src/HOL/Probability/Positive_Extended_Real.thy
 author hoelzl Fri Jan 14 14:21:48 2011 +0100 (2011-01-14) changeset 41544 c3b977fee8a3 parent 41413 64cd30d6b0b8 permissions -rw-r--r--
introduced integral syntax
1 (* Author: Johannes Hoelzl, TU Muenchen *)
3 header {* A type for positive real numbers with infinity *}
5 theory Positive_Extended_Real
6   imports Complex_Main "~~/src/HOL/Library/Nat_Bijection" Multivariate_Analysis
7 begin
9 lemma (in complete_lattice) Sup_start:
10   assumes *: "\<And>x. f x \<le> f 0"
11   shows "(SUP n. f n) = f 0"
12 proof (rule antisym)
13   show "f 0 \<le> (SUP n. f n)" by (rule le_SUPI) auto
14   show "(SUP n. f n) \<le> f 0" by (rule SUP_leI[OF *])
15 qed
17 lemma (in complete_lattice) Inf_start:
18   assumes *: "\<And>x. f 0 \<le> f x"
19   shows "(INF n. f n) = f 0"
20 proof (rule antisym)
21   show "(INF n. f n) \<le> f 0" by (rule INF_leI) simp
22   show "f 0 \<le> (INF n. f n)" by (rule le_INFI[OF *])
23 qed
25 lemma (in complete_lattice) Sup_mono_offset:
27   assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" and "0 \<le> k"
28   shows "(SUP n . f (k + n)) = (SUP n. f n)"
29 proof (rule antisym)
30   show "(SUP n. f (k + n)) \<le> (SUP n. f n)"
31     by (auto intro!: Sup_mono simp: SUPR_def)
32   { fix n :: 'b
33     have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
34     with * have "f n \<le> f (k + n)" by simp }
35   thus "(SUP n. f n) \<le> (SUP n. f (k + n))"
36     by (auto intro!: Sup_mono exI simp: SUPR_def)
37 qed
39 lemma (in complete_lattice) Sup_mono_offset_Suc:
40   assumes *: "\<And>x. f x \<le> f (Suc x)"
41   shows "(SUP n . f (Suc n)) = (SUP n. f n)"
42   unfolding Suc_eq_plus1
44   apply (rule Sup_mono_offset)
45   by (auto intro!: order.lift_Suc_mono_le[of "op \<le>" "op <" f, OF _ *]) default
47 lemma (in complete_lattice) Inf_mono_offset:
49   assumes *: "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x" and "0 \<le> k"
50   shows "(INF n . f (k + n)) = (INF n. f n)"
51 proof (rule antisym)
52   show "(INF n. f n) \<le> (INF n. f (k + n))"
53     by (auto intro!: Inf_mono simp: INFI_def)
54   { fix n :: 'b
55     have "0 + n \<le> k + n" using `0 \<le> k` by (rule add_right_mono)
56     with * have "f (k + n) \<le> f n" by simp }
57   thus "(INF n. f (k + n)) \<le> (INF n. f n)"
58     by (auto intro!: Inf_mono exI simp: INFI_def)
59 qed
61 lemma (in complete_lattice) isotone_converge:
62   fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y "
63   shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
64 proof -
65   have "\<And>n. (SUP m. f (n + m)) = (SUP n. f n)"
66     apply (rule Sup_mono_offset)
67     apply (rule assms)
68     by simp_all
69   moreover
70   { fix n have "(INF m. f (n + m)) = f n"
71       using Inf_start[of "\<lambda>m. f (n + m)"] assms by simp }
72   ultimately show ?thesis by simp
73 qed
75 lemma (in complete_lattice) antitone_converges:
76   fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>x y. x \<le> y \<Longrightarrow> f y \<le> f x"
77   shows "(INF n. SUP m. f (n + m)) = (SUP n. INF m. f (n + m))"
78 proof -
79   have "\<And>n. (INF m. f (n + m)) = (INF n. f n)"
80     apply (rule Inf_mono_offset)
81     apply (rule assms)
82     by simp_all
83   moreover
84   { fix n have "(SUP m. f (n + m)) = f n"
85       using Sup_start[of "\<lambda>m. f (n + m)"] assms by simp }
86   ultimately show ?thesis by simp
87 qed
89 lemma (in complete_lattice) lim_INF_le_lim_SUP:
90   fixes f :: "nat \<Rightarrow> 'a"
91   shows "(SUP n. INF m. f (n + m)) \<le> (INF n. SUP m. f (n + m))"
92 proof (rule SUP_leI, rule le_INFI)
93   fix i j show "(INF m. f (i + m)) \<le> (SUP m. f (j + m))"
94   proof (cases rule: le_cases)
95     assume "i \<le> j"
96     have "(INF m. f (i + m)) \<le> f (i + (j - i))" by (rule INF_leI) simp
97     also have "\<dots> = f (j + 0)" using `i \<le> j` by auto
98     also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
99     finally show ?thesis .
100   next
101     assume "j \<le> i"
102     have "(INF m. f (i + m)) \<le> f (i + 0)" by (rule INF_leI) simp
103     also have "\<dots> = f (j + (i - j))" using `j \<le> i` by auto
104     also have "\<dots> \<le> (SUP m. f (j + m))" by (rule le_SUPI) simp
105     finally show ?thesis .
106   qed
107 qed
109 text {*
111 We introduce the the positive real numbers as needed for measure theory.
113 *}
115 typedef pextreal = "(Some ` {0::real..}) \<union> {None}"
116   by (rule exI[of _ None]) simp
118 subsection "Introduce @{typ pextreal} similar to a datatype"
120 definition "Real x = Abs_pextreal (Some (sup 0 x))"
121 definition "\<omega> = Abs_pextreal None"
123 definition "pextreal_case f i x = (if x = \<omega> then i else f (THE r. 0 \<le> r \<and> x = Real r))"
125 definition "of_pextreal = pextreal_case (\<lambda>x. x) 0"
128   real_of_pextreal_def [code_unfold]: "real == of_pextreal"
130 lemma pextreal_Some[simp]: "0 \<le> x \<Longrightarrow> Some x \<in> pextreal"
131   unfolding pextreal_def by simp
133 lemma pextreal_Some_sup[simp]: "Some (sup 0 x) \<in> pextreal"
136 lemma pextreal_None[simp]: "None \<in> pextreal"
137   unfolding pextreal_def by simp
139 lemma Real_inj[simp]:
140   assumes  "0 \<le> x" and "0 \<le> y"
141   shows "Real x = Real y \<longleftrightarrow> x = y"
142   unfolding Real_def assms[THEN sup_absorb2]
143   using assms by (simp add: Abs_pextreal_inject)
145 lemma Real_neq_\<omega>[simp]:
146   "Real x = \<omega> \<longleftrightarrow> False"
147   "\<omega> = Real x \<longleftrightarrow> False"
148   by (simp_all add: Abs_pextreal_inject \<omega>_def Real_def)
150 lemma Real_neg: "x < 0 \<Longrightarrow> Real x = Real 0"
151   unfolding Real_def by (auto simp add: Abs_pextreal_inject intro!: sup_absorb1)
153 lemma pextreal_cases[case_names preal infinite, cases type: pextreal]:
154   assumes preal: "\<And>r. x = Real r \<Longrightarrow> 0 \<le> r \<Longrightarrow> P" and inf: "x = \<omega> \<Longrightarrow> P"
155   shows P
156 proof (cases x rule: pextreal.Abs_pextreal_cases)
157   case (Abs_pextreal y)
158   hence "y = None \<or> (\<exists>x \<ge> 0. y = Some x)"
159     unfolding pextreal_def by auto
160   thus P
161   proof (rule disjE)
162     assume "\<exists>x\<ge>0. y = Some x" then guess x ..
163     thus P by (simp add: preal[of x] Real_def Abs_pextreal(1) sup_absorb2)
164   qed (simp add: \<omega>_def Abs_pextreal(1) inf)
165 qed
167 lemma pextreal_case_\<omega>[simp]: "pextreal_case f i \<omega> = i"
168   unfolding pextreal_case_def by simp
170 lemma pextreal_case_Real[simp]: "pextreal_case f i (Real x) = (if 0 \<le> x then f x else f 0)"
171 proof (cases "0 \<le> x")
172   case True thus ?thesis unfolding pextreal_case_def by (auto intro: theI2)
173 next
174   case False
175   moreover have "(THE r. 0 \<le> r \<and> Real 0 = Real r) = 0"
176     by (auto intro!: the_equality)
177   ultimately show ?thesis unfolding pextreal_case_def by (simp add: Real_neg)
178 qed
180 lemma pextreal_case_cancel[simp]: "pextreal_case (\<lambda>c. i) i x = i"
181   by (cases x) simp_all
183 lemma pextreal_case_split:
184   "P (pextreal_case f i x) = ((x = \<omega> \<longrightarrow> P i) \<and> (\<forall>r\<ge>0. x = Real r \<longrightarrow> P (f r)))"
185   by (cases x) simp_all
187 lemma pextreal_case_split_asm:
188   "P (pextreal_case f i x) = (\<not> (x = \<omega> \<and> \<not> P i \<or> (\<exists>r. r \<ge> 0 \<and> x = Real r \<and> \<not> P (f r))))"
189   by (cases x) auto
191 lemma pextreal_case_cong[cong]:
192   assumes eq: "x = x'" "i = i'" and cong: "\<And>r. 0 \<le> r \<Longrightarrow> f r = f' r"
193   shows "pextreal_case f i x = pextreal_case f' i' x'"
194   unfolding eq using cong by (cases x') simp_all
196 lemma real_Real[simp]: "real (Real x) = (if 0 \<le> x then x else 0)"
197   unfolding real_of_pextreal_def of_pextreal_def by simp
199 lemma Real_real_image:
200   assumes "\<omega> \<notin> A" shows "Real ` real ` A = A"
201 proof safe
202   fix x assume "x \<in> A"
203   hence *: "x = Real (real x)"
204     using `\<omega> \<notin> A` by (cases x) auto
205   show "x \<in> Real ` real ` A"
206     using `x \<in> A` by (subst *) (auto intro!: imageI)
207 next
208   fix x assume "x \<in> A"
209   thus "Real (real x) \<in> A"
210     using `\<omega> \<notin> A` by (cases x) auto
211 qed
213 lemma real_pextreal_nonneg[simp, intro]: "0 \<le> real (x :: pextreal)"
214   unfolding real_of_pextreal_def of_pextreal_def
215   by (cases x) auto
217 lemma real_\<omega>[simp]: "real \<omega> = 0"
218   unfolding real_of_pextreal_def of_pextreal_def by simp
220 lemma pextreal_noteq_omega_Ex: "X \<noteq> \<omega> \<longleftrightarrow> (\<exists>r\<ge>0. X = Real r)" by (cases X) auto
222 subsection "@{typ pextreal} is a monoid for addition"
225 begin
227 definition "0 = Real 0"
228 definition "x + y = pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r + p)) \<omega> y) \<omega> x"
230 lemma pextreal_plus[simp]:
231   "Real r + Real p = (if 0 \<le> r then if 0 \<le> p then Real (r + p) else Real r else Real p)"
232   "x + 0 = x"
233   "0 + x = x"
234   "x + \<omega> = \<omega>"
235   "\<omega> + x = \<omega>"
236   by (simp_all add: plus_pextreal_def Real_neg zero_pextreal_def split: pextreal_case_split)
238 lemma \<omega>_neq_0[simp]:
239   "\<omega> = 0 \<longleftrightarrow> False"
240   "0 = \<omega> \<longleftrightarrow> False"
243 lemma Real_eq_0[simp]:
244   "Real r = 0 \<longleftrightarrow> r \<le> 0"
245   "0 = Real r \<longleftrightarrow> r \<le> 0"
246   by (auto simp add: Abs_pextreal_inject zero_pextreal_def Real_def sup_real_def)
248 lemma Real_0[simp]: "Real 0 = 0" by (simp add: zero_pextreal_def)
250 instance
251 proof
252   fix a :: pextreal
253   show "0 + a = a" by (cases a) simp_all
255   fix b show "a + b = b + a"
256     by (cases a, cases b) simp_all
258   fix c show "a + b + c = a + (b + c)"
259     by (cases a, cases b, cases c) simp_all
260 qed
261 end
263 lemma Real_minus_abs[simp]: "Real (- \<bar>x\<bar>) = 0"
264   by simp
266 lemma pextreal_plus_eq_\<omega>[simp]: "(a :: pextreal) + b = \<omega> \<longleftrightarrow> a = \<omega> \<or> b = \<omega>"
267   by (cases a, cases b) auto
270   "a + b = a + c \<longleftrightarrow> (a = \<omega> \<or> b = c)"
271   by (cases a, cases b, cases c, simp_all, cases c, simp_all)
274   "b + a = c + a \<longleftrightarrow> (a = \<omega> \<or> b = c)"
275   by (cases a, cases b, cases c, simp_all, cases c, simp_all)
277 lemma Real_eq_Real:
278   "Real a = Real b \<longleftrightarrow> (a = b \<or> (a \<le> 0 \<and> b \<le> 0))"
279 proof (cases "a \<le> 0 \<or> b \<le> 0")
280   case False with Real_inj[of a b] show ?thesis by auto
281 next
282   case True
283   thus ?thesis
284   proof
285     assume "a \<le> 0"
286     hence *: "Real a = 0" by simp
287     show ?thesis using `a \<le> 0` unfolding * by auto
288   next
289     assume "b \<le> 0"
290     hence *: "Real b = 0" by simp
291     show ?thesis using `b \<le> 0` unfolding * by auto
292   qed
293 qed
295 lemma real_pextreal_0[simp]: "real (0 :: pextreal) = 0"
296   unfolding zero_pextreal_def real_Real by simp
298 lemma real_of_pextreal_eq_0: "real X = 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
299   by (cases X) auto
301 lemma real_of_pextreal_eq: "real X = real Y \<longleftrightarrow>
302     (X = Y \<or> (X = 0 \<and> Y = \<omega>) \<or> (Y = 0 \<and> X = \<omega>))"
303   by (cases X, cases Y) (auto simp add: real_of_pextreal_eq_0)
305 lemma real_of_pextreal_add: "real X + real Y =
306     (if X = \<omega> then real Y else if Y = \<omega> then real X else real (X + Y))"
307   by (auto simp: pextreal_noteq_omega_Ex)
309 subsection "@{typ pextreal} is a monoid for multiplication"
311 instantiation pextreal :: comm_monoid_mult
312 begin
314 definition "1 = Real 1"
315 definition "x * y = (if x = 0 \<or> y = 0 then 0 else
316   pextreal_case (\<lambda>r. pextreal_case (\<lambda>p. Real (r * p)) \<omega> y) \<omega> x)"
318 lemma pextreal_times[simp]:
319   "Real r * Real p = (if 0 \<le> r \<and> 0 \<le> p then Real (r * p) else 0)"
320   "\<omega> * x = (if x = 0 then 0 else \<omega>)"
321   "x * \<omega> = (if x = 0 then 0 else \<omega>)"
322   "0 * x = 0"
323   "x * 0 = 0"
324   "1 = \<omega> \<longleftrightarrow> False"
325   "\<omega> = 1 \<longleftrightarrow> False"
326   by (auto simp add: times_pextreal_def one_pextreal_def)
328 lemma pextreal_one_mult[simp]:
329   "Real x + 1 = (if 0 \<le> x then Real (x + 1) else 1)"
330   "1 + Real x = (if 0 \<le> x then Real (1 + x) else 1)"
331   unfolding one_pextreal_def by simp_all
333 instance
334 proof
335   fix a :: pextreal show "1 * a = a"
336     by (cases a) (simp_all add: one_pextreal_def)
338   fix b show "a * b = b * a"
339     by (cases a, cases b) (simp_all add: mult_nonneg_nonneg)
341   fix c show "a * b * c = a * (b * c)"
342     apply (cases a, cases b, cases c)
343     apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
344     apply (cases b, cases c)
345     apply (simp_all add: mult_nonneg_nonneg not_le mult_pos_pos)
346     done
347 qed
348 end
350 lemma pextreal_mult_cancel_left:
351   "a * b = a * c \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
352   by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
354 lemma pextreal_mult_cancel_right:
355   "b * a = c * a \<longleftrightarrow> (a = 0 \<or> b = c \<or> (a = \<omega> \<and> b \<noteq> 0 \<and> c \<noteq> 0))"
356   by (cases a, cases b, cases c, auto simp: Real_eq_Real mult_le_0_iff, cases c, auto)
358 lemma Real_1[simp]: "Real 1 = 1" by (simp add: one_pextreal_def)
360 lemma real_pextreal_1[simp]: "real (1 :: pextreal) = 1"
361   unfolding one_pextreal_def real_Real by simp
363 lemma real_of_pextreal_mult: "real X * real Y = real (X * Y :: pextreal)"
364   by (cases X, cases Y) (auto simp: zero_le_mult_iff)
366 lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
367   shows "Real (x * y) = Real x * Real y" using assms by auto
369 lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
370 proof(cases "finite A")
371   case True thus ?thesis using assms
372   proof(induct A) case (insert x A)
373     have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
374     thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
375       apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
376       using insert by auto
377   qed auto
378 qed auto
380 subsection "@{typ pextreal} is a linear order"
382 instantiation pextreal :: linorder
383 begin
385 definition "x < y \<longleftrightarrow> pextreal_case (\<lambda>i. pextreal_case (\<lambda>j. i < j) True y) False x"
386 definition "x \<le> y \<longleftrightarrow> pextreal_case (\<lambda>j. pextreal_case (\<lambda>i. i \<le> j) False x) True y"
388 lemma pextreal_less[simp]:
389   "Real r < \<omega>"
390   "Real r < Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r < p else 0 < p)"
391   "\<omega> < x \<longleftrightarrow> False"
392   "0 < \<omega>"
393   "0 < Real r \<longleftrightarrow> 0 < r"
394   "x < 0 \<longleftrightarrow> False"
395   "0 < (1::pextreal)"
396   by (simp_all add: less_pextreal_def zero_pextreal_def one_pextreal_def del: Real_0 Real_1)
398 lemma pextreal_less_eq[simp]:
399   "x \<le> \<omega>"
400   "Real r \<le> Real p \<longleftrightarrow> (if 0 \<le> r \<and> 0 \<le> p then r \<le> p else r \<le> 0)"
401   "0 \<le> x"
402   by (simp_all add: less_eq_pextreal_def zero_pextreal_def del: Real_0)
404 lemma pextreal_\<omega>_less_eq[simp]:
405   "\<omega> \<le> x \<longleftrightarrow> x = \<omega>"
406   by (cases x) (simp_all add: not_le less_eq_pextreal_def)
408 lemma pextreal_less_eq_zero[simp]:
409   "(x::pextreal) \<le> 0 \<longleftrightarrow> x = 0"
410   by (cases x) (simp_all add: zero_pextreal_def del: Real_0)
412 instance
413 proof
414   fix x :: pextreal
415   show "x \<le> x" by (cases x) simp_all
416   fix y
417   show "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
418     by (cases x, cases y) auto
419   show "x \<le> y \<or> y \<le> x "
420     by (cases x, cases y) auto
421   { assume "x \<le> y" "y \<le> x" thus "x = y"
422       by (cases x, cases y) auto }
423   { fix z assume "x \<le> y" "y \<le> z"
424     thus "x \<le> z" by (cases x, cases y, cases z) auto }
425 qed
426 end
428 lemma pextreal_zero_lessI[intro]:
429   "(a :: pextreal) \<noteq> 0 \<Longrightarrow> 0 < a"
430   by (cases a) auto
432 lemma pextreal_less_omegaI[intro, simp]:
433   "a \<noteq> \<omega> \<Longrightarrow> a < \<omega>"
434   by (cases a) auto
436 lemma pextreal_plus_eq_0[simp]: "(a :: pextreal) + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
437   by (cases a, cases b) auto
439 lemma pextreal_le_add1[simp, intro]: "n \<le> n + (m::pextreal)"
440   by (cases n, cases m) simp_all
442 lemma pextreal_le_add2: "(n::pextreal) + m \<le> k \<Longrightarrow> m \<le> k"
443   by (cases n, cases m, cases k) simp_all
445 lemma pextreal_le_add3: "(n::pextreal) + m \<le> k \<Longrightarrow> n \<le> k"
446   by (cases n, cases m, cases k) simp_all
448 lemma pextreal_less_\<omega>: "x < \<omega> \<longleftrightarrow> x \<noteq> \<omega>"
449   by (cases x) auto
451 lemma pextreal_0_less_mult_iff[simp]:
452   fixes x y :: pextreal shows "0 < x * y \<longleftrightarrow> 0 < x \<and> 0 < y"
453   by (cases x, cases y) (auto simp: zero_less_mult_iff)
455 lemma pextreal_ord_one[simp]:
456   "Real p < 1 \<longleftrightarrow> p < 1"
457   "Real p \<le> 1 \<longleftrightarrow> p \<le> 1"
458   "1 < Real p \<longleftrightarrow> 1 < p"
459   "1 \<le> Real p \<longleftrightarrow> 1 \<le> p"
460   by (simp_all add: one_pextreal_def del: Real_1)
462 subsection {* @{text "x - y"} on @{typ pextreal} *}
464 instantiation pextreal :: minus
465 begin
466 definition "x - y = (if y < x then THE d. x = y + d else 0 :: pextreal)"
468 lemma minus_pextreal_eq:
469   "(x - y = (z :: pextreal)) \<longleftrightarrow> (if y < x then x = y + z else z = 0)"
470   (is "?diff \<longleftrightarrow> ?if")
471 proof
472   assume ?diff
473   thus ?if
474   proof (cases "y < x")
475     case True
476     then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
478     show ?thesis unfolding `?diff`[symmetric] if_P[OF True] minus_pextreal_def
479     proof (rule theI2[where Q="\<lambda>d. x = y + d"])
480       show "x = y + pextreal_case (\<lambda>r. Real (r - real y)) \<omega> x" (is "x = y + ?d")
481         using `y < x` p by (cases x) simp_all
483       fix d assume "x = y + d"
484       thus "d = ?d" using `y < x` p by (cases d, cases x) simp_all
485     qed simp
487 next
488   assume ?if
489   thus ?diff
490   proof (cases "y < x")
491     case True
492     then obtain p where p: "y = Real p" "0 \<le> p" by (cases y) auto
494     from True `?if` have "x = y + z" by simp
496     show ?thesis unfolding minus_pextreal_def if_P[OF True] unfolding `x = y + z`
497     proof (rule the_equality)
498       fix d :: pextreal assume "y + z = y + d"
499       thus "d = z" using `y < x` p
500         by (cases d, cases z) simp_all
501     qed simp
503 qed
505 instance ..
506 end
508 lemma pextreal_minus[simp]:
509   "Real r - Real p = (if 0 \<le> r \<and> p < r then if 0 \<le> p then Real (r - p) else Real r else 0)"
510   "(A::pextreal) - A = 0"
511   "\<omega> - Real r = \<omega>"
512   "Real r - \<omega> = 0"
513   "A - 0 = A"
514   "0 - A = 0"
515   by (auto simp: minus_pextreal_eq not_less)
517 lemma pextreal_le_epsilon:
518   fixes x y :: pextreal
519   assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
520   shows "x \<le> y"
521 proof (cases y)
522   case (preal r)
523   then obtain p where x: "x = Real p" "0 \<le> p"
524     using assms[of 1] by (cases x) auto
525   { fix e have "0 < e \<Longrightarrow> p \<le> r + e"
526       using assms[of "Real e"] preal x by auto }
527   hence "p \<le> r" by (rule field_le_epsilon)
528   thus ?thesis using preal x by auto
529 qed simp
531 instance pextreal :: "{ordered_comm_semiring, comm_semiring_1}"
532 proof
533   show "0 \<noteq> (1::pextreal)" unfolding zero_pextreal_def one_pextreal_def
534     by (simp del: Real_1 Real_0)
536   fix a :: pextreal
537   show "0 * a = 0" "a * 0 = 0" by simp_all
539   fix b c :: pextreal
540   show "(a + b) * c = a * c + b * c"
541     by (cases c, cases a, cases b)
542        (auto intro!: arg_cong[where f=Real] simp: field_simps not_le mult_le_0_iff mult_less_0_iff)
544   { assume "a \<le> b" thus "c + a \<le> c + b"
545      by (cases c, cases a, cases b) auto }
547   assume "a \<le> b" "0 \<le> c"
548   thus "c * a \<le> c * b"
549     apply (cases c, cases a, cases b)
550     by (auto simp: mult_left_mono mult_le_0_iff mult_less_0_iff not_le)
551 qed
553 lemma mult_\<omega>[simp]: "x * y = \<omega> \<longleftrightarrow> (x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0"
554   by (cases x, cases y) auto
556 lemma \<omega>_mult[simp]: "(\<omega> = x * y) = ((x = \<omega> \<or> y = \<omega>) \<and> x \<noteq> 0 \<and> y \<noteq> 0)"
557   by (cases x, cases y) auto
559 lemma pextreal_mult_0[simp]: "x * y = 0 \<longleftrightarrow> x = 0 \<or> (y::pextreal) = 0"
560   by (cases x, cases y) (auto simp: mult_le_0_iff)
562 lemma pextreal_mult_cancel:
563   fixes x y z :: pextreal
564   assumes "y \<le> z"
565   shows "x * y \<le> x * z"
566   using assms
567   by (cases x, cases y, cases z)
568      (auto simp: mult_le_cancel_left mult_le_0_iff mult_less_0_iff not_le)
570 lemma Real_power[simp]:
571   "Real x ^ n = (if x \<le> 0 then (if n = 0 then 1 else 0) else Real (x ^ n))"
572   by (induct n) auto
574 lemma Real_power_\<omega>[simp]:
575   "\<omega> ^ n = (if n = 0 then 1 else \<omega>)"
576   by (induct n) auto
578 lemma pextreal_of_nat[simp]: "of_nat m = Real (real m)"
579   by (induct m) (auto simp: real_of_nat_Suc one_pextreal_def simp del: Real_1)
581 lemma less_\<omega>_Ex_of_nat: "x < \<omega> \<longleftrightarrow> (\<exists>n. x < of_nat n)"
582 proof safe
583   assume "x < \<omega>"
584   then obtain r where "0 \<le> r" "x = Real r" by (cases x) auto
585   moreover obtain n where "r < of_nat n" using ex_less_of_nat by auto
586   ultimately show "\<exists>n. x < of_nat n" by (auto simp: real_eq_of_nat)
587 qed auto
589 lemma real_of_pextreal_mono:
590   fixes a b :: pextreal
591   assumes "b \<noteq> \<omega>" "a \<le> b"
592   shows "real a \<le> real b"
593 using assms by (cases b, cases a) auto
595 lemma setprod_pextreal_0:
596   "(\<Prod>i\<in>I. f i) = (0::pextreal) \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = 0)"
597 proof cases
598   assume "finite I" then show ?thesis
599   proof (induct I)
600     case (insert i I)
601     then show ?case by simp
602   qed simp
603 qed simp
605 lemma setprod_\<omega>:
606   "(\<Prod>i\<in>I. f i) = \<omega> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<omega>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
607 proof cases
608   assume "finite I" then show ?thesis
609   proof (induct I)
610     case (insert i I) then show ?case
611       by (auto simp: setprod_pextreal_0)
612   qed simp
613 qed simp
615 instance pextreal :: "semiring_char_0"
616 proof
617   fix m n
618   show "inj (of_nat::nat\<Rightarrow>pextreal)" by (auto intro!: inj_onI)
619 qed
621 subsection "@{typ pextreal} is a complete lattice"
623 instantiation pextreal :: lattice
624 begin
625 definition [simp]: "sup x y = (max x y :: pextreal)"
626 definition [simp]: "inf x y = (min x y :: pextreal)"
627 instance proof qed simp_all
628 end
630 instantiation pextreal :: complete_lattice
631 begin
633 definition "bot = Real 0"
634 definition "top = \<omega>"
636 definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: pextreal)"
637 definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: pextreal)"
639 lemma pextreal_complete_Sup:
640   fixes S :: "pextreal set" assumes "S \<noteq> {}"
641   shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
642 proof (cases "\<exists>x\<ge>0. \<forall>a\<in>S. a \<le> Real x")
643   case False
644   hence *: "\<And>x. x\<ge>0 \<Longrightarrow> \<exists>a\<in>S. \<not>a \<le> Real x" by simp
645   show ?thesis
646   proof (safe intro!: exI[of _ \<omega>])
647     fix y assume **: "\<forall>z\<in>S. z \<le> y"
648     show "\<omega> \<le> y" unfolding pextreal_\<omega>_less_eq
649     proof (rule ccontr)
650       assume "y \<noteq> \<omega>"
651       then obtain x where [simp]: "y = Real x" and "0 \<le> x" by (cases y) auto
652       from *[OF `0 \<le> x`] show False using ** by auto
653     qed
654   qed simp
655 next
656   case True then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> Real y" and "0 \<le> y" by auto
657   from y[of \<omega>] have "\<omega> \<notin> S" by auto
659   with `S \<noteq> {}` obtain x where "x \<in> S" and "x \<noteq> \<omega>" by auto
661   have bound: "\<forall>x\<in>real ` S. x \<le> y"
662   proof
663     fix z assume "z \<in> real ` S" then guess a ..
664     with y[of a] `\<omega> \<notin> S` `0 \<le> y` show "z \<le> y" by (cases a) auto
665   qed
666   with reals_complete2[of "real ` S"] `x \<in> S`
667   obtain s where s: "\<forall>y\<in>S. real y \<le> s" "\<forall>z. ((\<forall>y\<in>S. real y \<le> z) \<longrightarrow> s \<le> z)"
668     by auto
670   show ?thesis
671   proof (safe intro!: exI[of _ "Real s"])
672     fix z assume "z \<in> S" thus "z \<le> Real s"
673       using s `\<omega> \<notin> S` by (cases z) auto
674   next
675     fix z assume *: "\<forall>y\<in>S. y \<le> z"
676     show "Real s \<le> z"
677     proof (cases z)
678       case (preal u)
679       { fix v assume "v \<in> S"
680         hence "v \<le> Real u" using * preal by auto
681         hence "real v \<le> u" using `\<omega> \<notin> S` `0 \<le> u` by (cases v) auto }
682       hence "s \<le> u" using s(2) by auto
683       thus "Real s \<le> z" using preal by simp
684     qed simp
685   qed
686 qed
688 lemma pextreal_complete_Inf:
689   fixes S :: "pextreal set" assumes "S \<noteq> {}"
690   shows "\<exists>x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
691 proof (cases "S = {\<omega>}")
692   case True thus ?thesis by (auto intro!: exI[of _ \<omega>])
693 next
694   case False with `S \<noteq> {}` have "S - {\<omega>} \<noteq> {}" by auto
695   hence not_empty: "\<exists>x. x \<in> uminus ` real ` (S - {\<omega>})" by auto
696   have bounds: "\<exists>x. \<forall>y\<in>uminus ` real ` (S - {\<omega>}). y \<le> x" by (auto intro!: exI[of _ 0])
697   from reals_complete2[OF not_empty bounds]
698   obtain s where s: "\<And>y. y\<in>S - {\<omega>} \<Longrightarrow> - real y \<le> s" "\<forall>z. ((\<forall>y\<in>S - {\<omega>}. - real y \<le> z) \<longrightarrow> s \<le> z)"
699     by auto
701   show ?thesis
702   proof (safe intro!: exI[of _ "Real (-s)"])
703     fix z assume "z \<in> S"
704     show "Real (-s) \<le> z"
705     proof (cases z)
706       case (preal r)
707       with s `z \<in> S` have "z \<in> S - {\<omega>}" by simp
708       hence "- r \<le> s" using preal s(1)[of z] by auto
709       hence "- s \<le> r" by (subst neg_le_iff_le[symmetric]) simp
710       thus ?thesis using preal by simp
711     qed simp
712   next
713     fix z assume *: "\<forall>y\<in>S. z \<le> y"
714     show "z \<le> Real (-s)"
715     proof (cases z)
716       case (preal u)
717       { fix v assume "v \<in> S-{\<omega>}"
718         hence "Real u \<le> v" using * preal by auto
719         hence "- real v \<le> - u" using `0 \<le> u` `v \<in> S - {\<omega>}` by (cases v) auto }
720       hence "u \<le> - s" using s(2) by (subst neg_le_iff_le[symmetric]) auto
721       thus "z \<le> Real (-s)" using preal by simp
722     next
723       case infinite
724       with * have "S = {\<omega>}" using `S \<noteq> {}` by auto
725       with `S - {\<omega>} \<noteq> {}` show ?thesis by simp
726     qed
727   qed
728 qed
730 instance
731 proof
732   fix x :: pextreal and A
734   show "bot \<le> x" by (cases x) (simp_all add: bot_pextreal_def)
735   show "x \<le> top" by (simp add: top_pextreal_def)
737   { assume "x \<in> A"
738     with pextreal_complete_Sup[of A]
739     obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
740     hence "x \<le> s" using `x \<in> A` by auto
741     also have "... = Sup A" using s unfolding Sup_pextreal_def
742       by (auto intro!: Least_equality[symmetric])
743     finally show "x \<le> Sup A" . }
745   { assume "x \<in> A"
746     with pextreal_complete_Inf[of A]
747     obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
748     hence "Inf A = i" unfolding Inf_pextreal_def
749       by (auto intro!: Greatest_equality)
750     also have "i \<le> x" using i `x \<in> A` by auto
751     finally show "Inf A \<le> x" . }
753   { assume *: "\<And>z. z \<in> A \<Longrightarrow> z \<le> x"
754     show "Sup A \<le> x"
755     proof (cases "A = {}")
756       case True
757       hence "Sup A = 0" unfolding Sup_pextreal_def
758         by (auto intro!: Least_equality)
759       thus "Sup A \<le> x" by simp
760     next
761       case False
762       with pextreal_complete_Sup[of A]
763       obtain s where s: "\<forall>y\<in>A. y \<le> s" "\<forall>z. (\<forall>y\<in>A. y \<le> z) \<longrightarrow> s \<le> z" by auto
764       hence "Sup A = s"
765         unfolding Sup_pextreal_def by (auto intro!: Least_equality)
766       also have "s \<le> x" using * s by auto
767       finally show "Sup A \<le> x" .
768     qed }
770   { assume *: "\<And>z. z \<in> A \<Longrightarrow> x \<le> z"
771     show "x \<le> Inf A"
772     proof (cases "A = {}")
773       case True
774       hence "Inf A = \<omega>" unfolding Inf_pextreal_def
775         by (auto intro!: Greatest_equality)
776       thus "x \<le> Inf A" by simp
777     next
778       case False
779       with pextreal_complete_Inf[of A]
780       obtain i where i: "\<forall>y\<in>A. i \<le> y" "\<forall>z. (\<forall>y\<in>A. z \<le> y) \<longrightarrow> z \<le> i" by auto
781       have "x \<le> i" using * i by auto
782       also have "i = Inf A" using i
783         unfolding Inf_pextreal_def by (auto intro!: Greatest_equality[symmetric])
784       finally show "x \<le> Inf A" .
785     qed }
786 qed
787 end
789 lemma Inf_pextreal_iff:
790   fixes z :: pextreal
791   shows "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x<y) \<longleftrightarrow> Inf X < y"
792   by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
793             order_less_le_trans)
795 lemma Inf_greater:
796   fixes z :: pextreal assumes "Inf X < z"
797   shows "\<exists>x \<in> X. x < z"
798 proof -
799   have "X \<noteq> {}" using assms by (auto simp: Inf_empty top_pextreal_def)
800   with assms show ?thesis
801     by (metis Inf_pextreal_iff mem_def not_leE)
802 qed
804 lemma Inf_close:
805   fixes e :: pextreal assumes "Inf X \<noteq> \<omega>" "0 < e"
806   shows "\<exists>x \<in> X. x < Inf X + e"
807 proof (rule Inf_greater)
808   show "Inf X < Inf X + e" using assms
809     by (cases "Inf X", cases e) auto
810 qed
812 lemma pextreal_SUPI:
813   fixes x :: pextreal
814   assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> x"
815   assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> f i \<le> y) \<Longrightarrow> x \<le> y"
816   shows "(SUP i:A. f i) = x"
817   unfolding SUPR_def Sup_pextreal_def
818   using assms by (auto intro!: Least_equality)
820 lemma Sup_pextreal_iff:
821   fixes z :: pextreal
822   shows "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> (\<exists>x\<in>X. y<x) \<longleftrightarrow> y < Sup X"
823   by (metis complete_lattice_class.Sup_least complete_lattice_class.Sup_upper less_le_not_le linear
824             order_less_le_trans)
826 lemma Sup_lesser:
827   fixes z :: pextreal assumes "z < Sup X"
828   shows "\<exists>x \<in> X. z < x"
829 proof -
830   have "X \<noteq> {}" using assms by (auto simp: Sup_empty bot_pextreal_def)
831   with assms show ?thesis
832     by (metis Sup_pextreal_iff mem_def not_leE)
833 qed
835 lemma Sup_eq_\<omega>: "\<omega> \<in> S \<Longrightarrow> Sup S = \<omega>"
836   unfolding Sup_pextreal_def
837   by (auto intro!: Least_equality)
839 lemma Sup_close:
840   assumes "0 < e" and S: "Sup S \<noteq> \<omega>" "S \<noteq> {}"
841   shows "\<exists>X\<in>S. Sup S < X + e"
842 proof cases
843   assume "Sup S = 0"
844   moreover obtain X where "X \<in> S" using `S \<noteq> {}` by auto
845   ultimately show ?thesis using `0 < e` by (auto intro!: bexI[OF _ `X\<in>S`])
846 next
847   assume "Sup S \<noteq> 0"
848   have "\<exists>X\<in>S. Sup S - e < X"
849   proof (rule Sup_lesser)
850     show "Sup S - e < Sup S" using `0 < e` `Sup S \<noteq> 0` `Sup S \<noteq> \<omega>`
851       by (cases e) (auto simp: pextreal_noteq_omega_Ex)
852   qed
853   then guess X .. note X = this
854   with `Sup S \<noteq> \<omega>` Sup_eq_\<omega> have "X \<noteq> \<omega>" by auto
855   thus ?thesis using `Sup S \<noteq> \<omega>` X unfolding pextreal_noteq_omega_Ex
856     by (cases e) (auto intro!: bexI[OF _ `X\<in>S`] simp: split: split_if_asm)
857 qed
859 lemma Sup_\<omega>: "(SUP i::nat. Real (real i)) = \<omega>"
860 proof (rule pextreal_SUPI)
861   fix y assume *: "\<And>i::nat. i \<in> UNIV \<Longrightarrow> Real (real i) \<le> y"
862   thus "\<omega> \<le> y"
863   proof (cases y)
864     case (preal r)
865     then obtain k :: nat where "r < real k"
866       using ex_less_of_nat by (auto simp: real_eq_of_nat)
867     with *[of k] preal show ?thesis by auto
868   qed simp
869 qed simp
871 lemma SUP_\<omega>: "(SUP i:A. f i) = \<omega> \<longleftrightarrow> (\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)"
872 proof
873   assume *: "(SUP i:A. f i) = \<omega>"
874   show "(\<forall>x<\<omega>. \<exists>i\<in>A. x < f i)" unfolding *[symmetric]
875   proof (intro allI impI)
876     fix x assume "x < SUPR A f" then show "\<exists>i\<in>A. x < f i"
877       unfolding less_SUP_iff by auto
878   qed
879 next
880   assume *: "\<forall>x<\<omega>. \<exists>i\<in>A. x < f i"
881   show "(SUP i:A. f i) = \<omega>"
882   proof (rule pextreal_SUPI)
883     fix y assume **: "\<And>i. i \<in> A \<Longrightarrow> f i \<le> y"
884     show "\<omega> \<le> y"
885     proof cases
886       assume "y < \<omega>"
887       from *[THEN spec, THEN mp, OF this]
888       obtain i where "i \<in> A" "\<not> (f i \<le> y)" by auto
889       with ** show ?thesis by auto
890     qed auto
891   qed auto
892 qed
894 subsubsection {* Equivalence between @{text "f ----> x"} and @{text SUP} on @{typ pextreal} *}
896 lemma monoseq_monoI: "mono f \<Longrightarrow> monoseq f"
897   unfolding mono_def monoseq_def by auto
899 lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
900   unfolding mono_def incseq_def by auto
902 lemma SUP_eq_LIMSEQ:
903   assumes "mono f" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
904   shows "(SUP n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
905 proof
906   assume x: "(SUP n. Real (f n)) = Real x"
907   { fix n
908     have "Real (f n) \<le> Real x" using x[symmetric] by (auto intro: le_SUPI)
909     hence "f n \<le> x" using assms by simp }
910   show "f ----> x"
911   proof (rule LIMSEQ_I)
912     fix r :: real assume "0 < r"
913     show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
914     proof (rule ccontr)
915       assume *: "\<not> ?thesis"
916       { fix N
917         from * obtain n where "N \<le> n" "r \<le> x - f n"
918           using `\<And>n. f n \<le> x` by (auto simp: not_less)
919         hence "f N \<le> f n" using `mono f` by (auto dest: monoD)
920         hence "f N \<le> x - r" using `r \<le> x - f n` by auto
921         hence "Real (f N) \<le> Real (x - r)" and "r \<le> x" using `0 \<le> f N` by auto }
922       hence "(SUP n. Real (f n)) \<le> Real (x - r)"
923         and "Real (x - r) < Real x" using `0 < r` by (auto intro: SUP_leI)
924       hence "(SUP n. Real (f n)) < Real x" by (rule le_less_trans)
925       thus False using x by auto
926     qed
927   qed
928 next
929   assume "f ----> x"
930   show "(SUP n. Real (f n)) = Real x"
931   proof (rule pextreal_SUPI)
932     fix n
933     from incseq_le[of f x] `mono f` `f ----> x`
934     show "Real (f n) \<le> Real x" using assms incseq_mono by auto
935   next
936     fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> Real (f n) \<le> y"
937     show "Real x \<le> y"
938     proof (cases y)
939       case (preal r)
940       with * have "\<exists>N. \<forall>n\<ge>N. f n \<le> r" using assms by fastsimp
941       from LIMSEQ_le_const2[OF `f ----> x` this]
942       show "Real x \<le> y" using `0 \<le> x` preal by auto
943     qed simp
944   qed
945 qed
947 lemma SUPR_bound:
948   assumes "\<forall>N. f N \<le> x"
949   shows "(SUP n. f n) \<le> x"
950   using assms by (simp add: SUPR_def Sup_le_iff)
952 lemma pextreal_less_eq_diff_eq_sum:
953   fixes x y z :: pextreal
954   assumes "y \<le> x" and "x \<noteq> \<omega>"
955   shows "z \<le> x - y \<longleftrightarrow> z + y \<le> x"
956   using assms
957   apply (cases z, cases y, cases x)
958   by (simp_all add: field_simps minus_pextreal_eq)
960 lemma Real_diff_less_omega: "Real r - x < \<omega>" by (cases x) auto
962 subsubsection {* Numbers on @{typ pextreal} *}
964 instantiation pextreal :: number
965 begin
966 definition [simp]: "number_of x = Real (number_of x)"
967 instance proof qed
968 end
970 subsubsection {* Division on @{typ pextreal} *}
972 instantiation pextreal :: inverse
973 begin
975 definition "inverse x = pextreal_case (\<lambda>x. if x = 0 then \<omega> else Real (inverse x)) 0 x"
976 definition [simp]: "x / y = x * inverse (y :: pextreal)"
978 instance proof qed
979 end
981 lemma pextreal_inverse[simp]:
982   "inverse 0 = \<omega>"
983   "inverse (Real x) = (if x \<le> 0 then \<omega> else Real (inverse x))"
984   "inverse \<omega> = 0"
985   "inverse (1::pextreal) = 1"
986   "inverse (inverse x) = x"
987   by (simp_all add: inverse_pextreal_def one_pextreal_def split: pextreal_case_split del: Real_1)
989 lemma pextreal_inverse_le_eq:
990   assumes "x \<noteq> 0" "x \<noteq> \<omega>"
991   shows "y \<le> z / x \<longleftrightarrow> x * y \<le> (z :: pextreal)"
992 proof -
993   from assms obtain r where r: "x = Real r" "0 < r" by (cases x) auto
994   { fix p q :: real assume "0 \<le> p" "0 \<le> q"
995     have "p \<le> q * inverse r \<longleftrightarrow> p \<le> q / r" by (simp add: divide_inverse)
996     also have "... \<longleftrightarrow> p * r \<le> q" using `0 < r` by (auto simp: field_simps)
997     finally have "p \<le> q * inverse r \<longleftrightarrow> p * r \<le> q" . }
998   with r show ?thesis
999     by (cases y, cases z, auto simp: zero_le_mult_iff field_simps)
1000 qed
1002 lemma inverse_antimono_strict:
1003   fixes x y :: pextreal
1004   assumes "x < y" shows "inverse y < inverse x"
1005   using assms by (cases x, cases y) auto
1007 lemma inverse_antimono:
1008   fixes x y :: pextreal
1009   assumes "x \<le> y" shows "inverse y \<le> inverse x"
1010   using assms by (cases x, cases y) auto
1012 lemma pextreal_inverse_\<omega>_iff[simp]: "inverse x = \<omega> \<longleftrightarrow> x = 0"
1013   by (cases x) auto
1015 subsection "Infinite sum over @{typ pextreal}"
1017 text {*
1019 The infinite sum over @{typ pextreal} has the nice property that it is always
1020 defined.
1022 *}
1024 definition psuminf :: "(nat \<Rightarrow> pextreal) \<Rightarrow> pextreal" (binder "\<Sum>\<^isub>\<infinity>" 10) where
1025   "(\<Sum>\<^isub>\<infinity> x. f x) = (SUP n. \<Sum>i<n. f i)"
1027 subsubsection {* Equivalence between @{text "\<Sum> n. f n"} and @{text "\<Sum>\<^isub>\<infinity> n. f n"} *}
1029 lemma setsum_Real:
1030   assumes "\<forall>x\<in>A. 0 \<le> f x"
1031   shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
1032 proof (cases "finite A")
1033   case True
1034   thus ?thesis using assms
1035   proof induct case (insert x s)
1036     hence "0 \<le> setsum f s" apply-apply(rule setsum_nonneg) by auto
1037     thus ?case using insert by auto
1038   qed auto
1039 qed simp
1041 lemma setsum_Real':
1042   assumes "\<forall>x. 0 \<le> f x"
1043   shows "(\<Sum>x\<in>A. Real (f x)) = Real (\<Sum>x\<in>A. f x)"
1044   apply(rule setsum_Real) using assms by auto
1046 lemma setsum_\<omega>:
1047   "(\<Sum>x\<in>P. f x) = \<omega> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<omega>))"
1048 proof safe
1049   assume *: "setsum f P = \<omega>"
1050   show "finite P"
1051   proof (rule ccontr) assume "infinite P" with * show False by auto qed
1052   show "\<exists>i\<in>P. f i = \<omega>"
1053   proof (rule ccontr)
1054     assume "\<not> ?thesis" hence "\<And>i. i\<in>P \<Longrightarrow> f i \<noteq> \<omega>" by auto
1055     from `finite P` this have "setsum f P \<noteq> \<omega>"
1056       by induct auto
1057     with * show False by auto
1058   qed
1059 next
1060   fix i assume "finite P" "i \<in> P" "f i = \<omega>"
1061   thus "setsum f P = \<omega>"
1062   proof induct
1063     case (insert x A)
1064     show ?case using insert by (cases "x = i") auto
1065   qed simp
1066 qed
1068 lemma real_of_pextreal_setsum:
1069   assumes "\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> \<omega>"
1070   shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
1071 proof cases
1072   assume "finite S"
1073   from this assms show ?thesis
1075 qed simp
1077 lemma setsum_0:
1078   fixes f :: "'a \<Rightarrow> pextreal" assumes "finite A"
1079   shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
1080   using assms by induct auto
1082 lemma suminf_imp_psuminf:
1083   assumes "f sums x" and "\<forall>n. 0 \<le> f n"
1084   shows "(\<Sum>\<^isub>\<infinity> x. Real (f x)) = Real x"
1085   unfolding psuminf_def setsum_Real'[OF assms(2)]
1086 proof (rule SUP_eq_LIMSEQ[THEN iffD2])
1087   show "mono (\<lambda>n. setsum f {..<n})" (is "mono ?S")
1088     unfolding mono_iff_le_Suc using assms by simp
1090   { fix n show "0 \<le> ?S n"
1091       using setsum_nonneg[of "{..<n}" f] assms by auto }
1093   thus "0 \<le> x" "?S ----> x"
1094     using `f sums x` LIMSEQ_le_const
1095     by (auto simp: atLeast0LessThan sums_def)
1096 qed
1098 lemma psuminf_equality:
1099   assumes "\<And>n. setsum f {..<n} \<le> x"
1100   and "\<And>y. y \<noteq> \<omega> \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> y) \<Longrightarrow> x \<le> y"
1101   shows "psuminf f = x"
1102   unfolding psuminf_def
1103 proof (safe intro!: pextreal_SUPI)
1104   fix n show "setsum f {..<n} \<le> x" using assms(1) .
1105 next
1106   fix y assume *: "\<forall>n. n \<in> UNIV \<longrightarrow> setsum f {..<n} \<le> y"
1107   show "x \<le> y"
1108   proof (cases "y = \<omega>")
1109     assume "y \<noteq> \<omega>" from assms(2)[OF this] *
1110     show "x \<le> y" by auto
1111   qed simp
1112 qed
1114 lemma psuminf_\<omega>:
1115   assumes "f i = \<omega>"
1116   shows "(\<Sum>\<^isub>\<infinity> x. f x) = \<omega>"
1117 proof (rule psuminf_equality)
1118   fix y assume *: "\<And>n. setsum f {..<n} \<le> y"
1119   have "setsum f {..<Suc i} = \<omega>"
1120     using assms by (simp add: setsum_\<omega>)
1121   thus "\<omega> \<le> y" using *[of "Suc i"] by auto
1122 qed simp
1124 lemma psuminf_imp_suminf:
1125   assumes "(\<Sum>\<^isub>\<infinity> x. f x) \<noteq> \<omega>"
1126   shows "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity> x. f x)"
1127 proof -
1128   have "\<forall>i. \<exists>r. f i = Real r \<and> 0 \<le> r"
1129   proof
1130     fix i show "\<exists>r. f i = Real r \<and> 0 \<le> r" using psuminf_\<omega> assms by (cases "f i") auto
1131   qed
1132   from choice[OF this] obtain r where f: "f = (\<lambda>i. Real (r i))"
1133     and pos: "\<forall>i. 0 \<le> r i"
1134     by (auto simp: fun_eq_iff)
1135   hence [simp]: "\<And>i. real (f i) = r i" by auto
1137   have "mono (\<lambda>n. setsum r {..<n})" (is "mono ?S")
1138     unfolding mono_iff_le_Suc using pos by simp
1140   { fix n have "0 \<le> ?S n"
1141       using setsum_nonneg[of "{..<n}" r] pos by auto }
1143   from assms obtain p where *: "(\<Sum>\<^isub>\<infinity> x. f x) = Real p" and "0 \<le> p"
1144     by (cases "(\<Sum>\<^isub>\<infinity> x. f x)") auto
1145   show ?thesis unfolding * using * pos `0 \<le> p` SUP_eq_LIMSEQ[OF `mono ?S` `\<And>n. 0 \<le> ?S n` `0 \<le> p`]
1146     by (simp add: f atLeast0LessThan sums_def psuminf_def setsum_Real'[OF pos] f)
1147 qed
1149 lemma psuminf_bound:
1150   assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x"
1151   shows "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x"
1152   using assms by (simp add: psuminf_def SUPR_def Sup_le_iff)
1155   assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x"
1156   shows "(\<Sum>\<^isub>\<infinity> n. f n) + y \<le> x"
1157 proof (cases "x = \<omega>")
1158   have "y \<le> x" using assms by (auto intro: pextreal_le_add2)
1159   assume "x \<noteq> \<omega>"
1160   note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
1162   have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y" using assms by (simp add: move_y)
1163   hence "(\<Sum>\<^isub>\<infinity> n. f n) \<le> x - y" by (rule psuminf_bound)
1164   thus ?thesis by (simp add: move_y)
1165 qed simp
1167 lemma psuminf_finite:
1168   assumes "\<forall>N\<ge>n. f N = 0"
1169   shows "(\<Sum>\<^isub>\<infinity> n. f n) = (\<Sum>N<n. f N)"
1170 proof (rule psuminf_equality)
1171   fix N
1172   show "setsum f {..<N} \<le> setsum f {..<n}"
1173   proof (cases rule: linorder_cases)
1174     assume "N < n" thus ?thesis by (auto intro!: setsum_mono3)
1175   next
1176     assume "n < N"
1177     hence *: "{..<N} = {..<n} \<union> {n..<N}" by auto
1178     moreover have "setsum f {n..<N} = 0"
1179       using assms by (auto intro!: setsum_0')
1180     ultimately show ?thesis unfolding *
1181       by (subst setsum_Un_disjoint) auto
1182   qed simp
1183 qed simp
1185 lemma psuminf_upper:
1186   shows "(\<Sum>n<N. f n) \<le> (\<Sum>\<^isub>\<infinity> n. f n)"
1187   unfolding psuminf_def SUPR_def
1188   by (auto intro: complete_lattice_class.Sup_upper image_eqI)
1190 lemma psuminf_le:
1191   assumes "\<And>N. f N \<le> g N"
1192   shows "psuminf f \<le> psuminf g"
1193 proof (safe intro!: psuminf_bound)
1194   fix n
1195   have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
1196   also have "... \<le> psuminf g" by (rule psuminf_upper)
1197   finally show "setsum f {..<n} \<le> psuminf g" .
1198 qed
1200 lemma psuminf_const[simp]: "psuminf (\<lambda>n. c) = (if c = 0 then 0 else \<omega>)" (is "_ = ?if")
1201 proof (rule psuminf_equality)
1202   fix y assume *: "\<And>n :: nat. (\<Sum>n<n. c) \<le> y" and "y \<noteq> \<omega>"
1203   then obtain r p where
1204     y: "y = Real r" "0 \<le> r" and
1205     c: "c = Real p" "0 \<le> p"
1206       using *[of 1] by (cases c, cases y) auto
1207   show "(if c = 0 then 0 else \<omega>) \<le> y"
1208   proof (cases "p = 0")
1209     assume "p = 0" with c show ?thesis by simp
1210   next
1211     assume "p \<noteq> 0"
1212     with * c y have **: "\<And>n :: nat. real n \<le> r / p"
1213       by (auto simp: zero_le_mult_iff field_simps)
1214     from ex_less_of_nat[of "r / p"] guess n ..
1215     with **[of n] show ?thesis by (simp add: real_eq_of_nat)
1216   qed
1217 qed (cases "c = 0", simp_all)
1219 lemma psuminf_add[simp]: "psuminf (\<lambda>n. f n + g n) = psuminf f + psuminf g"
1220 proof (rule psuminf_equality)
1221   fix n
1222   from psuminf_upper[of f n] psuminf_upper[of g n]
1223   show "(\<Sum>n<n. f n + g n) \<le> psuminf f + psuminf g"
1225 next
1226   fix y assume *: "\<And>n. (\<Sum>n<n. f n + g n) \<le> y" and "y \<noteq> \<omega>"
1227   { fix n m
1228     have **: "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y"
1229     proof (cases rule: linorder_le_cases)
1230       assume "n \<le> m"
1231       hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<m. f n) + (\<Sum>n<m. g n)"
1232         by (auto intro!: add_right_mono setsum_mono3)
1233       also have "... \<le> y"
1235       finally show ?thesis .
1236     next
1237       assume "m \<le> n"
1238       hence "(\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> (\<Sum>n<n. f n) + (\<Sum>n<n. g n)"
1239         by (auto intro!: add_left_mono setsum_mono3)
1240       also have "... \<le> y"
1242       finally show ?thesis .
1243     qed }
1244   hence "\<And>m. \<forall>n. (\<Sum>n<n. f n) + (\<Sum>n<m. g n) \<le> y" by simp
1246   have "\<forall>m. (\<Sum>n<m. g n) + psuminf f \<le> y" by (simp add: ac_simps)
1248   show "psuminf f + psuminf g \<le> y" by (simp add: ac_simps)
1249 qed
1251 lemma psuminf_0: "psuminf f = 0 \<longleftrightarrow> (\<forall>i. f i = 0)"
1252 proof safe
1253   assume "\<forall>i. f i = 0"
1254   hence "f = (\<lambda>i. 0)" by (simp add: fun_eq_iff)
1255   thus "psuminf f = 0" using psuminf_const by simp
1256 next
1257   fix i assume "psuminf f = 0"
1258   hence "(\<Sum>n<Suc i. f n) = 0" using psuminf_upper[of f "Suc i"] by simp
1259   thus "f i = 0" by simp
1260 qed
1262 lemma psuminf_cmult_right[simp]: "psuminf (\<lambda>n. c * f n) = c * psuminf f"
1263 proof (rule psuminf_equality)
1264   fix n show "(\<Sum>n<n. c * f n) \<le> c * psuminf f"
1265    by (auto simp: setsum_right_distrib[symmetric] intro: mult_left_mono psuminf_upper)
1266 next
1267   fix y
1268   assume "\<And>n. (\<Sum>n<n. c * f n) \<le> y"
1269   hence *: "\<And>n. c * (\<Sum>n<n. f n) \<le> y" by (auto simp add: setsum_right_distrib)
1270   thus "c * psuminf f \<le> y"
1271   proof (cases "c = \<omega> \<or> c = 0")
1272     assume "c = \<omega> \<or> c = 0"
1273     thus ?thesis
1274       using * by (fastsimp simp add: psuminf_0 setsum_0 split: split_if_asm)
1275   next
1276     assume "\<not> (c = \<omega> \<or> c = 0)"
1277     hence "c \<noteq> 0" "c \<noteq> \<omega>" by auto
1278     note rewrite_div = pextreal_inverse_le_eq[OF this, of _ y]
1279     hence "\<forall>n. (\<Sum>n<n. f n) \<le> y / c" using * by simp
1280     hence "psuminf f \<le> y / c" by (rule psuminf_bound)
1281     thus ?thesis using rewrite_div by simp
1282   qed
1283 qed
1285 lemma psuminf_cmult_left[simp]: "psuminf (\<lambda>n. f n * c) = psuminf f * c"
1286   using psuminf_cmult_right[of c f] by (simp add: ac_simps)
1288 lemma psuminf_half_series: "psuminf (\<lambda>n. (1/2)^Suc n) = 1"
1289   using suminf_imp_psuminf[OF power_half_series] by auto
1291 lemma setsum_pinfsum: "(\<Sum>\<^isub>\<infinity> n. \<Sum>m\<in>A. f n m) = (\<Sum>m\<in>A. (\<Sum>\<^isub>\<infinity> n. f n m))"
1292 proof (cases "finite A")
1293   assume "finite A"
1294   thus ?thesis by induct simp_all
1295 qed simp
1297 lemma psuminf_reindex:
1298   fixes f:: "nat \<Rightarrow> nat" assumes "bij f"
1299   shows "psuminf (g \<circ> f) = psuminf g"
1300 proof -
1301   have [intro, simp]: "\<And>A. inj_on f A" using `bij f` unfolding bij_def by (auto intro: subset_inj_on)
1302   have f[intro, simp]: "\<And>x. f (inv f x) = x"
1303     using `bij f` unfolding bij_def by (auto intro: surj_f_inv_f)
1304   show ?thesis
1305   proof (rule psuminf_equality)
1306     fix n
1307     have "setsum (g \<circ> f) {..<n} = setsum g (f ` {..<n})"
1309     also have "\<dots> \<le> setsum g {..Max (f ` {..<n})}"
1310       by (rule setsum_mono3) auto
1311     also have "\<dots> \<le> psuminf g" unfolding lessThan_Suc_atMost[symmetric] by (rule psuminf_upper)
1312     finally show "setsum (g \<circ> f) {..<n} \<le> psuminf g" .
1313   next
1314     fix y assume *: "\<And>n. setsum (g \<circ> f) {..<n} \<le> y"
1315     show "psuminf g \<le> y"
1316     proof (safe intro!: psuminf_bound)
1317       fix N
1318       have "setsum g {..<N} \<le> setsum g (f ` {..Max (inv f ` {..<N})})"
1319         by (rule setsum_mono3) (auto intro!: image_eqI[where f="f", OF f[symmetric]])
1320       also have "\<dots> = setsum (g \<circ> f) {..Max (inv f ` {..<N})}"
1322       also have "\<dots> \<le> y" unfolding lessThan_Suc_atMost[symmetric] by (rule *)
1323       finally show "setsum g {..<N} \<le> y" .
1324     qed
1325   qed
1326 qed
1328 lemma pextreal_mult_less_right:
1329   assumes "b * a < c * a" "0 < a" "a < \<omega>"
1330   shows "b < c"
1331   using assms
1332   by (cases a, cases b, cases c) (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
1334 lemma pextreal_\<omega>_eq_plus[simp]: "\<omega> = a + b \<longleftrightarrow> (a = \<omega> \<or> b = \<omega>)"
1335   by (cases a, cases b) auto
1337 lemma pextreal_of_nat_le_iff:
1338   "(of_nat k :: pextreal) \<le> of_nat m \<longleftrightarrow> k \<le> m" by auto
1340 lemma pextreal_of_nat_less_iff:
1341   "(of_nat k :: pextreal) < of_nat m \<longleftrightarrow> k < m" by auto
1344   assumes "\<forall>N. f N + y \<le> (x::pextreal)"
1345   shows "(SUP n. f n) + y \<le> x"
1346 proof (cases "x = \<omega>")
1347   have "y \<le> x" using assms by (auto intro: pextreal_le_add2)
1348   assume "x \<noteq> \<omega>"
1349   note move_y = pextreal_less_eq_diff_eq_sum[OF `y \<le> x` this]
1351   have "\<forall>N. f N \<le> x - y" using assms by (simp add: move_y)
1352   hence "(SUP n. f n) \<le> x - y" by (rule SUPR_bound)
1353   thus ?thesis by (simp add: move_y)
1354 qed simp
1357   fixes f g :: "nat \<Rightarrow> pextreal"
1358   assumes f: "\<forall>n. f n \<le> f (Suc n)" and g: "\<forall>n. g n \<le> g (Suc n)"
1359   shows "(SUP n. f n + g n) = (SUP n. f n) + (SUP n. g n)"
1360 proof (rule pextreal_SUPI)
1361   fix n :: nat from le_SUPI[of n UNIV f] le_SUPI[of n UNIV g]
1362   show "f n + g n \<le> (SUP n. f n) + (SUP n. g n)"
1364 next
1365   fix y assume *: "\<And>n. n \<in> UNIV \<Longrightarrow> f n + g n \<le> y"
1366   { fix n m
1367     have "f n + g m \<le> y"
1368     proof (cases rule: linorder_le_cases)
1369       assume "n \<le> m"
1370       hence "f n + g m \<le> f m + g m"
1371         using f lift_Suc_mono_le by (auto intro!: add_right_mono)
1372       also have "\<dots> \<le> y" using * by simp
1373       finally show ?thesis .
1374     next
1375       assume "m \<le> n"
1376       hence "f n + g m \<le> f n + g n"
1377         using g lift_Suc_mono_le by (auto intro!: add_left_mono)
1378       also have "\<dots> \<le> y" using * by simp
1379       finally show ?thesis .
1380     qed }
1381   hence "\<And>m. \<forall>n. f n + g m \<le> y" by simp
1383   have "\<forall>m. (g m) + (SUP n. f n) \<le> y" by (simp add: ac_simps)
1385   show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps)
1386 qed
1388 lemma SUPR_pextreal_setsum:
1389   fixes f :: "'x \<Rightarrow> nat \<Rightarrow> pextreal"
1390   assumes "\<And>i. i \<in> P \<Longrightarrow> \<forall>n. f i n \<le> f i (Suc n)"
1391   shows "(SUP n. \<Sum>i\<in>P. f i n) = (\<Sum>i\<in>P. SUP n. f i n)"
1392 proof cases
1393   assume "finite P" from this assms show ?thesis
1394   proof induct
1395     case (insert i P)
1396     thus ?case
1397       apply simp
1399       by (auto intro!: setsum_mono)
1400   qed simp
1401 qed simp
1403 lemma psuminf_SUP_eq:
1404   assumes "\<And>n i. f n i \<le> f (Suc n) i"
1405   shows "(\<Sum>\<^isub>\<infinity> i. SUP n::nat. f n i) = (SUP n::nat. \<Sum>\<^isub>\<infinity> i. f n i)"
1406 proof -
1407   { fix n :: nat
1408     have "(\<Sum>i<n. SUP k. f k i) = (SUP k. \<Sum>i<n. f k i)"
1409       using assms by (auto intro!: SUPR_pextreal_setsum[symmetric]) }
1410   note * = this
1411   show ?thesis
1412     unfolding psuminf_def
1413     unfolding *
1414     apply (subst SUP_commute) ..
1415 qed
1417 lemma psuminf_commute:
1418   shows "(\<Sum>\<^isub>\<infinity> i j. f i j) = (\<Sum>\<^isub>\<infinity> j i. f i j)"
1419 proof -
1420   have "(SUP n. \<Sum> i < n. SUP m. \<Sum> j < m. f i j) = (SUP n. SUP m. \<Sum> i < n. \<Sum> j < m. f i j)"
1421     apply (subst SUPR_pextreal_setsum)
1422     by auto
1423   also have "\<dots> = (SUP m n. \<Sum> j < m. \<Sum> i < n. f i j)"
1424     apply (subst SUP_commute)
1425     apply (subst setsum_commute)
1426     by auto
1427   also have "\<dots> = (SUP m. \<Sum> j < m. SUP n. \<Sum> i < n. f i j)"
1428     apply (subst SUPR_pextreal_setsum)
1429     by auto
1430   finally show ?thesis
1431     unfolding psuminf_def by auto
1432 qed
1434 lemma psuminf_2dimen:
1435   fixes f:: "nat * nat \<Rightarrow> pextreal"
1436   assumes fsums: "\<And>m. g m = (\<Sum>\<^isub>\<infinity> n. f (m,n))"
1437   shows "psuminf (f \<circ> prod_decode) = psuminf g"
1438 proof (rule psuminf_equality)
1439   fix n :: nat
1440   let ?P = "prod_decode ` {..<n}"
1441   have "setsum (f \<circ> prod_decode) {..<n} = setsum f ?P"
1442     by (auto simp: setsum_reindex inj_prod_decode)
1443   also have "\<dots> \<le> setsum f ({..Max (fst ` ?P)} \<times> {..Max (snd ` ?P)})"
1444   proof (safe intro!: setsum_mono3 Max_ge image_eqI)
1445     fix a b x assume "(a, b) = prod_decode x"
1446     from this[symmetric] show "a = fst (prod_decode x)" "b = snd (prod_decode x)"
1447       by simp_all
1448   qed simp_all
1449   also have "\<dots> = (\<Sum>m\<le>Max (fst ` ?P). (\<Sum>n\<le>Max (snd ` ?P). f (m,n)))"
1450     unfolding setsum_cartesian_product by simp
1451   also have "\<dots> \<le> (\<Sum>m\<le>Max (fst ` ?P). g m)"
1452     by (auto intro!: setsum_mono psuminf_upper simp del: setsum_lessThan_Suc
1453         simp: fsums lessThan_Suc_atMost[symmetric])
1454   also have "\<dots> \<le> psuminf g"
1455     by (auto intro!: psuminf_upper simp del: setsum_lessThan_Suc
1456         simp: lessThan_Suc_atMost[symmetric])
1457   finally show "setsum (f \<circ> prod_decode) {..<n} \<le> psuminf g" .
1458 next
1459   fix y assume *: "\<And>n. setsum (f \<circ> prod_decode) {..<n} \<le> y"
1460   have g: "g = (\<lambda>m. \<Sum>\<^isub>\<infinity> n. f (m,n))" unfolding fsums[symmetric] ..
1461   show "psuminf g \<le> y" unfolding g
1462   proof (rule psuminf_bound, unfold setsum_pinfsum[symmetric], safe intro!: psuminf_bound)
1463     fix N M :: nat
1464     let ?P = "{..<N} \<times> {..<M}"
1465     let ?M = "Max (prod_encode ` ?P)"
1466     have "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> (\<Sum>(m, n)\<in>?P. f (m, n))"
1467       unfolding setsum_commute[of _ _ "{..<M}"] unfolding setsum_cartesian_product ..
1468     also have "\<dots> \<le> (\<Sum>(m,n)\<in>(prod_decode ` {..?M}). f (m, n))"
1469       by (auto intro!: setsum_mono3 image_eqI[where f=prod_decode, OF prod_encode_inverse[symmetric]])
1470     also have "\<dots> \<le> y" using *[of "Suc ?M"]
1471       by (simp add: lessThan_Suc_atMost[symmetric] setsum_reindex
1472                inj_prod_decode del: setsum_lessThan_Suc)
1473     finally show "(\<Sum>n<M. \<Sum>m<N. f (m, n)) \<le> y" .
1474   qed
1475 qed
1477 lemma Real_max:
1478   assumes "x \<ge> 0" "y \<ge> 0"
1479   shows "Real (max x y) = max (Real x) (Real y)"
1480   using assms unfolding max_def by (auto simp add:not_le)
1482 lemma Real_real: "Real (real x) = (if x = \<omega> then 0 else x)"
1483   using assms by (cases x) auto
1485 lemma inj_on_real: "inj_on real (UNIV - {\<omega>})"
1486 proof (rule inj_onI)
1487   fix x y assume mem: "x \<in> UNIV - {\<omega>}" "y \<in> UNIV - {\<omega>}" and "real x = real y"
1488   thus "x = y" by (cases x, cases y) auto
1489 qed
1491 lemma inj_on_Real: "inj_on Real {0..}"
1492   by (auto intro!: inj_onI)
1494 lemma range_Real[simp]: "range Real = UNIV - {\<omega>}"
1495 proof safe
1496   fix x assume "x \<notin> range Real"
1497   thus "x = \<omega>" by (cases x) auto
1498 qed auto
1500 lemma image_Real[simp]: "Real ` {0..} = UNIV - {\<omega>}"
1501 proof safe
1502   fix x assume "x \<notin> Real ` {0..}"
1503   thus "x = \<omega>" by (cases x) auto
1504 qed auto
1506 lemma pextreal_SUP_cmult:
1507   fixes f :: "'a \<Rightarrow> pextreal"
1508   shows "(SUP i : R. z * f i) = z * (SUP i : R. f i)"
1509 proof (rule pextreal_SUPI)
1510   fix i assume "i \<in> R"
1511   from le_SUPI[OF this]
1512   show "z * f i \<le> z * (SUP i:R. f i)" by (rule pextreal_mult_cancel)
1513 next
1514   fix y assume "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y"
1515   hence *: "\<And>i. i\<in>R \<Longrightarrow> z * f i \<le> y" by auto
1516   show "z * (SUP i:R. f i) \<le> y"
1517   proof (cases "\<forall>i\<in>R. f i = 0")
1518     case True
1519     show ?thesis
1520     proof cases
1521       assume "R \<noteq> {}" hence "f ` R = {0}" using True by auto
1522       thus ?thesis by (simp add: SUPR_def)
1523     qed (simp add: SUPR_def Sup_empty bot_pextreal_def)
1524   next
1525     case False then obtain i where i: "i \<in> R" and f0: "f i \<noteq> 0" by auto
1526     show ?thesis
1527     proof (cases "z = 0 \<or> z = \<omega>")
1528       case True with f0 *[OF i] show ?thesis by auto
1529     next
1530       case False hence z: "z \<noteq> 0" "z \<noteq> \<omega>" by auto
1531       note div = pextreal_inverse_le_eq[OF this, symmetric]
1532       hence "\<And>i. i\<in>R \<Longrightarrow> f i \<le> y / z" using * by auto
1533       thus ?thesis unfolding div SUP_le_iff by simp
1534     qed
1535   qed
1536 qed
1538 instantiation pextreal :: topological_space
1539 begin
1541 definition "open A \<longleftrightarrow>
1542   (\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
1544 lemma open_omega: "open A \<Longrightarrow> \<omega> \<in> A \<Longrightarrow> (\<exists>x\<ge>0. {Real x<..} \<subseteq> A)"
1545   unfolding open_pextreal_def by auto
1547 lemma open_omegaD: assumes "open A" "\<omega> \<in> A" obtains x where "x\<ge>0" "{Real x<..} \<subseteq> A"
1548   using open_omega[OF assms] by auto
1550 lemma pextreal_openE: assumes "open A" obtains A' x where
1551   "open A'" "Real ` (A' \<inter> {0..}) = A - {\<omega>}"
1552   "x \<ge> 0" "\<omega> \<in> A \<Longrightarrow> {Real x<..} \<subseteq> A"
1553   using assms open_pextreal_def by auto
1555 instance
1556 proof
1557   let ?U = "UNIV::pextreal set"
1558   show "open ?U" unfolding open_pextreal_def
1559     by (auto intro!: exI[of _ "UNIV"] exI[of _ 0])
1560 next
1561   fix S T::"pextreal set" assume "open S" and "open T"
1562   from `open S`[THEN pextreal_openE] guess S' xS . note S' = this
1563   from `open T`[THEN pextreal_openE] guess T' xT . note T' = this
1565   from S'(1-3) T'(1-3)
1566   show "open (S \<inter> T)" unfolding open_pextreal_def
1567   proof (safe intro!: exI[of _ "S' \<inter> T'"] exI[of _ "max xS xT"])
1568     fix x assume *: "Real (max xS xT) < x" and "\<omega> \<in> S" "\<omega> \<in> T"
1569     from `\<omega> \<in> S`[THEN S'(4)] * show "x \<in> S"
1570       by (cases x, auto simp: max_def split: split_if_asm)
1571     from `\<omega> \<in> T`[THEN T'(4)] * show "x \<in> T"
1572       by (cases x, auto simp: max_def split: split_if_asm)
1573   next
1574     fix x assume x: "x \<notin> Real ` (S' \<inter> T' \<inter> {0..})"
1575     have *: "S' \<inter> T' \<inter> {0..} = (S' \<inter> {0..}) \<inter> (T' \<inter> {0..})" by auto
1576     assume "x \<in> T" "x \<in> S"
1577     with S'(2) T'(2) show "x = \<omega>"
1578       using x[unfolded *] inj_on_image_Int[OF inj_on_Real] by auto
1579   qed auto
1580 next
1581   fix K assume openK: "\<forall>S \<in> K. open (S:: pextreal set)"
1582   hence "\<forall>S\<in>K. \<exists>T. open T \<and> Real ` (T \<inter> {0..}) = S - {\<omega>}" by (auto simp: open_pextreal_def)
1583   from bchoice[OF this] guess T .. note T = this[rule_format]
1585   show "open (\<Union>K)" unfolding open_pextreal_def
1586   proof (safe intro!: exI[of _ "\<Union>(T ` K)"])
1587     fix x S assume "0 \<le> x" "x \<in> T S" "S \<in> K"
1588     with T[OF `S \<in> K`] show "Real x \<in> \<Union>K" by auto
1589   next
1590     fix x S assume x: "x \<notin> Real ` (\<Union>T ` K \<inter> {0..})" "S \<in> K" "x \<in> S"
1591     hence "x \<notin> Real ` (T S \<inter> {0..})"
1592       by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps)
1593     thus "x = \<omega>" using T[OF `S \<in> K`] `x \<in> S` by auto
1594   next
1595     fix S assume "\<omega> \<in> S" "S \<in> K"
1596     from openK[rule_format, OF `S \<in> K`, THEN pextreal_openE] guess S' x .
1597     from this(3, 4) `\<omega> \<in> S`
1598     show "\<exists>x\<ge>0. {Real x<..} \<subseteq> \<Union>K"
1599       by (auto intro!: exI[of _ x] bexI[OF _ `S \<in> K`])
1600   next
1601     from T[THEN conjunct1] show "open (\<Union>T`K)" by auto
1602   qed auto
1603 qed
1604 end
1606 lemma minus_omega[simp]: "x - \<omega> = 0" by (cases x) auto
1608 lemma open_pextreal_alt: "open A \<longleftrightarrow>
1609   (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A) \<and> (\<omega> \<in> A \<longrightarrow> (\<exists>x\<ge>0. {Real x <..} \<subseteq> A))"
1610 proof -
1611   have *: "(\<exists>T. open T \<and> (Real ` (T\<inter>{0..}) = A - {\<omega>})) \<longleftrightarrow>
1612     open (real ` (A - {\<omega>}) \<union> {..<0})"
1613   proof safe
1614     fix T assume "open T" and A: "Real ` (T \<inter> {0..}) = A - {\<omega>}"
1615     have *: "(\<lambda>x. real (Real x)) ` (T \<inter> {0..}) = T \<inter> {0..}"
1616       by auto
1617     have **: "T \<inter> {0..} \<union> {..<0} = T \<union> {..<0}" by auto
1618     show "open (real ` (A - {\<omega>}) \<union> {..<0})"
1619       unfolding A[symmetric] image_image * ** using `open T` by auto
1620   next
1621     assume "open (real ` (A - {\<omega>}) \<union> {..<0})"
1622     moreover have "Real ` ((real ` (A - {\<omega>}) \<union> {..<0}) \<inter> {0..}) = A - {\<omega>}"
1623       apply auto
1624       apply (case_tac xb)
1625       apply auto
1626       apply (case_tac x)
1627       apply (auto simp: image_iff)
1628       apply (erule_tac x="Real r" in ballE)
1629       apply auto
1630       done
1631     ultimately show "\<exists>T. open T \<and> Real ` (T \<inter> {0..}) = A - {\<omega>}" by auto
1632   qed
1633   also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>A. \<exists>e>0. {x-e <..< x+e} \<subseteq> A)"
1634   proof (intro iffI ballI open_subopen[THEN iffD2])
1635     fix x assume *: "\<forall>x\<in>A. \<exists>e>0. {x - e<..<x + e} \<subseteq> A" and x: "x \<in> real ` (A - {\<omega>}) \<union> {..<0}"
1636     show "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
1637     proof (cases rule: linorder_cases)
1638       assume "x < 0" then show ?thesis by (intro exI[of _ "{..<0}"]) auto
1639     next
1640       assume "x = 0" with x
1641       have "0 \<in> A"
1642         apply auto by (case_tac x) auto
1643       with * obtain e where "e > 0" "{0 - e<..<0 + e} \<subseteq> A" by auto
1644       then have "{..<e} \<subseteq> A" using `0 \<in> A`
1645         apply auto
1646         apply (case_tac "x = 0")
1647         by (auto dest!: pextreal_zero_lessI)
1648       then have *: "{..<e} \<subseteq> A - {\<omega>}" by auto
1649       def e' \<equiv> "if e = \<omega> then 1 else real e"
1650       then have "0 < e'" using `e > 0` by (cases e) auto
1651       have "{0..<e'} \<subseteq> real ` (A - {\<omega>})"
1652       proof (cases e)
1653         case infinite
1654         then have "{..<e} = UNIV - {\<omega>}" by auto
1655         then have A: "A - {\<omega>} = UNIV - {\<omega>}" using * by auto
1656         show ?thesis unfolding e'_def infinite A
1657           apply safe
1658           apply (rule_tac x="Real x" in image_eqI)
1659           apply auto
1660           done
1661       next
1662         case (preal r)
1663         then show ?thesis unfolding e'_def using *
1664           apply safe
1665           apply (rule_tac x="Real x" in image_eqI)
1666           by (auto simp: subset_eq)
1667       qed
1668       then have "{0..<e'} \<union> {..<0} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by auto
1669       moreover have "{0..<e'} \<union> {..<0} = {..<e'}" using `0 < e'` by auto
1670       ultimately have "{..<e'} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}" by simp
1671       then show ?thesis using `0 < e'` `x = 0` by auto
1672     next
1673       assume "0 < x"
1674       with x have "Real x \<in> A" apply auto by (case_tac x) auto
1675       with * obtain e where "0 < e" and e: "{Real x - e<..<Real x + e} \<subseteq> A" by auto
1676       show ?thesis
1677       proof cases
1678         assume "e < Real x"
1679         with `0 < e` obtain r where r: "e = Real r" "0 < r" by (cases e) auto
1680         then have "r < x" using `e < Real x` `0 < e` by (auto split: split_if_asm)
1681         then have "{x - r <..< x + r} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
1682           using e unfolding r
1683           apply (auto simp: subset_eq)
1684           apply (rule_tac x="Real xa" in image_eqI)
1685           by auto
1686         then show ?thesis using `0 < r` by (intro exI[of _ "{x - r <..< x + r}"]) auto
1687       next
1688         assume "\<not> e < Real x"
1689         moreover then have "Real x - e = 0" by (cases e) auto
1690         moreover have "\<And>z. 0 < z \<Longrightarrow>  z * 2 < 3 * x \<Longrightarrow> Real z < Real x + e"
1691           using `\<not> e < Real x` by (cases e) auto
1692         ultimately have "{0 <..< x + x / 2} \<subseteq> real ` (A - {\<omega>}) \<union> {..<0}"
1693           using e
1694           apply (auto simp: subset_eq)
1695           apply (erule_tac x="Real xa" in ballE)
1696           apply (auto simp: not_less)
1697           apply (rule_tac x="Real xa" in image_eqI)
1698           apply auto
1699           done
1700         moreover have "x \<in> {0 <..< x + x / 2}" using `0 < x` by auto
1701         ultimately show ?thesis by (intro exI[of _ "{0 <..< x + x / 2}"]) auto
1702       qed
1703     qed
1704   next
1705     fix x assume x: "x \<in> A" "open (real ` (A - {\<omega>}) \<union> {..<0})"
1706     then show "\<exists>e>0. {x - e<..<x + e} \<subseteq> A"
1707     proof (cases x)
1708       case infinite then show ?thesis by (intro exI[of _ 2]) auto
1709     next
1710       case (preal r)
1711       with `x \<in> A` have r: "r \<in> real ` (A - {\<omega>}) \<union> {..<0}" by force
1712       from x(2)[unfolded open_real, THEN bspec, OF r]
1713       obtain e where e: "0 < e" "\<And>x'. \<bar>x' - r\<bar> < e \<Longrightarrow> x' \<in> real ` (A - {\<omega>}) \<union> {..<0}"
1714         by auto
1715       show ?thesis using `0 < e` preal
1716       proof (auto intro!: exI[of _ "Real e"] simp: greaterThanLessThan_iff not_less
1717                   split: split_if_asm)
1718         fix z assume *: "Real (r - e) < z" "z < Real (r + e)"
1719         then obtain q where [simp]: "z = Real q" "0 \<le> q" by (cases z) auto
1720         with * have "r - e < q" "q < r + e" by (auto split: split_if_asm)
1721         with e(2)[of q] have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" by auto
1722         then show "z \<in> A" using `0 \<le> q` apply auto apply (case_tac x) apply auto done
1723       next
1724         fix z assume *: "0 < z" "z < Real (r + e)" "r \<le> e"
1725         then obtain q where [simp]: "z = Real q" and "0 < q" by (cases z) auto
1726         with * have "q < r + e" by (auto split: split_if_asm)
1727         moreover have "r - e < q" using `r \<le> e` `0 < q` by auto
1728         ultimately have "q \<in> real ` (A - {\<omega>}) \<union> {..<0}" using e(2)[of q] by auto
1729         then show "z \<in> A" using `0 < q` apply auto apply (case_tac x) apply auto done
1730       qed
1731     qed
1732   qed
1733   finally show ?thesis unfolding open_pextreal_def by simp
1734 qed
1736 lemma open_pextreal_lessThan[simp]:
1737   "open {..< a :: pextreal}"
1738 proof (cases a)
1739   case (preal x) thus ?thesis unfolding open_pextreal_def
1740   proof (safe intro!: exI[of _ "{..< x}"])
1741     fix y assume "y < Real x"
1742     moreover assume "y \<notin> Real ` ({..<x} \<inter> {0..})"
1743     ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
1744     thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
1745   qed auto
1746 next
1747   case infinite thus ?thesis
1748     unfolding open_pextreal_def by (auto intro!: exI[of _ UNIV])
1749 qed
1751 lemma open_pextreal_greaterThan[simp]:
1752   "open {a :: pextreal <..}"
1753 proof (cases a)
1754   case (preal x) thus ?thesis unfolding open_pextreal_def
1755   proof (safe intro!: exI[of _ "{x <..}"])
1756     fix y assume "Real x < y"
1757     moreover assume "y \<notin> Real ` ({x<..} \<inter> {0..})"
1758     ultimately have "y \<noteq> Real (real y)" using preal by (cases y) auto
1759     thus "y = \<omega>" by (auto simp: Real_real split: split_if_asm)
1760   qed auto
1761 next
1762   case infinite thus ?thesis
1763     unfolding open_pextreal_def by (auto intro!: exI[of _ "{}"])
1764 qed
1766 lemma pextreal_open_greaterThanLessThan[simp]: "open {a::pextreal <..< b}"
1767   unfolding greaterThanLessThan_def by auto
1769 lemma closed_pextreal_atLeast[simp, intro]: "closed {a :: pextreal ..}"
1770 proof -
1771   have "- {a ..} = {..< a}" by auto
1772   then show "closed {a ..}"
1773     unfolding closed_def using open_pextreal_lessThan by auto
1774 qed
1776 lemma closed_pextreal_atMost[simp, intro]: "closed {.. b :: pextreal}"
1777 proof -
1778   have "- {.. b} = {b <..}" by auto
1779   then show "closed {.. b}"
1780     unfolding closed_def using open_pextreal_greaterThan by auto
1781 qed
1783 lemma closed_pextreal_atLeastAtMost[simp, intro]:
1784   shows "closed {a :: pextreal .. b}"
1785   unfolding atLeastAtMost_def by auto
1787 lemma pextreal_dense:
1788   fixes x y :: pextreal assumes "x < y"
1789   shows "\<exists>z. x < z \<and> z < y"
1790 proof -
1791   from `x < y` obtain p where p: "x = Real p" "0 \<le> p" by (cases x) auto
1792   show ?thesis
1793   proof (cases y)
1794     case (preal r) with p `x < y` have "p < r" by auto
1795     with dense obtain z where "p < z" "z < r" by auto
1796     thus ?thesis using preal p by (auto intro!: exI[of _ "Real z"])
1797   next
1798     case infinite thus ?thesis using `x < y` p
1799       by (auto intro!: exI[of _ "Real p + 1"])
1800   qed
1801 qed
1803 instance pextreal :: t2_space
1804 proof
1805   fix x y :: pextreal assume "x \<noteq> y"
1806   let "?P x (y::pextreal)" = "\<exists> U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1808   { fix x y :: pextreal assume "x < y"
1809     from pextreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
1810     have "?P x y"
1811       apply (rule exI[of _ "{..<z}"])
1812       apply (rule exI[of _ "{z<..}"])
1813       using z by auto }
1814   note * = this
1816   from `x \<noteq> y`
1817   show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1818   proof (cases rule: linorder_cases)
1819     assume "x = y" with `x \<noteq> y` show ?thesis by simp
1820   next assume "x < y" from *[OF this] show ?thesis by auto
1821   next assume "y < x" from *[OF this] show ?thesis by auto
1822   qed
1823 qed
1825 definition (in complete_lattice) isoton :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> bool" (infix "\<up>" 50) where
1826   "A \<up> X \<longleftrightarrow> (\<forall>i. A i \<le> A (Suc i)) \<and> (SUP i. A i) = X"
1828 definition (in complete_lattice) antiton (infix "\<down>" 50) where
1829   "A \<down> X \<longleftrightarrow> (\<forall>i. A i \<ge> A (Suc i)) \<and> (INF i. A i) = X"
1831 lemma isotoneI[intro?]: "\<lbrakk> \<And>i. f i \<le> f (Suc i) ; (SUP i. f i) = F \<rbrakk> \<Longrightarrow> f \<up> F"
1832   unfolding isoton_def by auto
1834 lemma (in complete_lattice) isotonD[dest]:
1835   assumes "A \<up> X" shows "A i \<le> A (Suc i)" "(SUP i. A i) = X"
1836   using assms unfolding isoton_def by auto
1838 lemma isotonD'[dest]:
1839   assumes "(A::_=>_) \<up> X" shows "A i x \<le> A (Suc i) x" "(SUP i. A i) = X"
1840   using assms unfolding isoton_def le_fun_def by auto
1842 lemma isoton_mono_le:
1843   assumes "f \<up> x" "i \<le> j"
1844   shows "f i \<le> f j"
1845   using `f \<up> x`[THEN isotonD(1)] lift_Suc_mono_le[of f, OF _ `i \<le> j`] by auto
1847 lemma isoton_const:
1848   shows "(\<lambda> i. c) \<up> c"
1849 unfolding isoton_def by auto
1851 lemma isoton_cmult_right:
1852   assumes "f \<up> (x::pextreal)"
1853   shows "(\<lambda>i. c * f i) \<up> (c * x)"
1854   using assms unfolding isoton_def pextreal_SUP_cmult
1855   by (auto intro: pextreal_mult_cancel)
1857 lemma isoton_cmult_left:
1858   "f \<up> (x::pextreal) \<Longrightarrow> (\<lambda>i. f i * c) \<up> (x * c)"
1859   by (subst (1 2) mult_commute) (rule isoton_cmult_right)
1862   assumes "f \<up> (x::pextreal)" and "g \<up> y"
1863   shows "(\<lambda>i. f i + g i) \<up> (x + y)"
1864   using assms unfolding isoton_def
1867 lemma isoton_fun_expand:
1868   "f \<up> x \<longleftrightarrow> (\<forall>i. (\<lambda>j. f j i) \<up> (x i))"
1869 proof -
1870   have "\<And>i. {y. \<exists>f'\<in>range f. y = f' i} = range (\<lambda>j. f j i)"
1871     by auto
1872   with assms show ?thesis
1873     by (auto simp add: isoton_def le_fun_def Sup_fun_def SUPR_def)
1874 qed
1876 lemma isoton_indicator:
1877   assumes "f \<up> g"
1878   shows "(\<lambda>i x. f i x * indicator A x) \<up> (\<lambda>x. g x * indicator A x :: pextreal)"
1879   using assms unfolding isoton_fun_expand by (auto intro!: isoton_cmult_left)
1881 lemma isoton_setsum:
1882   fixes f :: "'a \<Rightarrow> nat \<Rightarrow> pextreal"
1883   assumes "finite A" "A \<noteq> {}"
1884   assumes "\<And> x. x \<in> A \<Longrightarrow> f x \<up> y x"
1885   shows "(\<lambda> i. (\<Sum> x \<in> A. f x i)) \<up> (\<Sum> x \<in> A. y x)"
1886 using assms
1887 proof (induct A rule:finite_ne_induct)
1888   case singleton thus ?case by auto
1889 next
1890   case (insert a A) note asms = this
1891   hence *: "(\<lambda> i. \<Sum> x \<in> A. f x i) \<up> (\<Sum> x \<in> A. y x)" by auto
1892   have **: "(\<lambda> i. f a i) \<up> y a" using asms by simp
1893   have "(\<lambda> i. f a i + (\<Sum> x \<in> A. f x i)) \<up> (y a + (\<Sum> x \<in> A. y x))"
1894     using * ** isoton_add by auto
1895   thus "(\<lambda> i. \<Sum> x \<in> insert a A. f x i) \<up> (\<Sum> x \<in> insert a A. y x)"
1896     using asms by fastsimp
1897 qed
1899 lemma isoton_Sup:
1900   assumes "f \<up> u"
1901   shows "f i \<le> u"
1902   using le_SUPI[of i UNIV f] assms
1903   unfolding isoton_def by auto
1905 lemma isoton_mono:
1906   assumes iso: "x \<up> a" "y \<up> b" and *: "\<And>n. x n \<le> y (N n)"
1907   shows "a \<le> b"
1908 proof -
1909   from iso have "a = (SUP n. x n)" "b = (SUP n. y n)"
1910     unfolding isoton_def by auto
1911   with * show ?thesis by (auto intro!: SUP_mono)
1912 qed
1914 lemma pextreal_le_mult_one_interval:
1915   fixes x y :: pextreal
1916   assumes "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
1917   shows "x \<le> y"
1918 proof (cases x, cases y)
1919   assume "x = \<omega>"
1920   with assms[of "1 / 2"]
1921   show "x \<le> y" by simp
1922 next
1923   fix r p assume *: "y = Real p" "x = Real r" and **: "0 \<le> r" "0 \<le> p"
1924   have "r \<le> p"
1925   proof (rule field_le_mult_one_interval)
1926     fix z :: real assume "0 < z" and "z < 1"
1927     with assms[of "Real z"]
1928     show "z * r \<le> p" using ** * by (auto simp: zero_le_mult_iff)
1929   qed
1930   thus "x \<le> y" using ** * by simp
1931 qed simp
1933 lemma pextreal_greater_0[intro]:
1934   fixes a :: pextreal
1935   assumes "a \<noteq> 0"
1936   shows "a > 0"
1937 using assms apply (cases a) by auto
1939 lemma pextreal_mult_strict_right_mono:
1940   assumes "a < b" and "0 < c" "c < \<omega>"
1941   shows "a * c < b * c"
1942   using assms
1943   by (cases a, cases b, cases c)
1944      (auto simp: zero_le_mult_iff pextreal_less_\<omega>)
1946 lemma minus_pextreal_eq2:
1947   fixes x y z :: pextreal
1948   assumes "y \<le> x" and "y \<noteq> \<omega>" shows "z = x - y \<longleftrightarrow> z + y = x"
1949   using assms
1950   apply (subst eq_commute)
1951   apply (subst minus_pextreal_eq)
1952   by (cases x, cases z, auto simp add: ac_simps not_less)
1954 lemma pextreal_diff_eq_diff_imp_eq:
1955   assumes "a \<noteq> \<omega>" "b \<le> a" "c \<le> a"
1956   assumes "a - b = a - c"
1957   shows "b = c"
1958   using assms
1959   by (cases a, cases b, cases c) (auto split: split_if_asm)
1961 lemma pextreal_inverse_eq_0: "inverse x = 0 \<longleftrightarrow> x = \<omega>"
1962   by (cases x) auto
1964 lemma pextreal_mult_inverse:
1965   "\<lbrakk> x \<noteq> \<omega> ; x \<noteq> 0 \<rbrakk> \<Longrightarrow> x * inverse x = 1"
1966   by (cases x) auto
1968 lemma pextreal_zero_less_diff_iff:
1969   fixes a b :: pextreal shows "0 < a - b \<longleftrightarrow> b < a"
1970   apply (cases a, cases b)
1971   apply (auto simp: pextreal_noteq_omega_Ex pextreal_less_\<omega>)
1972   apply (cases b)
1973   by auto
1975 lemma pextreal_less_Real_Ex:
1976   fixes a b :: pextreal shows "x < Real r \<longleftrightarrow> (\<exists>p\<ge>0. p < r \<and> x = Real p)"
1977   by (cases x) auto
1979 lemma open_Real: assumes "open S" shows "open (Real ` ({0..} \<inter> S))"
1980   unfolding open_pextreal_def apply(rule,rule,rule,rule assms) by auto
1982 lemma pextreal_zero_le_diff:
1983   fixes a b :: pextreal shows "a - b = 0 \<longleftrightarrow> a \<le> b"
1984   by (cases a, cases b, simp_all, cases b, auto)
1986 lemma lim_Real[simp]: assumes "\<forall>n. f n \<ge> 0" "m\<ge>0"
1987   shows "(\<lambda>n. Real (f n)) ----> Real m \<longleftrightarrow> (\<lambda>n. f n) ----> m" (is "?l = ?r")
1988 proof assume ?l show ?r unfolding Lim_sequentially
1989   proof safe fix e::real assume e:"e>0"
1990     note open_ball[of m e] note open_Real[OF this]
1991     note * = `?l`[unfolded tendsto_def,rule_format,OF this]
1992     have "eventually (\<lambda>x. Real (f x) \<in> Real ` ({0..} \<inter> ball m e)) sequentially"
1993       apply(rule *) unfolding image_iff using assms(2) e by auto
1994     thus "\<exists>N. \<forall>n\<ge>N. dist (f n) m < e" unfolding eventually_sequentially
1995       apply safe apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
1996     proof- fix n x assume "Real (f n) = Real x" "0 \<le> x"
1997       hence *:"f n = x" using assms(1) by auto
1998       assume "x \<in> ball m e" thus "dist (f n) m < e" unfolding *
2000     qed qed
2001 next assume ?r show ?l unfolding tendsto_def eventually_sequentially
2002   proof safe fix S assume S:"open S" "Real m \<in> S"
2003     guess T y using S(1) apply-apply(erule pextreal_openE) . note T=this
2004     have "m\<in>real ` (S - {\<omega>})" unfolding image_iff
2005       apply(rule_tac x="Real m" in bexI) using assms(2) S(2) by auto
2006     hence "m \<in> T" unfolding T(2)[THEN sym] by auto
2007     from `?r`[unfolded tendsto_def eventually_sequentially,rule_format,OF T(1) this]
2008     guess N .. note N=this[rule_format]
2009     show "\<exists>N. \<forall>n\<ge>N. Real (f n) \<in> S" apply(rule_tac x=N in exI)
2010     proof safe fix n assume n:"N\<le>n"
2011       have "f n \<in> real ` (S - {\<omega>})" using N[OF n] assms unfolding T(2)[THEN sym]
2012         unfolding image_iff apply-apply(rule_tac x="Real (f n)" in bexI)
2013         unfolding real_Real by auto
2014       then guess x unfolding image_iff .. note x=this
2015       show "Real (f n) \<in> S" unfolding x apply(subst Real_real) using x by auto
2016     qed
2017   qed
2018 qed
2020 lemma pextreal_INFI:
2021   fixes x :: pextreal
2022   assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"
2023   assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> f i) \<Longrightarrow> y \<le> x"
2024   shows "(INF i:A. f i) = x"
2025   unfolding INFI_def Inf_pextreal_def
2026   using assms by (auto intro!: Greatest_equality)
2028 lemma real_of_pextreal_less:"x < y \<Longrightarrow> y\<noteq>\<omega> \<Longrightarrow> real x < real y"
2029 proof- case goal1
2030   have *:"y = Real (real y)" "x = Real (real x)" using goal1 Real_real by auto
2031   show ?case using goal1 apply- apply(subst(asm) *(1))apply(subst(asm) *(2))
2032     unfolding pextreal_less by auto
2033 qed
2035 lemma not_less_omega[simp]:"\<not> x < \<omega> \<longleftrightarrow> x = \<omega>"
2036   by (metis antisym_conv3 pextreal_less(3))
2038 lemma Real_real': assumes "x\<noteq>\<omega>" shows "Real (real x) = x"
2039 proof- have *:"(THE r. 0 \<le> r \<and> x = Real r) = real x"
2040     apply(rule the_equality) using assms unfolding Real_real by auto
2041   have "Real (THE r. 0 \<le> r \<and> x = Real r) = x" unfolding *
2042     using assms unfolding Real_real by auto
2043   thus ?thesis unfolding real_of_pextreal_def of_pextreal_def
2044     unfolding pextreal_case_def using assms by auto
2045 qed
2047 lemma Real_less_plus_one:"Real x < Real (max (x + 1) 1)"
2048   unfolding pextreal_less by auto
2050 lemma Lim_omega: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
2051 proof assume ?r show ?l apply(rule topological_tendstoI)
2052     unfolding eventually_sequentially
2053   proof- fix S assume "open S" "\<omega> \<in> S"
2054     from open_omega[OF this] guess B .. note B=this
2055     from `?r`[rule_format,of "(max B 0)+1"] guess N .. note N=this
2056     show "\<exists>N. \<forall>n\<ge>N. f n \<in> S" apply(rule_tac x=N in exI)
2057     proof safe case goal1
2058       have "Real B < Real ((max B 0) + 1)" by auto
2059       also have "... \<le> f n" using goal1 N by auto
2060       finally show ?case using B by fastsimp
2061     qed
2062   qed
2063 next assume ?l show ?r
2064   proof fix B::real have "open {Real B<..}" "\<omega> \<in> {Real B<..}" by auto
2065     from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
2066     guess N .. note N=this
2067     show "\<exists>N. \<forall>n\<ge>N. Real B \<le> f n" apply(rule_tac x=N in exI) using N by auto
2068   qed
2069 qed
2071 lemma Lim_bounded_omgea: assumes lim:"f ----> l" and "\<And>n. f n \<le> Real B" shows "l \<noteq> \<omega>"
2072 proof(rule ccontr,unfold not_not) let ?B = "max (B + 1) 1" assume as:"l=\<omega>"
2073   from lim[unfolded this Lim_omega,rule_format,of "?B"]
2074   guess N .. note N=this[rule_format,OF le_refl]
2075   hence "Real ?B \<le> Real B" using assms(2)[of N] by(rule order_trans)
2076   hence "Real ?B < Real ?B" using Real_less_plus_one[of B] by(rule le_less_trans)
2077   thus False by auto
2078 qed
2080 lemma incseq_le_pextreal: assumes inc: "\<And>n m. n\<ge>m \<Longrightarrow> X n \<ge> X m"
2081   and lim: "X ----> (L::pextreal)" shows "X n \<le> L"
2082 proof(cases "L = \<omega>")
2083   case False have "\<forall>n. X n \<noteq> \<omega>"
2084   proof(rule ccontr,unfold not_all not_not,safe)
2085     case goal1 hence "\<forall>n\<ge>x. X n = \<omega>" using inc[of x] by auto
2086     hence "X ----> \<omega>" unfolding tendsto_def eventually_sequentially
2087       apply safe apply(rule_tac x=x in exI) by auto
2088     note Lim_unique[OF trivial_limit_sequentially this lim]
2089     with False show False by auto
2090   qed note * =this[rule_format]
2092   have **:"\<forall>m n. m \<le> n \<longrightarrow> Real (real (X m)) \<le> Real (real (X n))"
2093     unfolding Real_real using * inc by auto
2094   have "real (X n) \<le> real L" apply-apply(rule incseq_le) defer
2095     apply(subst lim_Real[THEN sym]) apply(rule,rule,rule)
2096     unfolding Real_real'[OF *] Real_real'[OF False]
2097     unfolding incseq_def using ** lim by auto
2098   hence "Real (real (X n)) \<le> Real (real L)" by auto
2099   thus ?thesis unfolding Real_real using * False by auto
2100 qed auto
2102 lemma SUP_Lim_pextreal: assumes "\<And>n m. n\<ge>m \<Longrightarrow> f n \<ge> f m" "f ----> l"
2103   shows "(SUP n. f n) = (l::pextreal)" unfolding SUPR_def Sup_pextreal_def
2104 proof (safe intro!: Least_equality)
2105   fix n::nat show "f n \<le> l" apply(rule incseq_le_pextreal)
2106     using assms by auto
2107 next fix y assume y:"\<forall>x\<in>range f. x \<le> y" show "l \<le> y"
2108   proof(rule ccontr,cases "y=\<omega>",unfold not_le)
2109     case False assume as:"y < l"
2110     have l:"l \<noteq> \<omega>" apply(rule Lim_bounded_omgea[OF assms(2), of "real y"])
2111       using False y unfolding Real_real by auto
2113     have yl:"real y < real l" using as apply-
2114       apply(subst(asm) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
2115       apply(subst(asm) Real_real'[THEN sym,OF `l\<noteq>\<omega>`])
2116       unfolding pextreal_less apply(subst(asm) if_P) by auto
2117     hence "y + (y - l) * Real (1 / 2) < l" apply-
2118       apply(subst Real_real'[THEN sym,OF `y\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `y\<noteq>\<omega>`])
2119       apply(subst Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) apply(subst(2) Real_real'[THEN sym,OF `l\<noteq>\<omega>`]) by auto
2120     hence *:"l \<in> {y + (y - l) / 2<..}" by auto
2121     have "open {y + (y-l)/2 <..}" by auto
2122     note topological_tendstoD[OF assms(2) this *]
2123     from this[unfolded eventually_sequentially] guess N .. note this[rule_format, of N]
2124     hence "y + (y - l) * Real (1 / 2) < y" using y[rule_format,of "f N"] by auto
2125     hence "Real (real y) + (Real (real y) - Real (real l)) * Real (1 / 2) < Real (real y)"
2126       unfolding Real_real using `y\<noteq>\<omega>` `l\<noteq>\<omega>` by auto
2127     thus False using yl by auto
2128   qed auto
2129 qed
2131 lemma Real_max':"Real x = Real (max x 0)"
2132 proof(cases "x < 0") case True
2133   hence *:"max x 0 = 0" by auto
2134   show ?thesis unfolding * using True by auto
2135 qed auto
2137 lemma lim_pextreal_increasing: assumes "\<forall>n m. n\<ge>m \<longrightarrow> f n \<ge> f m"
2138   obtains l where "f ----> (l::pextreal)"
2139 proof(cases "\<exists>B. \<forall>n. f n < Real B")
2140   case False thus thesis apply- apply(rule that[of \<omega>]) unfolding Lim_omega not_ex not_all
2141     apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
2142     apply(rule order_trans[OF _ assms[rule_format]]) by auto
2143 next case True then guess B .. note B = this[rule_format]
2144   hence *:"\<And>n. f n < \<omega>" apply-apply(rule less_le_trans,assumption) by auto
2145   have *:"\<And>n. f n \<noteq> \<omega>" proof- case goal1 show ?case using *[of n] by auto qed
2146   have B':"\<And>n. real (f n) \<le> max 0 B" proof- case goal1 thus ?case
2147       using B[of n] apply-apply(subst(asm) Real_real'[THEN sym]) defer
2148       apply(subst(asm)(2) Real_max') unfolding pextreal_less apply(subst(asm) if_P) using *[of n] by auto
2149   qed
2150   have "\<exists>l. (\<lambda>n. real (f n)) ----> l" apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
2151   proof safe show "bounded {real (f n) |n. True}"
2152       unfolding bounded_def apply(rule_tac x=0 in exI,rule_tac x="max 0 B" in exI)
2153       using B' unfolding dist_norm by auto
2154     fix n::nat have "Real (real (f n)) \<le> Real (real (f (Suc n)))"
2155       using assms[rule_format,of n "Suc n"] apply(subst Real_real)+
2156       using *[of n] *[of "Suc n"] by fastsimp
2157     thus "real (f n) \<le> real (f (Suc n))" by auto
2158   qed then guess l .. note l=this
2159   have "0 \<le> l" apply(rule LIMSEQ_le_const[OF l])
2160     by(rule_tac x=0 in exI,auto)
2162   thus ?thesis apply-apply(rule that[of "Real l"])
2163     using l apply-apply(subst(asm) lim_Real[THEN sym]) prefer 3
2164     unfolding Real_real using * by auto
2165 qed
2167 lemma setsum_neq_omega: assumes "finite s" "\<And>x. x \<in> s \<Longrightarrow> f x \<noteq> \<omega>"
2168   shows "setsum f s \<noteq> \<omega>" using assms
2169 proof induct case (insert x s)
2170   show ?case unfolding setsum.insert[OF insert(1-2)]
2171     using insert by auto
2172 qed auto
2175 lemma real_Real': "0 \<le> x \<Longrightarrow> real (Real x) = x"
2176   unfolding real_Real by auto
2178 lemma real_pextreal_pos[intro]:
2179   assumes "x \<noteq> 0" "x \<noteq> \<omega>"
2180   shows "real x > 0"
2181   apply(subst real_Real'[THEN sym,of 0]) defer
2182   apply(rule real_of_pextreal_less) using assms by auto
2184 lemma Lim_omega_gt: "f ----> \<omega> \<longleftrightarrow> (\<forall>B. \<exists>N. \<forall>n\<ge>N. f n > Real B)" (is "?l = ?r")
2185 proof assume ?l thus ?r unfolding Lim_omega apply safe
2186     apply(erule_tac x="max B 0 +1" in allE,safe)
2187     apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
2188     apply(rule_tac y="Real (max B 0 + 1)" in less_le_trans) by auto
2189 next assume ?r thus ?l unfolding Lim_omega apply safe
2190     apply(erule_tac x=B in allE,safe) apply(rule_tac x=N in exI,safe) by auto
2191 qed
2193 lemma pextreal_minus_le_cancel:
2194   fixes a b c :: pextreal
2195   assumes "b \<le> a"
2196   shows "c - a \<le> c - b"
2197   using assms by (cases a, cases b, cases c, simp, simp, simp, cases b, cases c, simp_all)
2199 lemma pextreal_minus_\<omega>[simp]: "x - \<omega> = 0" by (cases x) simp_all
2201 lemma pextreal_minus_mono[intro]: "a - x \<le> (a::pextreal)"
2202 proof- have "a - x \<le> a - 0"
2203     apply(rule pextreal_minus_le_cancel) by auto
2204   thus ?thesis by auto
2205 qed
2207 lemma pextreal_minus_eq_\<omega>[simp]: "x - y = \<omega> \<longleftrightarrow> (x = \<omega> \<and> y \<noteq> \<omega>)"
2208   by (cases x, cases y) (auto, cases y, auto)
2210 lemma pextreal_less_minus_iff:
2211   fixes a b c :: pextreal
2212   shows "a < b - c \<longleftrightarrow> c + a < b"
2213   by (cases c, cases a, cases b, auto)
2215 lemma pextreal_minus_less_iff:
2216   fixes a b c :: pextreal shows "a - c < b \<longleftrightarrow> (0 < b \<and> (c \<noteq> \<omega> \<longrightarrow> a < b + c))"
2217   by (cases c, cases a, cases b, auto)
2219 lemma pextreal_le_minus_iff:
2220   fixes a b c :: pextreal
2221   shows "a \<le> c - b \<longleftrightarrow> ((c \<le> b \<longrightarrow> a = 0) \<and> (b < c \<longrightarrow> a + b \<le> c))"
2222   by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex)
2224 lemma pextreal_minus_le_iff:
2225   fixes a b c :: pextreal
2226   shows "a - c \<le> b \<longleftrightarrow> (c \<le> a \<longrightarrow> a \<le> b + c)"
2227   by (cases a, cases c, cases b, auto simp: pextreal_noteq_omega_Ex)
2229 lemmas pextreal_minus_order = pextreal_minus_le_iff pextreal_minus_less_iff pextreal_le_minus_iff pextreal_less_minus_iff
2231 lemma pextreal_minus_strict_mono:
2232   assumes "a > 0" "x > 0" "a\<noteq>\<omega>"
2233   shows "a - x < (a::pextreal)"
2234   using assms by(cases x, cases a, auto)
2236 lemma pextreal_minus':
2237   "Real r - Real p = (if 0 \<le> r \<and> p \<le> r then if 0 \<le> p then Real (r - p) else Real r else 0)"
2238   by (auto simp: minus_pextreal_eq not_less)
2240 lemma pextreal_minus_plus:
2241   "x \<le> (a::pextreal) \<Longrightarrow> a - x + x = a"
2242   by (cases a, cases x) auto
2244 lemma pextreal_cancel_plus_minus: "b \<noteq> \<omega> \<Longrightarrow> a + b - b = a"
2245   by (cases a, cases b) auto
2247 lemma pextreal_minus_le_cancel_right:
2248   fixes a b c :: pextreal
2249   assumes "a \<le> b" "c \<le> a"
2250   shows "a - c \<le> b - c"
2251   using assms by (cases a, cases b, cases c, auto, cases c, auto)
2253 lemma real_of_pextreal_setsum':
2254   assumes "\<forall>x \<in> S. f x \<noteq> \<omega>"
2255   shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
2256 proof cases
2257   assume "finite S"
2258   from this assms show ?thesis
2260 qed simp
2262 lemma Lim_omega_pos: "f ----> \<omega> \<longleftrightarrow> (\<forall>B>0. \<exists>N. \<forall>n\<ge>N. f n \<ge> Real B)" (is "?l = ?r")
2263   unfolding Lim_omega apply safe defer
2264   apply(erule_tac x="max 1 B" in allE) apply safe defer
2265   apply(rule_tac x=N in exI,safe) apply(erule_tac x=n in allE,safe)
2266   apply(rule_tac y="Real (max 1 B)" in order_trans) by auto
2268 lemma pextreal_LimI_finite:
2269   assumes "x \<noteq> \<omega>" "\<And>r. 0 < r \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. u n < x + r \<and> x < u n + r"
2270   shows "u ----> x"
2271 proof (rule topological_tendstoI, unfold eventually_sequentially)
2272   fix S assume "open S" "x \<in> S"
2273   then obtain A where "open A" and A_eq: "Real ` (A \<inter> {0..}) = S - {\<omega>}" by (auto elim!: pextreal_openE)
2274   then have "x \<in> Real ` (A \<inter> {0..})" using `x \<in> S` `x \<noteq> \<omega>` by auto
2275   then have "real x \<in> A" by auto
2276   then obtain r where "0 < r" and dist: "\<And>y. dist y (real x) < r \<Longrightarrow> y \<in> A"
2277     using `open A` unfolding open_real_def by auto
2278   then obtain n where
2279     upper: "\<And>N. n \<le> N \<Longrightarrow> u N < x + Real r" and
2280     lower: "\<And>N. n \<le> N \<Longrightarrow> x < u N + Real r" using assms(2)[of "Real r"] by auto
2281   show "\<exists>N. \<forall>n\<ge>N. u n \<in> S"
2282   proof (safe intro!: exI[of _ n])
2283     fix N assume "n \<le> N"
2284     from upper[OF this] `x \<noteq> \<omega>` `0 < r`
2285     have "u N \<noteq> \<omega>" by (force simp: pextreal_noteq_omega_Ex)
2286     with `x \<noteq> \<omega>` `0 < r` lower[OF `n \<le> N`] upper[OF `n \<le> N`]
2287     have "dist (real (u N)) (real x) < r" "u N \<noteq> \<omega>"
2288       by (auto simp: pextreal_noteq_omega_Ex dist_real_def abs_diff_less_iff field_simps)
2289     from dist[OF this(1)]
2290     have "u N \<in> Real ` (A \<inter> {0..})" using `u N \<noteq> \<omega>`
2291       by (auto intro!: image_eqI[of _ _ "real (u N)"] simp: pextreal_noteq_omega_Ex Real_real)
2292     thus "u N \<in> S" using A_eq by simp
2293   qed
2294 qed
2296 lemma real_Real_max:"real (Real x) = max x 0"
2297   unfolding real_Real by auto
2299 lemma Sup_lim:
2300   assumes "\<forall>n. b n \<in> s" "b ----> (a::pextreal)"
2301   shows "a \<le> Sup s"
2302 proof(rule ccontr,unfold not_le)
2303   assume as:"Sup s < a" hence om:"Sup s \<noteq> \<omega>" by auto
2304   have s:"s \<noteq> {}" using assms by auto
2305   { presume *:"\<forall>n. b n < a \<Longrightarrow> False"
2306     show False apply(cases,rule *,assumption,unfold not_all not_less)
2307     proof- case goal1 then guess n .. note n=this
2308       thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]]
2309         using as by auto
2310     qed
2311   } assume b:"\<forall>n. b n < a"
2312   show False
2313   proof(cases "a = \<omega>")
2314     case False have *:"a - Sup s > 0"
2315       using False as by(auto simp: pextreal_zero_le_diff)
2316     have "(a - Sup s) / 2 \<le> a / 2" unfolding divide_pextreal_def
2317       apply(rule mult_right_mono) by auto
2318     also have "... = Real (real (a / 2))" apply(rule Real_real'[THEN sym])
2319       using False by auto
2320     also have "... < Real (real a)" unfolding pextreal_less using as False
2321       by(auto simp add: real_of_pextreal_mult[THEN sym])
2322     also have "... = a" apply(rule Real_real') using False by auto
2323     finally have asup:"a > (a - Sup s) / 2" .
2324     have "\<exists>n. a - b n < (a - Sup s) / 2"
2325     proof(rule ccontr,unfold not_ex not_less)
2326       case goal1
2327       have "(a - Sup s) * Real (1 / 2)  > 0"
2328         using * by auto
2329       hence "a - (a - Sup s) * Real (1 / 2) < a"
2330         apply-apply(rule pextreal_minus_strict_mono)
2331         using False * by auto
2332       hence *:"a \<in> {a - (a - Sup s) / 2<..}"using asup by auto
2333       note topological_tendstoD[OF assms(2) open_pextreal_greaterThan,OF *]
2334       from this[unfolded eventually_sequentially] guess n ..
2335       note n = this[rule_format,of n]
2336       have "b n + (a - Sup s) / 2 \<le> a"
2337         using add_right_mono[OF goal1[rule_format,of n],of "b n"]
2338         unfolding pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]]
2340       hence "b n \<le> a - (a - Sup s) / 2" unfolding pextreal_le_minus_iff
2341         using asup by auto
2342       hence "b n \<notin> {a - (a - Sup s) / 2<..}" by auto
2343       thus False using n by auto
2344     qed
2345     then guess n .. note n = this
2346     have "Sup s < a - (a - Sup s) / 2"
2347       using False as om by (cases a) (auto simp: pextreal_noteq_omega_Ex field_simps)
2348     also have "... \<le> b n"
2349     proof- note add_right_mono[OF less_imp_le[OF n],of "b n"]
2350       note this[unfolded pextreal_minus_plus[OF less_imp_le[OF b[rule_format]]]]
2351       hence "a - (a - Sup s) / 2 \<le> (a - Sup s) / 2 + b n - (a - Sup s) / 2"
2352         apply(rule pextreal_minus_le_cancel_right) using asup by auto
2353       also have "... = b n + (a - Sup s) / 2 - (a - Sup s) / 2"
2355       also have "... = b n" apply(subst pextreal_cancel_plus_minus)
2356       proof(rule ccontr,unfold not_not) case goal1
2357         show ?case using asup unfolding goal1 by auto
2358       qed auto
2359       finally show ?thesis .
2360     qed
2361     finally show False
2362       using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of n]] by auto
2363   next case True
2364     from assms(2)[unfolded True Lim_omega_gt,rule_format,of "real (Sup s)"]
2365     guess N .. note N = this[rule_format,of N]
2366     thus False using complete_lattice_class.Sup_upper[OF assms(1)[rule_format,of N]]
2367       unfolding Real_real using om by auto
2368   qed qed
2370 lemma Sup_mono_lim:
2371   assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> b ----> (a::pextreal)"
2372   shows "Sup A \<le> Sup B"
2373   unfolding Sup_le_iff apply(rule) apply(drule assms[rule_format]) apply safe
2374   apply(rule_tac b=b in Sup_lim) by auto
2377   assumes "x \<noteq> \<omega>" "a < b"
2378   shows "x + a < x + b"
2379   using assms by (cases a, cases b, cases x) auto
2381 lemma SUPR_lim:
2382   assumes "\<forall>n. b n \<in> B" "(\<lambda>n. f (b n)) ----> (f a::pextreal)"
2383   shows "f a \<le> SUPR B f"
2384   unfolding SUPR_def apply(rule Sup_lim[of "\<lambda>n. f (b n)"])
2385   using assms by auto
2387 lemma SUP_\<omega>_imp:
2388   assumes "(SUP i. f i) = \<omega>"
2389   shows "\<exists>i. Real x < f i"
2390 proof (rule ccontr)
2391   assume "\<not> ?thesis" hence "\<And>i. f i \<le> Real x" by (simp add: not_less)
2392   hence "(SUP i. f i) \<le> Real x" unfolding SUP_le_iff by auto
2393   with assms show False by auto
2394 qed
2396 lemma SUPR_mono_lim:
2397   assumes "\<forall>a\<in>A. \<exists>b. \<forall>n. b n \<in> B \<and> (\<lambda>n. f (b n)) ----> (f a::pextreal)"
2398   shows "SUPR A f \<le> SUPR B f"
2399   unfolding SUPR_def apply(rule Sup_mono_lim)
2400   apply safe apply(drule assms[rule_format],safe)
2401   apply(rule_tac x="\<lambda>n. f (b n)" in exI) by auto
2403 lemma real_0_imp_eq_0:
2404   assumes "x \<noteq> \<omega>" "real x = 0"
2405   shows "x = 0"
2406 using assms by (cases x) auto
2408 lemma SUPR_mono:
2409   assumes "\<forall>a\<in>A. \<exists>b\<in>B. f b \<ge> f a"
2410   shows "SUPR A f \<le> SUPR B f"
2411   unfolding SUPR_def apply(rule Sup_mono)
2412   using assms by auto
2415   fixes x :: real
2416   fixes a b :: pextreal
2417   assumes "x \<ge> 0" "a < b"
2418   shows "a + Real x < b + Real x"
2419 using assms by (cases a, cases b) auto
2422   fixes x :: real
2423   fixes a b :: pextreal
2424   assumes "x \<ge> 0" "a \<le> b"
2425   shows "a + Real x \<le> b + Real x"
2426 using assms by (cases a, cases b) auto
2428 lemma le_imp_less_pextreal:
2429   fixes x :: pextreal
2430   assumes "x > 0" "a + x \<le> b" "a \<noteq> \<omega>"
2431   shows "a < b"
2432 using assms by (cases x, cases a, cases b) auto
2434 lemma pextreal_INF_minus:
2435   fixes f :: "nat \<Rightarrow> pextreal"
2436   assumes "c \<noteq> \<omega>"
2437   shows "(INF i. c - f i) = c - (SUP i. f i)"
2438 proof (cases "SUP i. f i")
2439   case infinite
2440   from `c \<noteq> \<omega>` obtain x where [simp]: "c = Real x" by (cases c) auto
2441   from SUP_\<omega>_imp[OF infinite] obtain i where "Real x < f i" by auto
2442   have "(INF i. c - f i) \<le> c - f i"
2443     by (auto intro!: complete_lattice_class.INF_leI)
2444   also have "\<dots> = 0" using `Real x < f i` by (auto simp: minus_pextreal_eq)
2445   finally show ?thesis using infinite by auto
2446 next
2447   case (preal r)
2448   from `c \<noteq> \<omega>` obtain x where c: "c = Real x" by (cases c) auto
2450   show ?thesis unfolding c
2451   proof (rule pextreal_INFI)
2452     fix i have "f i \<le> (SUP i. f i)" by (rule le_SUPI) simp
2453     thus "Real x - (SUP i. f i) \<le> Real x - f i" by (rule pextreal_minus_le_cancel)
2454   next
2455     fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> y \<le> Real x - f i"
2456     from this[of 0] obtain p where p: "y = Real p" "0 \<le> p"
2457       by (cases "f 0", cases y, auto split: split_if_asm)
2458     hence "\<And>i. Real p \<le> Real x - f i" using * by auto
2459     hence *: "\<And>i. Real x \<le> f i \<Longrightarrow> Real p = 0"
2460       "\<And>i. f i < Real x \<Longrightarrow> Real p + f i \<le> Real x"
2461       unfolding pextreal_le_minus_iff by auto
2462     show "y \<le> Real x - (SUP i. f i)" unfolding p pextreal_le_minus_iff
2463     proof safe
2464       assume x_less: "Real x \<le> (SUP i. f i)"
2465       show "Real p = 0"
2466       proof (rule ccontr)
2467         assume "Real p \<noteq> 0"
2468         hence "0 < Real p" by auto
2469         from Sup_close[OF this, of "range f"]
2470         obtain i where e: "(SUP i. f i) < f i + Real p"
2471           using preal unfolding SUPR_def by auto
2472         hence "Real x \<le> f i + Real p" using x_less by auto
2473         show False
2474         proof cases
2475           assume "\<forall>i. f i < Real x"
2476           hence "Real p + f i \<le> Real x" using * by auto
2477           hence "f i + Real p \<le> (SUP i. f i)" using x_less by (auto simp: field_simps)
2478           thus False using e by auto
2479         next
2480           assume "\<not> (\<forall>i. f i < Real x)"
2481           then obtain i where "Real x \<le> f i" by (auto simp: not_less)
2482           from *(1)[OF this] show False using `Real p \<noteq> 0` by auto
2483         qed
2484       qed
2485     next
2486       have "\<And>i. f i \<le> (SUP i. f i)" by (rule complete_lattice_class.le_SUPI) auto
2487       also assume "(SUP i. f i) < Real x"
2488       finally have "\<And>i. f i < Real x" by auto
2489       hence *: "\<And>i. Real p + f i \<le> Real x" using * by auto
2490       have "Real p \<le> Real x" using *[of 0] by (cases "f 0") (auto split: split_if_asm)
2492       have SUP_eq: "(SUP i. f i) \<le> Real x - Real p"
2493       proof (rule SUP_leI)
2494         fix i show "f i \<le> Real x - Real p" unfolding pextreal_le_minus_iff
2495         proof safe
2496           assume "Real x \<le> Real p"
2497           with *[of i] show "f i = 0"
2498             by (cases "f i") (auto split: split_if_asm)
2499         next
2500           assume "Real p < Real x"
2501           show "f i + Real p \<le> Real x" using * by (auto simp: field_simps)
2502         qed
2503       qed
2505       show "Real p + (SUP i. f i) \<le> Real x"
2506       proof cases
2507         assume "Real x \<le> Real p"
2508         with `Real p \<le> Real x` have [simp]: "Real p = Real x" by (rule antisym)
2509         { fix i have "f i = 0" using *[of i] by (cases "f i") (auto split: split_if_asm) }
2510         hence "(SUP i. f i) \<le> 0" by (auto intro!: SUP_leI)
2511         thus ?thesis by simp
2512       next
2513         assume "\<not> Real x \<le> Real p" hence "Real p < Real x" unfolding not_le .
2514         with SUP_eq show ?thesis unfolding pextreal_le_minus_iff by (auto simp: field_simps)
2515       qed
2516     qed
2517   qed
2518 qed
2520 lemma pextreal_SUP_minus:
2521   fixes f :: "nat \<Rightarrow> pextreal"
2522   shows "(SUP i. c - f i) = c - (INF i. f i)"
2523 proof (rule pextreal_SUPI)
2524   fix i have "(INF i. f i) \<le> f i" by (rule INF_leI) simp
2525   thus "c - f i \<le> c - (INF i. f i)" by (rule pextreal_minus_le_cancel)
2526 next
2527   fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c - f i \<le> y"
2528   show "c - (INF i. f i) \<le> y"
2529   proof (cases y)
2530     case (preal p)
2532     show ?thesis unfolding pextreal_minus_le_iff preal
2533     proof safe
2534       assume INF_le_x: "(INF i. f i) \<le> c"
2535       from * have *: "\<And>i. f i \<le> c \<Longrightarrow> c \<le> Real p + f i"
2536         unfolding pextreal_minus_le_iff preal by auto
2538       have INF_eq: "c - Real p \<le> (INF i. f i)"
2539       proof (rule le_INFI)
2540         fix i show "c - Real p \<le> f i" unfolding pextreal_minus_le_iff
2541         proof safe
2542           assume "Real p \<le> c"
2543           show "c \<le> f i + Real p"
2544           proof cases
2545             assume "f i \<le> c" from *[OF this]
2546             show ?thesis by (simp add: field_simps)
2547           next
2548             assume "\<not> f i \<le> c"
2549             hence "c \<le> f i" by auto
2550             also have "\<dots> \<le> f i + Real p" by auto
2551             finally show ?thesis .
2552           qed
2553         qed
2554       qed
2556       show "c \<le> Real p + (INF i. f i)"
2557       proof cases
2558         assume "Real p \<le> c"
2559         with INF_eq show ?thesis unfolding pextreal_minus_le_iff by (auto simp: field_simps)
2560       next
2561         assume "\<not> Real p \<le> c"
2562         hence "c \<le> Real p" by auto
2563         also have "Real p \<le> Real p + (INF i. f i)" by auto
2564         finally show ?thesis .
2565       qed
2566     qed
2567   qed simp
2568 qed
2570 lemma pextreal_le_minus_imp_0:
2571   fixes a b :: pextreal
2572   shows "a \<le> a - b \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a \<noteq> \<omega> \<Longrightarrow> b = 0"
2573   by (cases a, cases b, auto split: split_if_asm)
2575 lemma lim_INF_eq_lim_SUP:
2576   fixes X :: "nat \<Rightarrow> real"
2577   assumes "\<And>i. 0 \<le> X i" and "0 \<le> x"
2578   and lim_INF: "(SUP n. INF m. Real (X (n + m))) = Real x" (is "(SUP n. ?INF n) = _")
2579   and lim_SUP: "(INF n. SUP m. Real (X (n + m))) = Real x" (is "(INF n. ?SUP n) = _")
2580   shows "X ----> x"
2581 proof (rule LIMSEQ_I)
2582   fix r :: real assume "0 < r"
2583   hence "0 \<le> r" by auto
2584   from Sup_close[of "Real r" "range ?INF"]
2585   obtain n where inf: "Real x < ?INF n + Real r"
2586     unfolding SUPR_def lim_INF[unfolded SUPR_def] using `0 < r` by auto
2588   from Inf_close[of "range ?SUP" "Real r"]
2589   obtain n' where sup: "?SUP n' < Real x + Real r"
2590     unfolding INFI_def lim_SUP[unfolded INFI_def] using `0 < r` by auto
2592   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
2593   proof (safe intro!: exI[of _ "max n n'"])
2594     fix m assume "max n n' \<le> m" hence "n \<le> m" "n' \<le> m" by auto
2596     note inf
2597     also have "?INF n + Real r \<le> Real (X (n + (m - n))) + Real r"
2598       by (rule le_add_Real, auto simp: `0 \<le> r` intro: INF_leI)
2599     finally have up: "x < X m + r"
2600       using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n \<le> m` by auto
2602     have "Real (X (n' + (m - n'))) \<le> ?SUP n'"
2603       by (auto simp: `0 \<le> r` intro: le_SUPI)
2604     also note sup
2605     finally have down: "X m < x + r"
2606       using `0 \<le> X m` `0 \<le> x` `0 \<le> r` `n' \<le> m` by auto
2608     show "norm (X m - x) < r" using up down by auto
2609   qed
2610 qed
2612 lemma Sup_countable_SUPR:
2613   assumes "Sup A \<noteq> \<omega>" "A \<noteq> {}"
2614   shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f"
2615 proof -
2616   have "\<And>n. 0 < 1 / (of_nat n :: pextreal)" by auto
2617   from Sup_close[OF this assms]
2618   have "\<forall>n. \<exists>x. x \<in> A \<and> Sup A < x + 1 / of_nat n" by blast
2619   from choice[OF this] obtain f where "range f \<subseteq> A" and
2620     epsilon: "\<And>n. Sup A < f n + 1 / of_nat n" by blast
2621   have "SUPR UNIV f = Sup A"
2622   proof (rule pextreal_SUPI)
2623     fix i show "f i \<le> Sup A" using `range f \<subseteq> A`
2624       by (auto intro!: complete_lattice_class.Sup_upper)
2625   next
2626     fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y"
2627     show "Sup A \<le> y"
2628     proof (rule pextreal_le_epsilon)
2629       fix e :: pextreal assume "0 < e"
2630       show "Sup A \<le> y + e"
2631       proof (cases e)
2632         case (preal r)
2633         hence "0 < r" using `0 < e` by auto
2634         then obtain n where *: "inverse (of_nat n) < r" "0 < n"
2635           using ex_inverse_of_nat_less by auto
2636         have "Sup A \<le> f n + 1 / of_nat n" using epsilon[of n] by auto
2637         also have "1 / of_nat n \<le> e" using preal * by (auto simp: real_eq_of_nat)
2638         with bound have "f n + 1 / of_nat n \<le> y + e" by (rule add_mono) simp
2639         finally show "Sup A \<le> y + e" .
2640       qed simp
2641     qed
2642   qed
2643   with `range f \<subseteq> A` show ?thesis by (auto intro!: exI[of _ f])
2644 qed
2646 lemma SUPR_countable_SUPR:
2647   assumes "SUPR A g \<noteq> \<omega>" "A \<noteq> {}"
2648   shows "\<exists> f::nat \<Rightarrow> pextreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f"
2649 proof -
2650   have "Sup (g`A) \<noteq> \<omega>" "g`A \<noteq> {}" using assms unfolding SUPR_def by auto
2651   from Sup_countable_SUPR[OF this]
2652   show ?thesis unfolding SUPR_def .
2653 qed
2655 lemma pextreal_setsum_subtractf:
2656   assumes "\<And>i. i\<in>A \<Longrightarrow> g i \<le> f i" and "\<And>i. i\<in>A \<Longrightarrow> f i \<noteq> \<omega>"
2657   shows "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
2658 proof cases
2659   assume "finite A" from this assms show ?thesis
2660   proof induct
2661     case (insert x A)
2662     hence hyp: "(\<Sum>i\<in>A. f i - g i) = (\<Sum>i\<in>A. f i) - (\<Sum>i\<in>A. g i)"
2663       by auto
2664     { fix i assume *: "i \<in> insert x A"
2665       hence "g i \<le> f i" using insert by simp
2666       also have "f i < \<omega>" using * insert by (simp add: pextreal_less_\<omega>)
2667       finally have "g i \<noteq> \<omega>" by (simp add: pextreal_less_\<omega>) }
2668     hence "setsum g A \<noteq> \<omega>" "g x \<noteq> \<omega>" by (auto simp: setsum_\<omega>)
2669     moreover have "setsum f A \<noteq> \<omega>" "f x \<noteq> \<omega>" using insert by (auto simp: setsum_\<omega>)
2670     moreover have "g x \<le> f x" using insert by auto
2671     moreover have "(\<Sum>i\<in>A. g i) \<le> (\<Sum>i\<in>A. f i)" using insert by (auto intro!: setsum_mono)
2672     ultimately show ?case using `finite A` `x \<notin> A` hyp
2673       by (auto simp: pextreal_noteq_omega_Ex)
2674   qed simp
2675 qed simp
2677 lemma real_of_pextreal_diff:
2678   "y \<le> x \<Longrightarrow> x \<noteq> \<omega> \<Longrightarrow> real x - real y = real (x - y)"
2679   by (cases x, cases y) auto
2681 lemma psuminf_minus:
2682   assumes ord: "\<And>i. g i \<le> f i" and fin: "psuminf g \<noteq> \<omega>" "psuminf f \<noteq> \<omega>"
2683   shows "(\<Sum>\<^isub>\<infinity> i. f i - g i) = psuminf f - psuminf g"
2684 proof -
2685   have [simp]: "\<And>i. f i \<noteq> \<omega>" using fin by (auto intro: psuminf_\<omega>)
2686   from fin have "(\<lambda>x. real (f x)) sums real (\<Sum>\<^isub>\<infinity>x. f x)"
2687     and "(\<lambda>x. real (g x)) sums real (\<Sum>\<^isub>\<infinity>x. g x)"
2688     by (auto intro: psuminf_imp_suminf)
2689   from sums_diff[OF this]
2690   have "(\<lambda>n. real (f n - g n)) sums (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))" using fin ord
2691     by (subst (asm) (1 2) real_of_pextreal_diff) (auto simp: psuminf_\<omega> psuminf_le)
2692   hence "(\<Sum>\<^isub>\<infinity> i. Real (real (f i - g i))) = Real (real ((\<Sum>\<^isub>\<infinity>x. f x) - (\<Sum>\<^isub>\<infinity>x. g x)))"
2693     by (rule suminf_imp_psuminf) simp
2694   thus ?thesis using fin by (simp add: Real_real psuminf_\<omega>)
2695 qed
2697 lemma INF_eq_LIMSEQ:
2698   assumes "mono (\<lambda>i. - f i)" and "\<And>n. 0 \<le> f n" and "0 \<le> x"
2699   shows "(INF n. Real (f n)) = Real x \<longleftrightarrow> f ----> x"
2700 proof
2701   assume x: "(INF n. Real (f n)) = Real x"
2702   { fix n
2703     have "Real x \<le> Real (f n)" using x[symmetric] by (auto intro: INF_leI)
2704     hence "x \<le> f n" using assms by simp
2705     hence "\<bar>f n - x\<bar> = f n - x" by auto }
2706   note abs_eq = this
2707   show "f ----> x"
2708   proof (rule LIMSEQ_I)
2709     fix r :: real assume "0 < r"
2710     show "\<exists>no. \<forall>n\<ge>no. norm (f n - x) < r"
2711     proof (rule ccontr)
2712       assume *: "\<not> ?thesis"
2713       { fix N
2714         from * obtain n where *: "N \<le> n" "r \<le> f n - x"
2715           using abs_eq by (auto simp: not_less)
2716         hence "x + r \<le> f n" by auto
2717         also have "f n \<le> f N" using `mono (\<lambda>i. - f i)` * by (auto dest: monoD)
2718         finally have "Real (x + r) \<le> Real (f N)" using `0 \<le> f N` by auto }
2719       hence "Real x < Real (x + r)"
2720         and "Real (x + r) \<le> (INF n. Real (f n))" using `0 < r` `0 \<le> x` by (auto intro: le_INFI)
2721       hence "Real x < (INF n. Real (f n))" by (rule less_le_trans)
2722       thus False using x by auto
2723     qed
2724   qed
2725 next
2726   assume "f ----> x"
2727   show "(INF n. Real (f n)) = Real x"
2728   proof (rule pextreal_INFI)
2729     fix n
2730     from decseq_le[OF _ `f ----> x`] assms
2731     show "Real x \<le> Real (f n)" unfolding decseq_eq_incseq incseq_mono by auto
2732   next
2733     fix y assume *: "\<And>n. n\<in>UNIV \<Longrightarrow> y \<le> Real (f n)"
2734     thus "y \<le> Real x"
2735     proof (cases y)
2736       case (preal r)
2737       with * have "\<exists>N. \<forall>n\<ge>N. r \<le> f n" using assms by fastsimp
2738       from LIMSEQ_le_const[OF `f ----> x` this]
2739       show "y \<le> Real x" using `0 \<le> x` preal by auto
2740     qed simp
2741   qed
2742 qed
2744 lemma INFI_bound:
2745   assumes "\<forall>N. x \<le> f N"
2746   shows "x \<le> (INF n. f n)"
2747   using assms by (simp add: INFI_def le_Inf_iff)
2749 lemma LIMSEQ_imp_lim_INF:
2750   assumes pos: "\<And>i. 0 \<le> X i" and lim: "X ----> x"
2751   shows "(SUP n. INF m. Real (X (n + m))) = Real x"
2752 proof -
2753   have "0 \<le> x" using assms by (auto intro!: LIMSEQ_le_const)
2755   have "\<And>n. (INF m. Real (X (n + m))) \<le> Real (X (n + 0))" by (rule INF_leI) simp
2756   also have "\<And>n. Real (X (n + 0)) < \<omega>" by simp
2757   finally have "\<forall>n. \<exists>r\<ge>0. (INF m. Real (X (n + m))) = Real r"
2758     by (auto simp: pextreal_less_\<omega> pextreal_noteq_omega_Ex)
2759   from choice[OF this] obtain r where r: "\<And>n. (INF m. Real (X (n + m))) = Real (r n)" "\<And>n. 0 \<le> r n"
2760     by auto
2762   show ?thesis unfolding r
2763   proof (subst SUP_eq_LIMSEQ)
2764     show "mono r" unfolding mono_def
2765     proof safe
2766       fix x y :: nat assume "x \<le> y"
2767       have "Real (r x) \<le> Real (r y)" unfolding r(1)[symmetric] using pos
2768       proof (safe intro!: INF_mono bexI)
2769         fix m have "x + (m + y - x) = y + m"
2770           using `x \<le> y` by auto
2771         thus "Real (X (x + (m + y - x))) \<le> Real (X (y + m))" by simp
2772       qed simp
2773       thus "r x \<le> r y" using r by auto
2774     qed
2775     show "\<And>n. 0 \<le> r n" by fact
2776     show "0 \<le> x" by fact
2777     show "r ----> x"
2778     proof (rule LIMSEQ_I)
2779       fix e :: real assume "0 < e"
2780       hence "0 < e/2" by auto
2781       from LIMSEQ_D[OF lim this] obtain N where *: "\<And>n. N \<le> n \<Longrightarrow> \<bar>X n - x\<bar> < e/2"
2782         by auto
2783       show "\<exists>N. \<forall>n\<ge>N. norm (r n - x) < e"
2784       proof (safe intro!: exI[of _ N])
2785         fix n assume "N \<le> n"
2786         show "norm (r n - x) < e"
2787         proof cases
2788           assume "r n < x"
2789           have "x - r n \<le> e/2"
2790           proof cases
2791             assume e: "e/2 \<le> x"
2792             have "Real (x - e/2) \<le> Real (r n)" unfolding r(1)[symmetric]
2793             proof (rule le_INFI)
2794               fix m show "Real (x - e / 2) \<le> Real (X (n + m))"
2795                 using *[of "n + m"] `N \<le> n`
2796                 using pos by (auto simp: field_simps abs_real_def split: split_if_asm)
2797             qed
2798             with e show ?thesis using pos `0 \<le> x` r(2) by auto
2799           next
2800             assume "\<not> e/2 \<le> x" hence "x - e/2 < 0" by auto
2801             with `0 \<le> r n` show ?thesis by auto
2802           qed
2803           with `r n < x` show ?thesis by simp
2804         next
2805           assume e: "\<not> r n < x"
2806           have "Real (r n) \<le> Real (X (n + 0))" unfolding r(1)[symmetric]
2807             by (rule INF_leI) simp
2808           hence "r n - x \<le> X n - x" using r pos by auto
2809           also have "\<dots> < e/2" using *[OF `N \<le> n`] by (auto simp: field_simps abs_real_def split: split_if_asm)
2810           finally have "r n - x < e" using `0 < e` by auto
2811           with e show ?thesis by auto
2812         qed
2813       qed
2814     qed
2815   qed
2816 qed
2818 lemma real_of_pextreal_strict_mono_iff:
2819   "real a < real b \<longleftrightarrow> (b \<noteq> \<omega> \<and> ((a = \<omega> \<and> 0 < b) \<or> (a < b)))"
2820 proof (cases a)
2821   case infinite thus ?thesis by (cases b) auto
2822 next
2823   case preal thus ?thesis by (cases b) auto
2824 qed
2826 lemma real_of_pextreal_mono_iff:
2827   "real a \<le> real b \<longleftrightarrow> (a = \<omega> \<or> (b \<noteq> \<omega> \<and> a \<le> b) \<or> (b = \<omega> \<and> a = 0))"
2828 proof (cases a)
2829   case infinite thus ?thesis by (cases b) auto
2830 next
2831   case preal thus ?thesis by (cases b)  auto
2832 qed
2834 lemma ex_pextreal_inverse_of_nat_Suc_less:
2835   fixes e :: pextreal assumes "0 < e" shows "\<exists>n. inverse (of_nat (Suc n)) < e"
2836 proof (cases e)
2837   case (preal r)
2838   with `0 < e` ex_inverse_of_nat_Suc_less[of r]
2839   obtain n where "inverse (of_nat (Suc n)) < r" by auto
2840   with preal show ?thesis
2841     by (auto simp: real_eq_of_nat[symmetric])
2842 qed auto
2844 lemma Lim_eq_Sup_mono:
2845   fixes u :: "nat \<Rightarrow> pextreal" assumes "mono u"
2846   shows "u ----> (SUP i. u i)"
2847 proof -
2848   from lim_pextreal_increasing[of u] `mono u`
2849   obtain l where l: "u ----> l" unfolding mono_def by auto
2850   from SUP_Lim_pextreal[OF _ this] `mono u`
2851   have "(SUP i. u i) = l" unfolding mono_def by auto
2852   with l show ?thesis by simp
2853 qed
2855 lemma isotone_Lim:
2856   fixes x :: pextreal assumes "u \<up> x"
2857   shows "u ----> x" (is ?lim) and "mono u" (is ?mono)
2858 proof -
2859   show ?mono using assms unfolding mono_iff_le_Suc isoton_def by auto
2860   from Lim_eq_Sup_mono[OF this] `u \<up> x`
2861   show ?lim unfolding isoton_def by simp
2862 qed
2864 lemma isoton_iff_Lim_mono:
2865   fixes u :: "nat \<Rightarrow> pextreal"
2866   shows "u \<up> x \<longleftrightarrow> (mono u \<and> u ----> x)"
2867 proof safe
2868   assume "mono u" and x: "u ----> x"
2869   with SUP_Lim_pextreal[OF _ x]
2870   show "u \<up> x" unfolding isoton_def
2871     using `mono u`[unfolded mono_def]
2872     using `mono u`[unfolded mono_iff_le_Suc]
2873     by auto
2874 qed (auto dest: isotone_Lim)
2876 lemma pextreal_inverse_inverse[simp]:
2877   fixes x :: pextreal
2878   shows "inverse (inverse x) = x"
2879   by (cases x) auto
2881 lemma atLeastAtMost_omega_eq_atLeast:
2882   shows "{a .. \<omega>} = {a ..}"
2883 by auto
2885 lemma atLeast0AtMost_eq_atMost: "{0 :: pextreal .. a} = {.. a}" by auto
2887 lemma greaterThan_omega_Empty: "{\<omega> <..} = {}" by auto
2889 lemma lessThan_0_Empty: "{..< 0 :: pextreal} = {}" by auto
2891 lemma real_of_pextreal_inverse[simp]:
2892   fixes X :: pextreal
2893   shows "real (inverse X) = 1 / real X"
2894   by (cases X) (auto simp: inverse_eq_divide)
2896 lemma real_of_pextreal_le_0[simp]: "real (X :: pextreal) \<le> 0 \<longleftrightarrow> (X = 0 \<or> X = \<omega>)"
2897   by (cases X) auto
2899 lemma real_of_pextreal_less_0[simp]: "\<not> (real (X :: pextreal) < 0)"
2900   by (cases X) auto
2902 lemma abs_real_of_pextreal[simp]: "\<bar>real (X :: pextreal)\<bar> = real X"
2903   by simp
2905 lemma zero_less_real_of_pextreal: "0 < real (X :: pextreal) \<longleftrightarrow> X \<noteq> 0 \<and> X \<noteq> \<omega>"
2906   by (cases X) auto
2908 end