--- a/src/HOL/Library/Extended_Reals.thy Mon Mar 14 14:37:46 2011 +0100
+++ b/src/HOL/Library/Extended_Reals.thy Mon Mar 14 14:37:47 2011 +0100
@@ -1,13 +1,22 @@
-(* Title: Extended_Reals.thy
- Author: Johannes Hölzl, Robert Himmelmann, Armin Heller; TU München
+(* Title: src/HOL/Library/Extended_Reals.thy
+ Author: Johannes Hölzl; TU München
+ Author: Robert Himmelmann; TU München
+ Author: Armin Heller; TU München
Author: Bogdan Grechuk; University of Edinburgh *)
header {* Extended real number line *}
theory Extended_Reals
- imports Topology_Euclidean_Space
+ imports Complex_Main
begin
+text {*
+
+For more lemmas about the extended real numbers go to
+ @{text "src/HOL/Multivaraite_Analysis/Extended_Real_Limits.thy"}
+
+*}
+
lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
proof
assume "{x..} = UNIV"
@@ -20,11 +29,11 @@
lemma SUPR_pair:
"(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
- by (rule antisym) (auto intro!: SUP_leI le_SUPI_trans)
+ by (rule antisym) (auto intro!: SUP_leI le_SUPI2)
lemma INFI_pair:
"(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
- by (rule antisym) (auto intro!: le_INFI INF_leI_trans)
+ by (rule antisym) (auto intro!: le_INFI INF_leI2)
subsection {* Definition and basic properties *}
@@ -306,7 +315,7 @@
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))"
proof (cases x)
case (real r) then show ?thesis
- using real_arch_lt[of r] by simp
+ using reals_Archimedean2[of r] by simp
qed simp_all
lemma extreal_add_mono:
@@ -1371,9 +1380,9 @@
shows "(SUP i:A. f i) = (SUP j:B. g j)"
proof (intro antisym)
show "(SUP i:A. f i) \<le> (SUP j:B. g j)"
- using assms by (metis SUP_leI le_SUPI_trans)
+ using assms by (metis SUP_leI le_SUPI2)
show "(SUP i:B. g i) \<le> (SUP j:A. f j)"
- using assms by (metis SUP_leI le_SUPI_trans)
+ using assms by (metis SUP_leI le_SUPI2)
qed
lemma SUP_extreal_le_addI:
@@ -1662,7 +1671,7 @@
then show "open (Union K)" unfolding open_extreal_def
proof (intro conjI impI)
show "open (extreal -` \<Union>K)"
- using *[unfolded choice_iff] by (auto simp: vimage_Union)
+ using *[THEN choice] by (auto simp: vimage_Union)
qed ((metis UnionE Union_upper subset_trans *)+)
qed
end
@@ -1673,15 +1682,6 @@
lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)"
unfolding open_extreal_def by auto
-lemma continuous_on_extreal[intro, simp]: "continuous_on A extreal"
- unfolding continuous_on_topological open_extreal_def by auto
-
-lemma continuous_at_extreal[intro, simp]: "continuous (at x) extreal"
- using continuous_on_eq_continuous_at[of UNIV] by auto
-
-lemma continuous_within_extreal[intro, simp]: "x \<in> A \<Longrightarrow> continuous (at x within A) extreal"
- using continuous_on_eq_continuous_within[of A] by auto
-
lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}"
proof -
have "\<And>x. extreal -` {..<extreal x} = {..< x}"
@@ -1727,15 +1727,15 @@
obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
proof-
from `open S` have "open (extreal -` S)" by (rule extreal_openE)
- then obtain e where "0 < e" and e: "ball (real x) e \<subseteq> extreal -` S"
- using assms unfolding open_contains_ball by force
+ then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S"
+ using assms unfolding open_dist by force
show thesis
proof (intro that subsetI)
show "0 < extreal e" using `0 < e` by auto
fix y assume "y \<in> {x - extreal e<..<x + extreal e}"
- with assms obtain t where "y = extreal t" "t \<in> ball (real x) e"
- by (cases y) (auto simp: ball_def dist_real_def)
- then show "y \<in> S" using e by auto
+ with assms obtain t where "y = extreal t" "dist t (real x) < e"
+ apply (cases y) by (auto simp: dist_real_def)
+ then show "y \<in> S" using e[of t] by auto
qed
qed
@@ -1748,150 +1748,6 @@
show thesis by auto
qed
-lemma extreal_open_uminus:
- fixes S :: "extreal set"
- assumes "open S"
- shows "open (uminus ` S)"
- unfolding open_extreal_def
-proof (intro conjI impI)
- obtain x y where S: "open (extreal -` S)"
- "\<infinity> \<in> S \<Longrightarrow> {extreal x<..} \<subseteq> S" "-\<infinity> \<in> S \<Longrightarrow> {..< extreal y} \<subseteq> S"
- using `open S` unfolding open_extreal_def by auto
- have "extreal -` uminus ` S = uminus ` (extreal -` S)"
- proof safe
- fix x y assume "extreal x = - y" "y \<in> S"
- then show "x \<in> uminus ` extreal -` S" by (cases y) auto
- next
- fix x assume "extreal x \<in> S"
- then show "- x \<in> extreal -` uminus ` S"
- by (auto intro: image_eqI[of _ _ "extreal x"])
- qed
- then show "open (extreal -` uminus ` S)"
- using S by (auto intro: open_negations)
- { assume "\<infinity> \<in> uminus ` S"
- then have "-\<infinity> \<in> S" by (metis image_iff extreal_uminus_uminus)
- then have "uminus ` {..<extreal y} \<subseteq> uminus ` S" using S by (intro image_mono) auto
- then show "\<exists>x. {extreal x<..} \<subseteq> uminus ` S" using extreal_uminus_lessThan by auto }
- { assume "-\<infinity> \<in> uminus ` S"
- then have "\<infinity> : S" by (metis image_iff extreal_uminus_uminus)
- then have "uminus ` {extreal x<..} <= uminus ` S" using S by (intro image_mono) auto
- then show "\<exists>y. {..<extreal y} <= uminus ` S" using extreal_uminus_greaterThan by auto }
-qed
-
-lemma extreal_uminus_complement:
- fixes S :: "extreal set"
- shows "uminus ` (- S) = - uminus ` S"
- by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)
-
-lemma extreal_closed_uminus:
- fixes S :: "extreal set"
- assumes "closed S"
- shows "closed (uminus ` S)"
-using assms unfolding closed_def
-using extreal_open_uminus[of "- S"] extreal_uminus_complement by auto
-
-lemma not_open_extreal_singleton:
- "\<not> (open {a :: extreal})"
-proof(rule ccontr)
- assume "\<not> \<not> open {a}" hence a: "open {a}" by auto
- show False
- proof (cases a)
- case MInf
- then obtain y where "{..<extreal y} <= {a}" using a open_MInfty2[of "{a}"] by auto
- hence "extreal(y - 1):{a}" apply (subst subsetD[of "{..<extreal y}"]) by auto
- then show False using `a=(-\<infinity>)` by auto
- next
- case PInf
- then obtain y where "{extreal y<..} <= {a}" using a open_PInfty2[of "{a}"] by auto
- hence "extreal(y+1):{a}" apply (subst subsetD[of "{extreal y<..}"]) by auto
- then show False using `a=\<infinity>` by auto
- next
- case (real r) then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
- from extreal_open_cont_interval[OF a singletonI this] guess e . note e = this
- then obtain b where b_def: "a<b & b<a+e"
- using fin extreal_between extreal_dense[of a "a+e"] by auto
- then have "b: {a-e <..< a+e}" using fin extreal_between[of a e] e by auto
- then show False using b_def e by auto
- qed
-qed
-
-lemma extreal_closed_contains_Inf:
- fixes S :: "extreal set"
- assumes "closed S" "S ~= {}"
- shows "Inf S : S"
-proof(rule ccontr)
- assume "Inf S \<notin> S" hence a: "open (-S)" "Inf S:(- S)" using assms by auto
- show False
- proof (cases "Inf S")
- case MInf hence "(-\<infinity>) : - S" using a by auto
- then obtain y where "{..<extreal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
- hence "extreal y <= Inf S" by (metis Compl_anti_mono Compl_lessThan atLeast_iff
- complete_lattice_class.Inf_greatest double_complement set_rev_mp)
- then show False using MInf by auto
- next
- case PInf then have "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
- then show False by (metis `Inf S ~: S` insert_code mem_def PInf)
- next
- case (real r) then have fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>" by simp
- from extreal_open_cont_interval[OF a this] guess e . note e = this
- { fix x assume "x:S" hence "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
- hence *: "x>Inf S-e" using e by (metis fin extreal_between(1) order_less_le_trans)
- { assume "x<Inf S+e" hence "x:{Inf S-e <..< Inf S+e}" using * by auto
- hence False using e `x:S` by auto
- } hence "x>=Inf S+e" by (metis linorder_le_less_linear)
- } hence "Inf S + e <= Inf S" by (metis le_Inf_iff)
- then show False using real e by (cases e) auto
- qed
-qed
-
-lemma extreal_closed_contains_Sup:
- fixes S :: "extreal set"
- assumes "closed S" "S ~= {}"
- shows "Sup S : S"
-proof-
- have "closed (uminus ` S)" by (metis assms(1) extreal_closed_uminus)
- hence "Inf (uminus ` S) : uminus ` S" using assms extreal_closed_contains_Inf[of "uminus ` S"] by auto
- hence "- Sup S : uminus ` S" using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
- thus ?thesis by (metis imageI extreal_uminus_uminus extreal_minus_minus_image)
-qed
-
-lemma extreal_open_closed_aux:
- fixes S :: "extreal set"
- assumes "open S" "closed S"
- assumes S: "(-\<infinity>) ~: S"
- shows "S = {}"
-proof(rule ccontr)
- assume "S ~= {}"
- hence *: "(Inf S):S" by (metis assms(2) extreal_closed_contains_Inf)
- { assume "Inf S=(-\<infinity>)" hence False using * assms(3) by auto }
- moreover
- { assume "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
- hence False by (metis assms(1) not_open_extreal_singleton) }
- moreover
- { assume fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>"
- from extreal_open_cont_interval[OF assms(1) * fin] guess e . note e = this
- then obtain b where b_def: "Inf S-e<b & b<Inf S"
- using fin extreal_between[of "Inf S" e] extreal_dense[of "Inf S-e"] by auto
- hence "b: {Inf S-e <..< Inf S+e}" using e fin extreal_between[of "Inf S" e] by auto
- hence "b:S" using e by auto
- hence False using b_def by (metis complete_lattice_class.Inf_lower leD)
- } ultimately show False by auto
-qed
-
-
-lemma extreal_open_closed:
- fixes S :: "extreal set"
- shows "(open S & closed S) <-> (S = {} | S = UNIV)"
-proof-
-{ assume lhs: "open S & closed S"
- { assume "(-\<infinity>) ~: S" hence "S={}" using lhs extreal_open_closed_aux by auto }
- moreover
- { assume "(-\<infinity>) : S" hence "(- S)={}" using lhs extreal_open_closed_aux[of "-S"] by auto }
- ultimately have "S = {} | S = UNIV" by auto
-} thus ?thesis by auto
-qed
-
-
instance extreal :: t2_space
proof
fix x y :: extreal assume "x ~= y"
@@ -2055,7 +1911,7 @@
moreover
{ assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
hence "l=(-\<infinity>)" using assms
- Lim_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
+ tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
hence ?thesis by auto }
moreover
{ assume "EX B. C = extreal B"
@@ -2143,67 +1999,6 @@
"x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
-lemma extreal_open_affinity_pos:
- assumes "open S" and m: "m \<noteq> \<infinity>" "0 < m" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
- shows "open ((\<lambda>x. m * x + t) ` S)"
-proof -
- obtain r where r[simp]: "m = extreal r" using m by (cases m) auto
- obtain p where p[simp]: "t = extreal p" using t by auto
- have "r \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" using m by auto
- from `open S`[THEN extreal_openE] guess l u . note T = this
- let ?f = "(\<lambda>x. m * x + t)"
- show ?thesis unfolding open_extreal_def
- proof (intro conjI impI exI subsetI)
- have "extreal -` ?f ` S = (\<lambda>x. r * x + p) ` (extreal -` S)"
- proof safe
- fix x y assume "extreal y = m * x + t" "x \<in> S"
- then show "y \<in> (\<lambda>x. r * x + p) ` extreal -` S"
- using `r \<noteq> 0` by (cases x) (auto intro!: image_eqI[of _ _ "real x"] split: split_if_asm)
- qed force
- then show "open (extreal -` ?f ` S)"
- using open_affinity[OF T(1) `r \<noteq> 0`] by (auto simp: ac_simps)
- next
- assume "\<infinity> \<in> ?f`S" with `0 < r` have "\<infinity> \<in> S" by auto
- fix x assume "x \<in> {extreal (r * l + p)<..}"
- then have [simp]: "extreal (r * l + p) < x" by auto
- show "x \<in> ?f`S"
- proof (rule image_eqI)
- show "x = m * ((x - t) / m) + t"
- using m t by (cases rule: extreal3_cases[of m x t]) auto
- have "extreal l < (x - t)/m"
- using m t by (simp add: extreal_less_divide_pos extreal_less_minus)
- then show "(x - t)/m \<in> S" using T(2)[OF `\<infinity> \<in> S`] by auto
- qed
- next
- assume "-\<infinity> \<in> ?f`S" with `0 < r` have "-\<infinity> \<in> S" by auto
- fix x assume "x \<in> {..<extreal (r * u + p)}"
- then have [simp]: "x < extreal (r * u + p)" by auto
- show "x \<in> ?f`S"
- proof (rule image_eqI)
- show "x = m * ((x - t) / m) + t"
- using m t by (cases rule: extreal3_cases[of m x t]) auto
- have "(x - t)/m < extreal u"
- using m t by (simp add: extreal_divide_less_pos extreal_minus_less)
- then show "(x - t)/m \<in> S" using T(3)[OF `-\<infinity> \<in> S`] by auto
- qed
- qed
-qed
-
-lemma extreal_open_affinity:
- assumes "open S" and m: "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" and t: "\<bar>t\<bar> \<noteq> \<infinity>"
- shows "open ((\<lambda>x. m * x + t) ` S)"
-proof cases
- assume "0 < m" then show ?thesis
- using extreal_open_affinity_pos[OF `open S` _ _ t, of m] m by auto
-next
- assume "\<not> 0 < m" then
- have "0 < -m" using `m \<noteq> 0` by (cases m) auto
- then have m: "-m \<noteq> \<infinity>" "0 < -m" using `\<bar>m\<bar> \<noteq> \<infinity>`
- by (auto simp: extreal_uminus_eq_reorder)
- from extreal_open_affinity_pos[OF extreal_open_uminus[OF `open S`] m t]
- show ?thesis unfolding image_image by simp
-qed
-
lemma extreal_divide_eq:
"b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
by (cases rule: extreal3_cases[of a b c])
@@ -2212,54 +2007,9 @@
lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
by (cases a) auto
-lemma extreal_lim_mult:
- fixes X :: "'a \<Rightarrow> extreal"
- assumes lim: "(X ---> L) net" and a: "\<bar>a\<bar> \<noteq> \<infinity>"
- shows "((\<lambda>i. a * X i) ---> a * L) net"
-proof cases
- assume "a \<noteq> 0"
- show ?thesis
- proof (rule topological_tendstoI)
- fix S assume "open S" "a * L \<in> S"
- have "a * L / a = L"
- using `a \<noteq> 0` a by (cases rule: extreal2_cases[of a L]) auto
- then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
- using `a * L \<in> S` by (force simp: image_iff)
- moreover have "open ((\<lambda>x. x / a) ` S)"
- using extreal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 0` a
- by (auto simp: extreal_divide_eq extreal_inverse_eq_0 divide_extreal_def ac_simps)
- note * = lim[THEN topological_tendstoD, OF this L]
- { fix x from a `a \<noteq> 0` have "a * (x / a) = x"
- by (cases rule: extreal2_cases[of a x]) auto }
- note this[simp]
- show "eventually (\<lambda>x. a * X x \<in> S) net"
- by (rule eventually_mono[OF _ *]) auto
- qed
-qed auto
-
lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
by (cases x) auto
-lemma extreal_lim_uminus:
- fixes X :: "'a \<Rightarrow> extreal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
- using extreal_lim_mult[of X L net "extreal (-1)"]
- extreal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "extreal (-1)"]
- by (auto simp add: algebra_simps)
-
-lemma Lim_bounded2_extreal:
- assumes lim:"f ----> (l :: extreal)"
- and ge: "ALL n>=N. f n >= C"
- shows "l>=C"
-proof-
-def g == "(%i. -(f i))"
-{ fix n assume "n>=N" hence "g n <= -C" using assms extreal_minus_le_minus g_def by auto }
-hence "ALL n>=N. g n <= -C" by auto
-moreover have limg: "g ----> (-l)" using g_def extreal_lim_uminus lim by auto
-ultimately have "-l <= -C" using Lim_bounded_extreal[of g "-l" _ "-C"] by auto
-from this show ?thesis using extreal_minus_le_minus by auto
-qed
-
-
lemma extreal_LimI_finite:
assumes "\<bar>x\<bar> \<noteq> \<infinity>"
assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
@@ -2409,108 +2159,6 @@
qed
qed auto
-
-lemma extreal_open_atLeast: "open {x..} \<longleftrightarrow> x = -\<infinity>"
-proof
- assume "x = -\<infinity>" then have "{x..} = UNIV" by auto
- then show "open {x..}" by auto
-next
- assume "open {x..}"
- then have "open {x..} \<and> closed {x..}" by auto
- then have "{x..} = UNIV" unfolding extreal_open_closed by auto
- then show "x = -\<infinity>" by (simp add: bot_extreal_def atLeast_eq_UNIV_iff)
-qed
-
-lemma extreal_open_mono_set:
- fixes S :: "extreal set"
- defines "a \<equiv> Inf S"
- shows "(open S \<and> mono S) \<longleftrightarrow> (S = UNIV \<or> S = {a <..})"
- by (metis Inf_UNIV a_def atLeast_eq_UNIV_iff extreal_open_atLeast
- extreal_open_closed mono_set_iff open_extreal_greaterThan)
-
-lemma extreal_closed_mono_set:
- fixes S :: "extreal set"
- shows "(closed S \<and> mono S) \<longleftrightarrow> (S = {} \<or> S = {Inf S ..})"
- by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_extreal_atLeast
- extreal_open_closed mono_empty mono_set_iff open_extreal_greaterThan)
-
-lemma extreal_Liminf_Sup_monoset:
- fixes f :: "'a => extreal"
- shows "Liminf net f = Sup {l. \<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
- unfolding Liminf_Sup
-proof (intro arg_cong[where f="\<lambda>P. Sup (Collect P)"] ext iffI allI impI)
- fix l S assume ev: "\<forall>y<l. eventually (\<lambda>x. y < f x) net" and "open S" "mono S" "l \<in> S"
- then have "S = UNIV \<or> S = {Inf S <..}"
- using extreal_open_mono_set[of S] by auto
- then show "eventually (\<lambda>x. f x \<in> S) net"
- proof
- assume S: "S = {Inf S<..}"
- then have "Inf S < l" using `l \<in> S` by auto
- then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
- then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
- qed auto
-next
- fix l y assume S: "\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net" "y < l"
- have "eventually (\<lambda>x. f x \<in> {y <..}) net"
- using `y < l` by (intro S[rule_format]) auto
- then show "eventually (\<lambda>x. y < f x) net" by auto
-qed
-
-lemma extreal_Limsup_Inf_monoset:
- fixes f :: "'a => extreal"
- shows "Limsup net f = Inf {l. \<forall>S. open S \<longrightarrow> mono (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
- unfolding Limsup_Inf
-proof (intro arg_cong[where f="\<lambda>P. Inf (Collect P)"] ext iffI allI impI)
- fix l S assume ev: "\<forall>y>l. eventually (\<lambda>x. f x < y) net" and "open S" "mono (uminus`S)" "l \<in> S"
- then have "open (uminus`S) \<and> mono (uminus`S)" by (simp add: extreal_open_uminus)
- then have "S = UNIV \<or> S = {..< Sup S}"
- unfolding extreal_open_mono_set extreal_Inf_uminus_image_eq extreal_image_uminus_shift by simp
- then show "eventually (\<lambda>x. f x \<in> S) net"
- proof
- assume S: "S = {..< Sup S}"
- then have "l < Sup S" using `l \<in> S` by auto
- then have "eventually (\<lambda>x. f x < Sup S) net" using ev by auto
- then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
- qed auto
-next
- fix l y assume S: "\<forall>S. open S \<longrightarrow> mono (uminus`S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net" "l < y"
- have "eventually (\<lambda>x. f x \<in> {..< y}) net"
- using `l < y` by (intro S[rule_format]) auto
- then show "eventually (\<lambda>x. f x < y) net" by auto
-qed
-
-lemma open_uminus_iff: "open (uminus ` S) \<longleftrightarrow> open (S::extreal set)"
- using extreal_open_uminus[of S] extreal_open_uminus[of "uminus`S"] by auto
-
-lemma extreal_Limsup_uminus:
- fixes f :: "'a => extreal"
- shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
-proof -
- { fix P l have "(\<exists>x. (l::extreal) = -x \<and> P x) \<longleftrightarrow> P (-l)" by (auto intro!: exI[of _ "-l"]) }
- note Ex_cancel = this
- { fix P :: "extreal set \<Rightarrow> bool" have "(\<forall>S. P S) \<longleftrightarrow> (\<forall>S. P (uminus`S))"
- apply auto by (erule_tac x="uminus`S" in allE) (auto simp: image_image) }
- note add_uminus_image = this
- { fix x S have "(x::extreal) \<in> uminus`S \<longleftrightarrow> -x\<in>S" by (auto intro!: image_eqI[of _ _ "-x"]) }
- note remove_uminus_image = this
- show ?thesis
- unfolding extreal_Limsup_Inf_monoset extreal_Liminf_Sup_monoset
- unfolding extreal_Inf_uminus_image_eq[symmetric] image_Collect Ex_cancel
- by (subst add_uminus_image) (simp add: open_uminus_iff remove_uminus_image)
-qed
-
-lemma extreal_Liminf_uminus:
- fixes f :: "'a => extreal"
- shows "Liminf net (\<lambda>x. - (f x)) = -(Limsup net f)"
- using extreal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
-
-lemma extreal_Lim_uminus:
- fixes f :: "'a \<Rightarrow> extreal" shows "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x) ---> - f0) net"
- using
- extreal_lim_mult[of f f0 net "- 1"]
- extreal_lim_mult[of "\<lambda>x. - (f x)" "-f0" net "- 1"]
- by (auto simp: extreal_uminus_reorder)
-
lemma lim_imp_Liminf:
fixes f :: "'a \<Rightarrow> extreal"
assumes ntriv: "\<not> trivial_limit net"
@@ -2539,14 +2187,6 @@
qed
qed
-lemma lim_imp_Limsup:
- fixes f :: "'a => extreal"
- assumes "\<not> trivial_limit net"
- assumes lim: "(f ---> f0) net"
- shows "Limsup net f = f0"
- using extreal_Lim_uminus[of f f0] lim_imp_Liminf[of net "(%x. -(f x))" "-f0"]
- extreal_Liminf_uminus[of net f] assms by simp
-
lemma extreal_Liminf_le_Limsup:
fixes f :: "'a \<Rightarrow> extreal"
assumes ntriv: "\<not> trivial_limit net"
@@ -2566,59 +2206,6 @@
qed
qed
-lemma Liminf_PInfty:
- fixes f :: "'a \<Rightarrow> extreal"
- assumes "\<not> trivial_limit net"
- shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
-proof (intro lim_imp_Liminf iffI assms)
- assume rhs: "Liminf net f = \<infinity>"
- { fix S assume "open S & \<infinity> : S"
- then obtain m where "{extreal m<..} <= S" using open_PInfty2 by auto
- moreover have "eventually (\<lambda>x. f x \<in> {extreal m<..}) net"
- using rhs unfolding Liminf_Sup top_extreal_def[symmetric] Sup_eq_top_iff
- by (auto elim!: allE[where x="extreal m"] simp: top_extreal_def)
- ultimately have "eventually (%x. f x : S) net" apply (subst eventually_mono) by auto
- } then show "(f ---> \<infinity>) net" unfolding tendsto_def by auto
-qed
-
-lemma Limsup_MInfty:
- fixes f :: "'a \<Rightarrow> extreal"
- assumes "\<not> trivial_limit net"
- shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
- using assms extreal_Lim_uminus[of f "-\<infinity>"] Liminf_PInfty[of _ "\<lambda>x. - (f x)"]
- extreal_Liminf_uminus[of _ f] by (auto simp: extreal_uminus_eq_reorder)
-
-lemma extreal_Liminf_eq_Limsup:
- fixes f :: "'a \<Rightarrow> extreal"
- assumes ntriv: "\<not> trivial_limit net"
- assumes lim: "Liminf net f = f0" "Limsup net f = f0"
- shows "(f ---> f0) net"
-proof (cases f0)
- case PInf then show ?thesis using Liminf_PInfty[OF ntriv] lim by auto
-next
- case MInf then show ?thesis using Limsup_MInfty[OF ntriv] lim by auto
-next
- case (real r)
- show "(f ---> f0) net"
- proof (rule topological_tendstoI)
- fix S assume "open S""f0 \<in> S"
- then obtain a b where "a < Liminf net f" "Limsup net f < b" "{a<..<b} \<subseteq> S"
- using extreal_open_cont_interval2[of S f0] real lim by auto
- then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
- unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
- by (auto intro!: eventually_conj simp add: greaterThanLessThan_iff)
- with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
- by (rule_tac eventually_mono) auto
- qed
-qed
-
-lemma extreal_Liminf_eq_Limsup_iff:
- fixes f :: "'a \<Rightarrow> extreal"
- assumes "\<not> trivial_limit net"
- shows "(f ---> f0) net \<longleftrightarrow> Liminf net f = f0 \<and> Limsup net f = f0"
- by (metis assms extreal_Liminf_eq_Limsup lim_imp_Liminf lim_imp_Limsup)
-
-
lemma Liminf_mono:
fixes f g :: "'a => extreal"
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
@@ -2700,22 +2287,6 @@
qed (rule order_refl)
qed
-lemma limsup_INFI_SUPR:
- fixes f :: "nat \<Rightarrow> extreal"
- shows "limsup f = (INF n. SUP m:{n..}. f m)"
- using extreal_Limsup_uminus[of sequentially "\<lambda>x. - f x"]
- by (simp add: liminf_SUPR_INFI extreal_INFI_uminus extreal_SUPR_uminus)
-
-lemma liminf_PInfty:
- fixes X :: "nat => extreal"
- shows "X ----> \<infinity> <-> liminf X = \<infinity>"
-by (metis Liminf_PInfty trivial_limit_sequentially)
-
-lemma limsup_MInfty:
- fixes X :: "nat => extreal"
- shows "X ----> (-\<infinity>) <-> limsup X = (-\<infinity>)"
-by (metis Limsup_MInfty trivial_limit_sequentially)
-
lemma tail_same_limsup:
fixes X Y :: "nat => extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
@@ -2744,29 +2315,10 @@
lemma
fixes X :: "nat \<Rightarrow> extreal"
- shows incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
- and decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
+ shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
+ and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
unfolding incseq_def decseq_def by auto
-lemma extreal_lim_mono:
- fixes X Y :: "nat => extreal"
- assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
- assumes "X ----> x" "Y ----> y"
- shows "x <= y"
- by (metis extreal_Liminf_eq_Limsup_iff[OF trivial_limit_sequentially] assms liminf_mono)
-
-lemma incseq_le_extreal:
- fixes X :: "nat \<Rightarrow> extreal"
- assumes inc: "incseq X" and lim: "X ----> L"
- shows "X N \<le> L"
- using inc
- by (intro extreal_lim_mono[of N, OF _ Lim_const lim]) (simp add: incseq_def)
-
-lemma decseq_ge_extreal: assumes dec: "decseq X"
- and lim: "X ----> (L::extreal)" shows "X N >= L"
- using dec
- by (intro extreal_lim_mono[of N, OF _ lim Lim_const]) (simp add: decseq_def)
-
lemma liminf_bounded:
fixes X Y :: "nat \<Rightarrow> extreal"
assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
@@ -2798,27 +2350,6 @@
} then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
qed
-lemma liminf_bounded_open:
- fixes x :: "nat \<Rightarrow> extreal"
- shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))"
- (is "_ \<longleftrightarrow> ?P x0")
-proof
- assume "?P x0" then show "x0 \<le> liminf x"
- unfolding extreal_Liminf_Sup_monoset eventually_sequentially
- by (intro complete_lattice_class.Sup_upper) auto
-next
- assume "x0 \<le> liminf x"
- { fix S :: "extreal set" assume om: "open S & mono S & x0:S"
- { assume "S = UNIV" hence "EX N. (ALL n>=N. x n : S)" by auto }
- moreover
- { assume "~(S=UNIV)"
- then obtain B where B_def: "S = {B<..}" using om extreal_open_mono_set by auto
- hence "B<x0" using om by auto
- hence "EX N. ALL n>=N. x n : S" unfolding B_def using `x0 \<le> liminf x` liminf_bounded_iff by auto
- } ultimately have "EX N. (ALL n>=N. x n : S)" by auto
- } then show "?P x0" by auto
-qed
-
lemma liminf_subseq_mono:
fixes X :: "nat \<Rightarrow> extreal"
assumes "subseq r"
@@ -2832,116 +2363,9 @@
then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
qed
-lemma limsup_subseq_mono:
- fixes X :: "nat \<Rightarrow> extreal"
- assumes "subseq r"
- shows "limsup (X \<circ> r) \<le> limsup X"
-proof-
- have "(\<lambda>n. - X n) \<circ> r = (\<lambda>n. - (X \<circ> r) n)" by (simp add: fun_eq_iff)
- then have "- limsup X \<le> - limsup (X \<circ> r)"
- using liminf_subseq_mono[of r "(%n. - X n)"]
- extreal_Liminf_uminus[of sequentially X]
- extreal_Liminf_uminus[of sequentially "X o r"] assms by auto
- then show ?thesis by auto
-qed
-
-lemma bounded_abs:
- assumes "(a::real)<=x" "x<=b"
- shows "abs x <= max (abs a) (abs b)"
-by (metis abs_less_iff assms leI le_max_iff_disj less_eq_real_def less_le_not_le less_minus_iff minus_minus)
-
-
-lemma bounded_increasing_convergent2: fixes f::"nat => real"
- assumes "ALL n. f n <= B" "ALL n m. n>=m --> f n >= f m"
- shows "EX l. (f ---> l) sequentially"
-proof-
-def N == "max (abs (f 0)) (abs B)"
-{ fix n have "abs (f n) <= N" unfolding N_def apply (subst bounded_abs) using assms by auto }
-hence "bounded {f n| n::nat. True}" unfolding bounded_real by auto
-from this show ?thesis apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
- using assms by auto
-qed
-
lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x"
using assms by auto
-lemma lim_extreal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
- obtains l where "f ----> (l::extreal)"
-proof(cases "f = (\<lambda>x. - \<infinity>)")
- case True then show thesis using Lim_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
-next
- case False
- from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
- have "ALL n>=N. f n >= f N" using assms by auto
- hence minf: "ALL n>=N. f n > (-\<infinity>)" using N_def by auto
- def Y == "(%n. (if n>=N then f n else f N))"
- hence incy: "!!n m. n>=m ==> Y n >= Y m" using assms by auto
- from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
- show thesis
- proof(cases "EX B. ALL n. f n < extreal B")
- case False thus thesis apply- apply(rule that[of \<infinity>]) unfolding Lim_PInfty not_ex not_all
- apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
- apply(rule order_trans[OF _ assms[rule_format]]) by auto
- next case True then guess B ..
- hence "ALL n. Y n < extreal B" using Y_def by auto note B = this[rule_format]
- { fix n have "Y n < \<infinity>" using B[of n] apply (subst less_le_trans) by auto
- hence "Y n ~= \<infinity> & Y n ~= (-\<infinity>)" using minfy by auto
- } hence *: "ALL n. \<bar>Y n\<bar> \<noteq> \<infinity>" by auto
- { fix n have "real (Y n) < B" proof- case goal1 thus ?case
- using B[of n] apply-apply(subst(asm) extreal_real'[THEN sym]) defer defer
- unfolding extreal_less using * by auto
- qed
- }
- hence B': "ALL n. (real (Y n) <= B)" using less_imp_le by auto
- have "EX l. (%n. real (Y n)) ----> l"
- apply(rule bounded_increasing_convergent2)
- proof safe show "!!n. real (Y n) <= B" using B' by auto
- fix n m::nat assume "n<=m"
- hence "extreal (real (Y n)) <= extreal (real (Y m))"
- using incy[rule_format,of n m] apply(subst extreal_real)+
- using *[rule_format, of n] *[rule_format, of m] by auto
- thus "real (Y n) <= real (Y m)" by auto
- qed then guess l .. note l=this
- have "Y ----> extreal l" using l apply-apply(subst(asm) lim_extreal[THEN sym])
- unfolding extreal_real using * by auto
- thus thesis apply-apply(rule that[of "extreal l"])
- apply (subst tail_same_limit[of Y _ N]) using Y_def by auto
- qed
-qed
-
-lemma lim_extreal_decreasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n <= f m"
- obtains l where "f ----> (l::extreal)"
-proof -
- from lim_extreal_increasing[of "\<lambda>x. - f x"] assms
- obtain l where "(\<lambda>x. - f x) ----> l" by auto
- from extreal_lim_mult[OF this, of "- 1"] show thesis
- by (intro that[of "-l"]) (simp add: extreal_uminus_eq_reorder)
-qed
-
-lemma compact_extreal:
- fixes X :: "nat \<Rightarrow> extreal"
- shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
-proof -
- obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
- using seq_monosub[of X] unfolding comp_def by auto
- then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
- by (auto simp add: monoseq_def)
- then obtain l where "(X\<circ>r) ----> l"
- using lim_extreal_increasing[of "X \<circ> r"] lim_extreal_decreasing[of "X \<circ> r"] by auto
- then show ?thesis using `subseq r` by auto
-qed
-
-lemma extreal_Sup_lim:
- assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
- shows "a \<le> Sup s"
-by (metis Lim_bounded_extreal assms complete_lattice_class.Sup_upper)
-
-lemma extreal_Inf_lim:
- assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
- shows "Inf s \<le> a"
-by (metis Lim_bounded2_extreal assms complete_lattice_class.Inf_lower)
-
-
lemma extreal_le_extreal_bounded:
fixes x y z :: extreal
assumes "z \<le> y"
@@ -2975,289 +2399,6 @@
with z[THEN bspec, of w] show "x \<le> z" by auto
qed auto
-lemma SUP_Lim_extreal:
- fixes X :: "nat \<Rightarrow> extreal" assumes "incseq X" "X ----> l" shows "(SUP n. X n) = l"
-proof (rule extreal_SUPI)
- fix n from assms show "X n \<le> l"
- by (intro incseq_le_extreal) (simp add: incseq_def)
-next
- fix y assume "\<And>n. n \<in> UNIV \<Longrightarrow> X n \<le> y"
- with extreal_Sup_lim[OF _ `X ----> l`, of "{..y}"]
- show "l \<le> y" by auto
-qed
-
-lemma LIMSEQ_extreal_SUPR:
- fixes X :: "nat \<Rightarrow> extreal" assumes "incseq X" shows "X ----> (SUP n. X n)"
-proof (rule lim_extreal_increasing)
- fix n m :: nat assume "m \<le> n" then show "X m \<le> X n"
- using `incseq X` by (simp add: incseq_def)
-next
- fix l assume "X ----> l"
- with SUP_Lim_extreal[of X, OF assms this] show ?thesis by simp
-qed
-
-lemma INF_Lim_extreal: "decseq X \<Longrightarrow> X ----> l \<Longrightarrow> (INF n. X n) = (l::extreal)"
- using SUP_Lim_extreal[of "\<lambda>i. - X i" "- l"]
- by (simp add: extreal_SUPR_uminus extreal_lim_uminus)
-
-lemma LIMSEQ_extreal_INFI: "decseq X \<Longrightarrow> X ----> (INF n. X n :: extreal)"
- using LIMSEQ_extreal_SUPR[of "\<lambda>i. - X i"]
- by (simp add: extreal_SUPR_uminus extreal_lim_uminus)
-
-lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
- unfolding mono_def incseq_def by auto
-
-lemma SUP_eq_LIMSEQ:
- assumes "mono f"
- shows "(SUP n. extreal (f n)) = extreal x \<longleftrightarrow> f ----> x"
-proof
- have inc: "incseq (\<lambda>i. extreal (f i))"
- using `mono f` unfolding mono_def incseq_def by auto
- { assume "f ----> x"
- then have "(\<lambda>i. extreal (f i)) ----> extreal x" by auto
- from SUP_Lim_extreal[OF inc this]
- show "(SUP n. extreal (f n)) = extreal x" . }
- { assume "(SUP n. extreal (f n)) = extreal x"
- with LIMSEQ_extreal_SUPR[OF inc]
- show "f ----> x" by auto }
-qed
-
-lemma Liminf_within:
- fixes f :: "'a::metric_space => extreal"
- shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
-proof-
-let ?l="(SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
-{ fix T assume T_def: "open T & mono T & ?l:T"
- have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
- proof-
- { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
- moreover
- { assume "~(T=UNIV)"
- then obtain B where "T={B<..}" using T_def extreal_open_mono_set[of T] by auto
- hence "B<?l" using T_def by auto
- then obtain d where d_def: "0<d & B<(INF y:(S Int ball x d - {x}). f y)"
- unfolding less_SUP_iff by auto
- { fix y assume "y:S & 0 < dist y x & dist y x < d"
- hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
- hence "f y:T" using d_def INF_leI[of y "S Int ball x d - {x}" f] `T={B<..}` by auto
- } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
- } ultimately show ?thesis by auto
- qed
-}
-moreover
-{ fix z
- assume a: "ALL T. open T --> mono T --> z : T -->
- (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
- { fix B assume "B<z"
- then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> B < f y)"
- using a[rule_format, of "{B<..}"] mono_greaterThan by auto
- { fix y assume "y:(S Int ball x d - {x})"
- hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
- by (metis dist_eq_0_iff real_less_def zero_le_dist)
- hence "B <= f y" using d_def by auto
- } hence "B <= INFI (S Int ball x d - {x}) f" apply (subst le_INFI) by auto
- also have "...<=?l" apply (subst le_SUPI) using d_def by auto
- finally have "B<=?l" by auto
- } hence "z <= ?l" using extreal_le_extreal[of z "?l"] by auto
-}
-ultimately show ?thesis unfolding extreal_Liminf_Sup_monoset eventually_within
- apply (subst extreal_SupI[of _ "(SUP e:{0<..}. INFI (S Int ball x e - {x}) f)"]) by auto
-qed
-
-lemma Limsup_within:
- fixes f :: "'a::metric_space => extreal"
- shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
-proof-
-let ?l="(INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
-{ fix T assume T_def: "open T & mono (uminus ` T) & ?l:T"
- have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
- proof-
- { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
- moreover
- { assume "~(T=UNIV)" hence "~(uminus ` T = UNIV)"
- by (metis Int_UNIV_right Int_absorb1 image_mono extreal_minus_minus_image subset_UNIV)
- hence "uminus ` T = {Inf (uminus ` T)<..}" using T_def extreal_open_mono_set[of "uminus ` T"]
- extreal_open_uminus[of T] by auto
- then obtain B where "T={..<B}"
- unfolding extreal_Inf_uminus_image_eq extreal_uminus_lessThan[symmetric]
- unfolding inj_image_eq_iff[OF extreal_inj_on_uminus] by simp
- hence "?l<B" using T_def by auto
- then obtain d where d_def: "0<d & (SUP y:(S Int ball x d - {x}). f y)<B"
- unfolding INF_less_iff by auto
- { fix y assume "y:S & 0 < dist y x & dist y x < d"
- hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
- hence "f y:T" using d_def le_SUPI[of y "S Int ball x d - {x}" f] `T={..<B}` by auto
- } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
- } ultimately show ?thesis by auto
- qed
-}
-moreover
-{ fix z
- assume a: "ALL T. open T --> mono (uminus ` T) --> z : T -->
- (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
- { fix B assume "z<B"
- then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> f y<B)"
- using a[rule_format, of "{..<B}"] by auto
- { fix y assume "y:(S Int ball x d - {x})"
- hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
- by (metis dist_eq_0_iff real_less_def zero_le_dist)
- hence "f y <= B" using d_def by auto
- } hence "SUPR (S Int ball x d - {x}) f <= B" apply (subst SUP_leI) by auto
- moreover have "?l<=SUPR (S Int ball x d - {x}) f" apply (subst INF_leI) using d_def by auto
- ultimately have "?l<=B" by auto
- } hence "?l <= z" using extreal_ge_extreal[of z "?l"] by auto
-}
-ultimately show ?thesis unfolding extreal_Limsup_Inf_monoset eventually_within
- apply (subst extreal_InfI) by auto
-qed
-
-
-lemma Liminf_within_UNIV:
- fixes f :: "'a::metric_space => extreal"
- shows "Liminf (at x) f = Liminf (at x within UNIV) f"
-by (metis within_UNIV)
-
-
-lemma Liminf_at:
- fixes f :: "'a::metric_space => extreal"
- shows "Liminf (at x) f = (SUP e:{0<..}. INF y:(ball x e - {x}). f y)"
-using Liminf_within[of x UNIV f] Liminf_within_UNIV[of x f] by auto
-
-
-lemma Limsup_within_UNIV:
- fixes f :: "'a::metric_space => extreal"
- shows "Limsup (at x) f = Limsup (at x within UNIV) f"
-by (metis within_UNIV)
-
-
-lemma Limsup_at:
- fixes f :: "'a::metric_space => extreal"
- shows "Limsup (at x) f = (INF e:{0<..}. SUP y:(ball x e - {x}). f y)"
-using Limsup_within[of x UNIV f] Limsup_within_UNIV[of x f] by auto
-
-lemma Lim_within_constant:
- fixes f :: "'a::metric_space => 'b::topological_space"
- assumes "ALL y:S. f y = C"
- shows "(f ---> C) (at x within S)"
-unfolding tendsto_def eventually_within
-by (metis assms(1) linorder_le_less_linear n_not_Suc_n real_of_nat_le_zero_cancel_iff)
-
-lemma Liminf_within_constant:
- fixes f :: "'a::metric_space => extreal"
- assumes "ALL y:S. f y = C"
- assumes "~trivial_limit (at x within S)"
- shows "Liminf (at x within S) f = C"
-by (metis Lim_within_constant assms lim_imp_Liminf)
-
-lemma Limsup_within_constant:
- fixes f :: "'a::metric_space => extreal"
- assumes "ALL y:S. f y = C"
- assumes "~trivial_limit (at x within S)"
- shows "Limsup (at x within S) f = C"
-by (metis Lim_within_constant assms lim_imp_Limsup)
-
-lemma islimpt_punctured:
-"x islimpt S = x islimpt (S-{x})"
-unfolding islimpt_def by blast
-
-
-lemma islimpt_in_closure:
-"(x islimpt S) = (x:closure(S-{x}))"
-unfolding closure_def using islimpt_punctured by blast
-
-
-lemma not_trivial_limit_within:
- "~trivial_limit (at x within S) = (x:closure(S-{x}))"
-using islimpt_in_closure by (metis trivial_limit_within)
-
-
-lemma not_trivial_limit_within_ball:
- "(~trivial_limit (at x within S)) = (ALL e>0. S Int ball x e - {x} ~= {})"
- (is "?lhs = ?rhs")
-proof-
-{ assume "?lhs"
- { fix e :: real assume "e>0"
- then obtain y where "y:(S-{x}) & dist y x < e"
- using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
- hence "y : (S Int ball x e - {x})" unfolding ball_def by (simp add: dist_commute)
- hence "S Int ball x e - {x} ~= {}" by blast
- } hence "?rhs" by auto
-}
-moreover
-{ assume "?rhs"
- { fix e :: real assume "e>0"
- then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
- hence "y:(S-{x}) & dist y x < e" unfolding ball_def by (simp add: dist_commute)
- hence "EX y:(S-{x}). dist y x < e" by auto
- } hence "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
-} ultimately show ?thesis by auto
-qed
-
-
-lemma liminf_extreal_cminus:
- fixes f :: "nat \<Rightarrow> extreal" assumes "c \<noteq> -\<infinity>"
- shows "liminf (\<lambda>x. c - f x) = c - limsup f"
-proof (cases c)
- case PInf then show ?thesis by (simp add: Liminf_const)
-next
- case (real r) then show ?thesis
- unfolding liminf_SUPR_INFI limsup_INFI_SUPR
- apply (subst INFI_extreal_cminus)
- apply auto
- apply (subst SUPR_extreal_cminus)
- apply auto
- done
-qed (insert `c \<noteq> -\<infinity>`, simp)
-
-subsubsection {* Continuity *}
-
-lemma continuous_imp_tendsto:
- assumes "continuous (at x0) f"
- assumes "x ----> x0"
- shows "(f o x) ----> (f x0)"
-proof-
-{ fix S assume "open S & (f x0):S"
- from this obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
- using assms continuous_at_open by metis
- hence "(EX N. ALL n>=N. x n : T)" using assms tendsto_explicit T_def by auto
- hence "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
-} from this show ?thesis using tendsto_explicit[of "f o x" "f x0"] by auto
-qed
-
-
-lemma continuous_at_sequentially2:
-fixes f :: "'a::metric_space => 'b:: topological_space"
-shows "continuous (at x0) f <-> (ALL x. (x ----> x0) --> (f o x) ----> (f x0))"
-proof-
-{ assume "~(continuous (at x0) f)"
- from this obtain T where T_def:
- "open T & f x0 : T & (ALL S. (open S & x0 : S) --> (EX x':S. f x' ~: T))"
- using continuous_at_open[of x0 f] by metis
- def X == "{x'. f x' ~: T}" hence "x0 islimpt X" unfolding islimpt_def using T_def by auto
- from this obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
- using islimpt_sequential[of x0 X] by auto
- hence "~(f o x) ----> (f x0)" unfolding tendsto_explicit using X_def T_def by auto
- hence "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
-}
-from this show ?thesis using continuous_imp_tendsto by auto
-qed
-
-lemma continuous_at_of_extreal:
- fixes x0 :: extreal
- assumes "\<bar>x0\<bar> \<noteq> \<infinity>"
- shows "continuous (at x0) real"
-proof-
-{ fix T assume T_def: "open T & real x0 : T"
- def S == "extreal ` T"
- hence "extreal (real x0) : S" using T_def by auto
- hence "x0 : S" using assms extreal_real by auto
- moreover have "open S" using open_extreal S_def T_def by auto
- moreover have "ALL y:S. real y : T" using S_def T_def by auto
- ultimately have "EX S. x0 : S & open S & (ALL y:S. real y : T)" by auto
-} from this show ?thesis unfolding continuous_at_open by blast
-qed
-
-
lemma real_extreal_id: "real o extreal = id"
proof-
{ fix x have "(real o extreal) x = id x" by auto }
@@ -3265,133 +2406,9 @@
qed
-lemma continuous_at_iff_extreal:
-fixes f :: "'a::t2_space => real"
-shows "continuous (at x0) f <-> continuous (at x0) (extreal o f)"
-proof-
-{ assume "continuous (at x0) f" hence "continuous (at x0) (extreal o f)"
- using continuous_at_extreal continuous_at_compose[of x0 f extreal] by auto
-}
-moreover
-{ assume "continuous (at x0) (extreal o f)"
- hence "continuous (at x0) (real o (extreal o f))"
- using continuous_at_of_extreal by (intro continuous_at_compose[of x0 "extreal o f"]) auto
- moreover have "real o (extreal o f) = f" using real_extreal_id by (simp add: o_assoc)
- ultimately have "continuous (at x0) f" by auto
-} ultimately show ?thesis by auto
-qed
-
-
-lemma continuous_on_iff_extreal:
-fixes f :: "'a::t2_space => real"
-fixes A assumes "open A"
-shows "continuous_on A f <-> continuous_on A (extreal o f)"
- using continuous_at_iff_extreal assms by (auto simp add: continuous_on_eq_continuous_at)
-
-
lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
by (metis range_extreal open_extreal open_UNIV)
-lemma continuous_on_real: "continuous_on (UNIV-{\<infinity>,(-\<infinity>)}) real"
- using continuous_at_of_extreal continuous_on_eq_continuous_at open_image_extreal by auto
-
-
-lemma continuous_on_iff_real:
- fixes f :: "'a::t2_space => extreal"
- assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
- shows "continuous_on A f \<longleftrightarrow> continuous_on A (real \<circ> f)"
-proof-
- have "f ` A <= UNIV-{\<infinity>,(-\<infinity>)}" using assms by force
- hence *: "continuous_on (f ` A) real"
- using continuous_on_real by (simp add: continuous_on_subset)
-have **: "continuous_on ((real o f) ` A) extreal"
- using continuous_on_extreal continuous_on_subset[of "UNIV" "extreal" "(real o f) ` A"] by blast
-{ assume "continuous_on A f" hence "continuous_on A (real o f)"
- apply (subst continuous_on_compose) using * by auto
-}
-moreover
-{ assume "continuous_on A (real o f)"
- hence "continuous_on A (extreal o (real o f))"
- apply (subst continuous_on_compose) using ** by auto
- hence "continuous_on A f"
- apply (subst continuous_on_eq[of A "extreal o (real o f)" f])
- using assms extreal_real by auto
-}
-ultimately show ?thesis by auto
-qed
-
-
-lemma continuous_at_const:
- fixes f :: "'a::t2_space => extreal"
- assumes "ALL x. (f x = C)"
- shows "ALL x. continuous (at x) f"
-unfolding continuous_at_open using assms t1_space by auto
-
-
-lemma closure_contains_Inf:
- fixes S :: "real set"
- assumes "S ~= {}" "EX B. ALL x:S. B<=x"
- shows "Inf S : closure S"
-proof-
-have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] assms by metis
-{ fix e assume "e>(0 :: real)"
- from this obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
- moreover hence "x > Inf S - e" using * by auto
- ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
- hence "EX x:S. abs (x - Inf S) < e" using x_def by auto
-} from this show ?thesis apply (subst closure_approachable) unfolding dist_norm by auto
-qed
-
-
-lemma closed_contains_Inf:
- fixes S :: "real set"
- assumes "S ~= {}" "EX B. ALL x:S. B<=x"
- assumes "closed S"
- shows "Inf S : S"
-by (metis closure_contains_Inf closure_closed assms)
-
-
-lemma mono_closed_real:
- fixes S :: "real set"
- assumes mono: "ALL y z. y:S & y<=z --> z:S"
- assumes "closed S"
- shows "S = {} | S = UNIV | (EX a. S = {a ..})"
-proof-
-{ assume "S ~= {}"
- { assume ex: "EX B. ALL x:S. B<=x"
- hence *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
- hence "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
- hence "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
- hence "S = {Inf S ..}" by auto
- hence "EX a. S = {a ..}" by auto
- }
- moreover
- { assume "~(EX B. ALL x:S. B<=x)"
- hence nex: "ALL B. EX x:S. x<B" by (simp add: not_le)
- { fix y obtain x where "x:S & x < y" using nex by auto
- hence "y:S" using mono[rule_format, of x y] by auto
- } hence "S = UNIV" by auto
- } ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
-} from this show ?thesis by blast
-qed
-
-
-lemma mono_closed_extreal:
- fixes S :: "real set"
- assumes mono: "ALL y z. y:S & y<=z --> z:S"
- assumes "closed S"
- shows "EX a. S = {x. a <= extreal x}"
-proof-
-{ assume "S = {}" hence ?thesis apply(rule_tac x=PInfty in exI) by auto }
-moreover
-{ assume "S = UNIV" hence ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
-moreover
-{ assume "EX a. S = {a ..}"
- from this obtain a where "S={a ..}" by auto
- hence ?thesis apply(rule_tac x="extreal a" in exI) by auto
-} ultimately show ?thesis using mono_closed_real[of S] assms by auto
-qed
-
lemma extreal_le_distrib:
fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
by (cases rule: extreal3_cases[of a b c])
@@ -3418,296 +2435,4 @@
"[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
by (metis sup_extreal_def sup_least)
-subsection {* Sums *}
-
-lemma setsum_extreal[simp]:
- "(\<Sum>x\<in>A. extreal (f x)) = extreal (\<Sum>x\<in>A. f x)"
-proof cases
- assume "finite A" then show ?thesis by induct auto
-qed simp
-
-lemma setsum_Pinfty: "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
-proof safe
- assume *: "setsum f P = \<infinity>"
- show "finite P"
- proof (rule ccontr) assume "infinite P" with * show False by auto qed
- show "\<exists>i\<in>P. f i = \<infinity>"
- proof (rule ccontr)
- assume "\<not> ?thesis" then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>" by auto
- from `finite P` this have "setsum f P \<noteq> \<infinity>"
- by induct auto
- with * show False by auto
- qed
-next
- fix i assume "finite P" "i \<in> P" "f i = \<infinity>"
- thus "setsum f P = \<infinity>"
- proof induct
- case (insert x A)
- show ?case using insert by (cases "x = i") auto
- qed simp
-qed
-
-lemma setsum_Inf:
- shows "\<bar>setsum f A\<bar> = \<infinity> \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>))"
-proof
- assume *: "\<bar>setsum f A\<bar> = \<infinity>"
- have "finite A" by (rule ccontr) (insert *, auto)
- moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
- proof (rule ccontr)
- assume "\<not> ?thesis" then have "\<forall>i\<in>A. \<exists>r. f i = extreal r" by auto
- from bchoice[OF this] guess r ..
- with * show False by (auto simp: setsum_extreal)
- qed
- ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)" by auto
-next
- assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
- then obtain i where "finite A" "i \<in> A" "\<bar>f i\<bar> = \<infinity>" by auto
- then show "\<bar>setsum f A\<bar> = \<infinity>"
- proof induct
- case (insert j A) then show ?case
- by (cases rule: extreal3_cases[of "f i" "f j" "setsum f A"]) auto
- qed simp
-qed
-
-lemma setsum_of_pextreal:
- assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
- shows "(\<Sum>x\<in>S. real (f x)) = real (setsum f S)"
-proof -
- have "\<forall>x\<in>S. \<exists>r. f x = extreal r"
- proof
- fix x assume "x \<in> S"
- from assms[OF this] show "\<exists>r. f x = extreal r" by (cases "f x") auto
- qed
- from bchoice[OF this] guess r ..
- then show ?thesis by simp
-qed
-
-lemma setsum_extreal_0:
- fixes f :: "'a \<Rightarrow> extreal" assumes "finite A" "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
- shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
-proof
- assume *: "(\<Sum>x\<in>A. f x) = 0"
- then have "(\<Sum>x\<in>A. f x) \<noteq> \<infinity>" by auto
- then have "\<forall>i\<in>A. \<bar>f i\<bar> \<noteq> \<infinity>" using assms by (force simp: setsum_Pinfty)
- then have "\<forall>i\<in>A. \<exists>r. f i = extreal r" by auto
- from bchoice[OF this] * assms show "\<forall>i\<in>A. f i = 0"
- using setsum_nonneg_eq_0_iff[of A "\<lambda>i. real (f i)"] by auto
-qed (rule setsum_0')
-
-
-lemma setsum_extreal_right_distrib:
- fixes f :: "'a \<Rightarrow> extreal" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
- shows "r * setsum f A = (\<Sum>n\<in>A. r * f n)"
-proof cases
- assume "finite A" then show ?thesis using assms
- by induct (auto simp: extreal_right_distrib setsum_nonneg)
-qed simp
-
-lemma setsum_real_of_extreal:
- assumes "\<And>x. x \<in> A \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
- shows "real (\<Sum>x\<in>A. f x) = (\<Sum>x\<in>A. real (f x))"
-proof cases
- assume "finite A" from this assms show ?thesis
- proof induct
- case (insert a A) then show ?case
- by (simp add: real_of_extreal_add setsum_Inf)
- qed simp
-qed simp
-
-lemma sums_extreal_positive:
- fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" shows "f sums (SUP n. \<Sum>i<n. f i)"
-proof -
- have "incseq (\<lambda>i. \<Sum>j=0..<i. f j)"
- using extreal_add_mono[OF _ assms] by (auto intro!: incseq_SucI)
- from LIMSEQ_extreal_SUPR[OF this]
- show ?thesis unfolding sums_def by (simp add: atLeast0LessThan)
-qed
-
-lemma summable_extreal_pos:
- fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" shows "summable f"
- using sums_extreal_positive[of f, OF assms] unfolding summable_def by auto
-
-lemma suminf_extreal_eq_SUPR:
- fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i"
- shows "(\<Sum>x. f x) = (SUP n. \<Sum>i<n. f i)"
- using sums_extreal_positive[of f, OF assms, THEN sums_unique] by simp
-
-lemma sums_extreal:
- "(\<lambda>x. extreal (f x)) sums extreal x \<longleftrightarrow> f sums x"
- unfolding sums_def by simp
-
-lemma suminf_bound:
- fixes f :: "nat \<Rightarrow> extreal"
- assumes "\<forall>N. (\<Sum>n<N. f n) \<le> x" and pos: "\<And>n. 0 \<le> f n"
- shows "suminf f \<le> x"
-proof (rule Lim_bounded_extreal)
- have "summable f" using pos[THEN summable_extreal_pos] .
- then show "(\<lambda>N. \<Sum>n<N. f n) ----> suminf f"
- by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
- show "\<forall>n\<ge>0. setsum f {..<n} \<le> x"
- using assms by auto
-qed
-
-lemma suminf_bound_add:
- fixes f :: "nat \<Rightarrow> extreal"
- assumes "\<forall>N. (\<Sum>n<N. f n) + y \<le> x" and pos: "\<And>n. 0 \<le> f n" and "y \<noteq> -\<infinity>"
- shows "suminf f + y \<le> x"
-proof (cases y)
- case (real r) then have "\<forall>N. (\<Sum>n<N. f n) \<le> x - y"
- using assms by (simp add: extreal_le_minus)
- then have "(\<Sum> n. f n) \<le> x - y" using pos by (rule suminf_bound)
- then show "(\<Sum> n. f n) + y \<le> x"
- using assms real by (simp add: extreal_le_minus)
-qed (insert assms, auto)
-
-lemma sums_finite:
- assumes "\<forall>N\<ge>n. f N = 0"
- shows "f sums (\<Sum>N<n. f N)"
-proof -
- { fix i have "(\<Sum>N<i + n. f N) = (\<Sum>N<n. f N)"
- by (induct i) (insert assms, auto) }
- note this[simp]
- show ?thesis unfolding sums_def
- by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan)
-qed
-
-lemma suminf_finite:
- fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}" assumes "\<forall>N\<ge>n. f N = 0"
- shows "suminf f = (\<Sum>N<n. f N)"
- using sums_finite[OF assms, THEN sums_unique] by simp
-
-lemma suminf_extreal_0[simp]: "(\<Sum>i. 0) = (0::'a::{comm_monoid_add,t2_space})"
- using suminf_finite[of 0 "\<lambda>x. 0"] by simp
-
-lemma suminf_upper:
- fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>n. 0 \<le> f n"
- shows "(\<Sum>n<N. f n) \<le> (\<Sum>n. f n)"
- unfolding suminf_extreal_eq_SUPR[OF assms] SUPR_def
- by (auto intro: complete_lattice_class.Sup_upper image_eqI)
-
-lemma suminf_0_le:
- fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>n. 0 \<le> f n"
- shows "0 \<le> (\<Sum>n. f n)"
- using suminf_upper[of f 0, OF assms] by simp
-
-lemma suminf_le_pos:
- fixes f g :: "nat \<Rightarrow> extreal"
- assumes "\<And>N. f N \<le> g N" "\<And>N. 0 \<le> f N"
- shows "suminf f \<le> suminf g"
-proof (safe intro!: suminf_bound)
- fix n { fix N have "0 \<le> g N" using assms(2,1)[of N] by auto }
- have "setsum f {..<n} \<le> setsum g {..<n}" using assms by (auto intro: setsum_mono)
- also have "... \<le> suminf g" using `\<And>N. 0 \<le> g N` by (rule suminf_upper)
- finally show "setsum f {..<n} \<le> suminf g" .
-qed (rule assms(2))
-
-lemma suminf_half_series_extreal: "(\<Sum>n. (1/2 :: extreal)^Suc n) = 1"
- using sums_extreal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
- by (simp add: one_extreal_def)
-
-lemma suminf_add_extreal:
- fixes f g :: "nat \<Rightarrow> extreal"
- assumes "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
- shows "(\<Sum>i. f i + g i) = suminf f + suminf g"
- apply (subst (1 2 3) suminf_extreal_eq_SUPR)
- unfolding setsum_addf
- by (intro assms extreal_add_nonneg_nonneg SUPR_extreal_add_pos incseq_setsumI setsum_nonneg ballI)+
-
-lemma suminf_cmult_extreal:
- fixes f g :: "nat \<Rightarrow> extreal"
- assumes "\<And>i. 0 \<le> f i" "0 \<le> a"
- shows "(\<Sum>i. a * f i) = a * suminf f"
- by (auto simp: setsum_extreal_right_distrib[symmetric] assms
- extreal_zero_le_0_iff setsum_nonneg suminf_extreal_eq_SUPR
- intro!: SUPR_extreal_cmult )
-
-lemma suminf_PInfty:
- assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
- shows "f i \<noteq> \<infinity>"
-proof -
- from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
- have "(\<Sum>i<Suc i. f i) \<noteq> \<infinity>" by auto
- then show ?thesis
- unfolding setsum_Pinfty by simp
-qed
-
-lemma suminf_PInfty_fun:
- assumes "\<And>i. 0 \<le> f i" "suminf f \<noteq> \<infinity>"
- shows "\<exists>f'. f = (\<lambda>x. extreal (f' x))"
-proof -
- have "\<forall>i. \<exists>r. f i = extreal r"
- proof
- fix i show "\<exists>r. f i = extreal r"
- using suminf_PInfty[OF assms] assms(1)[of i] by (cases "f i") auto
- qed
- from choice[OF this] show ?thesis by auto
-qed
-
-lemma summable_extreal:
- assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. extreal (f i)) \<noteq> \<infinity>"
- shows "summable f"
-proof -
- have "0 \<le> (\<Sum>i. extreal (f i))"
- using assms by (intro suminf_0_le) auto
- with assms obtain r where r: "(\<Sum>i. extreal (f i)) = extreal r"
- by (cases "\<Sum>i. extreal (f i)") auto
- from summable_extreal_pos[of "\<lambda>x. extreal (f x)"]
- have "summable (\<lambda>x. extreal (f x))" using assms by auto
- from summable_sums[OF this]
- have "(\<lambda>x. extreal (f x)) sums (\<Sum>x. extreal (f x))" by auto
- then show "summable f"
- unfolding r sums_extreal summable_def ..
-qed
-
-lemma suminf_extreal:
- assumes "\<And>i. 0 \<le> f i" "(\<Sum>i. extreal (f i)) \<noteq> \<infinity>"
- shows "(\<Sum>i. extreal (f i)) = extreal (suminf f)"
-proof (rule sums_unique[symmetric])
- from summable_extreal[OF assms]
- show "(\<lambda>x. extreal (f x)) sums (extreal (suminf f))"
- unfolding sums_extreal using assms by (intro summable_sums summable_extreal)
-qed
-
-lemma suminf_extreal_minus:
- fixes f g :: "nat \<Rightarrow> extreal"
- assumes ord: "\<And>i. g i \<le> f i" "\<And>i. 0 \<le> g i" and fin: "suminf f \<noteq> \<infinity>" "suminf g \<noteq> \<infinity>"
- shows "(\<Sum>i. f i - g i) = suminf f - suminf g"
-proof -
- { fix i have "0 \<le> f i" using ord[of i] by auto }
- moreover
- from suminf_PInfty_fun[OF `\<And>i. 0 \<le> f i` fin(1)] guess f' .. note this[simp]
- from suminf_PInfty_fun[OF `\<And>i. 0 \<le> g i` fin(2)] guess g' .. note this[simp]
- { fix i have "0 \<le> f i - g i" using ord[of i] by (auto simp: extreal_le_minus_iff) }
- moreover
- have "suminf (\<lambda>i. f i - g i) \<le> suminf f"
- using assms by (auto intro!: suminf_le_pos simp: field_simps)
- then have "suminf (\<lambda>i. f i - g i) \<noteq> \<infinity>" using fin by auto
- ultimately show ?thesis using assms `\<And>i. 0 \<le> f i`
- apply simp
- by (subst (1 2 3) suminf_extreal)
- (auto intro!: suminf_diff[symmetric] summable_extreal)
-qed
-
-lemma suminf_extreal_PInf[simp]:
- "(\<Sum>x. \<infinity>) = \<infinity>"
-proof -
- have "(\<Sum>i<Suc 0. \<infinity>) \<le> (\<Sum>x. \<infinity>)" by (rule suminf_upper) auto
- then show ?thesis by simp
-qed
-
-lemma summable_real_of_extreal:
- assumes f: "\<And>i. 0 \<le> f i" and fin: "(\<Sum>i. f i) \<noteq> \<infinity>"
- shows "summable (\<lambda>i. real (f i))"
-proof (rule summable_def[THEN iffD2])
- have "0 \<le> (\<Sum>i. f i)" using assms by (auto intro: suminf_0_le)
- with fin obtain r where r: "extreal r = (\<Sum>i. f i)" by (cases "(\<Sum>i. f i)") auto
- { fix i have "f i \<noteq> \<infinity>" using f by (intro suminf_PInfty[OF _ fin]) auto
- then have "\<bar>f i\<bar> \<noteq> \<infinity>" using f[of i] by auto }
- note fin = this
- have "(\<lambda>i. extreal (real (f i))) sums (\<Sum>i. extreal (real (f i)))"
- using f by (auto intro!: summable_extreal_pos summable_sums simp: extreal_le_real_iff zero_extreal_def)
- also have "\<dots> = extreal r" using fin r by (auto simp: extreal_real)
- finally show "\<exists>r. (\<lambda>i. real (f i)) sums r" by (auto simp: sums_extreal)
-qed
-
end