/Extended_Reals.thy
2 Author: Johannes Hölzl, TU München
3 Author: Robert Himmelmann, TU München
4 Author: Armin Heller, TU München
5 Author: Bogdan Grechuk, University of Edinburgh
8 header {* Extended real number line *}
16 For more lemmas about the extended real numbers go to
17 @{text "src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"}
21 lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
26 assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less)
27 then show False using `{x..} = UNIV` by simp
32 "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
33 by (rule antisym) (auto intro!: SUP_leI le_SUPI2)
36 "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
37 by (rule antisym) (auto intro!: le_INFI INF_leI2)
39 subsection {* Definition and basic properties *}
41 datatype extreal = extreal real | PInfty | MInfty
44 PInfty ("\<infinity>")
46 notation (HTML output)
47 PInfty ("\<infinity>")
49 declare [[coercion "extreal :: real \<Rightarrow> extreal"]]
51 instantiation extreal :: uminus
53 fun uminus_extreal where
54 "- (extreal r) = extreal (- r)"
55 | "- \<infinity> = MInfty"
56 | "- MInfty = \<infinity>"
60 lemma inj_extreal[simp]: "inj_on extreal A"
61 unfolding inj_on_def by auto
63 lemma MInfty_neq_PInfty[simp]:
64 "\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all
66 lemma MInfty_neq_extreal[simp]:
67 "extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all
69 lemma MInfinity_cases[simp]:
70 "(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z"
73 lemma extreal_uminus_uminus[simp]:
74 fixes a :: extreal shows "- (- a) = a"
77 lemma MInfty_eq[simp, code_post]:
78 "MInfty = - \<infinity>" by simp
80 declare uminus_extreal.simps(2)[code_inline, simp del]
82 lemma extreal_cases[case_names real PInf MInf, cases type: extreal]:
83 assumes "\<And>r. x = extreal r \<Longrightarrow> P"
84 assumes "x = \<infinity> \<Longrightarrow> P"
85 assumes "x = -\<infinity> \<Longrightarrow> P"
87 using assms by (cases x) auto
89 lemmas extreal2_cases = extreal_cases[case_product extreal_cases]
90 lemmas extreal3_cases = extreal2_cases[case_product extreal_cases]
92 lemma extreal_uminus_eq_iff[simp]:
93 fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b"
94 by (cases rule: extreal2_cases[of a b]) simp_all
96 function of_extreal :: "extreal \<Rightarrow> real" where
97 "of_extreal (extreal r) = r" |
98 "of_extreal \<infinity> = 0" |
99 "of_extreal (-\<infinity>) = 0"
100 by (auto intro: extreal_cases)
101 termination proof qed (rule wf_empty)
104 real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal"
106 lemma real_of_extreal[simp]:
107 "real (- x :: extreal) = - (real x)"
108 "real (extreal r) = r"
109 "real \<infinity> = 0"
110 by (cases x) (simp_all add: real_of_extreal_def)
112 lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}"
114 fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>"
115 then show "x = -\<infinity>" by (cases x) auto
118 lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)"
120 fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
123 instantiation extreal :: number
125 definition [simp]: "number_of x = extreal (number_of x)"
129 instantiation extreal :: abs
131 function abs_extreal where
132 "\<bar>extreal r\<bar> = extreal \<bar>r\<bar>"
133 | "\<bar>-\<infinity>\<bar> = \<infinity>"
134 | "\<bar>\<infinity>\<bar> = \<infinity>"
135 by (auto intro: extreal_cases)
136 termination proof qed (rule wf_empty)
140 lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
143 lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P"
146 lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>"
149 subsubsection "Addition"
151 instantiation extreal :: comm_monoid_add
154 definition "0 = extreal 0"
156 function plus_extreal where
157 "extreal r + extreal p = extreal (r + p)" |
158 "\<infinity> + a = \<infinity>" |
159 "a + \<infinity> = \<infinity>" |
160 "extreal r + -\<infinity> = - \<infinity>" |
161 "-\<infinity> + extreal p = -\<infinity>" |
162 "-\<infinity> + -\<infinity> = -\<infinity>"
165 moreover then obtain a b where "x = (a, b)" by (cases x) auto
167 by (cases rule: extreal2_cases[of a b]) auto
169 termination proof qed (rule wf_empty)
171 lemma Infty_neq_0[simp]:
172 "\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>"
173 "-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>"
174 by (simp_all add: zero_extreal_def)
176 lemma extreal_eq_0[simp]:
177 "extreal r = 0 \<longleftrightarrow> r = 0"
178 "0 = extreal r \<longleftrightarrow> r = 0"
179 unfolding zero_extreal_def by simp_all
183 fix a :: extreal show "0 + a = a"
184 by (cases a) (simp_all add: zero_extreal_def)
185 fix b :: extreal show "a + b = b + a"
186 by (cases rule: extreal2_cases[of a b]) simp_all
187 fix c :: extreal show "a + b + c = a + (b + c)"
188 by (cases rule: extreal3_cases[of a b c]) simp_all
192 lemma real_of_extreal_0[simp]: "real (0::extreal) = 0"
193 unfolding real_of_extreal_def zero_extreal_def by simp
195 lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)"
196 unfolding zero_extreal_def abs_extreal.simps by simp
198 lemma extreal_uminus_zero[simp]:
200 by (simp add: zero_extreal_def)
202 lemma extreal_uminus_zero_iff[simp]:
203 fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0"
204 by (cases a) simp_all
206 lemma extreal_plus_eq_PInfty[simp]:
207 shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
208 by (cases rule: extreal2_cases[of a b]) auto
210 lemma extreal_plus_eq_MInfty[simp]:
211 shows "a + b = -\<infinity> \<longleftrightarrow>
212 (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
213 by (cases rule: extreal2_cases[of a b]) auto
215 lemma extreal_add_cancel_left:
216 assumes "a \<noteq> -\<infinity>"
217 shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
218 using assms by (cases rule: extreal3_cases[of a b c]) auto
220 lemma extreal_add_cancel_right:
221 assumes "a \<noteq> -\<infinity>"
222 shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
223 using assms by (cases rule: extreal3_cases[of a b c]) auto
226 "extreal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)"
227 by (cases x) simp_all
229 lemma real_of_extreal_add:
231 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)"
232 by (cases rule: extreal2_cases[of a b]) auto
234 subsubsection "Linear order on @{typ extreal}"
236 instantiation extreal :: linorder
239 function less_extreal where
240 "extreal x < extreal y \<longleftrightarrow> x < y" |
241 " \<infinity> < a \<longleftrightarrow> False" |
242 " a < -\<infinity> \<longleftrightarrow> False" |
243 "extreal x < \<infinity> \<longleftrightarrow> True" |
244 " -\<infinity> < extreal r \<longleftrightarrow> True" |
245 " -\<infinity> < \<infinity> \<longleftrightarrow> True"
248 moreover then obtain a b where "x = (a,b)" by (cases x) auto
249 ultimately show P by (cases rule: extreal2_cases[of a b]) auto
251 termination by (relation "{}") simp
253 definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y"
255 lemma extreal_infty_less[simp]:
256 "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
257 "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
258 by (cases x, simp_all) (cases x, simp_all)
260 lemma extreal_infty_less_eq[simp]:
261 "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
262 "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
263 by (auto simp add: less_eq_extreal_def)
265 lemma extreal_less[simp]:
266 "extreal r < 0 \<longleftrightarrow> (r < 0)"
267 "0 < extreal r \<longleftrightarrow> (0 < r)"
270 by (simp_all add: zero_extreal_def)
272 lemma extreal_less_eq[simp]:
273 "x \<le> \<infinity>"
274 "-\<infinity> \<le> x"
275 "extreal r \<le> extreal p \<longleftrightarrow> r \<le> p"
276 "extreal r \<le> 0 \<longleftrightarrow> r \<le> 0"
277 "0 \<le> extreal r \<longleftrightarrow> 0 \<le> r"
278 by (auto simp add: less_eq_extreal_def zero_extreal_def)
280 lemma extreal_infty_less_eq2:
281 "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>"
282 "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>"
287 fix x :: extreal show "x \<le> x"
288 by (cases x) simp_all
289 fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
290 by (cases rule: extreal2_cases[of x y]) auto
291 show "x \<le> y \<or> y \<le> x "
292 by (cases rule: extreal2_cases[of x y]) auto
293 { assume "x \<le> y" "y \<le> x" then show "x = y"
294 by (cases rule: extreal2_cases[of x y]) auto }
295 { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
296 by (cases rule: extreal3_cases[of x y z]) auto }
300 instance extreal :: ordered_ab_semigroup_add
302 fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b"
303 by (cases rule: extreal3_cases[of a b c]) auto
306 lemma real_of_extreal_positive_mono:
307 "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y"
308 by (cases rule: extreal2_cases[of x y]) auto
310 lemma extreal_MInfty_lessI[intro, simp]:
311 "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
314 lemma extreal_less_PInfty[intro, simp]:
315 "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
318 lemma extreal_less_extreal_Ex:
320 shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)"
323 lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))"
325 case (real r) then show ?thesis
326 using reals_Archimedean2[of r] by simp
329 lemma extreal_add_mono:
330 fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
333 apply (cases rule: extreal3_cases[of b c d], auto)
334 apply (cases rule: extreal3_cases[of b c d], auto)
337 lemma extreal_minus_le_minus[simp]:
338 fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
339 by (cases rule: extreal2_cases[of a b]) auto
341 lemma extreal_minus_less_minus[simp]:
342 fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a"
343 by (cases rule: extreal2_cases[of a b]) auto
345 lemma extreal_le_real_iff:
346 "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))"
349 lemma real_le_extreal_iff:
350 "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))"
353 lemma extreal_less_real_iff:
354 "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))"
357 lemma real_less_extreal_iff:
358 "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
361 lemma real_of_extreal_pos:
362 fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto
364 lemmas real_of_extreal_ord_simps =
365 extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff
367 lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x"
370 lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x"
373 lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>"
376 lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)"
379 lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>"
382 lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)"
385 lemma extreal_0_le_uminus_iff[simp]:
386 fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0"
387 by (cases rule: extreal2_cases[of a]) auto
389 lemma extreal_uminus_le_0_iff[simp]:
390 fixes a :: extreal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a"
391 by (cases rule: extreal2_cases[of a]) auto
394 fixes x y :: extreal assumes "x < y"
395 shows "EX z. x < z & z < y"
397 { assume a: "x = (-\<infinity>)"
398 { assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) }
400 { assume "y ~= \<infinity>"
401 with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
402 hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"])
403 } ultimately have ?thesis by auto
406 { assume "x ~= (-\<infinity>)"
407 with `x < y` obtain p where p: "x = extreal p" by (cases x) auto
408 { assume "y = \<infinity>" hence ?thesis using `x < y` p
409 by (auto intro!: exI[of _ "extreal (p + 1)"]) }
411 { assume "y ~= \<infinity>"
412 with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
413 with p `x < y` have "p < r" by auto
414 with dense obtain z where "p < z" "z < r" by auto
415 hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"])
416 } ultimately have ?thesis by auto
417 } ultimately show ?thesis by auto
420 lemma extreal_dense2:
421 fixes x y :: extreal assumes "x < y"
422 shows "EX z. x < extreal z & extreal z < y"
423 by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3))
425 lemma extreal_add_strict_mono:
426 fixes a b c d :: extreal
427 assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d"
428 shows "a + c < b + d"
429 using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto
431 lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b"
432 by (cases rule: extreal2_cases[of b c]) auto
434 lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto
436 lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)"
437 by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus)
439 lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)"
440 by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus)
442 lemmas extreal_uminus_reorder =
443 extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder
446 fixes x :: extreal assumes "\<And>B. x \<le> extreal B" shows "x = - \<infinity>"
448 case (real r) with assms[of "r - 1"] show ?thesis by auto
449 next case PInf with assms[of 0] show ?thesis by auto
450 next case MInf then show ?thesis by simp
454 fixes x :: extreal assumes "\<And>B. x \<ge> extreal B" shows "x = \<infinity>"
456 case (real r) with assms[of "r + 1"] show ?thesis by auto
457 next case MInf with assms[of 0] show ?thesis by auto
458 next case PInf then show ?thesis by simp
462 shows extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)"
463 and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)"
464 by (simp_all add: min_def max_def)
466 lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)"
467 by (auto simp: zero_extreal_def)
470 fixes f :: "nat \<Rightarrow> extreal"
471 shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
472 and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
473 unfolding decseq_def incseq_def by auto
475 lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))"
476 unfolding incseq_def by auto
478 lemma extreal_add_nonneg_nonneg:
479 fixes a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
480 using add_mono[of 0 a 0 b] by simp
482 lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)"
485 lemma incseq_setsumI:
486 fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
487 assumes "\<And>i. 0 \<le> f i"
488 shows "incseq (\<lambda>i. setsum f {..< i})"
489 proof (intro incseq_SucI)
490 fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n"
491 using assms by (rule add_left_mono)
492 then show "setsum f {..< n} \<le> setsum f {..< Suc n}"
496 lemma incseq_setsumI2:
497 fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}"
498 assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
499 shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
500 using assms unfolding incseq_def by (auto intro: setsum_mono)
502 subsubsection "Multiplication"
504 instantiation extreal :: "{comm_monoid_mult, sgn}"
507 definition "1 = extreal 1"
509 function sgn_extreal where
510 "sgn (extreal r) = extreal (sgn r)"
511 | "sgn \<infinity> = 1"
512 | "sgn (-\<infinity>) = -1"
513 by (auto intro: extreal_cases)
514 termination proof qed (rule wf_empty)
516 function times_extreal where
517 "extreal r * extreal p = extreal (r * p)" |
518 "extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
519 "\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
520 "extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
521 "-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
522 "\<infinity> * \<infinity> = \<infinity>" |
523 "-\<infinity> * \<infinity> = -\<infinity>" |
524 "\<infinity> * -\<infinity> = -\<infinity>" |
525 "-\<infinity> * -\<infinity> = \<infinity>"
528 moreover then obtain a b where "x = (a, b)" by (cases x) auto
529 ultimately show P by (cases rule: extreal2_cases[of a b]) auto
531 termination by (relation "{}") simp
535 fix a :: extreal show "1 * a = a"
536 by (cases a) (simp_all add: one_extreal_def)
537 fix b :: extreal show "a * b = b * a"
538 by (cases rule: extreal2_cases[of a b]) simp_all
539 fix c :: extreal show "a * b * c = a * (b * c)"
540 by (cases rule: extreal3_cases[of a b c])
541 (simp_all add: zero_extreal_def zero_less_mult_iff)
545 lemma real_of_extreal_le_1:
546 fixes a :: extreal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1"
547 by (cases a) (auto simp: one_extreal_def)
549 lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)"
550 unfolding one_extreal_def by simp
552 lemma extreal_mult_zero[simp]:
553 fixes a :: extreal shows "a * 0 = 0"
554 by (cases a) (simp_all add: zero_extreal_def)
556 lemma extreal_zero_mult[simp]:
557 fixes a :: extreal shows "0 * a = 0"
558 by (cases a) (simp_all add: zero_extreal_def)
560 lemma extreal_m1_less_0[simp]:
562 by (simp add: zero_extreal_def one_extreal_def)
564 lemma extreal_zero_m1[simp]:
565 "1 \<noteq> (0::extreal)"
566 by (simp add: zero_extreal_def one_extreal_def)
568 lemma extreal_times_0[simp]:
569 fixes x :: extreal shows "0 * x = 0"
570 by (cases x) (auto simp: zero_extreal_def)
572 lemma extreal_times[simp]:
573 "1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1"
574 "1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1"
575 by (auto simp add: times_extreal_def one_extreal_def)
577 lemma extreal_plus_1[simp]:
578 "1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)"
579 "1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>"
580 unfolding one_extreal_def by auto
582 lemma extreal_zero_times[simp]:
583 fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
584 by (cases rule: extreal2_cases[of a b]) auto
586 lemma extreal_mult_eq_PInfty[simp]:
587 shows "a * b = \<infinity> \<longleftrightarrow>
588 (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)"
589 by (cases rule: extreal2_cases[of a b]) auto
591 lemma extreal_mult_eq_MInfty[simp]:
592 shows "a * b = -\<infinity> \<longleftrightarrow>
593 (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)"
594 by (cases rule: extreal2_cases[of a b]) auto
596 lemma extreal_0_less_1[simp]: "0 < (1::extreal)"
597 by (simp_all add: zero_extreal_def one_extreal_def)
599 lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)"
600 by (simp_all add: zero_extreal_def one_extreal_def)
602 lemma extreal_mult_minus_left[simp]:
603 fixes a b :: extreal shows "-a * b = - (a * b)"
604 by (cases rule: extreal2_cases[of a b]) auto
606 lemma extreal_mult_minus_right[simp]:
607 fixes a b :: extreal shows "a * -b = - (a * b)"
608 by (cases rule: extreal2_cases[of a b]) auto
610 lemma extreal_mult_infty[simp]:
611 "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
614 lemma extreal_infty_mult[simp]:
615 "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
618 lemma extreal_mult_strict_right_mono:
619 assumes "a < b" and "0 < c" "c < \<infinity>"
620 shows "a * c < b * c"
622 by (cases rule: extreal3_cases[of a b c])
623 (auto simp: zero_le_mult_iff extreal_less_PInfty)
625 lemma extreal_mult_strict_left_mono:
626 "\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b"
627 using extreal_mult_strict_right_mono by (simp add: mult_commute[of c])
629 lemma extreal_mult_right_mono:
630 fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
632 apply (cases "c = 0") apply simp
633 by (cases rule: extreal3_cases[of a b c])
634 (auto simp: zero_le_mult_iff extreal_less_PInfty)
636 lemma extreal_mult_left_mono:
637 fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
638 using extreal_mult_right_mono by (simp add: mult_commute[of c])
640 lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)"
641 by (simp add: one_extreal_def zero_extreal_def)
643 lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)"
644 by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg)
646 lemma extreal_right_distrib:
647 fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b"
648 by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
650 lemma extreal_left_distrib:
651 fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r"
652 by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps)
654 lemma extreal_mult_le_0_iff:
656 shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
657 by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff)
659 lemma extreal_zero_le_0_iff:
661 shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
662 by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
664 lemma extreal_mult_less_0_iff:
666 shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
667 by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff)
669 lemma extreal_zero_less_0_iff:
671 shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
672 by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
674 lemma extreal_distrib:
675 fixes a b c :: extreal
676 assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>"
677 shows "(a + b) * c = a * c + b * c"
679 by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
681 lemma extreal_le_epsilon:
683 assumes "ALL e. 0 < e --> x <= y + e"
686 { assume a: "EX r. y = extreal r"
687 from this obtain r where r_def: "y = extreal r" by auto
688 { assume "x=(-\<infinity>)" hence ?thesis by auto }
690 { assume "~(x=(-\<infinity>))"
691 from this obtain p where p_def: "x = extreal p"
692 using a assms[rule_format, of 1] by (cases x) auto
693 { fix e have "0 < e --> p <= r + e"
694 using assms[rule_format, of "extreal e"] p_def r_def by auto }
695 hence "p <= r" apply (subst field_le_epsilon) by auto
696 hence ?thesis using r_def p_def by auto
697 } ultimately have ?thesis by blast
700 { assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
701 using assms[rule_format, of 1] by (cases x) auto
702 } ultimately show ?thesis by (cases y) auto
706 lemma extreal_le_epsilon2:
708 assumes "ALL e. 0 < e --> x <= y + extreal e"
711 { fix e :: extreal assume "e>0"
712 { assume "e=\<infinity>" hence "x<=y+e" by auto }
714 { assume "e~=\<infinity>"
715 from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto
716 hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
717 } ultimately have "x<=y+e" by blast
718 } from this show ?thesis using extreal_le_epsilon by auto
721 lemma extreal_le_real:
723 assumes "ALL z. x <= extreal z --> y <= extreal z"
725 by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1)
726 extreal_less_eq(2) order_refl uminus_extreal.simps(2))
728 lemma extreal_le_extreal:
730 assumes "\<And>B. B < x \<Longrightarrow> B <= y"
732 by (metis assms extreal_dense leD linorder_le_less_linear)
734 lemma extreal_ge_extreal:
736 assumes "ALL B. B>x --> B >= y"
738 by (metis assms extreal_dense leD linorder_le_less_linear)
740 lemma setprod_extreal_0:
741 fixes f :: "'a \<Rightarrow> extreal"
742 shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))"
745 then show ?thesis by (induct A) auto
748 lemma setprod_extreal_pos:
749 fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)"
751 assume "finite I" from this pos show ?thesis by induct auto
755 assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
756 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)"
758 assume "finite I" from this assms show ?thesis
761 then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos)
762 from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto
763 also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0"
764 using setprod_extreal_pos[of I f] pos
765 by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto
766 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)"
767 using insert by (auto simp: setprod_extreal_0)
772 lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)"
774 assume "finite A" then show ?thesis
775 by induct (auto simp: one_extreal_def)
776 qed (simp add: one_extreal_def)
778 subsubsection {* Power *}
780 instantiation extreal :: power
782 primrec power_extreal where
783 "power_extreal x 0 = 1" |
784 "power_extreal x (Suc n) = x * x ^ n"
788 lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)"
789 by (induct n) (auto simp: one_extreal_def)
791 lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)"
792 by (induct n) (auto simp: one_extreal_def)
794 lemma extreal_power_uminus[simp]:
796 shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
797 by (induct n) (auto simp: one_extreal_def)
799 lemma extreal_power_number_of[simp]:
800 "(number_of num :: extreal) ^ n = extreal (number_of num ^ n)"
801 by (induct n) (auto simp: one_extreal_def)
803 lemma zero_le_power_extreal[simp]:
804 fixes a :: extreal assumes "0 \<le> a"
805 shows "0 \<le> a ^ n"
806 using assms by (induct n) (auto simp: extreal_zero_le_0_iff)
808 subsubsection {* Subtraction *}
810 lemma extreal_minus_minus_image[simp]:
811 fixes S :: "extreal set"
812 shows "uminus ` uminus ` S = S"
813 by (auto simp: image_iff)
815 lemma extreal_uminus_lessThan[simp]:
816 fixes a :: extreal shows "uminus ` {..<a} = {-a<..}"
817 proof (safe intro!: image_eqI)
818 fix x assume "-a < x"
819 then have "- x < - (- a)" by (simp del: extreal_uminus_uminus)
820 then show "- x < a" by simp
823 lemma extreal_uminus_greaterThan[simp]:
824 "uminus ` {(a::extreal)<..} = {..<-a}"
825 by (metis extreal_uminus_lessThan extreal_uminus_uminus
826 extreal_minus_minus_image)
828 instantiation extreal :: minus
830 definition "x - y = x + -(y::extreal)"
834 lemma extreal_minus[simp]:
835 "extreal r - extreal p = extreal (r - p)"
836 "-\<infinity> - extreal r = -\<infinity>"
837 "extreal r - \<infinity> = -\<infinity>"
838 "\<infinity> - x = \<infinity>"
839 "-\<infinity> - \<infinity> = -\<infinity>"
843 by (simp_all add: minus_extreal_def)
845 lemma extreal_x_minus_x[simp]:
846 "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)"
847 by (cases x) simp_all
849 lemma extreal_eq_minus_iff:
850 fixes x y z :: extreal
851 shows "x = z - y \<longleftrightarrow>
852 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
853 (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
854 (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
855 (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
856 by (cases rule: extreal3_cases[of x y z]) auto
858 lemma extreal_eq_minus:
859 fixes x y z :: extreal
860 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
861 by (auto simp: extreal_eq_minus_iff)
863 lemma extreal_less_minus_iff:
864 fixes x y z :: extreal
865 shows "x < z - y \<longleftrightarrow>
866 (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
867 (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
868 (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
869 by (cases rule: extreal3_cases[of x y z]) auto
871 lemma extreal_less_minus:
872 fixes x y z :: extreal
873 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
874 by (auto simp: extreal_less_minus_iff)
876 lemma extreal_le_minus_iff:
877 fixes x y z :: extreal
878 shows "x \<le> z - y \<longleftrightarrow>
879 (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
880 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
881 by (cases rule: extreal3_cases[of x y z]) auto
883 lemma extreal_le_minus:
884 fixes x y z :: extreal
885 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
886 by (auto simp: extreal_le_minus_iff)
888 lemma extreal_minus_less_iff:
889 fixes x y z :: extreal
890 shows "x - y < z \<longleftrightarrow>
891 y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and>
892 (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
893 by (cases rule: extreal3_cases[of x y z]) auto
895 lemma extreal_minus_less:
896 fixes x y z :: extreal
897 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
898 by (auto simp: extreal_minus_less_iff)
900 lemma extreal_minus_le_iff:
901 fixes x y z :: extreal
902 shows "x - y \<le> z \<longleftrightarrow>
903 (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
904 (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
905 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
906 by (cases rule: extreal3_cases[of x y z]) auto
908 lemma extreal_minus_le:
909 fixes x y z :: extreal
910 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
911 by (auto simp: extreal_minus_le_iff)
913 lemma extreal_minus_eq_minus_iff:
914 fixes a b c :: extreal
915 shows "a - b = a - c \<longleftrightarrow>
916 b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
917 by (cases rule: extreal3_cases[of a b c]) auto
919 lemma extreal_add_le_add_iff:
920 "c + a \<le> c + b \<longleftrightarrow>
921 a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
922 by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
924 lemma extreal_mult_le_mult_iff:
925 "\<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)"
926 by (cases rule: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
928 lemma extreal_minus_mono:
929 fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C"
930 shows "A - C \<le> B - D"
932 by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all
934 lemma real_of_extreal_minus:
935 "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)"
936 by (cases rule: extreal2_cases[of a b]) auto
938 lemma extreal_diff_positive:
939 fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a"
940 by (cases rule: extreal2_cases[of a b]) auto
942 lemma extreal_between:
944 assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e"
945 shows "x - e < x" "x < x + e"
946 using assms apply (cases x, cases e) apply auto
947 using assms by (cases x, cases e) auto
949 subsubsection {* Division *}
951 instantiation extreal :: inverse
954 function inverse_extreal where
955 "inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" |
956 "inverse \<infinity> = 0" |
957 "inverse (-\<infinity>) = 0"
958 by (auto intro: extreal_cases)
959 termination by (relation "{}") simp
961 definition "x / y = x * inverse (y :: extreal)"
966 lemma real_of_extreal_inverse[simp]:
968 shows "real (inverse a) = 1 / real a"
969 by (cases a) (auto simp: inverse_eq_divide)
971 lemma extreal_inverse[simp]:
972 "inverse 0 = \<infinity>"
973 "inverse (1::extreal) = 1"
974 by (simp_all add: one_extreal_def zero_extreal_def)
976 lemma extreal_divide[simp]:
977 "extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))"
978 unfolding divide_extreal_def by (auto simp: divide_real_def)
980 lemma extreal_divide_same[simp]:
981 "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)"
983 (simp_all add: divide_real_def divide_extreal_def one_extreal_def)
985 lemma extreal_inv_inv[simp]:
986 "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
989 lemma extreal_inverse_minus[simp]:
990 "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
991 by (cases x) simp_all
993 lemma extreal_uminus_divide[simp]:
994 fixes x y :: extreal shows "- x / y = - (x / y)"
995 unfolding divide_extreal_def by simp
997 lemma extreal_divide_Infty[simp]:
998 "x / \<infinity> = 0" "x / -\<infinity> = 0"
999 unfolding divide_extreal_def by simp_all
1001 lemma extreal_divide_one[simp]:
1002 "x / 1 = (x::extreal)"
1003 unfolding divide_extreal_def by simp
1005 lemma extreal_divide_extreal[simp]:
1006 "\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
1007 unfolding divide_extreal_def by simp
1009 lemma zero_le_divide_extreal[simp]:
1010 fixes a :: extreal assumes "0 \<le> a" "0 \<le> b"
1011 shows "0 \<le> a / b"
1012 using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff)
1014 lemma extreal_le_divide_pos:
1015 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
1016 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
1018 lemma extreal_divide_le_pos:
1019 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
1020 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
1022 lemma extreal_le_divide_neg:
1023 "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
1024 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
1026 lemma extreal_divide_le_neg:
1027 "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
1028 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
1030 lemma extreal_inverse_antimono_strict:
1031 fixes x y :: extreal
1032 shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
1033 by (cases rule: extreal2_cases[of x y]) auto
1035 lemma extreal_inverse_antimono:
1036 fixes x y :: extreal
1037 shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
1038 by (cases rule: extreal2_cases[of x y]) auto
1040 lemma inverse_inverse_Pinfty_iff[simp]:
1041 "inverse x = \<infinity> \<longleftrightarrow> x = 0"
1044 lemma extreal_inverse_eq_0:
1045 "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
1048 lemma extreal_0_gt_inverse:
1049 fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
1052 lemma extreal_mult_less_right:
1053 assumes "b * a < c * a" "0 < a" "a < \<infinity>"
1056 by (cases rule: extreal3_cases[of a b c])
1057 (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
1059 lemma extreal_power_divide:
1060 "y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n"
1061 by (cases rule: extreal2_cases[of x y])
1062 (auto simp: one_extreal_def zero_extreal_def power_divide not_le
1063 power_less_zero_eq zero_le_power_iff)
1065 lemma extreal_le_mult_one_interval:
1066 fixes x y :: extreal
1067 assumes y: "y \<noteq> -\<infinity>"
1068 assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
1071 case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_extreal_def)
1073 case (real r) note r = this
1076 case (real p) note p = this
1078 proof (rule field_le_mult_one_interval)
1079 fix z :: real assume "0 < z" and "z < 1"
1080 with z[of "extreal z"]
1081 show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def)
1083 then show "x \<le> y" using p r by simp
1084 qed (insert y, simp_all)
1087 subsection "Complete lattice"
1089 instantiation extreal :: lattice
1091 definition [simp]: "sup x y = (max x y :: extreal)"
1092 definition [simp]: "inf x y = (min x y :: extreal)"
1093 instance proof qed simp_all
1096 instantiation extreal :: complete_lattice
1099 definition "bot = -\<infinity>"
1100 definition "top = \<infinity>"
1102 definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)"
1103 definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)"
1105 lemma extreal_complete_Sup:
1106 fixes S :: "extreal set" assumes "S \<noteq> {}"
1107 shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
1109 assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x"
1110 then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto
1111 then have "\<infinity> \<notin> S" by force
1114 assume "S = {-\<infinity>}"
1115 then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
1117 assume "S \<noteq> {-\<infinity>}"
1118 with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
1119 with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
1120 by (auto simp: real_of_extreal_ord_simps)
1121 with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
1123 "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
1126 proof (safe intro!: exI[of _ "extreal s"])
1127 fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s"
1131 using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto
1134 fix z assume *: "\<forall>y\<in>S. y \<le> z"
1135 with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z"
1138 with * have "s \<le> u"
1139 by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps)
1140 then show ?thesis using real by simp
1145 assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)"
1147 proof (safe intro!: exI[of _ \<infinity>])
1148 fix y assume **: "\<forall>z\<in>S. z \<le> y"
1149 with * show "\<infinity> \<le> y"
1151 case MInf with * ** show ?thesis by (force simp: not_le)
1156 lemma extreal_complete_Inf:
1157 fixes S :: "extreal set" assumes "S ~= {}"
1158 shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
1160 def S1 == "uminus ` S"
1161 hence "S1 ~= {}" using assms by auto
1162 from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
1163 using extreal_complete_Sup[of S1] by auto
1164 { fix z assume "ALL y:S. z <= y"
1165 hence "ALL y:S1. y <= -z" unfolding S1_def by auto
1166 hence "x <= -z" using x_def by auto
1168 apply (subst extreal_uminus_uminus[symmetric])
1169 unfolding extreal_minus_le_minus . }
1170 moreover have "(ALL y:S. -x <= y)"
1171 using x_def unfolding S1_def
1173 apply (subst (3) extreal_uminus_uminus[symmetric])
1174 unfolding extreal_minus_le_minus by simp
1175 ultimately show ?thesis by auto
1178 lemma extreal_complete_uminus_eq:
1179 fixes S :: "extreal set"
1180 shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
1181 \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
1182 by simp (metis extreal_minus_le_minus extreal_uminus_uminus)
1184 lemma extreal_Sup_uminus_image_eq:
1185 fixes S :: "extreal set"
1186 shows "Sup (uminus ` S) = - Inf S"
1189 moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)"
1190 by (rule the_equality) (auto intro!: extreal_bot)
1191 moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)"
1192 by (rule some_equality) (auto intro!: extreal_top)
1193 ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def
1194 Least_def Greatest_def GreatestM_def by simp
1196 assume "S \<noteq> {}"
1197 with extreal_complete_Sup[of "uminus`S"]
1198 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)"
1199 unfolding extreal_complete_uminus_eq by auto
1200 show "Sup (uminus ` S) = - Inf S"
1201 unfolding Inf_extreal_def Greatest_def GreatestM_def
1202 proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
1203 show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
1205 fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
1206 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)"
1207 unfolding extreal_complete_uminus_eq by simp
1208 then show "Sup (uminus ` S) = -x'"
1209 unfolding Sup_extreal_def extreal_uminus_eq_iff
1210 by (intro Least_equality) auto
1216 { fix x :: extreal and A
1217 show "bot <= x" by (cases x) (simp_all add: bot_extreal_def)
1218 show "x <= top" by (simp add: top_extreal_def) }
1220 { fix x :: extreal and A assume "x : A"
1221 with extreal_complete_Sup[of A]
1222 obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
1223 hence "x <= s" using `x : A` by auto
1224 also have "... = Sup A" using s unfolding Sup_extreal_def
1225 by (auto intro!: Least_equality[symmetric])
1226 finally show "x <= Sup A" . }
1229 { fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)"
1231 proof (cases "A = {}")
1233 hence "Sup A = -\<infinity>" unfolding Sup_extreal_def
1234 by (auto intro!: Least_equality)
1235 thus "Sup A <= x" by simp
1238 with extreal_complete_Sup[of A]
1239 obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
1241 unfolding Sup_extreal_def by (auto intro!: Least_equality)
1242 also have "s <= x" using * s by auto
1243 finally show "Sup A <= x" .
1247 { fix x :: extreal and A assume "x \<in> A"
1248 with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
1249 unfolding extreal_Sup_uminus_image_eq by simp }
1251 { fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)"
1252 with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
1253 unfolding extreal_Sup_uminus_image_eq by force }
1257 lemma extreal_SUPR_uminus:
1258 fixes f :: "'a => extreal"
1259 shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
1260 unfolding SUPR_def INFI_def
1261 using extreal_Sup_uminus_image_eq[of "f`R"]
1262 by (simp add: image_image)
1264 lemma extreal_INFI_uminus:
1265 fixes f :: "'a => extreal"
1266 shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
1267 using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
1269 lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)"
1270 using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
1272 lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)"
1273 by (auto intro!: inj_onI)
1275 lemma extreal_image_uminus_shift:
1276 fixes X Y :: "extreal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y"
1278 assume "uminus ` X = Y"
1279 then have "uminus ` uminus ` X = uminus ` Y"
1280 by (simp add: inj_image_eq_iff)
1281 then show "X = uminus ` Y" by (simp add: image_image)
1282 qed (simp add: image_image)
1284 lemma Inf_extreal_iff:
1286 shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
1287 by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
1288 order_less_le_trans)
1290 lemma Sup_eq_MInfty:
1291 fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
1293 assume a: "Sup S = -\<infinity>"
1294 with complete_lattice_class.Sup_upper[of _ S]
1295 show "S={} \<or> S={-\<infinity>}" by auto
1297 assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
1298 unfolding Sup_extreal_def by (auto intro!: Least_equality)
1301 lemma Inf_eq_PInfty:
1302 fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
1303 using Sup_eq_MInfty[of "uminus`S"]
1304 unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp
1306 lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>"
1307 unfolding Inf_extreal_def
1308 by (auto intro!: Greatest_equality)
1310 lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>"
1311 unfolding Sup_extreal_def
1312 by (auto intro!: Least_equality)
1316 assumes "!!i. i : A ==> f i <= x"
1317 assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
1318 shows "(SUP i:A. f i) = x"
1319 unfolding SUPR_def Sup_extreal_def
1320 using assms by (auto intro!: Least_equality)
1324 assumes "!!i. i : A ==> f i >= x"
1325 assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
1326 shows "(INF i:A. f i) = x"
1327 unfolding INFI_def Inf_extreal_def
1328 using assms by (auto intro!: Greatest_equality)
1330 lemma Sup_extreal_close:
1332 assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
1333 shows "\<exists>x\<in>S. Sup S - e < x"
1334 using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])
1336 lemma Inf_extreal_close:
1337 fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e"
1338 shows "\<exists>x\<in>X. x < Inf X + e"
1339 proof (rule Inf_less_iff[THEN iffD1])
1340 show "Inf X < Inf X + e" using assms
1344 lemma Sup_eq_top_iff:
1345 fixes A :: "'a::{complete_lattice, linorder} set"
1346 shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
1348 assume *: "Sup A = top"
1349 show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
1350 proof (intro allI impI)
1351 fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
1352 unfolding less_Sup_iff by auto
1355 assume *: "\<forall>x<top. \<exists>i\<in>A. x < i"
1358 assume "Sup A \<noteq> top"
1359 with top_greatest[of "Sup A"]
1360 have "Sup A < top" unfolding le_less by auto
1361 then have "Sup A < Sup A"
1362 using * unfolding less_Sup_iff by auto
1363 then show False by auto
1367 lemma SUP_eq_top_iff:
1368 fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
1369 shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
1370 unfolding SUPR_def Sup_eq_top_iff by auto
1372 lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>"
1374 { fix x assume "x \<noteq> \<infinity>"
1375 then have "\<exists>k::nat. x < extreal (real k)"
1377 case MInf then show ?thesis by (intro exI[of _ 0]) auto
1380 moreover obtain k :: nat where "r < real k"
1381 using ex_less_of_nat by (auto simp: real_eq_of_nat)
1382 ultimately show ?thesis by auto
1385 using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"]
1386 by (auto simp: top_extreal_def)
1389 lemma extreal_le_Sup:
1391 shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
1392 (is "?lhs <-> ?rhs")
1395 { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
1396 from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto
1397 from this obtain i where "i : A & y <= f i" using `?rhs` by auto
1398 hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto
1399 hence False using y_def by auto
1400 } hence "?lhs" by auto
1403 { assume "?lhs" hence "?rhs"
1404 by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff
1405 inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
1406 } ultimately show ?thesis by auto
1409 lemma extreal_Inf_le:
1411 shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
1412 (is "?lhs <-> ?rhs")
1415 { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
1416 from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto
1417 from this obtain i where "i : A & f i <= y" using `?rhs` by auto
1418 hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto
1419 hence False using y_def by auto
1420 } hence "?lhs" by auto
1423 { assume "?lhs" hence "?rhs"
1424 by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff
1425 inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
1426 } ultimately show ?thesis by auto
1431 assumes "(INF i:A. f i) < x"
1432 shows "EX i. i : A & f i <= x"
1434 assume "~ (EX i. i : A & f i <= x)"
1435 hence "ALL i:A. f i > x" by auto
1436 hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto
1437 thus False using assms by auto
1441 assumes "ALL e:A. f e = g e"
1442 shows "(INF e:A. f e) = (INF e:A. g e)"
1444 have "f ` A = g ` A" unfolding image_def using assms by auto
1445 thus ?thesis unfolding INFI_def by auto
1449 assumes "ALL e:A. f e = g e"
1450 shows "(SUP e:A. f e) = (SUP e:A. g e)"
1452 have "f ` A = g ` A" unfolding image_def using assms by auto
1453 thus ?thesis unfolding SUPR_def by auto
1457 assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j"
1458 assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i"
1459 shows "(SUP i:A. f i) = (SUP j:B. g j)"
1460 proof (intro antisym)
1461 show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
1462 using assms by (metis SUP_leI le_SUPI2)
1463 show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
1464 using assms by (metis SUP_leI le_SUPI2)
1467 lemma SUP_extreal_le_addI:
1468 assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
1469 shows "SUPR UNIV f + y \<le> z"
1472 then have "\<And>i. f i \<le> z - y" using assms by (simp add: extreal_le_minus_iff)
1473 then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI)
1474 then show ?thesis using real by (simp add: extreal_le_minus_iff)
1475 qed (insert assms, auto)
1477 lemma SUPR_extreal_add:
1478 fixes f g :: "nat \<Rightarrow> extreal"
1479 assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
1480 shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
1481 proof (rule extreal_SUPI)
1482 fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y"
1483 have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos
1484 unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD)
1487 have "f i + g j \<le> f i + g (max i j)"
1488 using `incseq g`[THEN incseqD] by (rule add_left_mono) auto
1489 also have "\<dots> \<le> f (max i j) + g (max i j)"
1490 using `incseq f`[THEN incseqD] by (rule add_right_mono) auto
1491 also have "\<dots> \<le> y" using * by auto
1492 finally have "f i + g j \<le> y" . }
1493 then have "SUPR UNIV f + g j \<le> y"
1494 using assms(4)[of j] by (intro SUP_extreal_le_addI) auto
1495 then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) }
1496 then have "SUPR UNIV g + SUPR UNIV f \<le> y"
1497 using f by (rule SUP_extreal_le_addI)
1498 then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
1499 qed (auto intro!: add_mono le_SUPI)
1501 lemma SUPR_extreal_add_pos:
1502 fixes f g :: "nat \<Rightarrow> extreal"
1503 assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
1504 shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g"
1505 proof (intro SUPR_extreal_add inc)
1506 fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto
1509 lemma SUPR_extreal_setsum:
1510 fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal"
1511 assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
1512 shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))"
1514 assume "finite A" then show ?thesis using assms
1515 by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos)
1518 lemma SUPR_extreal_cmult:
1519 fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c"
1520 shows "(SUP i. c * f i) = c * SUPR UNIV f"
1521 proof (rule extreal_SUPI)
1522 fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto
1523 then show "c * f i \<le> c * SUPR UNIV f"
1524 using `0 \<le> c` by (rule extreal_mult_left_mono)
1526 fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y"
1527 show "c * SUPR UNIV f \<le> y"
1529 assume c: "0 < c \<and> c \<noteq> \<infinity>"
1530 with * have "SUPR UNIV f \<le> y / c"
1531 by (intro SUP_leI) (auto simp: extreal_le_divide_pos)
1533 by (auto simp: extreal_le_divide_pos)
1535 { assume "c = \<infinity>" have ?thesis
1537 assume "\<forall>i. f i = 0"
1538 moreover then have "range f = {0}" by auto
1539 ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def)
1541 assume "\<not> (\<forall>i. f i = 0)"
1542 then obtain i where "f i \<noteq> 0" by auto
1543 with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm)
1545 moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)"
1546 ultimately show ?thesis using * `0 \<le> c` by auto
1551 fixes f :: "'a \<Rightarrow> extreal"
1552 assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i"
1553 shows "(SUP i:A. f i) = \<infinity>"
1554 unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def]
1557 fix x assume "x \<noteq> \<infinity>"
1558 show "\<exists>i\<in>A. x < f i"
1560 case PInf with `x \<noteq> \<infinity>` show ?thesis by simp
1562 case MInf with assms[of "0"] show ?thesis by force
1565 with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < extreal (real n)" by auto
1566 moreover from assms[of n] guess i ..
1567 ultimately show ?thesis
1568 by (auto intro!: bexI[of _ i])
1572 lemma Sup_countable_SUPR:
1573 assumes "A \<noteq> {}"
1574 shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
1575 proof (cases "Sup A")
1577 have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
1579 fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x"
1580 using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def)
1582 then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)"
1583 by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff)
1585 from choice[OF this] guess f .. note f = this
1586 have "SUPR UNIV f = Sup A"
1587 proof (rule extreal_SUPI)
1588 fix i show "f i \<le> Sup A" using f
1589 by (auto intro!: complete_lattice_class.Sup_upper)
1591 fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
1592 show "Sup A \<le> y"
1593 proof (rule extreal_le_epsilon, intro allI impI)
1594 fix e :: extreal assume "0 < e"
1595 show "Sup A \<le> y + e"
1598 hence "0 < r" using `0 < e` by auto
1599 then obtain n ::nat where *: "1 / real n < r" "0 < n"
1600 using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
1601 have "Sup A \<le> f n + 1 / extreal (real n)" using f[THEN spec, of n] by auto
1602 also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def )
1603 with bound have "f n + 1 / extreal (real n) \<le> y + e" by (rule add_mono) simp
1604 finally show "Sup A \<le> y + e" .
1605 qed (insert `0 < e`, auto)
1608 with f show ?thesis by (auto intro!: exI[of _ f])
1611 from `A \<noteq> {}` obtain x where "x \<in> A" by auto
1614 assume "\<infinity> \<in> A"
1615 moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper)
1616 ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"])
1618 assume "\<infinity> \<notin> A"
1619 have "\<exists>x\<in>A. 0 \<le> x"
1620 by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear)
1621 then obtain x where "x \<in> A" "0 \<le> x" by auto
1622 have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f"
1624 assume "\<not> ?thesis"
1625 then have "\<exists>n::nat. Sup A \<le> x + extreal (real n)"
1626 by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le)
1627 then show False using `x \<in> A` `\<infinity> \<notin> A` PInf
1630 from choice[OF this] guess f .. note f = this
1631 have "SUPR UNIV f = \<infinity>"
1632 proof (rule SUP_PInfty)
1633 fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i"
1634 using f[THEN spec, of n] `0 \<le> x`
1635 by (cases rule: extreal2_cases[of "f n" x]) (auto intro!: exI[of _ n])
1637 then show ?thesis using f PInf by (auto intro!: exI[of _ f])
1641 with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty)
1642 then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"])
1645 lemma SUPR_countable_SUPR:
1646 "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
1647 using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def)
1650 lemma Sup_extreal_cadd:
1651 fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1652 shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A"
1653 proof (rule antisym)
1654 have *: "\<And>a::extreal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A"
1655 by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
1656 then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
1657 show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
1659 case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant)
1662 then have **: "op + (- a) ` op + a ` A = A"
1663 by (auto simp: image_iff ac_simps zero_extreal_def[symmetric])
1664 from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding **
1665 by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto
1666 qed (insert `a \<noteq> -\<infinity>`, auto)
1669 lemma Sup_extreal_cminus:
1670 fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1671 shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A"
1672 using Sup_extreal_cadd[of "uminus ` A" a] assms
1673 by (simp add: comp_def image_image minus_extreal_def
1674 extreal_Sup_uminus_image_eq)
1676 lemma SUPR_extreal_cminus:
1677 fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>"
1678 shows "(SUP x:A. a - f x) = a - (INF x:A. f x)"
1679 using Sup_extreal_cminus[of "f`A" a] assms
1680 unfolding SUPR_def INFI_def image_image by auto
1682 lemma Inf_extreal_cminus:
1683 fixes A :: "extreal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
1684 shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A"
1686 { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto }
1687 moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A"
1688 by (auto simp: image_image)
1689 ultimately show ?thesis
1690 using Sup_extreal_cminus[of "uminus ` A" "-a"] assms
1691 by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq)
1694 lemma INFI_extreal_cminus:
1695 fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>"
1696 shows "(INF x:A. a - f x) = a - (SUP x:A. f x)"
1697 using Inf_extreal_cminus[of "f`A" a] assms
1698 unfolding SUPR_def INFI_def image_image
1701 lemma uminus_extreal_add_uminus_uminus:
1702 fixes a b :: extreal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b"
1703 by (cases rule: extreal2_cases[of a b]) auto
1705 lemma INFI_extreal_add:
1706 assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
1707 shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g"
1709 have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
1710 using assms unfolding INF_less_iff by auto
1711 { fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i"
1712 by (rule uminus_extreal_add_uminus_uminus) }
1713 then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
1715 also have "\<dots> = INFI UNIV f + INFI UNIV g"
1716 unfolding extreal_INFI_uminus
1717 using assms INF_less
1718 by (subst SUPR_extreal_add)
1719 (auto simp: extreal_SUPR_uminus intro!: uminus_extreal_add_uminus_uminus)
1720 finally show ?thesis .
1723 subsection "Limits on @{typ extreal}"
1725 subsubsection "Topological space"
1727 instantiation extreal :: topological_space
1730 definition "open A \<longleftrightarrow> open (extreal -` A)
1731 \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A))
1732 \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))"
1734 lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)"
1735 unfolding open_extreal_def by auto
1737 lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)"
1738 unfolding open_extreal_def by auto
1740 lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A"
1741 using open_PInfty[OF assms] by auto
1743 lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A"
1744 using open_MInfty[OF assms] by auto
1746 lemma extreal_openE: assumes "open A" obtains x y where
1747 "open (extreal -` A)"
1748 "\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A"
1749 "-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A"
1750 using assms open_extreal_def by auto
1754 let ?U = "UNIV::extreal set"
1755 show "open ?U" unfolding open_extreal_def
1756 by (auto intro!: exI[of _ 0])
1758 fix S T::"extreal set" assume "open S" and "open T"
1759 from `open S`[THEN extreal_openE] guess xS yS .
1760 moreover from `open T`[THEN extreal_openE] guess xT yT .
1762 "open (extreal -` (S \<inter> T))"
1763 "\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T"
1764 "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T"
1766 then show "open (S Int T)" unfolding open_extreal_def by blast
1768 fix K :: "extreal set set" assume "\<forall>S\<in>K. open S"
1769 then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and>
1770 (\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)"
1771 by (auto simp: open_extreal_def)
1772 then show "open (Union K)" unfolding open_extreal_def
1773 proof (intro conjI impI)
1774 show "open (extreal -` \<Union>K)"
1775 using *[THEN choice] by (auto simp: vimage_Union)
1776 qed ((metis UnionE Union_upper subset_trans *)+)
1780 lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)"
1781 by (auto simp: inj_vimage_image_eq open_extreal_def)
1783 lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)"
1784 unfolding open_extreal_def by auto
1786 lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}"
1788 have "\<And>x. extreal -` {..<extreal x} = {..< x}"
1789 "extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto
1790 then show ?thesis by (cases a) (auto simp: open_extreal_def)
1793 lemma open_extreal_greaterThan[intro, simp]:
1794 "open {a :: extreal <..}"
1796 have "\<And>x. extreal -` {extreal x<..} = {x<..}"
1797 "extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto
1798 then show ?thesis by (cases a) (auto simp: open_extreal_def)
1801 lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}"
1802 unfolding greaterThanLessThan_def by auto
1804 lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}"
1806 have "- {a ..} = {..< a}" by auto
1807 then show "closed {a ..}"
1808 unfolding closed_def using open_extreal_lessThan by auto
1811 lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}"
1813 have "- {.. b} = {b <..}" by auto
1814 then show "closed {.. b}"
1815 unfolding closed_def using open_extreal_greaterThan by auto
1818 lemma closed_extreal_atLeastAtMost[simp, intro]:
1819 shows "closed {a :: extreal .. b}"
1820 unfolding atLeastAtMost_def by auto
1822 lemma closed_extreal_singleton:
1823 "closed {a :: extreal}"
1824 by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost)
1826 lemma extreal_open_cont_interval:
1827 assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
1828 obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
1830 from `open S` have "open (extreal -` S)" by (rule extreal_openE)
1831 then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S"
1832 using assms unfolding open_dist by force
1834 proof (intro that subsetI)
1835 show "0 < extreal e" using `0 < e` by auto
1836 fix y assume "y \<in> {x - extreal e<..<x + extreal e}"
1837 with assms obtain t where "y = extreal t" "dist t (real x) < e"
1838 apply (cases y) by (auto simp: dist_real_def)
1839 then show "y \<in> S" using e[of t] by auto
1843 lemma extreal_open_cont_interval2:
1844 assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>"
1845 obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
1847 guess e using extreal_open_cont_interval[OF assms] .
1848 with that[of "x-e" "x+e"] extreal_between[OF x, of e]
1852 instance extreal :: t2_space
1854 fix x y :: extreal assume "x ~= y"
1855 let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
1857 { fix x y :: extreal assume "x < y"
1858 from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
1860 apply (rule exI[of _ "{..<z}"])
1861 apply (rule exI[of _ "{z<..}"])
1866 show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
1867 proof (cases rule: linorder_cases)
1868 assume "x = y" with `x ~= y` show ?thesis by simp
1869 next assume "x < y" from *[OF this] show ?thesis by auto
1870 next assume "y < x" from *[OF this] show ?thesis by auto
1874 subsubsection {* Convergent sequences *}
1876 lemma lim_extreal[simp]:
1877 "((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
1878 proof (intro iffI topological_tendstoI)
1879 fix S assume "?l" "open S" "x \<in> S"
1880 then show "eventually (\<lambda>x. f x \<in> S) net"
1881 using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`]
1882 by (simp add: inj_image_mem_iff)
1884 fix S assume "?r" "open S" "extreal x \<in> S"
1885 show "eventually (\<lambda>x. extreal (f x) \<in> S) net"
1886 using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`]
1887 using `extreal x \<in> S` by auto
1890 lemma lim_real_of_extreal[simp]:
1891 assumes lim: "(f ---> extreal x) net"
1892 shows "((\<lambda>x. real (f x)) ---> x) net"
1893 proof (intro topological_tendstoI)
1894 fix S assume "open S" "x \<in> S"
1895 then have S: "open S" "extreal x \<in> extreal ` S"
1896 by (simp_all add: inj_image_mem_iff)
1897 have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto
1898 from this lim[THEN topological_tendstoD, OF open_extreal, OF S]
1899 show "eventually (\<lambda>x. real (f x) \<in> S) net"
1900 by (rule eventually_mono)
1903 lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r")
1904 proof assume ?r show ?l apply(rule topological_tendstoI)
1905 unfolding eventually_sequentially
1906 proof- fix S assume "open S" "\<infinity> : S"
1907 from open_PInfty[OF this] guess B .. note B=this
1908 from `?r`[rule_format,of "B+1"] guess N .. note N=this
1909 show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
1910 proof safe case goal1
1911 have "extreal B < extreal (B + 1)" by auto
1912 also have "... <= f n" using goal1 N by auto
1913 finally show ?case using B by fastsimp
1916 next assume ?l show ?r
1917 proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto
1918 from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
1919 guess N .. note N=this
1920 show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto
1925 lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r")
1926 proof assume ?r show ?l apply(rule topological_tendstoI)
1927 unfolding eventually_sequentially
1928 proof- fix S assume "open S" "(-\<infinity>) : S"
1929 from open_MInfty[OF this] guess B .. note B=this
1930 from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
1931 show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
1932 proof safe case goal1
1933 have "extreal (B - 1) >= f n" using goal1 N by auto
1934 also have "... < extreal B" by auto
1935 finally show ?case using B by fastsimp
1938 next assume ?l show ?r
1939 proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto
1940 from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
1941 guess N .. note N=this
1942 show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto
1947 lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>"
1948 proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
1949 from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
1950 guess N .. note N=this[rule_format,OF le_refl]
1951 hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans)
1952 hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto
1957 lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)"
1958 proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
1959 from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
1960 guess N .. note N=this[rule_format,OF le_refl]
1961 hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast
1966 lemma tendsto_explicit:
1967 "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
1968 unfolding tendsto_def eventually_sequentially by auto
1971 lemma tendsto_obtains_N:
1972 assumes "f ----> f0"
1973 assumes "open S" "f0 : S"
1974 obtains N where "ALL n>=N. f n : S"
1975 using tendsto_explicit[of f f0] assms by auto
1978 lemma tail_same_limit:
1980 assumes "X ----> L" "ALL n>=N. X n = Y n"
1983 { fix S assume "open S" and "L:S"
1984 from this obtain N1 where "ALL n>=N1. X n : S"
1985 using assms unfolding tendsto_def eventually_sequentially by auto
1986 hence "ALL n>=max N N1. Y n : S" using assms by auto
1987 hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
1989 thus ?thesis using tendsto_explicit by auto
1993 lemma Lim_bounded_PInfty2:
1994 assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B"
1995 shows "l ~= \<infinity>"
1997 def g == "(%n. if n>=N then f n else extreal B)"
1998 hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
1999 moreover have "!!n. g n <= extreal B" using g_def assms by auto
2000 ultimately show ?thesis using Lim_bounded_PInfty by auto
2003 lemma Lim_bounded_extreal:
2004 assumes lim:"f ----> (l :: extreal)"
2005 and "ALL n>=M. f n <= C"
2008 { assume "l=(-\<infinity>)" hence ?thesis by auto }
2010 { assume "~(l=(-\<infinity>))"
2011 { assume "C=\<infinity>" hence ?thesis by auto }
2013 { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
2014 hence "l=(-\<infinity>)" using assms
2015 tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
2016 hence ?thesis by auto }
2018 { assume "EX B. C = extreal B"
2019 from this obtain B where B_def: "C=extreal B" by auto
2020 hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
2021 from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
2022 from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}"
2023 apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto
2024 { fix n assume "n>=N"
2025 hence "EX r. extreal r = f n" using N_def by (cases "f n") auto
2026 } from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis
2027 hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
2028 hence *: "(%n. g n) ----> m" using m_def by auto
2029 { fix n assume "n>=max N M"
2030 hence "extreal (g n) <= extreal B" using assms g_def B_def by auto
2031 hence "g n <= B" by auto
2032 } hence "EX N. ALL n>=N. g n <= B" by blast
2033 hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
2034 hence ?thesis using m_def B_def by auto
2035 } ultimately have ?thesis by (cases C) auto
2036 } ultimately show ?thesis by blast
2039 lemma real_of_extreal_mult[simp]:
2040 fixes a b :: extreal shows "real (a * b) = real a * real b"
2041 by (cases rule: extreal2_cases[of a b]) auto
2043 lemma real_of_extreal_eq_0:
2044 "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
2047 lemma tendsto_extreal_realD:
2048 fixes f :: "'a \<Rightarrow> extreal"
2049 assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net"
2050 shows "(f ---> x) net"
2051 proof (intro topological_tendstoI)
2052 fix S assume S: "open S" "x \<in> S"
2053 with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
2054 from tendsto[THEN topological_tendstoD, OF this]
2055 show "eventually (\<lambda>x. f x \<in> S) net"
2056 by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0)
2059 lemma tendsto_extreal_realI:
2060 fixes f :: "'a \<Rightarrow> extreal"
2061 assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net"
2062 shows "((\<lambda>x. extreal (real (f x))) ---> x) net"
2063 proof (intro topological_tendstoI)
2064 fix S assume "open S" "x \<in> S"
2065 with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
2066 from tendsto[THEN topological_tendstoD, OF this]
2067 show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net"
2068 by (elim eventually_elim1) (auto simp: extreal_real)
2071 lemma extreal_mult_cancel_left:
2072 fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow>
2073 ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
2074 by (cases rule: extreal3_cases[of a b c])
2075 (simp_all add: zero_less_mult_iff)
2077 lemma extreal_inj_affinity:
2078 assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>"
2079 shows "inj_on (\<lambda>x. m * x + t) A"
2081 by (cases rule: extreal2_cases[of m t])
2082 (auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left)
2084 lemma extreal_PInfty_eq_plus[simp]:
2085 shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
2086 by (cases rule: extreal2_cases[of a b]) auto
2088 lemma extreal_MInfty_eq_plus[simp]:
2089 shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
2090 by (cases rule: extreal2_cases[of a b]) auto
2092 lemma extreal_less_divide_pos:
2093 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
2094 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
2096 lemma extreal_divide_less_pos:
2097 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
2098 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
2100 lemma extreal_divide_eq:
2101 "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
2102 by (cases rule: extreal3_cases[of a b c])
2103 (simp_all add: field_simps)
2105 lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
2108 lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
2111 lemma extreal_LimI_finite:
2112 assumes "\<bar>x\<bar> \<noteq> \<infinity>"
2113 assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
2115 proof (rule topological_tendstoI, unfold eventually_sequentially)
2116 obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto
2117 fix S assume "open S" "x : S"
2118 then have "open (extreal -` S)" unfolding open_extreal_def by auto
2119 with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S"
2120 unfolding open_real_def rx_def by auto
2122 upper: "!!N. n <= N ==> u N < x + extreal r" and
2123 lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto
2124 show "EX N. ALL n>=N. u n : S"
2125 proof (safe intro!: exI[of _ n])
2126 fix N assume "n <= N"
2127 from upper[OF this] lower[OF this] assms `0 < r`
2128 have "u N ~: {\<infinity>,(-\<infinity>)}" by auto
2129 from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto
2130 hence "rx < ra + r" and "ra < rx + r"
2131 using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
2132 hence "dist (real (u N)) rx < r"
2134 by (auto simp: dist_real_def abs_diff_less_iff field_simps)
2135 from dist[OF this] show "u N : S" using `u N ~: {\<infinity>, -\<infinity>}`
2136 by (auto simp: extreal_real split: split_if_asm)
2140 lemma extreal_LimI_finite_iff:
2141 assumes "\<bar>x\<bar> \<noteq> \<infinity>"
2142 shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
2143 (is "?lhs <-> ?rhs")
2145 assume lim: "u ----> x"
2146 { fix r assume "(r::extreal)>0"
2147 from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
2148 apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
2149 using lim extreal_between[of x r] assms `r>0` by auto
2150 hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
2151 using extreal_minus_less[of r x] by (cases r) auto
2152 } then show "?rhs" by auto
2154 assume ?rhs then show "u ----> x"
2155 using extreal_LimI_finite[of x] assms by auto
2159 subsubsection {* @{text Liminf} and @{text Limsup} *}
2162 "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
2165 "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
2168 fixes f :: "'a => 'b::{complete_lattice, linorder}"
2169 shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
2170 by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
2173 fixes f :: "'a => 'b::{complete_lattice, linorder}"
2174 shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
2175 by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
2179 assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
2180 assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
2182 unfolding Sup_extreal_def
2183 using assms by (auto intro!: Least_equality)
2187 assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
2188 assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
2190 unfolding Inf_extreal_def
2191 using assms by (auto intro!: Greatest_equality)
2194 fixes c :: "'a::{complete_lattice, linorder}"
2195 assumes ntriv: "\<not> trivial_limit net"
2196 shows "Limsup net (\<lambda>x. c) = c"
2197 unfolding Limsup_Inf
2198 proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
2199 fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
2202 assume "\<not> c \<le> x" then have "x < c" by auto
2203 then show False using ntriv * by (auto simp: trivial_limit_def)
2208 fixes c :: "'a::{complete_lattice, linorder}"
2209 assumes ntriv: "\<not> trivial_limit net"
2210 shows "Liminf net (\<lambda>x. c) = c"
2211 unfolding Liminf_Sup
2212 proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
2213 fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
2216 assume "\<not> x \<le> c" then have "c < x" by auto
2217 then show False using ntriv * by (auto simp: trivial_limit_def)
2222 fixes S :: "('a::order) set"
2223 shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
2224 by (auto simp: mono_def mem_def)
2226 lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
2227 lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
2228 lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
2229 lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
2232 fixes S :: "'a::{linorder,complete_lattice} set"
2233 defines "a \<equiv> Inf S"
2234 shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
2237 then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
2242 using mono[OF _ `a \<in> S`]
2243 by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
2245 assume "a \<notin> S"
2248 fix x assume "x \<in> S"
2249 then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
2250 then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
2252 fix x assume "a < x"
2253 then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
2254 with mono[of y x] show "x \<in> S" by auto
2260 lemma lim_imp_Liminf:
2261 fixes f :: "'a \<Rightarrow> extreal"
2262 assumes ntriv: "\<not> trivial_limit net"
2263 assumes lim: "(f ---> f0) net"
2264 shows "Liminf net f = f0"
2265 unfolding Liminf_Sup
2266 proof (safe intro!: extreal_SupI)
2267 fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
2269 proof (rule extreal_le_extreal)
2270 fix B assume "B < y"
2272 then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
2273 using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
2274 by (auto intro: eventually_conj)
2275 also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
2276 finally have False using ntriv[unfolded trivial_limit_def] by auto
2277 } then show "B \<le> f0" by (metis linorder_le_less_linear)
2280 fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
2282 proof (safe intro!: *[rule_format])
2283 fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
2284 using lim[THEN topological_tendstoD, of "{y <..}"] by auto
2288 lemma extreal_Liminf_le_Limsup:
2289 fixes f :: "'a \<Rightarrow> extreal"
2290 assumes ntriv: "\<not> trivial_limit net"
2291 shows "Liminf net f \<le> Limsup net f"
2292 unfolding Limsup_Inf Liminf_Sup
2293 proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least)
2294 fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
2297 assume "\<not> u \<le> v"
2298 then obtain t where "t < u" "v < t"
2299 using extreal_dense[of v u] by (auto simp: not_le)
2300 then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
2301 using * by (auto intro: eventually_conj)
2302 also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
2303 finally show False using ntriv by (auto simp: trivial_limit_def)
2308 fixes f g :: "'a => extreal"
2309 assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
2310 shows "Liminf net f \<le> Liminf net g"
2311 unfolding Liminf_Sup
2312 proof (safe intro!: Sup_mono bexI)
2313 fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
2314 then have "eventually (\<lambda>x. y < f x) net" by auto
2315 then show "eventually (\<lambda>x. y < g x) net"
2316 by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
2320 fixes f g :: "'a \<Rightarrow> extreal"
2321 assumes "eventually (\<lambda>x. f x = g x) net"
2322 shows "Liminf net f = Liminf net g"
2323 by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
2325 lemma Liminf_mono_all:
2326 fixes f g :: "'a \<Rightarrow> extreal"
2327 assumes "\<And>x. f x \<le> g x"
2328 shows "Liminf net f \<le> Liminf net g"
2329 using assms by (intro Liminf_mono always_eventually) auto
2332 fixes f g :: "'a \<Rightarrow> extreal"
2333 assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
2334 shows "Limsup net f \<le> Limsup net g"
2335 unfolding Limsup_Inf
2336 proof (safe intro!: Inf_mono bexI)
2337 fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
2338 then have "eventually (\<lambda>x. g x < y) net" by auto
2339 then show "eventually (\<lambda>x. f x < y) net"
2340 by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
2343 lemma Limsup_mono_all:
2344 fixes f g :: "'a \<Rightarrow> extreal"
2345 assumes "\<And>x. f x \<le> g x"
2346 shows "Limsup net f \<le> Limsup net g"
2347 using assms by (intro Limsup_mono always_eventually) auto
2350 fixes f g :: "'a \<Rightarrow> extreal"
2351 assumes "eventually (\<lambda>x. f x = g x) net"
2352 shows "Limsup net f = Limsup net g"
2353 by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
2355 abbreviation "liminf \<equiv> Liminf sequentially"
2357 abbreviation "limsup \<equiv> Limsup sequentially"
2359 lemma (in complete_lattice) less_INFD:
2360 assumes "y < INFI A f"" i \<in> A" shows "y < f i"
2363 also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI)
2364 finally show "y < f i" .
2367 lemma liminf_SUPR_INFI:
2368 fixes f :: "nat \<Rightarrow> extreal"
2369 shows "liminf f = (SUP n. INF m:{n..}. f m)"
2370 unfolding Liminf_Sup eventually_sequentially
2371 proof (safe intro!: antisym complete_lattice_class.Sup_least)
2372 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)"
2373 proof (rule extreal_le_extreal)
2374 fix y assume "y < x"
2375 with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
2376 then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
2377 also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto
2378 finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
2381 show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
2382 proof (unfold SUPR_def, safe intro!: Sup_mono bexI)
2383 fix y n assume "y < INFI {n..} f"
2384 from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
2385 qed (rule order_refl)
2388 lemma tail_same_limsup:
2389 fixes X Y :: "nat => extreal"
2390 assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
2391 shows "limsup X = limsup Y"
2392 using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
2394 lemma tail_same_liminf:
2395 fixes X Y :: "nat => extreal"
2396 assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
2397 shows "liminf X = liminf Y"
2398 using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
2401 fixes X Y :: "nat \<Rightarrow> extreal"
2402 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
2403 shows "liminf X \<le> liminf Y"
2404 using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
2407 fixes X Y :: "nat => extreal"
2408 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
2409 shows "limsup X \<le> limsup Y"
2410 using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
2412 declare trivial_limit_sequentially[simp]
2415 fixes X :: "nat \<Rightarrow> extreal"
2416 shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
2417 and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
2418 unfolding incseq_def decseq_def by auto
2420 lemma liminf_bounded:
2421 fixes X Y :: "nat \<Rightarrow> extreal"
2422 assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
2423 shows "C \<le> liminf X"
2424 using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
2426 lemma limsup_bounded:
2427 fixes X Y :: "nat => extreal"
2428 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
2429 shows "limsup X \<le> C"
2430 using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
2432 lemma liminf_bounded_iff:
2433 fixes x :: "nat \<Rightarrow> extreal"
2434 shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
2436 fix B assume "B < C" "C \<le> liminf x"
2437 then have "B < liminf x" by auto
2438 then obtain N where "B < (INF m:{N..}. x m)"
2439 unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto
2440 from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
2442 assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
2443 { fix B assume "B<C"
2444 then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
2445 hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto
2446 also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp
2447 finally have "B \<le> liminf x" .
2448 } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
2451 lemma liminf_subseq_mono:
2452 fixes X :: "nat \<Rightarrow> extreal"
2454 shows "liminf X \<le> liminf (X \<circ> r) "
2456 have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
2457 proof (safe intro!: INF_mono)
2458 fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
2459 using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
2461 then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
2464 lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x"
2467 lemma extreal_le_extreal_bounded:
2468 fixes x y z :: extreal
2470 assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y"
2472 proof (rule extreal_le_extreal)
2473 fix B assume "B < x"
2476 assume "z < B" from *[OF this `B < x`] show "B \<le> y" .
2478 assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto
2482 lemma fixes x y :: extreal
2483 shows Sup_atMost[simp]: "Sup {.. y} = y"
2484 and Sup_lessThan[simp]: "Sup {..< y} = y"
2485 and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y"
2486 and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y"
2487 and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y"
2488 by (auto simp: Sup_extreal_def intro!: Least_equality
2489 intro: extreal_le_extreal extreal_le_extreal_bounded[of x])
2491 lemma Sup_greaterThanlessThan[simp]:
2492 fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y"
2493 unfolding Sup_extreal_def
2494 proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y])
2495 fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z"
2496 from extreal_dense[OF `x < y`] guess w .. note w = this
2497 with z[THEN bspec, of w] show "x \<le> z" by auto
2500 lemma real_extreal_id: "real o extreal = id"
2502 { fix x have "(real o extreal) x = id x" by auto }
2503 from this show ?thesis using ext by blast
2506 lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
2507 by (metis range_extreal open_extreal open_UNIV)
2509 lemma extreal_le_distrib:
2510 fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
2511 by (cases rule: extreal3_cases[of a b c])
2512 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
2514 lemma extreal_pos_distrib:
2515 fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
2516 using assms by (cases rule: extreal3_cases[of a b c])
2517 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
2519 lemma extreal_pos_le_distrib:
2520 fixes a b c :: extreal
2522 shows "c * (a + b) <= c * a + c * b"
2523 using assms by (cases rule: extreal3_cases[of a b c])
2524 (auto simp add: field_simps)
2526 lemma extreal_max_mono:
2527 "[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d"
2528 by (metis sup_extreal_def sup_mono)
2531 lemma extreal_max_least:
2532 "[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
2533 by (metis sup_extreal_def sup_least)