author | huffman |
Fri, 12 Aug 2011 09:17:24 -0700 | |
changeset 44170 | 510ac30f44c0 |
parent 44142 | 8e27e0177518 |
child 44520 | 316256709a8c |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Extended_Real.thy |
41983 | 2 |
Author: Johannes Hölzl, TU München |
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Author: Robert Himmelmann, TU München |
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Author: Armin Heller, TU München |
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Author: Bogdan Grechuk, University of Edinburgh |
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*) |
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header {* Extended real number line *} |
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theory Extended_Real |
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imports Complex_Main Extended_Nat |
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begin |
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text {* |
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|
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For more lemmas about the extended real numbers go to |
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@{text "src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"} |
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|
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*} |
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lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot" |
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proof |
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assume "{x..} = UNIV" |
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show "x = bot" |
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proof (rule ccontr) |
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assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less) |
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then show False using `{x..} = UNIV` by simp |
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qed |
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qed auto |
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lemma SUPR_pair: |
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"(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))" |
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by (rule antisym) (auto intro!: SUP_leI le_SUPI2) |
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lemma INFI_pair: |
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"(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))" |
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by (rule antisym) (auto intro!: le_INFI INF_leI2) |
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subsection {* Definition and basic properties *} |
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||
43920 | 41 |
datatype ereal = ereal real | PInfty | MInfty |
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|
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instantiation ereal :: uminus |
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begin |
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fun uminus_ereal where |
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"- (ereal r) = ereal (- r)" |
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| "- PInfty = MInfty" |
48 |
| "- MInfty = PInfty" |
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instance .. |
50 |
end |
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instantiation ereal :: infinity |
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begin |
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definition "(\<infinity>::ereal) = PInfty" |
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instance .. |
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56 |
end |
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43924 | 58 |
definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)" |
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declare [[coercion "ereal :: real \<Rightarrow> ereal"]] |
61 |
declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]] |
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62 |
declare [[coercion "(\<lambda>n. ereal (of_nat n)) :: nat \<Rightarrow> ereal"]] |
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43920 | 64 |
lemma ereal_uminus_uminus[simp]: |
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fixes a :: ereal shows "- (- a) = a" |
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by (cases a) simp_all |
67 |
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43923 | 68 |
lemma |
69 |
shows PInfty_eq_infinity[simp]: "PInfty = \<infinity>" |
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and MInfty_eq_minfinity[simp]: "MInfty = - \<infinity>" |
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and MInfty_neq_PInfty[simp]: "\<infinity> \<noteq> - (\<infinity>::ereal)" "- \<infinity> \<noteq> (\<infinity>::ereal)" |
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and MInfty_neq_ereal[simp]: "ereal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> ereal r" |
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and PInfty_neq_ereal[simp]: "ereal r \<noteq> \<infinity>" "\<infinity> \<noteq> ereal r" |
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and PInfty_cases[simp]: "(case \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = y" |
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and MInfty_cases[simp]: "(case - \<infinity> of ereal r \<Rightarrow> f r | PInfty \<Rightarrow> y | MInfty \<Rightarrow> z) = z" |
|
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by (simp_all add: infinity_ereal_def) |
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41973 | 77 |
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43933 | 78 |
declare |
79 |
PInfty_eq_infinity[code_post] |
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80 |
MInfty_eq_minfinity[code_post] |
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81 |
||
82 |
lemma [code_unfold]: |
|
83 |
"\<infinity> = PInfty" |
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84 |
"-PInfty = MInfty" |
|
85 |
by simp_all |
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86 |
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43923 | 87 |
lemma inj_ereal[simp]: "inj_on ereal A" |
88 |
unfolding inj_on_def by auto |
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41973 | 89 |
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43920 | 90 |
lemma ereal_cases[case_names real PInf MInf, cases type: ereal]: |
91 |
assumes "\<And>r. x = ereal r \<Longrightarrow> P" |
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41973 | 92 |
assumes "x = \<infinity> \<Longrightarrow> P" |
93 |
assumes "x = -\<infinity> \<Longrightarrow> P" |
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94 |
shows P |
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95 |
using assms by (cases x) auto |
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43920 | 97 |
lemmas ereal2_cases = ereal_cases[case_product ereal_cases] |
98 |
lemmas ereal3_cases = ereal2_cases[case_product ereal_cases] |
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41973 | 99 |
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43920 | 100 |
lemma ereal_uminus_eq_iff[simp]: |
101 |
fixes a b :: ereal shows "-a = -b \<longleftrightarrow> a = b" |
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by (cases rule: ereal2_cases[of a b]) simp_all |
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41973 | 103 |
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43920 | 104 |
function of_ereal :: "ereal \<Rightarrow> real" where |
105 |
"of_ereal (ereal r) = r" | |
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106 |
"of_ereal \<infinity> = 0" | |
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107 |
"of_ereal (-\<infinity>) = 0" |
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108 |
by (auto intro: ereal_cases) |
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termination proof qed (rule wf_empty) |
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111 |
defs (overloaded) |
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43920 | 112 |
real_of_ereal_def [code_unfold]: "real \<equiv> of_ereal" |
41973 | 113 |
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lemma real_of_ereal[simp]: |
115 |
"real (- x :: ereal) = - (real x)" |
|
116 |
"real (ereal r) = r" |
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43923 | 117 |
"real (\<infinity>::ereal) = 0" |
43920 | 118 |
by (cases x) (simp_all add: real_of_ereal_def) |
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|
43920 | 120 |
lemma range_ereal[simp]: "range ereal = UNIV - {\<infinity>, -\<infinity>}" |
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proof safe |
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fix x assume "x \<notin> range ereal" "x \<noteq> \<infinity>" |
41973 | 123 |
then show "x = -\<infinity>" by (cases x) auto |
124 |
qed auto |
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lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)" |
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proof safe |
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fix x :: ereal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto |
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qed auto |
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instantiation ereal :: number |
41973 | 132 |
begin |
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definition [simp]: "number_of x = ereal (number_of x)" |
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instance proof qed |
135 |
end |
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instantiation ereal :: abs |
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begin |
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function abs_ereal where |
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"\<bar>ereal r\<bar> = ereal \<bar>r\<bar>" |
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| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)" |
142 |
| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)" |
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by (auto intro: ereal_cases) |
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termination proof qed (rule wf_empty) |
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instance .. |
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146 |
end |
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lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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by (cases x) auto |
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lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x :: ereal\<bar> \<noteq> \<infinity> ; \<And>r. x = ereal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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by (cases x) auto |
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lemma abs_ereal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::ereal\<bar>" |
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by (cases x) auto |
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subsubsection "Addition" |
158 |
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instantiation ereal :: comm_monoid_add |
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begin |
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definition "0 = ereal 0" |
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|
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function plus_ereal where |
165 |
"ereal r + ereal p = ereal (r + p)" | |
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"\<infinity> + a = (\<infinity>::ereal)" | |
167 |
"a + \<infinity> = (\<infinity>::ereal)" | |
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"ereal r + -\<infinity> = - \<infinity>" | |
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"-\<infinity> + ereal p = -(\<infinity>::ereal)" | |
170 |
"-\<infinity> + -\<infinity> = -(\<infinity>::ereal)" |
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41973 | 171 |
proof - |
172 |
case (goal1 P x) |
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173 |
moreover then obtain a b where "x = (a, b)" by (cases x) auto |
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174 |
ultimately show P |
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by (cases rule: ereal2_cases[of a b]) auto |
41973 | 176 |
qed auto |
177 |
termination proof qed (rule wf_empty) |
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178 |
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179 |
lemma Infty_neq_0[simp]: |
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"(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)" |
181 |
"-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)" |
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by (simp_all add: zero_ereal_def) |
41973 | 183 |
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lemma ereal_eq_0[simp]: |
185 |
"ereal r = 0 \<longleftrightarrow> r = 0" |
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186 |
"0 = ereal r \<longleftrightarrow> r = 0" |
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187 |
unfolding zero_ereal_def by simp_all |
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41973 | 188 |
|
189 |
instance |
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190 |
proof |
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fix a :: ereal show "0 + a = a" |
192 |
by (cases a) (simp_all add: zero_ereal_def) |
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193 |
fix b :: ereal show "a + b = b + a" |
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194 |
by (cases rule: ereal2_cases[of a b]) simp_all |
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fix c :: ereal show "a + b + c = a + (b + c)" |
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196 |
by (cases rule: ereal3_cases[of a b c]) simp_all |
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41973 | 197 |
qed |
198 |
end |
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199 |
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lemma real_of_ereal_0[simp]: "real (0::ereal) = 0" |
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unfolding real_of_ereal_def zero_ereal_def by simp |
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lemma abs_ereal_zero[simp]: "\<bar>0\<bar> = (0::ereal)" |
204 |
unfolding zero_ereal_def abs_ereal.simps by simp |
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lemma ereal_uminus_zero[simp]: |
207 |
"- 0 = (0::ereal)" |
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208 |
by (simp add: zero_ereal_def) |
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41973 | 209 |
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43920 | 210 |
lemma ereal_uminus_zero_iff[simp]: |
211 |
fixes a :: ereal shows "-a = 0 \<longleftrightarrow> a = 0" |
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41973 | 212 |
by (cases a) simp_all |
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lemma ereal_plus_eq_PInfty[simp]: |
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fixes a b :: ereal shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
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by (cases rule: ereal2_cases[of a b]) auto |
41973 | 217 |
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43920 | 218 |
lemma ereal_plus_eq_MInfty[simp]: |
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fixes a b :: ereal shows "a + b = -\<infinity> \<longleftrightarrow> |
41973 | 220 |
(a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>" |
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by (cases rule: ereal2_cases[of a b]) auto |
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lemma ereal_add_cancel_left: |
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fixes a b :: ereal assumes "a \<noteq> -\<infinity>" |
41973 | 225 |
shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)" |
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using assms by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 227 |
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lemma ereal_add_cancel_right: |
43923 | 229 |
fixes a b :: ereal assumes "a \<noteq> -\<infinity>" |
41973 | 230 |
shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)" |
43920 | 231 |
using assms by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 232 |
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lemma ereal_real: |
234 |
"ereal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)" |
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41973 | 235 |
by (cases x) simp_all |
236 |
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lemma real_of_ereal_add: |
238 |
fixes a b :: ereal |
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shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)" |
43920 | 240 |
by (cases rule: ereal2_cases[of a b]) auto |
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241 |
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subsubsection "Linear order on @{typ ereal}" |
41973 | 243 |
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43920 | 244 |
instantiation ereal :: linorder |
41973 | 245 |
begin |
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43920 | 247 |
function less_ereal where |
43923 | 248 |
" ereal x < ereal y \<longleftrightarrow> x < y" | |
249 |
"(\<infinity>::ereal) < a \<longleftrightarrow> False" | |
|
250 |
" a < -(\<infinity>::ereal) \<longleftrightarrow> False" | |
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251 |
"ereal x < \<infinity> \<longleftrightarrow> True" | |
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252 |
" -\<infinity> < ereal r \<longleftrightarrow> True" | |
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253 |
" -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> True" |
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41973 | 254 |
proof - |
255 |
case (goal1 P x) |
|
256 |
moreover then obtain a b where "x = (a,b)" by (cases x) auto |
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43920 | 257 |
ultimately show P by (cases rule: ereal2_cases[of a b]) auto |
41973 | 258 |
qed simp_all |
259 |
termination by (relation "{}") simp |
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260 |
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43920 | 261 |
definition "x \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y" |
41973 | 262 |
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43920 | 263 |
lemma ereal_infty_less[simp]: |
43923 | 264 |
fixes x :: ereal |
265 |
shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)" |
|
266 |
"-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)" |
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41973 | 267 |
by (cases x, simp_all) (cases x, simp_all) |
268 |
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43920 | 269 |
lemma ereal_infty_less_eq[simp]: |
43923 | 270 |
fixes x :: ereal |
271 |
shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>" |
|
41973 | 272 |
"x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>" |
43920 | 273 |
by (auto simp add: less_eq_ereal_def) |
41973 | 274 |
|
43920 | 275 |
lemma ereal_less[simp]: |
276 |
"ereal r < 0 \<longleftrightarrow> (r < 0)" |
|
277 |
"0 < ereal r \<longleftrightarrow> (0 < r)" |
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43923 | 278 |
"0 < (\<infinity>::ereal)" |
279 |
"-(\<infinity>::ereal) < 0" |
|
43920 | 280 |
by (simp_all add: zero_ereal_def) |
41973 | 281 |
|
43920 | 282 |
lemma ereal_less_eq[simp]: |
43923 | 283 |
"x \<le> (\<infinity>::ereal)" |
284 |
"-(\<infinity>::ereal) \<le> x" |
|
43920 | 285 |
"ereal r \<le> ereal p \<longleftrightarrow> r \<le> p" |
286 |
"ereal r \<le> 0 \<longleftrightarrow> r \<le> 0" |
|
287 |
"0 \<le> ereal r \<longleftrightarrow> 0 \<le> r" |
|
288 |
by (auto simp add: less_eq_ereal_def zero_ereal_def) |
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41973 | 289 |
|
43920 | 290 |
lemma ereal_infty_less_eq2: |
43923 | 291 |
"a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)" |
292 |
"a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)" |
|
41973 | 293 |
by simp_all |
294 |
||
295 |
instance |
|
296 |
proof |
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43920 | 297 |
fix x :: ereal show "x \<le> x" |
41973 | 298 |
by (cases x) simp_all |
43920 | 299 |
fix y :: ereal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
300 |
by (cases rule: ereal2_cases[of x y]) auto |
|
41973 | 301 |
show "x \<le> y \<or> y \<le> x " |
43920 | 302 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 303 |
{ assume "x \<le> y" "y \<le> x" then show "x = y" |
43920 | 304 |
by (cases rule: ereal2_cases[of x y]) auto } |
41973 | 305 |
{ fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z" |
43920 | 306 |
by (cases rule: ereal3_cases[of x y z]) auto } |
41973 | 307 |
qed |
308 |
end |
|
309 |
||
43920 | 310 |
instance ereal :: ordered_ab_semigroup_add |
41978 | 311 |
proof |
43920 | 312 |
fix a b c :: ereal assume "a \<le> b" then show "c + a \<le> c + b" |
313 |
by (cases rule: ereal3_cases[of a b c]) auto |
|
41978 | 314 |
qed |
315 |
||
43920 | 316 |
lemma real_of_ereal_positive_mono: |
43923 | 317 |
fixes x y :: ereal shows "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y" |
43920 | 318 |
by (cases rule: ereal2_cases[of x y]) auto |
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|
319 |
|
43920 | 320 |
lemma ereal_MInfty_lessI[intro, simp]: |
43923 | 321 |
fixes a :: ereal shows "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a" |
41973 | 322 |
by (cases a) auto |
323 |
||
43920 | 324 |
lemma ereal_less_PInfty[intro, simp]: |
43923 | 325 |
fixes a :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>" |
41973 | 326 |
by (cases a) auto |
327 |
||
43920 | 328 |
lemma ereal_less_ereal_Ex: |
329 |
fixes a b :: ereal |
|
330 |
shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)" |
|
41973 | 331 |
by (cases x) auto |
332 |
||
43920 | 333 |
lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < ereal (real n))" |
41979
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|
334 |
proof (cases x) |
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|
335 |
case (real r) then show ?thesis |
41980
28b51effc5ed
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hoelzl
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|
336 |
using reals_Archimedean2[of r] by simp |
41979
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|
337 |
qed simp_all |
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|
338 |
|
43920 | 339 |
lemma ereal_add_mono: |
340 |
fixes a b c d :: ereal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d" |
|
41973 | 341 |
using assms |
342 |
apply (cases a) |
|
43920 | 343 |
apply (cases rule: ereal3_cases[of b c d], auto) |
344 |
apply (cases rule: ereal3_cases[of b c d], auto) |
|
41973 | 345 |
done |
346 |
||
43920 | 347 |
lemma ereal_minus_le_minus[simp]: |
348 |
fixes a b :: ereal shows "- a \<le> - b \<longleftrightarrow> b \<le> a" |
|
349 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 350 |
|
43920 | 351 |
lemma ereal_minus_less_minus[simp]: |
352 |
fixes a b :: ereal shows "- a < - b \<longleftrightarrow> b < a" |
|
353 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 354 |
|
43920 | 355 |
lemma ereal_le_real_iff: |
356 |
"x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))" |
|
41973 | 357 |
by (cases y) auto |
358 |
||
43920 | 359 |
lemma real_le_ereal_iff: |
360 |
"real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))" |
|
41973 | 361 |
by (cases y) auto |
362 |
||
43920 | 363 |
lemma ereal_less_real_iff: |
364 |
"x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))" |
|
41973 | 365 |
by (cases y) auto |
366 |
||
43920 | 367 |
lemma real_less_ereal_iff: |
368 |
"real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))" |
|
41973 | 369 |
by (cases y) auto |
370 |
||
43920 | 371 |
lemma real_of_ereal_pos: |
372 |
fixes x :: ereal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto |
|
41979
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|
373 |
|
43920 | 374 |
lemmas real_of_ereal_ord_simps = |
375 |
ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff |
|
41973 | 376 |
|
43920 | 377 |
lemma abs_ereal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x" |
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|
378 |
by (cases x) auto |
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changeset
|
379 |
|
43920 | 380 |
lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x" |
42950
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changeset
|
381 |
by (cases x) auto |
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move lemmas to Extended_Reals and Extended_Real_Limits
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changeset
|
382 |
|
43920 | 383 |
lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>" |
42950
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changeset
|
384 |
by (cases x) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
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parents:
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diff
changeset
|
385 |
|
43923 | 386 |
lemma real_of_ereal_le_0[simp]: "real (x :: ereal) \<le> 0 \<longleftrightarrow> (x \<le> 0 \<or> x = \<infinity>)" |
387 |
by (cases x) auto |
|
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
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diff
changeset
|
388 |
|
43923 | 389 |
lemma abs_real_of_ereal[simp]: "\<bar>real (x :: ereal)\<bar> = real \<bar>x\<bar>" |
390 |
by (cases x) auto |
|
42950
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diff
changeset
|
391 |
|
43923 | 392 |
lemma zero_less_real_of_ereal: |
393 |
fixes x :: ereal shows "0 < real x \<longleftrightarrow> (0 < x \<and> x \<noteq> \<infinity>)" |
|
394 |
by (cases x) auto |
|
42950
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changeset
|
395 |
|
43920 | 396 |
lemma ereal_0_le_uminus_iff[simp]: |
397 |
fixes a :: ereal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0" |
|
398 |
by (cases rule: ereal2_cases[of a]) auto |
|
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
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changeset
|
399 |
|
43920 | 400 |
lemma ereal_uminus_le_0_iff[simp]: |
401 |
fixes a :: ereal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
|
402 |
by (cases rule: ereal2_cases[of a]) auto |
|
42950
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parents:
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diff
changeset
|
403 |
|
43923 | 404 |
lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> ereal z < y" |
405 |
using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto |
|
406 |
||
43920 | 407 |
lemma ereal_dense: |
408 |
fixes x y :: ereal assumes "x < y" |
|
43923 | 409 |
shows "\<exists>z. x < z \<and> z < y" |
410 |
using ereal_dense2[OF `x < y`] by blast |
|
41973 | 411 |
|
43920 | 412 |
lemma ereal_add_strict_mono: |
413 |
fixes a b c d :: ereal |
|
41979
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lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
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diff
changeset
|
414 |
assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
415 |
shows "a + c < b + d" |
43920 | 416 |
using assms by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto |
41979
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lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
417 |
|
43923 | 418 |
lemma ereal_less_add: |
419 |
fixes a b c :: ereal shows "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b" |
|
43920 | 420 |
by (cases rule: ereal2_cases[of b c]) auto |
41979
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lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
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diff
changeset
|
421 |
|
43920 | 422 |
lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)" by auto |
41979
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lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
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diff
changeset
|
423 |
|
43920 | 424 |
lemma ereal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::ereal)" |
425 |
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
426 |
|
43920 | 427 |
lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::ereal)" |
428 |
by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
429 |
|
43920 | 430 |
lemmas ereal_uminus_reorder = |
431 |
ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder |
|
41979
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lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
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diff
changeset
|
432 |
|
43920 | 433 |
lemma ereal_bot: |
434 |
fixes x :: ereal assumes "\<And>B. x \<le> ereal B" shows "x = - \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
435 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
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parents:
41978
diff
changeset
|
436 |
case (real r) with assms[of "r - 1"] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
437 |
next case PInf with assms[of 0] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
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41978
diff
changeset
|
438 |
next case MInf then show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
439 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
440 |
|
43920 | 441 |
lemma ereal_top: |
442 |
fixes x :: ereal assumes "\<And>B. x \<ge> ereal B" shows "x = \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
443 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
444 |
case (real r) with assms[of "r + 1"] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
445 |
next case MInf with assms[of 0] show ?thesis by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
446 |
next case PInf then show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
447 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
448 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
449 |
lemma |
43920 | 450 |
shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)" |
451 |
and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
452 |
by (simp_all add: min_def max_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
453 |
|
43920 | 454 |
lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)" |
455 |
by (auto simp: zero_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
456 |
|
41978 | 457 |
lemma |
43920 | 458 |
fixes f :: "nat \<Rightarrow> ereal" |
41978 | 459 |
shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f" |
460 |
and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f" |
|
461 |
unfolding decseq_def incseq_def by auto |
|
462 |
||
43920 | 463 |
lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))" |
42950
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move lemmas to Extended_Reals and Extended_Real_Limits
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diff
changeset
|
464 |
unfolding incseq_def by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
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42600
diff
changeset
|
465 |
|
43920 | 466 |
lemma ereal_add_nonneg_nonneg: |
467 |
fixes a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b" |
|
41978 | 468 |
using add_mono[of 0 a 0 b] by simp |
469 |
||
470 |
lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)" |
|
471 |
by auto |
|
472 |
||
473 |
lemma incseq_setsumI: |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
474 |
fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
41978 | 475 |
assumes "\<And>i. 0 \<le> f i" |
476 |
shows "incseq (\<lambda>i. setsum f {..< i})" |
|
477 |
proof (intro incseq_SucI) |
|
478 |
fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n" |
|
479 |
using assms by (rule add_left_mono) |
|
480 |
then show "setsum f {..< n} \<le> setsum f {..< Suc n}" |
|
481 |
by auto |
|
482 |
qed |
|
483 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
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41978
diff
changeset
|
484 |
lemma incseq_setsumI2: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
485 |
fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
486 |
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
487 |
shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
488 |
using assms unfolding incseq_def by (auto intro: setsum_mono) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
489 |
|
41973 | 490 |
subsubsection "Multiplication" |
491 |
||
43920 | 492 |
instantiation ereal :: "{comm_monoid_mult, sgn}" |
41973 | 493 |
begin |
494 |
||
43920 | 495 |
definition "1 = ereal 1" |
41973 | 496 |
|
43920 | 497 |
function sgn_ereal where |
498 |
"sgn (ereal r) = ereal (sgn r)" |
|
43923 | 499 |
| "sgn (\<infinity>::ereal) = 1" |
500 |
| "sgn (-\<infinity>::ereal) = -1" |
|
43920 | 501 |
by (auto intro: ereal_cases) |
41976 | 502 |
termination proof qed (rule wf_empty) |
503 |
||
43920 | 504 |
function times_ereal where |
505 |
"ereal r * ereal p = ereal (r * p)" | |
|
506 |
"ereal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
|
507 |
"\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
|
508 |
"ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
|
509 |
"-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
|
43923 | 510 |
"(\<infinity>::ereal) * \<infinity> = \<infinity>" | |
511 |
"-(\<infinity>::ereal) * \<infinity> = -\<infinity>" | |
|
512 |
"(\<infinity>::ereal) * -\<infinity> = -\<infinity>" | |
|
513 |
"-(\<infinity>::ereal) * -\<infinity> = \<infinity>" |
|
41973 | 514 |
proof - |
515 |
case (goal1 P x) |
|
516 |
moreover then obtain a b where "x = (a, b)" by (cases x) auto |
|
43920 | 517 |
ultimately show P by (cases rule: ereal2_cases[of a b]) auto |
41973 | 518 |
qed simp_all |
519 |
termination by (relation "{}") simp |
|
520 |
||
521 |
instance |
|
522 |
proof |
|
43920 | 523 |
fix a :: ereal show "1 * a = a" |
524 |
by (cases a) (simp_all add: one_ereal_def) |
|
525 |
fix b :: ereal show "a * b = b * a" |
|
526 |
by (cases rule: ereal2_cases[of a b]) simp_all |
|
527 |
fix c :: ereal show "a * b * c = a * (b * c)" |
|
528 |
by (cases rule: ereal3_cases[of a b c]) |
|
529 |
(simp_all add: zero_ereal_def zero_less_mult_iff) |
|
41973 | 530 |
qed |
531 |
end |
|
532 |
||
43920 | 533 |
lemma real_of_ereal_le_1: |
534 |
fixes a :: ereal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1" |
|
535 |
by (cases a) (auto simp: one_ereal_def) |
|
42950
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parents:
42600
diff
changeset
|
536 |
|
43920 | 537 |
lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (1::ereal)" |
538 |
unfolding one_ereal_def by simp |
|
41976 | 539 |
|
43920 | 540 |
lemma ereal_mult_zero[simp]: |
541 |
fixes a :: ereal shows "a * 0 = 0" |
|
542 |
by (cases a) (simp_all add: zero_ereal_def) |
|
41973 | 543 |
|
43920 | 544 |
lemma ereal_zero_mult[simp]: |
545 |
fixes a :: ereal shows "0 * a = 0" |
|
546 |
by (cases a) (simp_all add: zero_ereal_def) |
|
41973 | 547 |
|
43920 | 548 |
lemma ereal_m1_less_0[simp]: |
549 |
"-(1::ereal) < 0" |
|
550 |
by (simp add: zero_ereal_def one_ereal_def) |
|
41973 | 551 |
|
43920 | 552 |
lemma ereal_zero_m1[simp]: |
553 |
"1 \<noteq> (0::ereal)" |
|
554 |
by (simp add: zero_ereal_def one_ereal_def) |
|
41973 | 555 |
|
43920 | 556 |
lemma ereal_times_0[simp]: |
557 |
fixes x :: ereal shows "0 * x = 0" |
|
558 |
by (cases x) (auto simp: zero_ereal_def) |
|
41973 | 559 |
|
43920 | 560 |
lemma ereal_times[simp]: |
43923 | 561 |
"1 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1" |
562 |
"1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 1" |
|
43920 | 563 |
by (auto simp add: times_ereal_def one_ereal_def) |
41973 | 564 |
|
43920 | 565 |
lemma ereal_plus_1[simp]: |
566 |
"1 + ereal r = ereal (r + 1)" "ereal r + 1 = ereal (r + 1)" |
|
43923 | 567 |
"1 + -(\<infinity>::ereal) = -\<infinity>" "-(\<infinity>::ereal) + 1 = -\<infinity>" |
43920 | 568 |
unfolding one_ereal_def by auto |
41973 | 569 |
|
43920 | 570 |
lemma ereal_zero_times[simp]: |
571 |
fixes a b :: ereal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
|
572 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 573 |
|
43920 | 574 |
lemma ereal_mult_eq_PInfty[simp]: |
43923 | 575 |
shows "a * b = (\<infinity>::ereal) \<longleftrightarrow> |
41973 | 576 |
(a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)" |
43920 | 577 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 578 |
|
43920 | 579 |
lemma ereal_mult_eq_MInfty[simp]: |
43923 | 580 |
shows "a * b = -(\<infinity>::ereal) \<longleftrightarrow> |
41973 | 581 |
(a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)" |
43920 | 582 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 583 |
|
43920 | 584 |
lemma ereal_0_less_1[simp]: "0 < (1::ereal)" |
585 |
by (simp_all add: zero_ereal_def one_ereal_def) |
|
41973 | 586 |
|
43920 | 587 |
lemma ereal_zero_one[simp]: "0 \<noteq> (1::ereal)" |
588 |
by (simp_all add: zero_ereal_def one_ereal_def) |
|
41973 | 589 |
|
43920 | 590 |
lemma ereal_mult_minus_left[simp]: |
591 |
fixes a b :: ereal shows "-a * b = - (a * b)" |
|
592 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 593 |
|
43920 | 594 |
lemma ereal_mult_minus_right[simp]: |
595 |
fixes a b :: ereal shows "a * -b = - (a * b)" |
|
596 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 597 |
|
43920 | 598 |
lemma ereal_mult_infty[simp]: |
43923 | 599 |
"a * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
41973 | 600 |
by (cases a) auto |
601 |
||
43920 | 602 |
lemma ereal_infty_mult[simp]: |
43923 | 603 |
"(\<infinity>::ereal) * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
41973 | 604 |
by (cases a) auto |
605 |
||
43920 | 606 |
lemma ereal_mult_strict_right_mono: |
43923 | 607 |
assumes "a < b" and "0 < c" "c < (\<infinity>::ereal)" |
41973 | 608 |
shows "a * c < b * c" |
609 |
using assms |
|
43920 | 610 |
by (cases rule: ereal3_cases[of a b c]) |
44142 | 611 |
(auto simp: zero_le_mult_iff) |
41973 | 612 |
|
43920 | 613 |
lemma ereal_mult_strict_left_mono: |
43923 | 614 |
"\<lbrakk> a < b ; 0 < c ; c < (\<infinity>::ereal)\<rbrakk> \<Longrightarrow> c * a < c * b" |
43920 | 615 |
using ereal_mult_strict_right_mono by (simp add: mult_commute[of c]) |
41973 | 616 |
|
43920 | 617 |
lemma ereal_mult_right_mono: |
618 |
fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c" |
|
41973 | 619 |
using assms |
620 |
apply (cases "c = 0") apply simp |
|
43920 | 621 |
by (cases rule: ereal3_cases[of a b c]) |
44142 | 622 |
(auto simp: zero_le_mult_iff) |
41973 | 623 |
|
43920 | 624 |
lemma ereal_mult_left_mono: |
625 |
fixes a b c :: ereal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b" |
|
626 |
using ereal_mult_right_mono by (simp add: mult_commute[of c]) |
|
41973 | 627 |
|
43920 | 628 |
lemma zero_less_one_ereal[simp]: "0 \<le> (1::ereal)" |
629 |
by (simp add: one_ereal_def zero_ereal_def) |
|
41978 | 630 |
|
43920 | 631 |
lemma ereal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: ereal)" |
632 |
by (cases rule: ereal2_cases[of a b]) (auto simp: mult_nonneg_nonneg) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
633 |
|
43920 | 634 |
lemma ereal_right_distrib: |
635 |
fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b" |
|
636 |
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
637 |
|
43920 | 638 |
lemma ereal_left_distrib: |
639 |
fixes r a b :: ereal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r" |
|
640 |
by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
641 |
|
43920 | 642 |
lemma ereal_mult_le_0_iff: |
643 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
644 |
shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)" |
43920 | 645 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
646 |
|
43920 | 647 |
lemma ereal_zero_le_0_iff: |
648 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
649 |
shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)" |
43920 | 650 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
651 |
|
43920 | 652 |
lemma ereal_mult_less_0_iff: |
653 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
654 |
shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)" |
43920 | 655 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
656 |
|
43920 | 657 |
lemma ereal_zero_less_0_iff: |
658 |
fixes a b :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
659 |
shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)" |
43920 | 660 |
by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
661 |
|
43920 | 662 |
lemma ereal_distrib: |
663 |
fixes a b c :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
664 |
assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
665 |
shows "(a + b) * c = a * c + b * c" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
666 |
using assms |
43920 | 667 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
668 |
|
43920 | 669 |
lemma ereal_le_epsilon: |
670 |
fixes x y :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
671 |
assumes "ALL e. 0 < e --> x <= y + e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
672 |
shows "x <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
673 |
proof- |
43920 | 674 |
{ assume a: "EX r. y = ereal r" |
675 |
from this obtain r where r_def: "y = ereal r" by auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
676 |
{ assume "x=(-\<infinity>)" hence ?thesis by auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
677 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
678 |
{ assume "~(x=(-\<infinity>))" |
43920 | 679 |
from this obtain p where p_def: "x = ereal p" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
680 |
using a assms[rule_format, of 1] by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
681 |
{ fix e have "0 < e --> p <= r + e" |
43920 | 682 |
using assms[rule_format, of "ereal e"] p_def r_def by auto } |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
683 |
hence "p <= r" apply (subst field_le_epsilon) by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
684 |
hence ?thesis using r_def p_def by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
685 |
} ultimately have ?thesis by blast |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
686 |
} |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
687 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
688 |
{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
689 |
using assms[rule_format, of 1] by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
690 |
} ultimately show ?thesis by (cases y) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
691 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
692 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
693 |
|
43920 | 694 |
lemma ereal_le_epsilon2: |
695 |
fixes x y :: ereal |
|
696 |
assumes "ALL e. 0 < e --> x <= y + ereal e" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
697 |
shows "x <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
698 |
proof- |
43920 | 699 |
{ fix e :: ereal assume "e>0" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
700 |
{ assume "e=\<infinity>" hence "x<=y+e" by auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
701 |
moreover |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
702 |
{ assume "e~=\<infinity>" |
43920 | 703 |
from this obtain r where "e = ereal r" using `e>0` apply (cases e) by auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
704 |
hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
705 |
} ultimately have "x<=y+e" by blast |
43920 | 706 |
} from this show ?thesis using ereal_le_epsilon by auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
707 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
708 |
|
43920 | 709 |
lemma ereal_le_real: |
710 |
fixes x y :: ereal |
|
711 |
assumes "ALL z. x <= ereal z --> y <= ereal z" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
712 |
shows "y <= x" |
44142 | 713 |
by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
714 |
|
43920 | 715 |
lemma ereal_le_ereal: |
716 |
fixes x y :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
717 |
assumes "\<And>B. B < x \<Longrightarrow> B <= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
718 |
shows "x <= y" |
43920 | 719 |
by (metis assms ereal_dense leD linorder_le_less_linear) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
720 |
|
43920 | 721 |
lemma ereal_ge_ereal: |
722 |
fixes x y :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
723 |
assumes "ALL B. B>x --> B >= y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
724 |
shows "x >= y" |
43920 | 725 |
by (metis assms ereal_dense leD linorder_le_less_linear) |
41978 | 726 |
|
43920 | 727 |
lemma setprod_ereal_0: |
728 |
fixes f :: "'a \<Rightarrow> ereal" |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
729 |
shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
730 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
731 |
assume "finite A" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
732 |
then show ?thesis by (induct A) auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
733 |
qed auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
734 |
|
43920 | 735 |
lemma setprod_ereal_pos: |
736 |
fixes f :: "'a \<Rightarrow> ereal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)" |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
737 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
738 |
assume "finite I" from this pos show ?thesis by induct auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
739 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
740 |
|
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
741 |
lemma setprod_PInf: |
43923 | 742 |
fixes f :: "'a \<Rightarrow> ereal" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
743 |
assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
744 |
shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
745 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
746 |
assume "finite I" from this assms show ?thesis |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
747 |
proof (induct I) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
748 |
case (insert i I) |
43920 | 749 |
then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_ereal_pos) |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
750 |
from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
751 |
also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0" |
43920 | 752 |
using setprod_ereal_pos[of I f] pos |
753 |
by (cases rule: ereal2_cases[of "f i" "setprod f I"]) auto |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
754 |
also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)" |
43920 | 755 |
using insert by (auto simp: setprod_ereal_0) |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
756 |
finally show ?case . |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
757 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
758 |
qed simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
759 |
|
43920 | 760 |
lemma setprod_ereal: "(\<Prod>i\<in>A. ereal (f i)) = ereal (setprod f A)" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
761 |
proof cases |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
762 |
assume "finite A" then show ?thesis |
43920 | 763 |
by induct (auto simp: one_ereal_def) |
764 |
qed (simp add: one_ereal_def) |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
765 |
|
41978 | 766 |
subsubsection {* Power *} |
767 |
||
43920 | 768 |
instantiation ereal :: power |
41978 | 769 |
begin |
43920 | 770 |
primrec power_ereal where |
771 |
"power_ereal x 0 = 1" | |
|
772 |
"power_ereal x (Suc n) = x * x ^ n" |
|
41978 | 773 |
instance .. |
774 |
end |
|
775 |
||
43920 | 776 |
lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)" |
777 |
by (induct n) (auto simp: one_ereal_def) |
|
41978 | 778 |
|
43923 | 779 |
lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)" |
43920 | 780 |
by (induct n) (auto simp: one_ereal_def) |
41978 | 781 |
|
43920 | 782 |
lemma ereal_power_uminus[simp]: |
783 |
fixes x :: ereal |
|
41978 | 784 |
shows "(- x) ^ n = (if even n then x ^ n else - (x^n))" |
43920 | 785 |
by (induct n) (auto simp: one_ereal_def) |
41978 | 786 |
|
43920 | 787 |
lemma ereal_power_number_of[simp]: |
788 |
"(number_of num :: ereal) ^ n = ereal (number_of num ^ n)" |
|
789 |
by (induct n) (auto simp: one_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
790 |
|
43920 | 791 |
lemma zero_le_power_ereal[simp]: |
792 |
fixes a :: ereal assumes "0 \<le> a" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
793 |
shows "0 \<le> a ^ n" |
43920 | 794 |
using assms by (induct n) (auto simp: ereal_zero_le_0_iff) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
795 |
|
41973 | 796 |
subsubsection {* Subtraction *} |
797 |
||
43920 | 798 |
lemma ereal_minus_minus_image[simp]: |
799 |
fixes S :: "ereal set" |
|
41973 | 800 |
shows "uminus ` uminus ` S = S" |
801 |
by (auto simp: image_iff) |
|
802 |
||
43920 | 803 |
lemma ereal_uminus_lessThan[simp]: |
804 |
fixes a :: ereal shows "uminus ` {..<a} = {-a<..}" |
|
41973 | 805 |
proof (safe intro!: image_eqI) |
806 |
fix x assume "-a < x" |
|
43920 | 807 |
then have "- x < - (- a)" by (simp del: ereal_uminus_uminus) |
41973 | 808 |
then show "- x < a" by simp |
809 |
qed auto |
|
810 |
||
43920 | 811 |
lemma ereal_uminus_greaterThan[simp]: |
812 |
"uminus ` {(a::ereal)<..} = {..<-a}" |
|
813 |
by (metis ereal_uminus_lessThan ereal_uminus_uminus |
|
814 |
ereal_minus_minus_image) |
|
41973 | 815 |
|
43920 | 816 |
instantiation ereal :: minus |
41973 | 817 |
begin |
43920 | 818 |
definition "x - y = x + -(y::ereal)" |
41973 | 819 |
instance .. |
820 |
end |
|
821 |
||
43920 | 822 |
lemma ereal_minus[simp]: |
823 |
"ereal r - ereal p = ereal (r - p)" |
|
824 |
"-\<infinity> - ereal r = -\<infinity>" |
|
825 |
"ereal r - \<infinity> = -\<infinity>" |
|
43923 | 826 |
"(\<infinity>::ereal) - x = \<infinity>" |
827 |
"-(\<infinity>::ereal) - \<infinity> = -\<infinity>" |
|
41973 | 828 |
"x - -y = x + y" |
829 |
"x - 0 = x" |
|
830 |
"0 - x = -x" |
|
43920 | 831 |
by (simp_all add: minus_ereal_def) |
41973 | 832 |
|
43920 | 833 |
lemma ereal_x_minus_x[simp]: |
43923 | 834 |
"x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)" |
41973 | 835 |
by (cases x) simp_all |
836 |
||
43920 | 837 |
lemma ereal_eq_minus_iff: |
838 |
fixes x y z :: ereal |
|
41973 | 839 |
shows "x = z - y \<longleftrightarrow> |
41976 | 840 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and> |
41973 | 841 |
(y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and> |
842 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and> |
|
843 |
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)" |
|
43920 | 844 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 845 |
|
43920 | 846 |
lemma ereal_eq_minus: |
847 |
fixes x y z :: ereal |
|
41976 | 848 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z" |
43920 | 849 |
by (auto simp: ereal_eq_minus_iff) |
41973 | 850 |
|
43920 | 851 |
lemma ereal_less_minus_iff: |
852 |
fixes x y z :: ereal |
|
41973 | 853 |
shows "x < z - y \<longleftrightarrow> |
854 |
(y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and> |
|
855 |
(y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and> |
|
41976 | 856 |
(\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)" |
43920 | 857 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 858 |
|
43920 | 859 |
lemma ereal_less_minus: |
860 |
fixes x y z :: ereal |
|
41976 | 861 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z" |
43920 | 862 |
by (auto simp: ereal_less_minus_iff) |
41973 | 863 |
|
43920 | 864 |
lemma ereal_le_minus_iff: |
865 |
fixes x y z :: ereal |
|
41973 | 866 |
shows "x \<le> z - y \<longleftrightarrow> |
867 |
(y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> |
|
41976 | 868 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)" |
43920 | 869 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 870 |
|
43920 | 871 |
lemma ereal_le_minus: |
872 |
fixes x y z :: ereal |
|
41976 | 873 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z" |
43920 | 874 |
by (auto simp: ereal_le_minus_iff) |
41973 | 875 |
|
43920 | 876 |
lemma ereal_minus_less_iff: |
877 |
fixes x y z :: ereal |
|
41973 | 878 |
shows "x - y < z \<longleftrightarrow> |
879 |
y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> |
|
880 |
(y \<noteq> \<infinity> \<longrightarrow> x < z + y)" |
|
43920 | 881 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 882 |
|
43920 | 883 |
lemma ereal_minus_less: |
884 |
fixes x y z :: ereal |
|
41976 | 885 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y" |
43920 | 886 |
by (auto simp: ereal_minus_less_iff) |
41973 | 887 |
|
43920 | 888 |
lemma ereal_minus_le_iff: |
889 |
fixes x y z :: ereal |
|
41973 | 890 |
shows "x - y \<le> z \<longleftrightarrow> |
891 |
(y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
892 |
(y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
41976 | 893 |
(\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)" |
43920 | 894 |
by (cases rule: ereal3_cases[of x y z]) auto |
41973 | 895 |
|
43920 | 896 |
lemma ereal_minus_le: |
897 |
fixes x y z :: ereal |
|
41976 | 898 |
shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y" |
43920 | 899 |
by (auto simp: ereal_minus_le_iff) |
41973 | 900 |
|
43920 | 901 |
lemma ereal_minus_eq_minus_iff: |
902 |
fixes a b c :: ereal |
|
41973 | 903 |
shows "a - b = a - c \<longleftrightarrow> |
904 |
b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)" |
|
43920 | 905 |
by (cases rule: ereal3_cases[of a b c]) auto |
41973 | 906 |
|
43920 | 907 |
lemma ereal_add_le_add_iff: |
43923 | 908 |
fixes a b c :: ereal |
909 |
shows "c + a \<le> c + b \<longleftrightarrow> |
|
41973 | 910 |
a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)" |
43920 | 911 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps) |
41973 | 912 |
|
43920 | 913 |
lemma ereal_mult_le_mult_iff: |
43923 | 914 |
fixes a b c :: ereal |
915 |
shows "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
43920 | 916 |
by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left) |
41973 | 917 |
|
43920 | 918 |
lemma ereal_minus_mono: |
919 |
fixes A B C D :: ereal assumes "A \<le> B" "D \<le> C" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
920 |
shows "A - C \<le> B - D" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
921 |
using assms |
43920 | 922 |
by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
923 |
|
43920 | 924 |
lemma real_of_ereal_minus: |
43923 | 925 |
fixes a b :: ereal |
926 |
shows "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)" |
|
43920 | 927 |
by (cases rule: ereal2_cases[of a b]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
928 |
|
43920 | 929 |
lemma ereal_diff_positive: |
930 |
fixes a b :: ereal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a" |
|
931 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
932 |
|
43920 | 933 |
lemma ereal_between: |
934 |
fixes x e :: ereal |
|
41976 | 935 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e" |
41973 | 936 |
shows "x - e < x" "x < x + e" |
937 |
using assms apply (cases x, cases e) apply auto |
|
938 |
using assms by (cases x, cases e) auto |
|
939 |
||
940 |
subsubsection {* Division *} |
|
941 |
||
43920 | 942 |
instantiation ereal :: inverse |
41973 | 943 |
begin |
944 |
||
43920 | 945 |
function inverse_ereal where |
946 |
"inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))" | |
|
43923 | 947 |
"inverse (\<infinity>::ereal) = 0" | |
948 |
"inverse (-\<infinity>::ereal) = 0" |
|
43920 | 949 |
by (auto intro: ereal_cases) |
41973 | 950 |
termination by (relation "{}") simp |
951 |
||
43920 | 952 |
definition "x / y = x * inverse (y :: ereal)" |
41973 | 953 |
|
954 |
instance proof qed |
|
955 |
end |
|
956 |
||
43920 | 957 |
lemma real_of_ereal_inverse[simp]: |
958 |
fixes a :: ereal |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
959 |
shows "real (inverse a) = 1 / real a" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
960 |
by (cases a) (auto simp: inverse_eq_divide) |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
961 |
|
43920 | 962 |
lemma ereal_inverse[simp]: |
43923 | 963 |
"inverse (0::ereal) = \<infinity>" |
43920 | 964 |
"inverse (1::ereal) = 1" |
965 |
by (simp_all add: one_ereal_def zero_ereal_def) |
|
41973 | 966 |
|
43920 | 967 |
lemma ereal_divide[simp]: |
968 |
"ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))" |
|
969 |
unfolding divide_ereal_def by (auto simp: divide_real_def) |
|
41973 | 970 |
|
43920 | 971 |
lemma ereal_divide_same[simp]: |
43923 | 972 |
fixes x :: ereal shows "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)" |
41973 | 973 |
by (cases x) |
43920 | 974 |
(simp_all add: divide_real_def divide_ereal_def one_ereal_def) |
41973 | 975 |
|
43920 | 976 |
lemma ereal_inv_inv[simp]: |
43923 | 977 |
fixes x :: ereal shows "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)" |
41973 | 978 |
by (cases x) auto |
979 |
||
43920 | 980 |
lemma ereal_inverse_minus[simp]: |
43923 | 981 |
fixes x :: ereal shows "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)" |
41973 | 982 |
by (cases x) simp_all |
983 |
||
43920 | 984 |
lemma ereal_uminus_divide[simp]: |
985 |
fixes x y :: ereal shows "- x / y = - (x / y)" |
|
986 |
unfolding divide_ereal_def by simp |
|
41973 | 987 |
|
43920 | 988 |
lemma ereal_divide_Infty[simp]: |
43923 | 989 |
fixes x :: ereal shows "x / \<infinity> = 0" "x / -\<infinity> = 0" |
43920 | 990 |
unfolding divide_ereal_def by simp_all |
41973 | 991 |
|
43920 | 992 |
lemma ereal_divide_one[simp]: |
993 |
"x / 1 = (x::ereal)" |
|
994 |
unfolding divide_ereal_def by simp |
|
41973 | 995 |
|
43920 | 996 |
lemma ereal_divide_ereal[simp]: |
997 |
"\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)" |
|
998 |
unfolding divide_ereal_def by simp |
|
41973 | 999 |
|
43920 | 1000 |
lemma zero_le_divide_ereal[simp]: |
1001 |
fixes a :: ereal assumes "0 \<le> a" "0 \<le> b" |
|
41978 | 1002 |
shows "0 \<le> a / b" |
43920 | 1003 |
using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff) |
41978 | 1004 |
|
43920 | 1005 |
lemma ereal_le_divide_pos: |
43923 | 1006 |
fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z" |
43920 | 1007 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1008 |
|
43920 | 1009 |
lemma ereal_divide_le_pos: |
43923 | 1010 |
fixes x y z :: ereal shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y" |
43920 | 1011 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1012 |
|
43920 | 1013 |
lemma ereal_le_divide_neg: |
43923 | 1014 |
fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y" |
43920 | 1015 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1016 |
|
43920 | 1017 |
lemma ereal_divide_le_neg: |
43923 | 1018 |
fixes x y z :: ereal shows "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z" |
43920 | 1019 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 1020 |
|
43920 | 1021 |
lemma ereal_inverse_antimono_strict: |
1022 |
fixes x y :: ereal |
|
41973 | 1023 |
shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x" |
43920 | 1024 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 1025 |
|
43920 | 1026 |
lemma ereal_inverse_antimono: |
1027 |
fixes x y :: ereal |
|
41973 | 1028 |
shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x" |
43920 | 1029 |
by (cases rule: ereal2_cases[of x y]) auto |
41973 | 1030 |
|
1031 |
lemma inverse_inverse_Pinfty_iff[simp]: |
|
43923 | 1032 |
fixes x :: ereal shows "inverse x = \<infinity> \<longleftrightarrow> x = 0" |
41973 | 1033 |
by (cases x) auto |
1034 |
||
43920 | 1035 |
lemma ereal_inverse_eq_0: |
43923 | 1036 |
fixes x :: ereal shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>" |
41973 | 1037 |
by (cases x) auto |
1038 |
||
43920 | 1039 |
lemma ereal_0_gt_inverse: |
1040 |
fixes x :: ereal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1041 |
by (cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1042 |
|
43920 | 1043 |
lemma ereal_mult_less_right: |
43923 | 1044 |
fixes a b c :: ereal |
41973 | 1045 |
assumes "b * a < c * a" "0 < a" "a < \<infinity>" |
1046 |
shows "b < c" |
|
1047 |
using assms |
|
43920 | 1048 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 1049 |
(auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff) |
1050 |
||
43920 | 1051 |
lemma ereal_power_divide: |
43923 | 1052 |
fixes x y :: ereal shows "y \<noteq> 0 \<Longrightarrow> (x / y) ^ n = x^n / y^n" |
43920 | 1053 |
by (cases rule: ereal2_cases[of x y]) |
1054 |
(auto simp: one_ereal_def zero_ereal_def power_divide not_le |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1055 |
power_less_zero_eq zero_le_power_iff) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1056 |
|
43920 | 1057 |
lemma ereal_le_mult_one_interval: |
1058 |
fixes x y :: ereal |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1059 |
assumes y: "y \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1060 |
assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1061 |
shows "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1062 |
proof (cases x) |
43920 | 1063 |
case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_ereal_def) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1064 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1065 |
case (real r) note r = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1066 |
show "x \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1067 |
proof (cases y) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1068 |
case (real p) note p = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1069 |
have "r \<le> p" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1070 |
proof (rule field_le_mult_one_interval) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1071 |
fix z :: real assume "0 < z" and "z < 1" |
43920 | 1072 |
with z[of "ereal z"] |
1073 |
show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1074 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1075 |
then show "x \<le> y" using p r by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1076 |
qed (insert y, simp_all) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1077 |
qed simp |
41978 | 1078 |
|
41973 | 1079 |
subsection "Complete lattice" |
1080 |
||
43920 | 1081 |
instantiation ereal :: lattice |
41973 | 1082 |
begin |
43920 | 1083 |
definition [simp]: "sup x y = (max x y :: ereal)" |
1084 |
definition [simp]: "inf x y = (min x y :: ereal)" |
|
41973 | 1085 |
instance proof qed simp_all |
1086 |
end |
|
1087 |
||
43920 | 1088 |
instantiation ereal :: complete_lattice |
41973 | 1089 |
begin |
1090 |
||
43923 | 1091 |
definition "bot = (-\<infinity>::ereal)" |
1092 |
definition "top = (\<infinity>::ereal)" |
|
41973 | 1093 |
|
43923 | 1094 |
definition "Sup S = (LEAST z. \<forall>x\<in>S. x \<le> z :: ereal)" |
1095 |
definition "Inf S = (GREATEST z. \<forall>x\<in>S. z \<le> x :: ereal)" |
|
41973 | 1096 |
|
43920 | 1097 |
lemma ereal_complete_Sup: |
1098 |
fixes S :: "ereal set" assumes "S \<noteq> {}" |
|
41973 | 1099 |
shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)" |
1100 |
proof cases |
|
43920 | 1101 |
assume "\<exists>x. \<forall>a\<in>S. a \<le> ereal x" |
1102 |
then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> ereal y" by auto |
|
41973 | 1103 |
then have "\<infinity> \<notin> S" by force |
1104 |
show ?thesis |
|
1105 |
proof cases |
|
1106 |
assume "S = {-\<infinity>}" |
|
1107 |
then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"]) |
|
1108 |
next |
|
1109 |
assume "S \<noteq> {-\<infinity>}" |
|
1110 |
with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto |
|
1111 |
with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y" |
|
43920 | 1112 |
by (auto simp: real_of_ereal_ord_simps) |
41973 | 1113 |
with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}` |
1114 |
obtain s where s: |
|
1115 |
"\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z" |
|
1116 |
by auto |
|
1117 |
show ?thesis |
|
43920 | 1118 |
proof (safe intro!: exI[of _ "ereal s"]) |
1119 |
fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> ereal s" |
|
41973 | 1120 |
proof (cases z) |
1121 |
case (real r) |
|
1122 |
then show ?thesis |
|
43920 | 1123 |
using s(1)[rule_format, of z] `z \<in> S` `z = ereal r` by auto |
41973 | 1124 |
qed auto |
1125 |
next |
|
1126 |
fix z assume *: "\<forall>y\<in>S. y \<le> z" |
|
43920 | 1127 |
with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "ereal s \<le> z" |
41973 | 1128 |
proof (cases z) |
1129 |
case (real u) |
|
1130 |
with * have "s \<le> u" |
|
43920 | 1131 |
by (intro s(2)[of u]) (auto simp: real_of_ereal_ord_simps) |
41973 | 1132 |
then show ?thesis using real by simp |
1133 |
qed auto |
|
1134 |
qed |
|
1135 |
qed |
|
1136 |
next |
|
43920 | 1137 |
assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> ereal x)" |
41973 | 1138 |
show ?thesis |
1139 |
proof (safe intro!: exI[of _ \<infinity>]) |
|
1140 |
fix y assume **: "\<forall>z\<in>S. z \<le> y" |
|
1141 |
with * show "\<infinity> \<le> y" |
|
1142 |
proof (cases y) |
|
1143 |
case MInf with * ** show ?thesis by (force simp: not_le) |
|
1144 |
qed auto |
|
1145 |
qed simp |
|
1146 |
qed |
|
1147 |
||
43920 | 1148 |
lemma ereal_complete_Inf: |
1149 |
fixes S :: "ereal set" assumes "S ~= {}" |
|
41973 | 1150 |
shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)" |
1151 |
proof- |
|
1152 |
def S1 == "uminus ` S" |
|
1153 |
hence "S1 ~= {}" using assms by auto |
|
1154 |
from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)" |
|
43920 | 1155 |
using ereal_complete_Sup[of S1] by auto |
41973 | 1156 |
{ fix z assume "ALL y:S. z <= y" |
1157 |
hence "ALL y:S1. y <= -z" unfolding S1_def by auto |
|
1158 |
hence "x <= -z" using x_def by auto |
|
1159 |
hence "z <= -x" |
|
43920 | 1160 |
apply (subst ereal_uminus_uminus[symmetric]) |
1161 |
unfolding ereal_minus_le_minus . } |
|
41973 | 1162 |
moreover have "(ALL y:S. -x <= y)" |
1163 |
using x_def unfolding S1_def |
|
1164 |
apply simp |
|
43920 | 1165 |
apply (subst (3) ereal_uminus_uminus[symmetric]) |
1166 |
unfolding ereal_minus_le_minus by simp |
|
41973 | 1167 |
ultimately show ?thesis by auto |
1168 |
qed |
|
1169 |
||
43920 | 1170 |
lemma ereal_complete_uminus_eq: |
1171 |
fixes S :: "ereal set" |
|
41973 | 1172 |
shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z) |
1173 |
\<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
43920 | 1174 |
by simp (metis ereal_minus_le_minus ereal_uminus_uminus) |
41973 | 1175 |
|
43920 | 1176 |
lemma ereal_Sup_uminus_image_eq: |
1177 |
fixes S :: "ereal set" |
|
41973 | 1178 |
shows "Sup (uminus ` S) = - Inf S" |
1179 |
proof cases |
|
1180 |
assume "S = {}" |
|
43920 | 1181 |
moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::ereal)" |
1182 |
by (rule the_equality) (auto intro!: ereal_bot) |
|
1183 |
moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::ereal)" |
|
1184 |
by (rule some_equality) (auto intro!: ereal_top) |
|
1185 |
ultimately show ?thesis unfolding Inf_ereal_def Sup_ereal_def |
|
41973 | 1186 |
Least_def Greatest_def GreatestM_def by simp |
1187 |
next |
|
1188 |
assume "S \<noteq> {}" |
|
43920 | 1189 |
with ereal_complete_Sup[of "uminus`S"] |
41973 | 1190 |
obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
43920 | 1191 |
unfolding ereal_complete_uminus_eq by auto |
41973 | 1192 |
show "Sup (uminus ` S) = - Inf S" |
43920 | 1193 |
unfolding Inf_ereal_def Greatest_def GreatestM_def |
41973 | 1194 |
proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"]) |
1195 |
show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)" |
|
1196 |
using x . |
|
1197 |
fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')" |
|
1198 |
then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)" |
|
43920 | 1199 |
unfolding ereal_complete_uminus_eq by simp |
41973 | 1200 |
then show "Sup (uminus ` S) = -x'" |
43920 | 1201 |
unfolding Sup_ereal_def ereal_uminus_eq_iff |
41973 | 1202 |
by (intro Least_equality) auto |
1203 |
qed |
|
1204 |
qed |
|
1205 |
||
1206 |
instance |
|
1207 |
proof |
|
43920 | 1208 |
{ fix x :: ereal and A |
1209 |
show "bot <= x" by (cases x) (simp_all add: bot_ereal_def) |
|
1210 |
show "x <= top" by (simp add: top_ereal_def) } |
|
41973 | 1211 |
|
43920 | 1212 |
{ fix x :: ereal and A assume "x : A" |
1213 |
with ereal_complete_Sup[of A] |
|
41973 | 1214 |
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
1215 |
hence "x <= s" using `x : A` by auto |
|
43920 | 1216 |
also have "... = Sup A" using s unfolding Sup_ereal_def |
41973 | 1217 |
by (auto intro!: Least_equality[symmetric]) |
1218 |
finally show "x <= Sup A" . } |
|
1219 |
note le_Sup = this |
|
1220 |
||
43920 | 1221 |
{ fix x :: ereal and A assume *: "!!z. (z : A ==> z <= x)" |
41973 | 1222 |
show "Sup A <= x" |
1223 |
proof (cases "A = {}") |
|
1224 |
case True |
|
43920 | 1225 |
hence "Sup A = -\<infinity>" unfolding Sup_ereal_def |
41973 | 1226 |
by (auto intro!: Least_equality) |
1227 |
thus "Sup A <= x" by simp |
|
1228 |
next |
|
1229 |
case False |
|
43920 | 1230 |
with ereal_complete_Sup[of A] |
41973 | 1231 |
obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
1232 |
hence "Sup A = s" |
|
43920 | 1233 |
unfolding Sup_ereal_def by (auto intro!: Least_equality) |
41973 | 1234 |
also have "s <= x" using * s by auto |
1235 |
finally show "Sup A <= x" . |
|
1236 |
qed } |
|
1237 |
note Sup_le = this |
|
1238 |
||
43920 | 1239 |
{ fix x :: ereal and A assume "x \<in> A" |
41973 | 1240 |
with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x" |
43920 | 1241 |
unfolding ereal_Sup_uminus_image_eq by simp } |
41973 | 1242 |
|
43920 | 1243 |
{ fix x :: ereal and A assume *: "!!z. (z : A ==> x <= z)" |
41973 | 1244 |
with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A" |
43920 | 1245 |
unfolding ereal_Sup_uminus_image_eq by force } |
41973 | 1246 |
qed |
43941 | 1247 |
|
41973 | 1248 |
end |
1249 |
||
43941 | 1250 |
instance ereal :: complete_linorder .. |
1251 |
||
43920 | 1252 |
lemma ereal_SUPR_uminus: |
1253 |
fixes f :: "'a => ereal" |
|
41973 | 1254 |
shows "(SUP i : R. -(f i)) = -(INF i : R. f i)" |
1255 |
unfolding SUPR_def INFI_def |
|
43920 | 1256 |
using ereal_Sup_uminus_image_eq[of "f`R"] |
41973 | 1257 |
by (simp add: image_image) |
1258 |
||
43920 | 1259 |
lemma ereal_INFI_uminus: |
1260 |
fixes f :: "'a => ereal" |
|
41973 | 1261 |
shows "(INF i : R. -(f i)) = -(SUP i : R. f i)" |
43920 | 1262 |
using ereal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp |
41973 | 1263 |
|
43920 | 1264 |
lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::ereal set)" |
1265 |
using ereal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1266 |
|
43920 | 1267 |
lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)" |
41973 | 1268 |
by (auto intro!: inj_onI) |
1269 |
||
43920 | 1270 |
lemma ereal_image_uminus_shift: |
1271 |
fixes X Y :: "ereal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y" |
|
41973 | 1272 |
proof |
1273 |
assume "uminus ` X = Y" |
|
1274 |
then have "uminus ` uminus ` X = uminus ` Y" |
|
1275 |
by (simp add: inj_image_eq_iff) |
|
1276 |
then show "X = uminus ` Y" by (simp add: image_image) |
|
1277 |
qed (simp add: image_image) |
|
1278 |
||
43920 | 1279 |
lemma Inf_ereal_iff: |
1280 |
fixes z :: ereal |
|
41973 | 1281 |
shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y" |
1282 |
by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear |
|
1283 |
order_less_le_trans) |
|
1284 |
||
1285 |
lemma Sup_eq_MInfty: |
|
43920 | 1286 |
fixes S :: "ereal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}" |
41973 | 1287 |
proof |
1288 |
assume a: "Sup S = -\<infinity>" |
|
1289 |
with complete_lattice_class.Sup_upper[of _ S] |
|
1290 |
show "S={} \<or> S={-\<infinity>}" by auto |
|
1291 |
next |
|
1292 |
assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>" |
|
43920 | 1293 |
unfolding Sup_ereal_def by (auto intro!: Least_equality) |
41973 | 1294 |
qed |
1295 |
||
1296 |
lemma Inf_eq_PInfty: |
|
43920 | 1297 |
fixes S :: "ereal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}" |
41973 | 1298 |
using Sup_eq_MInfty[of "uminus`S"] |
43920 | 1299 |
unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp |
41973 | 1300 |
|
43923 | 1301 |
lemma Inf_eq_MInfty: |
1302 |
fixes S :: "ereal set" shows "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>" |
|
43920 | 1303 |
unfolding Inf_ereal_def |
41973 | 1304 |
by (auto intro!: Greatest_equality) |
1305 |
||
43923 | 1306 |
lemma Sup_eq_PInfty: |
1307 |
fixes S :: "ereal set" shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>" |
|
43920 | 1308 |
unfolding Sup_ereal_def |
41973 | 1309 |
by (auto intro!: Least_equality) |
1310 |
||
43920 | 1311 |
lemma ereal_SUPI: |
1312 |
fixes x :: ereal |
|
41973 | 1313 |
assumes "!!i. i : A ==> f i <= x" |
1314 |
assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y" |
|
1315 |
shows "(SUP i:A. f i) = x" |
|
43920 | 1316 |
unfolding SUPR_def Sup_ereal_def |
41973 | 1317 |
using assms by (auto intro!: Least_equality) |
1318 |
||
43920 | 1319 |
lemma ereal_INFI: |
1320 |
fixes x :: ereal |
|
41973 | 1321 |
assumes "!!i. i : A ==> f i >= x" |
1322 |
assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y" |
|
1323 |
shows "(INF i:A. f i) = x" |
|
43920 | 1324 |
unfolding INFI_def Inf_ereal_def |
41973 | 1325 |
using assms by (auto intro!: Greatest_equality) |
1326 |
||
43920 | 1327 |
lemma Sup_ereal_close: |
1328 |
fixes e :: ereal |
|
41976 | 1329 |
assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}" |
41973 | 1330 |
shows "\<exists>x\<in>S. Sup S - e < x" |
41976 | 1331 |
using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1]) |
41973 | 1332 |
|
43920 | 1333 |
lemma Inf_ereal_close: |
1334 |
fixes e :: ereal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e" |
|
41973 | 1335 |
shows "\<exists>x\<in>X. x < Inf X + e" |
1336 |
proof (rule Inf_less_iff[THEN iffD1]) |
|
1337 |
show "Inf X < Inf X + e" using assms |
|
41976 | 1338 |
by (cases e) auto |
41973 | 1339 |
qed |
1340 |
||
43920 | 1341 |
lemma SUP_nat_Infty: "(SUP i::nat. ereal (real i)) = \<infinity>" |
41973 | 1342 |
proof - |
43923 | 1343 |
{ fix x ::ereal assume "x \<noteq> \<infinity>" |
43920 | 1344 |
then have "\<exists>k::nat. x < ereal (real k)" |
41973 | 1345 |
proof (cases x) |
1346 |
case MInf then show ?thesis by (intro exI[of _ 0]) auto |
|
1347 |
next |
|
1348 |
case (real r) |
|
1349 |
moreover obtain k :: nat where "r < real k" |
|
1350 |
using ex_less_of_nat by (auto simp: real_eq_of_nat) |
|
1351 |
ultimately show ?thesis by auto |
|
1352 |
qed simp } |
|
1353 |
then show ?thesis |
|
43920 | 1354 |
using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. ereal (real n)"] |
1355 |
by (auto simp: top_ereal_def) |
|
41973 | 1356 |
qed |
1357 |
||
43920 | 1358 |
lemma ereal_le_Sup: |
1359 |
fixes x :: ereal |
|
41973 | 1360 |
shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))" |
1361 |
(is "?lhs <-> ?rhs") |
|
1362 |
proof- |
|
1363 |
{ assume "?rhs" |
|
1364 |
{ assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le) |
|
43920 | 1365 |
from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using ereal_dense by auto |
41973 | 1366 |
from this obtain i where "i : A & y <= f i" using `?rhs` by auto |
1367 |
hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto |
|
1368 |
hence False using y_def by auto |
|
1369 |
} hence "?lhs" by auto |
|
1370 |
} |
|
1371 |
moreover |
|
1372 |
{ assume "?lhs" hence "?rhs" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
1373 |
by (metis SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff |
41973 | 1374 |
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
1375 |
} ultimately show ?thesis by auto |
|
1376 |
qed |
|
1377 |
||
43920 | 1378 |
lemma ereal_Inf_le: |
1379 |
fixes x :: ereal |
|
41973 | 1380 |
shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))" |
1381 |
(is "?lhs <-> ?rhs") |
|
1382 |
proof- |
|
1383 |
{ assume "?rhs" |
|
1384 |
{ assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le) |
|
43920 | 1385 |
from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using ereal_dense by auto |
41973 | 1386 |
from this obtain i where "i : A & f i <= y" using `?rhs` by auto |
1387 |
hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto |
|
1388 |
hence False using y_def by auto |
|
1389 |
} hence "?lhs" by auto |
|
1390 |
} |
|
1391 |
moreover |
|
1392 |
{ assume "?lhs" hence "?rhs" |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
1393 |
by (metis le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff |
41973 | 1394 |
inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
1395 |
} ultimately show ?thesis by auto |
|
1396 |
qed |
|
1397 |
||
1398 |
lemma Inf_less: |
|
43920 | 1399 |
fixes x :: ereal |
41973 | 1400 |
assumes "(INF i:A. f i) < x" |
1401 |
shows "EX i. i : A & f i <= x" |
|
1402 |
proof(rule ccontr) |
|
1403 |
assume "~ (EX i. i : A & f i <= x)" |
|
1404 |
hence "ALL i:A. f i > x" by auto |
|
1405 |
hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto |
|
1406 |
thus False using assms by auto |
|
1407 |
qed |
|
1408 |
||
1409 |
lemma same_INF: |
|
1410 |
assumes "ALL e:A. f e = g e" |
|
1411 |
shows "(INF e:A. f e) = (INF e:A. g e)" |
|
1412 |
proof- |
|
1413 |
have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1414 |
thus ?thesis unfolding INFI_def by auto |
|
1415 |
qed |
|
1416 |
||
1417 |
lemma same_SUP: |
|
1418 |
assumes "ALL e:A. f e = g e" |
|
1419 |
shows "(SUP e:A. f e) = (SUP e:A. g e)" |
|
1420 |
proof- |
|
1421 |
have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1422 |
thus ?thesis unfolding SUPR_def by auto |
|
1423 |
qed |
|
1424 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1425 |
lemma SUPR_eq: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1426 |
assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1427 |
assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1428 |
shows "(SUP i:A. f i) = (SUP j:B. g j)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1429 |
proof (intro antisym) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1430 |
show "(SUP i:A. f i) \<le> (SUP j:B. g j)" |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1431 |
using assms by (metis SUP_leI le_SUPI2) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1432 |
show "(SUP i:B. g i) \<le> (SUP j:A. f j)" |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1433 |
using assms by (metis SUP_leI le_SUPI2) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1434 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1435 |
|
43920 | 1436 |
lemma SUP_ereal_le_addI: |
43923 | 1437 |
fixes f :: "'i \<Rightarrow> ereal" |
41978 | 1438 |
assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>" |
1439 |
shows "SUPR UNIV f + y \<le> z" |
|
1440 |
proof (cases y) |
|
1441 |
case (real r) |
|
43920 | 1442 |
then have "\<And>i. f i \<le> z - y" using assms by (simp add: ereal_le_minus_iff) |
41978 | 1443 |
then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI) |
43920 | 1444 |
then show ?thesis using real by (simp add: ereal_le_minus_iff) |
41978 | 1445 |
qed (insert assms, auto) |
1446 |
||
43920 | 1447 |
lemma SUPR_ereal_add: |
1448 |
fixes f g :: "nat \<Rightarrow> ereal" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1449 |
assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>" |
41978 | 1450 |
shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
43920 | 1451 |
proof (rule ereal_SUPI) |
41978 | 1452 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y" |
1453 |
have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos |
|
1454 |
unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD) |
|
1455 |
{ fix j |
|
1456 |
{ fix i |
|
1457 |
have "f i + g j \<le> f i + g (max i j)" |
|
1458 |
using `incseq g`[THEN incseqD] by (rule add_left_mono) auto |
|
1459 |
also have "\<dots> \<le> f (max i j) + g (max i j)" |
|
1460 |
using `incseq f`[THEN incseqD] by (rule add_right_mono) auto |
|
1461 |
also have "\<dots> \<le> y" using * by auto |
|
1462 |
finally have "f i + g j \<le> y" . } |
|
1463 |
then have "SUPR UNIV f + g j \<le> y" |
|
43920 | 1464 |
using assms(4)[of j] by (intro SUP_ereal_le_addI) auto |
41978 | 1465 |
then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) } |
1466 |
then have "SUPR UNIV g + SUPR UNIV f \<le> y" |
|
43920 | 1467 |
using f by (rule SUP_ereal_le_addI) |
41978 | 1468 |
then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps) |
1469 |
qed (auto intro!: add_mono le_SUPI) |
|
1470 |
||
43920 | 1471 |
lemma SUPR_ereal_add_pos: |
1472 |
fixes f g :: "nat \<Rightarrow> ereal" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1473 |
assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1474 |
shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
43920 | 1475 |
proof (intro SUPR_ereal_add inc) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1476 |
fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1477 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1478 |
|
43920 | 1479 |
lemma SUPR_ereal_setsum: |
1480 |
fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1481 |
assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1482 |
shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1483 |
proof cases |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1484 |
assume "finite A" then show ?thesis using assms |
43920 | 1485 |
by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_ereal_add_pos) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1486 |
qed simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1487 |
|
43920 | 1488 |
lemma SUPR_ereal_cmult: |
1489 |
fixes f :: "nat \<Rightarrow> ereal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c" |
|
41978 | 1490 |
shows "(SUP i. c * f i) = c * SUPR UNIV f" |
43920 | 1491 |
proof (rule ereal_SUPI) |
41978 | 1492 |
fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto |
1493 |
then show "c * f i \<le> c * SUPR UNIV f" |
|
43920 | 1494 |
using `0 \<le> c` by (rule ereal_mult_left_mono) |
41978 | 1495 |
next |
1496 |
fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y" |
|
1497 |
show "c * SUPR UNIV f \<le> y" |
|
1498 |
proof cases |
|
1499 |
assume c: "0 < c \<and> c \<noteq> \<infinity>" |
|
1500 |
with * have "SUPR UNIV f \<le> y / c" |
|
43920 | 1501 |
by (intro SUP_leI) (auto simp: ereal_le_divide_pos) |
41978 | 1502 |
with c show ?thesis |
43920 | 1503 |
by (auto simp: ereal_le_divide_pos) |
41978 | 1504 |
next |
1505 |
{ assume "c = \<infinity>" have ?thesis |
|
1506 |
proof cases |
|
1507 |
assume "\<forall>i. f i = 0" |
|
1508 |
moreover then have "range f = {0}" by auto |
|
1509 |
ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def) |
|
1510 |
next |
|
1511 |
assume "\<not> (\<forall>i. f i = 0)" |
|
1512 |
then obtain i where "f i \<noteq> 0" by auto |
|
1513 |
with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm) |
|
1514 |
qed } |
|
1515 |
moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)" |
|
1516 |
ultimately show ?thesis using * `0 \<le> c` by auto |
|
1517 |
qed |
|
1518 |
qed |
|
1519 |
||
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1520 |
lemma SUP_PInfty: |
43920 | 1521 |
fixes f :: "'a \<Rightarrow> ereal" |
1522 |
assumes "\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1523 |
shows "(SUP i:A. f i) = \<infinity>" |
43920 | 1524 |
unfolding SUPR_def Sup_eq_top_iff[where 'a=ereal, unfolded top_ereal_def] |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1525 |
apply simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1526 |
proof safe |
43923 | 1527 |
fix x :: ereal assume "x \<noteq> \<infinity>" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1528 |
show "\<exists>i\<in>A. x < f i" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1529 |
proof (cases x) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1530 |
case PInf with `x \<noteq> \<infinity>` show ?thesis by simp |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1531 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1532 |
case MInf with assms[of "0"] show ?thesis by force |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1533 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1534 |
case (real r) |
43920 | 1535 |
with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < ereal (real n)" by auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1536 |
moreover from assms[of n] guess i .. |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1537 |
ultimately show ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1538 |
by (auto intro!: bexI[of _ i]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1539 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1540 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1541 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1542 |
lemma Sup_countable_SUPR: |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1543 |
assumes "A \<noteq> {}" |
43920 | 1544 |
shows "\<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1545 |
proof (cases "Sup A") |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1546 |
case (real r) |
43920 | 1547 |
have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1548 |
proof |
43920 | 1549 |
fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / ereal (real n) < x" |
1550 |
using assms real by (intro Sup_ereal_close) (auto simp: one_ereal_def) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1551 |
then guess x .. |
43920 | 1552 |
then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / ereal (real n)" |
1553 |
by (auto intro!: exI[of _ x] simp: ereal_minus_less_iff) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1554 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1555 |
from choice[OF this] guess f .. note f = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1556 |
have "SUPR UNIV f = Sup A" |
43920 | 1557 |
proof (rule ereal_SUPI) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1558 |
fix i show "f i \<le> Sup A" using f |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1559 |
by (auto intro!: complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1560 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1561 |
fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1562 |
show "Sup A \<le> y" |
43920 | 1563 |
proof (rule ereal_le_epsilon, intro allI impI) |
1564 |
fix e :: ereal assume "0 < e" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1565 |
show "Sup A \<le> y + e" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1566 |
proof (cases e) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1567 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1568 |
hence "0 < r" using `0 < e` by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1569 |
then obtain n ::nat where *: "1 / real n < r" "0 < n" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1570 |
using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide) |
43920 | 1571 |
have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n] by auto |
1572 |
also have "1 / ereal (real n) \<le> e" using real * by (auto simp: one_ereal_def ) |
|
1573 |
with bound have "f n + 1 / ereal (real n) \<le> y + e" by (rule add_mono) simp |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1574 |
finally show "Sup A \<le> y + e" . |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1575 |
qed (insert `0 < e`, auto) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1576 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1577 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1578 |
with f show ?thesis by (auto intro!: exI[of _ f]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1579 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1580 |
case PInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1581 |
from `A \<noteq> {}` obtain x where "x \<in> A" by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1582 |
show ?thesis |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1583 |
proof cases |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1584 |
assume "\<infinity> \<in> A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1585 |
moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1586 |
ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1587 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1588 |
assume "\<infinity> \<notin> A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1589 |
have "\<exists>x\<in>A. 0 \<le> x" |
43920 | 1590 |
by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least ereal_infty_less_eq2 linorder_linear) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1591 |
then obtain x where "x \<in> A" "0 \<le> x" by auto |
43920 | 1592 |
have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + ereal (real n) \<le> f" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1593 |
proof (rule ccontr) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1594 |
assume "\<not> ?thesis" |
43920 | 1595 |
then have "\<exists>n::nat. Sup A \<le> x + ereal (real n)" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1596 |
by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1597 |
then show False using `x \<in> A` `\<infinity> \<notin> A` PInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1598 |
by(cases x) auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1599 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1600 |
from choice[OF this] guess f .. note f = this |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1601 |
have "SUPR UNIV f = \<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1602 |
proof (rule SUP_PInfty) |
43920 | 1603 |
fix n :: nat show "\<exists>i\<in>UNIV. ereal (real n) \<le> f i" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1604 |
using f[THEN spec, of n] `0 \<le> x` |
43920 | 1605 |
by (cases rule: ereal2_cases[of "f n" x]) (auto intro!: exI[of _ n]) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1606 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1607 |
then show ?thesis using f PInf by (auto intro!: exI[of _ f]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1608 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1609 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1610 |
case MInf |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1611 |
with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1612 |
then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"]) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1613 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1614 |
|
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1615 |
lemma SUPR_countable_SUPR: |
43920 | 1616 |
"A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1617 |
using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1618 |
|
43920 | 1619 |
lemma Sup_ereal_cadd: |
1620 |
fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1621 |
shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1622 |
proof (rule antisym) |
43920 | 1623 |
have *: "\<And>a::ereal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1624 |
by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1625 |
then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" . |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1626 |
show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1627 |
proof (cases a) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1628 |
case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1629 |
next |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1630 |
case (real r) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1631 |
then have **: "op + (- a) ` op + a ` A = A" |
43920 | 1632 |
by (auto simp: image_iff ac_simps zero_ereal_def[symmetric]) |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1633 |
from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding ** |
43920 | 1634 |
by (cases rule: ereal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1635 |
qed (insert `a \<noteq> -\<infinity>`, auto) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1636 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1637 |
|
43920 | 1638 |
lemma Sup_ereal_cminus: |
1639 |
fixes A :: "ereal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1640 |
shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A" |
43920 | 1641 |
using Sup_ereal_cadd[of "uminus ` A" a] assms |
1642 |
by (simp add: comp_def image_image minus_ereal_def |
|
1643 |
ereal_Sup_uminus_image_eq) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1644 |
|
43920 | 1645 |
lemma SUPR_ereal_cminus: |
43923 | 1646 |
fixes f :: "'i \<Rightarrow> ereal" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1647 |
fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1648 |
shows "(SUP x:A. a - f x) = a - (INF x:A. f x)" |
43920 | 1649 |
using Sup_ereal_cminus[of "f`A" a] assms |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1650 |
unfolding SUPR_def INFI_def image_image by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1651 |
|
43920 | 1652 |
lemma Inf_ereal_cminus: |
1653 |
fixes A :: "ereal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1654 |
shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1655 |
proof - |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1656 |
{ fix x have "-a - -x = -(a - x)" using assms by (cases x) auto } |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1657 |
moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A" |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1658 |
by (auto simp: image_image) |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1659 |
ultimately show ?thesis |
43920 | 1660 |
using Sup_ereal_cminus[of "uminus ` A" "-a"] assms |
1661 |
by (auto simp add: ereal_Sup_uminus_image_eq ereal_Inf_uminus_image_eq) |
|
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1662 |
qed |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1663 |
|
43920 | 1664 |
lemma INFI_ereal_cminus: |
43923 | 1665 |
fixes a :: ereal assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1666 |
shows "(INF x:A. a - f x) = a - (SUP x:A. f x)" |
43920 | 1667 |
using Inf_ereal_cminus[of "f`A" a] assms |
41979
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1668 |
unfolding SUPR_def INFI_def image_image |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1669 |
by auto |
b10ec1f5e9d5
lemmas about addition, SUP on countable sets and infinite sums for extreal
hoelzl
parents:
41978
diff
changeset
|
1670 |
|
43920 | 1671 |
lemma uminus_ereal_add_uminus_uminus: |
1672 |
fixes a b :: ereal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b" |
|
1673 |
by (cases rule: ereal2_cases[of a b]) auto |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1674 |
|
43920 | 1675 |
lemma INFI_ereal_add: |
43923 | 1676 |
fixes f :: "nat \<Rightarrow> ereal" |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1677 |
assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1678 |
shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1679 |
proof - |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1680 |
have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1681 |
using assms unfolding INF_less_iff by auto |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1682 |
{ fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i" |
43920 | 1683 |
by (rule uminus_ereal_add_uminus_uminus) } |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1684 |
then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))" |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1685 |
by simp |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1686 |
also have "\<dots> = INFI UNIV f + INFI UNIV g" |
43920 | 1687 |
unfolding ereal_INFI_uminus |
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1688 |
using assms INF_less |
43920 | 1689 |
by (subst SUPR_ereal_add) |
1690 |
(auto simp: ereal_SUPR_uminus intro!: uminus_ereal_add_uminus_uminus) |
|
42950
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1691 |
finally show ?thesis . |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1692 |
qed |
6e5c2a3c69da
move lemmas to Extended_Reals and Extended_Real_Limits
hoelzl
parents:
42600
diff
changeset
|
1693 |
|
43920 | 1694 |
subsection "Limits on @{typ ereal}" |
41973 | 1695 |
|
1696 |
subsubsection "Topological space" |
|
1697 |
||
43920 | 1698 |
instantiation ereal :: topological_space |
41973 | 1699 |
begin |
1700 |
||
43920 | 1701 |
definition "open A \<longleftrightarrow> open (ereal -` A) |
1702 |
\<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {ereal x <..} \<subseteq> A)) |
|
1703 |
\<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A))" |
|
41973 | 1704 |
|
43920 | 1705 |
lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)" |
1706 |
unfolding open_ereal_def by auto |
|
41973 | 1707 |
|
43920 | 1708 |
lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)" |
1709 |
unfolding open_ereal_def by auto |
|
41973 | 1710 |
|
43920 | 1711 |
lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{ereal x<..} \<subseteq> A" |
41973 | 1712 |
using open_PInfty[OF assms] by auto |
1713 |
||
43920 | 1714 |
lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<ereal x} \<subseteq> A" |
41973 | 1715 |
using open_MInfty[OF assms] by auto |
1716 |
||
43920 | 1717 |
lemma ereal_openE: assumes "open A" obtains x y where |
1718 |
"open (ereal -` A)" |
|
1719 |
"\<infinity> \<in> A \<Longrightarrow> {ereal x<..} \<subseteq> A" |
|
1720 |
"-\<infinity> \<in> A \<Longrightarrow> {..<ereal y} \<subseteq> A" |
|
1721 |
using assms open_ereal_def by auto |
|
41973 | 1722 |
|
1723 |
instance |
|
1724 |
proof |
|
43920 | 1725 |
let ?U = "UNIV::ereal set" |
1726 |
show "open ?U" unfolding open_ereal_def |
|
41975 | 1727 |
by (auto intro!: exI[of _ 0]) |
41973 | 1728 |
next |
43920 | 1729 |
fix S T::"ereal set" assume "open S" and "open T" |
1730 |
from `open S`[THEN ereal_openE] guess xS yS . |
|
1731 |
moreover from `open T`[THEN ereal_openE] guess xT yT . |
|
41975 | 1732 |
ultimately have |
43920 | 1733 |
"open (ereal -` (S \<inter> T))" |
1734 |
"\<infinity> \<in> S \<inter> T \<Longrightarrow> {ereal (max xS xT) <..} \<subseteq> S \<inter> T" |
|
1735 |
"-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< ereal (min yS yT)} \<subseteq> S \<inter> T" |
|
41975 | 1736 |
by auto |
43920 | 1737 |
then show "open (S Int T)" unfolding open_ereal_def by blast |
41973 | 1738 |
next |
43920 | 1739 |
fix K :: "ereal set set" assume "\<forall>S\<in>K. open S" |
1740 |
then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (ereal -` S) \<and> |
|
1741 |
(\<infinity> \<in> S \<longrightarrow> {ereal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< ereal y} \<subseteq> S)" |
|
1742 |
by (auto simp: open_ereal_def) |
|
1743 |
then show "open (Union K)" unfolding open_ereal_def |
|
41975 | 1744 |
proof (intro conjI impI) |
43920 | 1745 |
show "open (ereal -` \<Union>K)" |
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1746 |
using *[THEN choice] by (auto simp: vimage_Union) |
41975 | 1747 |
qed ((metis UnionE Union_upper subset_trans *)+) |
41973 | 1748 |
qed |
1749 |
end |
|
1750 |
||
43920 | 1751 |
lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)" |
1752 |
by (auto simp: inj_vimage_image_eq open_ereal_def) |
|
41976 | 1753 |
|
43920 | 1754 |
lemma open_ereal_vimage: "open S \<Longrightarrow> open (ereal -` S)" |
1755 |
unfolding open_ereal_def by auto |
|
41976 | 1756 |
|
43920 | 1757 |
lemma open_ereal_lessThan[intro, simp]: "open {..< a :: ereal}" |
41975 | 1758 |
proof - |
43920 | 1759 |
have "\<And>x. ereal -` {..<ereal x} = {..< x}" |
1760 |
"ereal -` {..< \<infinity>} = UNIV" "ereal -` {..< -\<infinity>} = {}" by auto |
|
1761 |
then show ?thesis by (cases a) (auto simp: open_ereal_def) |
|
41975 | 1762 |
qed |
1763 |
||
43920 | 1764 |
lemma open_ereal_greaterThan[intro, simp]: |
1765 |
"open {a :: ereal <..}" |
|
41975 | 1766 |
proof - |
43920 | 1767 |
have "\<And>x. ereal -` {ereal x<..} = {x<..}" |
1768 |
"ereal -` {\<infinity><..} = {}" "ereal -` {-\<infinity><..} = UNIV" by auto |
|
1769 |
then show ?thesis by (cases a) (auto simp: open_ereal_def) |
|
41975 | 1770 |
qed |
1771 |
||
43920 | 1772 |
lemma ereal_open_greaterThanLessThan[intro, simp]: "open {a::ereal <..< b}" |
41973 | 1773 |
unfolding greaterThanLessThan_def by auto |
1774 |
||
43920 | 1775 |
lemma closed_ereal_atLeast[simp, intro]: "closed {a :: ereal ..}" |
41973 | 1776 |
proof - |
1777 |
have "- {a ..} = {..< a}" by auto |
|
1778 |
then show "closed {a ..}" |
|
43920 | 1779 |
unfolding closed_def using open_ereal_lessThan by auto |
41973 | 1780 |
qed |
1781 |
||
43920 | 1782 |
lemma closed_ereal_atMost[simp, intro]: "closed {.. b :: ereal}" |
41973 | 1783 |
proof - |
1784 |
have "- {.. b} = {b <..}" by auto |
|
1785 |
then show "closed {.. b}" |
|
43920 | 1786 |
unfolding closed_def using open_ereal_greaterThan by auto |
41973 | 1787 |
qed |
1788 |
||
43920 | 1789 |
lemma closed_ereal_atLeastAtMost[simp, intro]: |
1790 |
shows "closed {a :: ereal .. b}" |
|
41973 | 1791 |
unfolding atLeastAtMost_def by auto |
1792 |
||
43920 | 1793 |
lemma closed_ereal_singleton: |
1794 |
"closed {a :: ereal}" |
|
1795 |
by (metis atLeastAtMost_singleton closed_ereal_atLeastAtMost) |
|
41973 | 1796 |
|
43920 | 1797 |
lemma ereal_open_cont_interval: |
43923 | 1798 |
fixes S :: "ereal set" |
41976 | 1799 |
assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 1800 |
obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S" |
1801 |
proof- |
|
43920 | 1802 |
from `open S` have "open (ereal -` S)" by (rule ereal_openE) |
1803 |
then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> ereal y \<in> S" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1804 |
using assms unfolding open_dist by force |
41975 | 1805 |
show thesis |
1806 |
proof (intro that subsetI) |
|
43920 | 1807 |
show "0 < ereal e" using `0 < e` by auto |
1808 |
fix y assume "y \<in> {x - ereal e<..<x + ereal e}" |
|
1809 |
with assms obtain t where "y = ereal t" "dist t (real x) < e" |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1810 |
apply (cases y) by (auto simp: dist_real_def) |
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1811 |
then show "y \<in> S" using e[of t] by auto |
41975 | 1812 |
qed |
41973 | 1813 |
qed |
1814 |
||
43920 | 1815 |
lemma ereal_open_cont_interval2: |
43923 | 1816 |
fixes S :: "ereal set" |
41976 | 1817 |
assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 1818 |
obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S" |
1819 |
proof- |
|
43920 | 1820 |
guess e using ereal_open_cont_interval[OF assms] . |
1821 |
with that[of "x-e" "x+e"] ereal_between[OF x, of e] |
|
41973 | 1822 |
show thesis by auto |
1823 |
qed |
|
1824 |
||
43920 | 1825 |
instance ereal :: t2_space |
41973 | 1826 |
proof |
43920 | 1827 |
fix x y :: ereal assume "x ~= y" |
1828 |
let "?P x (y::ereal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
41973 | 1829 |
|
43920 | 1830 |
{ fix x y :: ereal assume "x < y" |
1831 |
from ereal_dense[OF this] obtain z where z: "x < z" "z < y" by auto |
|
41973 | 1832 |
have "?P x y" |
1833 |
apply (rule exI[of _ "{..<z}"]) |
|
1834 |
apply (rule exI[of _ "{z<..}"]) |
|
1835 |
using z by auto } |
|
1836 |
note * = this |
|
1837 |
||
1838 |
from `x ~= y` |
|
1839 |
show "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
1840 |
proof (cases rule: linorder_cases) |
|
1841 |
assume "x = y" with `x ~= y` show ?thesis by simp |
|
1842 |
next assume "x < y" from *[OF this] show ?thesis by auto |
|
1843 |
next assume "y < x" from *[OF this] show ?thesis by auto |
|
1844 |
qed |
|
1845 |
qed |
|
1846 |
||
1847 |
subsubsection {* Convergent sequences *} |
|
1848 |
||
43920 | 1849 |
lemma lim_ereal[simp]: |
1850 |
"((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r") |
|
41973 | 1851 |
proof (intro iffI topological_tendstoI) |
1852 |
fix S assume "?l" "open S" "x \<in> S" |
|
1853 |
then show "eventually (\<lambda>x. f x \<in> S) net" |
|
43920 | 1854 |
using `?l`[THEN topological_tendstoD, OF open_ereal, OF `open S`] |
41973 | 1855 |
by (simp add: inj_image_mem_iff) |
1856 |
next |
|
43920 | 1857 |
fix S assume "?r" "open S" "ereal x \<in> S" |
1858 |
show "eventually (\<lambda>x. ereal (f x) \<in> S) net" |
|
1859 |
using `?r`[THEN topological_tendstoD, OF open_ereal_vimage, OF `open S`] |
|
1860 |
using `ereal x \<in> S` by auto |
|
41973 | 1861 |
qed |
1862 |
||
43920 | 1863 |
lemma lim_real_of_ereal[simp]: |
1864 |
assumes lim: "(f ---> ereal x) net" |
|
41973 | 1865 |
shows "((\<lambda>x. real (f x)) ---> x) net" |
1866 |
proof (intro topological_tendstoI) |
|
1867 |
fix S assume "open S" "x \<in> S" |
|
43920 | 1868 |
then have S: "open S" "ereal x \<in> ereal ` S" |
41973 | 1869 |
by (simp_all add: inj_image_mem_iff) |
43920 | 1870 |
have "\<forall>x. f x \<in> ereal ` S \<longrightarrow> real (f x) \<in> S" by auto |
1871 |
from this lim[THEN topological_tendstoD, OF open_ereal, OF S] |
|
41973 | 1872 |
show "eventually (\<lambda>x. real (f x) \<in> S) net" |
1873 |
by (rule eventually_mono) |
|
1874 |
qed |
|
1875 |
||
43920 | 1876 |
lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= ereal B)" (is "?l = ?r") |
43923 | 1877 |
proof |
1878 |
assume ?r |
|
1879 |
show ?l |
|
1880 |
apply(rule topological_tendstoI) |
|
41973 | 1881 |
unfolding eventually_sequentially |
43923 | 1882 |
proof- |
1883 |
fix S :: "ereal set" assume "open S" "\<infinity> : S" |
|
41973 | 1884 |
from open_PInfty[OF this] guess B .. note B=this |
1885 |
from `?r`[rule_format,of "B+1"] guess N .. note N=this |
|
1886 |
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1887 |
proof safe case goal1 |
|
43920 | 1888 |
have "ereal B < ereal (B + 1)" by auto |
41973 | 1889 |
also have "... <= f n" using goal1 N by auto |
1890 |
finally show ?case using B by fastsimp |
|
1891 |
qed |
|
1892 |
qed |
|
43923 | 1893 |
next |
1894 |
assume ?l |
|
1895 |
show ?r |
|
43920 | 1896 |
proof fix B::real have "open {ereal B<..}" "\<infinity> : {ereal B<..}" by auto |
41973 | 1897 |
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
1898 |
guess N .. note N=this |
|
43920 | 1899 |
show "EX N. ALL n>=N. ereal B <= f n" apply(rule_tac x=N in exI) using N by auto |
41973 | 1900 |
qed |
1901 |
qed |
|
1902 |
||
1903 |
||
43920 | 1904 |
lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= ereal B)" (is "?l = ?r") |
43923 | 1905 |
proof |
1906 |
assume ?r |
|
1907 |
show ?l |
|
1908 |
apply(rule topological_tendstoI) |
|
41973 | 1909 |
unfolding eventually_sequentially |
43923 | 1910 |
proof- |
1911 |
fix S :: "ereal set" |
|
1912 |
assume "open S" "(-\<infinity>) : S" |
|
41973 | 1913 |
from open_MInfty[OF this] guess B .. note B=this |
1914 |
from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this |
|
1915 |
show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1916 |
proof safe case goal1 |
|
43920 | 1917 |
have "ereal (B - 1) >= f n" using goal1 N by auto |
1918 |
also have "... < ereal B" by auto |
|
41973 | 1919 |
finally show ?case using B by fastsimp |
1920 |
qed |
|
1921 |
qed |
|
1922 |
next assume ?l show ?r |
|
43920 | 1923 |
proof fix B::real have "open {..<ereal B}" "(-\<infinity>) : {..<ereal B}" by auto |
41973 | 1924 |
from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
1925 |
guess N .. note N=this |
|
43920 | 1926 |
show "EX N. ALL n>=N. ereal B >= f n" apply(rule_tac x=N in exI) using N by auto |
41973 | 1927 |
qed |
1928 |
qed |
|
1929 |
||
1930 |
||
43920 | 1931 |
lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= ereal B" shows "l ~= \<infinity>" |
41973 | 1932 |
proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>" |
1933 |
from lim[unfolded this Lim_PInfty,rule_format,of "?B"] |
|
1934 |
guess N .. note N=this[rule_format,OF le_refl] |
|
43920 | 1935 |
hence "ereal ?B <= ereal B" using assms(2)[of N] by(rule order_trans) |
1936 |
hence "ereal ?B < ereal ?B" apply (rule le_less_trans) by auto |
|
41973 | 1937 |
thus False by auto |
1938 |
qed |
|
1939 |
||
1940 |
||
43920 | 1941 |
lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= ereal B" shows "l ~= (-\<infinity>)" |
41973 | 1942 |
proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)" |
1943 |
from lim[unfolded this Lim_MInfty,rule_format,of "?B"] |
|
1944 |
guess N .. note N=this[rule_format,OF le_refl] |
|
43920 | 1945 |
hence "ereal B <= ereal ?B" using assms(2)[of N] order_trans[of "ereal B" "f N" "ereal(B - 1)"] by blast |
41973 | 1946 |
thus False by auto |
1947 |
qed |
|
1948 |
||
1949 |
||
1950 |
lemma tendsto_explicit: |
|
1951 |
"f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))" |
|
1952 |
unfolding tendsto_def eventually_sequentially by auto |
|
1953 |
||
1954 |
||
1955 |
lemma tendsto_obtains_N: |
|
1956 |
assumes "f ----> f0" |
|
1957 |
assumes "open S" "f0 : S" |
|
1958 |
obtains N where "ALL n>=N. f n : S" |
|
1959 |
using tendsto_explicit[of f f0] assms by auto |
|
1960 |
||
1961 |
||
1962 |
lemma tail_same_limit: |
|
1963 |
fixes X Y N |
|
1964 |
assumes "X ----> L" "ALL n>=N. X n = Y n" |
|
1965 |
shows "Y ----> L" |
|
1966 |
proof- |
|
1967 |
{ fix S assume "open S" and "L:S" |
|
1968 |
from this obtain N1 where "ALL n>=N1. X n : S" |
|
1969 |
using assms unfolding tendsto_def eventually_sequentially by auto |
|
1970 |
hence "ALL n>=max N N1. Y n : S" using assms by auto |
|
1971 |
hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto |
|
1972 |
} |
|
1973 |
thus ?thesis using tendsto_explicit by auto |
|
1974 |
qed |
|
1975 |
||
1976 |
||
1977 |
lemma Lim_bounded_PInfty2: |
|
43920 | 1978 |
assumes lim:"f ----> l" and "ALL n>=N. f n <= ereal B" |
41973 | 1979 |
shows "l ~= \<infinity>" |
1980 |
proof- |
|
43920 | 1981 |
def g == "(%n. if n>=N then f n else ereal B)" |
41973 | 1982 |
hence "g ----> l" using tail_same_limit[of f l N g] lim by auto |
43920 | 1983 |
moreover have "!!n. g n <= ereal B" using g_def assms by auto |
41973 | 1984 |
ultimately show ?thesis using Lim_bounded_PInfty by auto |
1985 |
qed |
|
1986 |
||
43920 | 1987 |
lemma Lim_bounded_ereal: |
1988 |
assumes lim:"f ----> (l :: ereal)" |
|
41973 | 1989 |
and "ALL n>=M. f n <= C" |
1990 |
shows "l<=C" |
|
1991 |
proof- |
|
1992 |
{ assume "l=(-\<infinity>)" hence ?thesis by auto } |
|
1993 |
moreover |
|
1994 |
{ assume "~(l=(-\<infinity>))" |
|
1995 |
{ assume "C=\<infinity>" hence ?thesis by auto } |
|
1996 |
moreover |
|
1997 |
{ assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto |
|
1998 |
hence "l=(-\<infinity>)" using assms |
|
41980
28b51effc5ed
split Extended_Reals into parts for Library and Multivariate_Analysis
hoelzl
parents:
41979
diff
changeset
|
1999 |
tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto |
41973 | 2000 |
hence ?thesis by auto } |
2001 |
moreover |
|
43920 | 2002 |
{ assume "EX B. C = ereal B" |
2003 |
from this obtain B where B_def: "C=ereal B" by auto |
|
41973 | 2004 |
hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto |
43920 | 2005 |
from this obtain m where m_def: "ereal m=l" using `~(l=(-\<infinity>))` by (cases l) auto |
2006 |
from this obtain N where N_def: "ALL n>=N. f n : {ereal(m - 1) <..< ereal(m+1)}" |
|
2007 |
apply (subst tendsto_obtains_N[of f l "{ereal(m - 1) <..< ereal(m+1)}"]) using assms by auto |
|
41973 | 2008 |
{ fix n assume "n>=N" |
43920 | 2009 |
hence "EX r. ereal r = f n" using N_def by (cases "f n") auto |
2010 |
} from this obtain g where g_def: "ALL n>=N. ereal (g n) = f n" by metis |
|
2011 |
hence "(%n. ereal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto |
|
41973 | 2012 |
hence *: "(%n. g n) ----> m" using m_def by auto |
2013 |
{ fix n assume "n>=max N M" |
|
43920 | 2014 |
hence "ereal (g n) <= ereal B" using assms g_def B_def by auto |
41973 | 2015 |
hence "g n <= B" by auto |
2016 |
} hence "EX N. ALL n>=N. g n <= B" by blast |
|
2017 |
hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto |
|
2018 |
hence ?thesis using m_def B_def by auto |
|
2019 |
} ultimately have ?thesis by (cases C) auto |
|
2020 |
} ultimately show ?thesis by blast |
|
2021 |
qed |
|
2022 |
||
43920 | 2023 |
lemma real_of_ereal_mult[simp]: |
2024 |
fixes a b :: ereal shows "real (a * b) = real a * real b" |
|
2025 |
by (cases rule: ereal2_cases[of a b]) auto |
|
41973 | 2026 |
|
43920 | 2027 |
lemma real_of_ereal_eq_0: |
43923 | 2028 |
fixes x :: ereal shows "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0" |
41973 | 2029 |
by (cases x) auto |
2030 |
||
43920 | 2031 |
lemma tendsto_ereal_realD: |
2032 |
fixes f :: "'a \<Rightarrow> ereal" |
|
2033 |
assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. ereal (real (f x))) ---> x) net" |
|
41973 | 2034 |
shows "(f ---> x) net" |
2035 |
proof (intro topological_tendstoI) |
|
2036 |
fix S assume S: "open S" "x \<in> S" |
|
2037 |
with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto |
|
2038 |
from tendsto[THEN topological_tendstoD, OF this] |
|
2039 |
show "eventually (\<lambda>x. f x \<in> S) net" |
|
44142 | 2040 |
by (rule eventually_rev_mp) (auto simp: ereal_real) |
41973 | 2041 |
qed |
2042 |
||
43920 | 2043 |
lemma tendsto_ereal_realI: |
2044 |
fixes f :: "'a \<Rightarrow> ereal" |
|
41976 | 2045 |
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net" |
43920 | 2046 |
shows "((\<lambda>x. ereal (real (f x))) ---> x) net" |
41973 | 2047 |
proof (intro topological_tendstoI) |
2048 |
fix S assume "open S" "x \<in> S" |
|
2049 |
with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto |
|
2050 |
from tendsto[THEN topological_tendstoD, OF this] |
|
43920 | 2051 |
show "eventually (\<lambda>x. ereal (real (f x)) \<in> S) net" |
2052 |
by (elim eventually_elim1) (auto simp: ereal_real) |
|
41973 | 2053 |
qed |
2054 |
||
43920 | 2055 |
lemma ereal_mult_cancel_left: |
2056 |
fixes a b c :: ereal shows "a * b = a * c \<longleftrightarrow> |
|
41976 | 2057 |
((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)" |
43920 | 2058 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 2059 |
(simp_all add: zero_less_mult_iff) |
2060 |
||
43920 | 2061 |
lemma ereal_inj_affinity: |
43923 | 2062 |
fixes m t :: ereal |
41976 | 2063 |
assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>" |
41973 | 2064 |
shows "inj_on (\<lambda>x. m * x + t) A" |
2065 |
using assms |
|
43920 | 2066 |
by (cases rule: ereal2_cases[of m t]) |
2067 |
(auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left) |
|
41973 | 2068 |
|
43920 | 2069 |
lemma ereal_PInfty_eq_plus[simp]: |
43923 | 2070 |
fixes a b :: ereal |
41973 | 2071 |
shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
43920 | 2072 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 2073 |
|
43920 | 2074 |
lemma ereal_MInfty_eq_plus[simp]: |
43923 | 2075 |
fixes a b :: ereal |
41973 | 2076 |
shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)" |
43920 | 2077 |
by (cases rule: ereal2_cases[of a b]) auto |
41973 | 2078 |
|
43920 | 2079 |
lemma ereal_less_divide_pos: |
43923 | 2080 |
fixes x y :: ereal |
2081 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z" |
|
43920 | 2082 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 2083 |
|
43920 | 2084 |
lemma ereal_divide_less_pos: |
43923 | 2085 |
fixes x y z :: ereal |
2086 |
shows "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z" |
|
43920 | 2087 |
by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps) |
41973 | 2088 |
|
43920 | 2089 |
lemma ereal_divide_eq: |
43923 | 2090 |
fixes a b c :: ereal |
2091 |
shows "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c" |
|
43920 | 2092 |
by (cases rule: ereal3_cases[of a b c]) |
41973 | 2093 |
(simp_all add: field_simps) |
2094 |
||
43923 | 2095 |
lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal) \<noteq> -\<infinity>" |
41973 | 2096 |
by (cases a) auto |
2097 |
||
43920 | 2098 |
lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x" |
41973 | 2099 |
by (cases x) auto |
2100 |
||
43920 | 2101 |
lemma ereal_LimI_finite: |
43923 | 2102 |
fixes x :: ereal |
41976 | 2103 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 2104 |
assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r" |
2105 |
shows "u ----> x" |
|
2106 |
proof (rule topological_tendstoI, unfold eventually_sequentially) |
|
43920 | 2107 |
obtain rx where rx_def: "x=ereal rx" using assms by (cases x) auto |
41973 | 2108 |
fix S assume "open S" "x : S" |
43920 | 2109 |
then have "open (ereal -` S)" unfolding open_ereal_def by auto |
2110 |
with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> ereal y \<in> S" |
|
41975 | 2111 |
unfolding open_real_def rx_def by auto |
41973 | 2112 |
then obtain n where |
43920 | 2113 |
upper: "!!N. n <= N ==> u N < x + ereal r" and |
2114 |
lower: "!!N. n <= N ==> x < u N + ereal r" using assms(2)[of "ereal r"] by auto |
|
41973 | 2115 |
show "EX N. ALL n>=N. u n : S" |
2116 |
proof (safe intro!: exI[of _ n]) |
|
2117 |
fix N assume "n <= N" |
|
2118 |
from upper[OF this] lower[OF this] assms `0 < r` |
|
2119 |
have "u N ~: {\<infinity>,(-\<infinity>)}" by auto |
|
43920 | 2120 |
from this obtain ra where ra_def: "(u N) = ereal ra" by (cases "u N") auto |
41973 | 2121 |
hence "rx < ra + r" and "ra < rx + r" |
2122 |
using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto |
|
41975 | 2123 |
hence "dist (real (u N)) rx < r" |
41973 | 2124 |
using rx_def ra_def |
2125 |
by (auto simp: dist_real_def abs_diff_less_iff field_simps) |
|
41976 | 2126 |
from dist[OF this] show "u N : S" using `u N ~: {\<infinity>, -\<infinity>}` |
43920 | 2127 |
by (auto simp: ereal_real split: split_if_asm) |
41973 | 2128 |
qed |
2129 |
qed |
|
2130 |
||
43920 | 2131 |
lemma ereal_LimI_finite_iff: |
43923 | 2132 |
fixes x :: ereal |
41976 | 2133 |
assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
41973 | 2134 |
shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))" |
2135 |
(is "?lhs <-> ?rhs") |
|
41976 | 2136 |
proof |
2137 |
assume lim: "u ----> x" |
|
43920 | 2138 |
{ fix r assume "(r::ereal)>0" |
41973 | 2139 |
from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}" |
2140 |
apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"]) |
|
43920 | 2141 |
using lim ereal_between[of x r] assms `r>0` by auto |
41973 | 2142 |
hence "EX N. ALL n>=N. u n < x + r & x < u n + r" |
43920 | 2143 |
using ereal_minus_less[of r x] by (cases r) auto |
41976 | 2144 |
} then show "?rhs" by auto |
2145 |
next |
|
2146 |
assume ?rhs then show "u ----> x" |
|
43920 | 2147 |
using ereal_LimI_finite[of x] assms by auto |
41973 | 2148 |
qed |
2149 |
||
2150 |
||
2151 |
subsubsection {* @{text Liminf} and @{text Limsup} *} |
|
2152 |
||
2153 |
definition |
|
2154 |
"Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)" |
|
2155 |
||
2156 |
definition |
|
2157 |
"Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)" |
|
2158 |
||
2159 |
lemma Liminf_Sup: |
|
43941 | 2160 |
fixes f :: "'a => 'b::complete_linorder" |
41973 | 2161 |
shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}" |
2162 |
by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def) |
|
2163 |
||
2164 |
lemma Limsup_Inf: |
|
43941 | 2165 |
fixes f :: "'a => 'b::complete_linorder" |
41973 | 2166 |
shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}" |
2167 |
by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def) |
|
2168 |
||
43920 | 2169 |
lemma ereal_SupI: |
2170 |
fixes x :: ereal |
|
41973 | 2171 |
assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" |
2172 |
assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" |
|
2173 |
shows "Sup A = x" |
|
43920 | 2174 |
unfolding Sup_ereal_def |
41973 | 2175 |
using assms by (auto intro!: Least_equality) |
2176 |
||
43920 | 2177 |
lemma ereal_InfI: |
2178 |
fixes x :: ereal |
|
41973 | 2179 |
assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i" |
2180 |
assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x" |
|
2181 |
shows "Inf A = x" |
|
43920 | 2182 |
unfolding Inf_ereal_def |
41973 | 2183 |
using assms by (auto intro!: Greatest_equality) |
2184 |
||
2185 |
lemma Limsup_const: |
|
43941 | 2186 |
fixes c :: "'a::complete_linorder" |
41973 | 2187 |
assumes ntriv: "\<not> trivial_limit net" |
2188 |
shows "Limsup net (\<lambda>x. c) = c" |
|
2189 |
unfolding Limsup_Inf |
|
2190 |
proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower) |
|
2191 |
fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net" |
|
2192 |
show "c \<le> x" |
|
2193 |
proof (rule ccontr) |
|
2194 |
assume "\<not> c \<le> x" then have "x < c" by auto |
|
2195 |
then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2196 |
qed |
|
2197 |
qed auto |
|
2198 |
||
2199 |
lemma Liminf_const: |
|
43941 | 2200 |
fixes c :: "'a::complete_linorder" |
41973 | 2201 |
assumes ntriv: "\<not> trivial_limit net" |
2202 |
shows "Liminf net (\<lambda>x. c) = c" |
|
2203 |
unfolding Liminf_Sup |
|
2204 |
proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
|
2205 |
fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net" |
|
2206 |
show "x \<le> c" |
|
2207 |
proof (rule ccontr) |
|
2208 |
assume "\<not> x \<le> c" then have "c < x" by auto |
|
2209 |
then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2210 |
qed |
|
2211 |
qed auto |
|
2212 |
||
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2213 |
definition (in order) mono_set: |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2214 |
"mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)" |
41973 | 2215 |
|
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2216 |
lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2217 |
lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2218 |
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto |
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2219 |
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto |
41973 | 2220 |
|
43941 | 2221 |
lemma (in complete_linorder) mono_set_iff: |
2222 |
fixes S :: "'a set" |
|
41973 | 2223 |
defines "a \<equiv> Inf S" |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2224 |
shows "mono_set S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c") |
41973 | 2225 |
proof |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44142
diff
changeset
|
2226 |
assume "mono_set S" |
41973 | 2227 |
then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set) |
2228 |
show ?c |
|
2229 |
proof cases |
|
2230 |
assume "a \<in> S" |
|
2231 |
show ?c |
|
2232 |
using mono[OF _ `a \<in> S`] |
|
43941 | 2233 |
by (auto intro: Inf_lower simp: a_def) |
41973 | 2234 |
next |
2235 |
assume "a \<notin> S" |
|
2236 |
have "S = {a <..}" |
|
2237 |
proof safe |
|
2238 |
fix x assume "x \<in> S" |
|
43941 | 2239 |
then have "a \<le> x" unfolding a_def by (rule Inf_lower) |
41973 | 2240 |
then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto |
2241 |
next |
|
2242 |
fix x assume "a < x" |
|
2243 |
then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff .. |
|
2244 |
with mono[of y x] show "x \<in> S" by auto |
|
2245 |
qed |
|
2246 |
then show ?c .. |
|
2247 |
qed |
|
2248 |
qed auto |
|
2249 |
||
2250 |
lemma lim_imp_Liminf: |
|
43920 | 2251 |
fixes f :: "'a \<Rightarrow> ereal" |
41973 | 2252 |
assumes ntriv: "\<not> trivial_limit net" |
2253 |
assumes lim: "(f ---> f0) net" |
|
2254 |
shows "Liminf net f = f0" |
|
2255 |
unfolding Liminf_Sup |
|
43920 | 2256 |
proof (safe intro!: ereal_SupI) |
41973 | 2257 |
fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net" |
2258 |
show "y \<le> f0" |
|
43920 | 2259 |
proof (rule ereal_le_ereal) |
41973 | 2260 |
fix B assume "B < y" |
2261 |
{ assume "f0 < B" |
|
2262 |
then have "eventually (\<lambda>x. f x < B \<and> B < f x) net" |
|
2263 |
using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`] |
|
2264 |
by (auto intro: eventually_conj) |
|
2265 |
also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2266 |
finally have False using ntriv[unfolded trivial_limit_def] by auto |
|
2267 |
} then show "B \<le> f0" by (metis linorder_le_less_linear) |
|
2268 |
qed |
|
2269 |
next |
|
2270 |
fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y" |
|
2271 |
show "f0 \<le> y" |
|
2272 |
proof (safe intro!: *[rule_format]) |
|
2273 |
fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net" |
|
2274 |
using lim[THEN topological_tendstoD, of "{y <..}"] by auto |
|
2275 |
qed |
|
2276 |
qed |
|
2277 |
||
43920 | 2278 |
lemma ereal_Liminf_le_Limsup: |
2279 |
fixes f :: "'a \<Rightarrow> ereal" |
|
41973 | 2280 |
assumes ntriv: "\<not> trivial_limit net" |
2281 |
shows "Liminf net f \<le> Limsup net f" |
|
2282 |
unfolding Limsup_Inf Liminf_Sup |
|
2283 |
proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least) |
|
2284 |
fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net" |
|
2285 |
show "u \<le> v" |
|
2286 |
proof (rule ccontr) |
|
2287 |
assume "\<not> u \<le> v" |
|
2288 |
then obtain t where "t < u" "v < t" |
|
43920 | 2289 |
using ereal_dense[of v u] by (auto simp: not_le) |
41973 | 2290 |
then have "eventually (\<lambda>x. t < f x \<and> f x < t) net" |
2291 |
using * by (auto intro: eventually_conj) |
|
2292 |
also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2293 |
finally show False using ntriv by (auto simp: trivial_limit_def) |
|
2294 |
qed |
|
2295 |
qed |
|
2296 |
||
2297 |
lemma Liminf_mono: |
|
43920 | 2298 |
fixes f g :: "'a => ereal" |
41973 | 2299 |
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
2300 |
shows "Liminf net f \<le> Liminf net g" |
|
2301 |
unfolding Liminf_Sup |
|
2302 |
proof (safe intro!: Sup_mono bexI) |
|
2303 |
fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a" |
|
2304 |
then have "eventually (\<lambda>x. y < f x) net" by auto |
|
2305 |
then show "eventually (\<lambda>x. y < g x) net" |
|
2306 |
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2307 |
qed simp |
|
2308 |
||
2309 |
lemma Liminf_eq: |
|
43920 | 2310 |
fixes f g :: "'a \<Rightarrow> ereal" |
41973 | 2311 |
assumes "eventually (\<lambda>x. f x = g x) net" |
2312 |
shows "Liminf net f = Liminf net g" |
|
2313 |
by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto |
|
2314 |
||
2315 |
lemma Liminf_mono_all: |
|
43920 | 2316 |
fixes f g :: "'a \<Rightarrow> ereal" |
41973 | 2317 |
assumes "\<And>x. f x \<le> g x" |
2318 |
shows "Liminf net f \<le> Liminf net g" |
|
2319 |
using assms by (intro Liminf_mono always_eventually) auto |
|
2320 |
||
2321 |
lemma Limsup_mono: |
|
43920 | 2322 |
fixes f g :: "'a \<Rightarrow> ereal" |
41973 | 2323 |
assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
2324 |
shows "Limsup net f \<le> Limsup net g" |
|
2325 |
unfolding Limsup_Inf |
|
2326 |
proof (safe intro!: Inf_mono bexI) |
|
2327 |
fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y" |
|
2328 |
then have "eventually (\<lambda>x. g x < y) net" by auto |
|
2329 |
then show "eventually (\<lambda>x. f x < y) net" |
|
2330 |
by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2331 |
qed simp |
|
2332 |
||
2333 |
lemma Limsup_mono_all: |
|
43920 | 2334 |
fixes f g :: "'a \<Rightarrow> ereal" |
41973 | 2335 |
assumes "\<And>x. f x \<le> g x" |
2336 |
shows "Limsup net f \<le> Limsup net g" |
|
2337 |
using assms by (intro Limsup_mono always_eventually) auto |
|
2338 |
||
2339 |
lemma Limsup_eq: |
|
43920 | 2340 |
fixes f g :: "'a \<Rightarrow> ereal" |
41973 | 2341 |
assumes "eventually (\<lambda>x. f x = g x) net" |
2342 |
shows "Limsup net f = Limsup net g" |
|
2343 |
by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto |
|
2344 |
||
2345 |
abbreviation "liminf \<equiv> Liminf sequentially" |
|
2346 |
||
2347 |
abbreviation "limsup \<equiv> Limsup sequentially" |
|
2348 |
||
2349 |
lemma liminf_SUPR_INFI: |
|
43920 | 2350 |
fixes f :: "nat \<Rightarrow> ereal" |
41973 | 2351 |
shows "liminf f = (SUP n. INF m:{n..}. f m)" |
2352 |
unfolding Liminf_Sup eventually_sequentially |
|
2353 |
proof (safe intro!: antisym complete_lattice_class.Sup_least) |
|
2354 |
fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)" |
|
43920 | 2355 |
proof (rule ereal_le_ereal) |
41973 | 2356 |
fix y assume "y < x" |
2357 |
with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto |
|
2358 |
then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff) |
|
2359 |
also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto |
|
2360 |
finally show "y \<le> (SUP n. INF m:{n..}. f m)" . |
|
2361 |
qed |
|
2362 |
next |
|
2363 |
show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}" |
|
2364 |
proof (unfold SUPR_def, safe intro!: Sup_mono bexI) |
|
2365 |
fix y n assume "y < INFI {n..} f" |
|
2366 |
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto |
|
2367 |
qed (rule order_refl) |
|
2368 |
qed |
|
2369 |
||
2370 |
lemma tail_same_limsup: |
|
43920 | 2371 |
fixes X Y :: "nat => ereal" |
41973 | 2372 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
2373 |
shows "limsup X = limsup Y" |
|
2374 |
using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto |
|
2375 |
||
2376 |
lemma tail_same_liminf: |
|
43920 | 2377 |
fixes X Y :: "nat => ereal" |
41973 | 2378 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
2379 |
shows "liminf X = liminf Y" |
|
2380 |
using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto |
|
2381 |
||
2382 |
lemma liminf_mono: |
|
43920 | 2383 |
fixes X Y :: "nat \<Rightarrow> ereal" |
41973 | 2384 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
2385 |
shows "liminf X \<le> liminf Y" |
|
2386 |
using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto |
|
2387 |
||
2388 |
lemma limsup_mono: |
|
43920 | 2389 |
fixes X Y :: "nat => ereal" |
41973 | 2390 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
2391 |
shows "limsup X \<le> limsup Y" |
|
2392 |
using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto |
|
2393 |
||
2394 |
declare trivial_limit_sequentially[simp] |
|
2395 |
||
41978 | 2396 |
lemma |
43920 | 2397 |
fixes X :: "nat \<Rightarrow> ereal" |
2398 |
shows ereal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X" |
|
2399 |
and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X" |
|
41978 | 2400 |
unfolding incseq_def decseq_def by auto |
2401 |
||
41973 | 2402 |
lemma liminf_bounded: |
43920 | 2403 |
fixes X Y :: "nat \<Rightarrow> ereal" |
41973 | 2404 |
assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n" |
2405 |
shows "C \<le> liminf X" |
|
2406 |
using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp |
|
2407 |
||
2408 |
lemma limsup_bounded: |
|
43920 | 2409 |
fixes X Y :: "nat => ereal" |
41973 | 2410 |
assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C" |
2411 |
shows "limsup X \<le> C" |
|
2412 |
using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp |
|
2413 |
||
2414 |
lemma liminf_bounded_iff: |
|
43920 | 2415 |
fixes x :: "nat \<Rightarrow> ereal" |
41973 | 2416 |
shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs") |
2417 |
proof safe |
|
2418 |
fix B assume "B < C" "C \<le> liminf x" |
|
2419 |
then have "B < liminf x" by auto |
|
2420 |
then obtain N where "B < (INF m:{N..}. x m)" |
|
2421 |
unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto |
|
2422 |
from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto |
|
2423 |
next |
|
2424 |
assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n" |
|
2425 |
{ fix B assume "B<C" |
|
2426 |
then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto |
|
2427 |
hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto |
|
2428 |
also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp |
|
2429 |
finally have "B \<le> liminf x" . |
|
2430 |
} then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear) |
|
2431 |
qed |
|
2432 |
||
2433 |
lemma liminf_subseq_mono: |
|
43920 | 2434 |
fixes X :: "nat \<Rightarrow> ereal" |
41973 | 2435 |
assumes "subseq r" |
2436 |
shows "liminf X \<le> liminf (X \<circ> r) " |
|
2437 |
proof- |
|
2438 |
have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)" |
|
2439 |
proof (safe intro!: INF_mono) |
|
2440 |
fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m" |
|
2441 |
using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto |
|
2442 |
qed |
|
2443 |
then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def) |
|
2444 |
qed |
|
2445 |
||
43920 | 2446 |
lemma ereal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "ereal (real x) = x" |
41976 | 2447 |
using assms by auto |
41973 | 2448 |
|
43920 | 2449 |
lemma ereal_le_ereal_bounded: |
2450 |
fixes x y z :: ereal |
|
41978 | 2451 |
assumes "z \<le> y" |
2452 |
assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y" |
|
2453 |
shows "x \<le> y" |
|
43920 | 2454 |
proof (rule ereal_le_ereal) |
41978 | 2455 |
fix B assume "B < x" |
2456 |
show "B \<le> y" |
|
2457 |
proof cases |
|
2458 |
assume "z < B" from *[OF this `B < x`] show "B \<le> y" . |
|
41976 | 2459 |
next |
41978 | 2460 |
assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto |
41976 | 2461 |
qed |
41973 | 2462 |
qed |
2463 |
||
43920 | 2464 |
lemma fixes x y :: ereal |
41978 | 2465 |
shows Sup_atMost[simp]: "Sup {.. y} = y" |
2466 |
and Sup_lessThan[simp]: "Sup {..< y} = y" |
|
2467 |
and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y" |
|
2468 |
and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y" |
|
2469 |
and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y" |
|
43920 | 2470 |
by (auto simp: Sup_ereal_def intro!: Least_equality |
2471 |
intro: ereal_le_ereal ereal_le_ereal_bounded[of x]) |
|
41978 | 2472 |
|
2473 |
lemma Sup_greaterThanlessThan[simp]: |
|
43920 | 2474 |
fixes x y :: ereal assumes "x < y" shows "Sup { x <..< y} = y" |
2475 |
unfolding Sup_ereal_def |
|
2476 |
proof (intro Least_equality ereal_le_ereal_bounded[of _ _ y]) |
|
41978 | 2477 |
fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z" |
43920 | 2478 |
from ereal_dense[OF `x < y`] guess w .. note w = this |
41978 | 2479 |
with z[THEN bspec, of w] show "x \<le> z" by auto |
2480 |
qed auto |
|
2481 |
||
43920 | 2482 |
lemma real_ereal_id: "real o ereal = id" |
41973 | 2483 |
proof- |
43920 | 2484 |
{ fix x have "(real o ereal) x = id x" by auto } |
41973 | 2485 |
from this show ?thesis using ext by blast |
2486 |
qed |
|
2487 |
||
43923 | 2488 |
lemma open_image_ereal: "open(UNIV-{ \<infinity> , (-\<infinity> :: ereal)})" |
43920 | 2489 |
by (metis range_ereal open_ereal open_UNIV) |
41973 | 2490 |
|
43920 | 2491 |
lemma ereal_le_distrib: |
2492 |
fixes a b c :: ereal shows "c * (a + b) \<le> c * a + c * b" |
|
2493 |
by (cases rule: ereal3_cases[of a b c]) |
|
41973 | 2494 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
2495 |
||
43920 | 2496 |
lemma ereal_pos_distrib: |
2497 |
fixes a b c :: ereal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b" |
|
2498 |
using assms by (cases rule: ereal3_cases[of a b c]) |
|
41973 | 2499 |
(auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
2500 |
||
43920 | 2501 |
lemma ereal_pos_le_distrib: |
2502 |
fixes a b c :: ereal |
|
41973 | 2503 |
assumes "c>=0" |
2504 |
shows "c * (a + b) <= c * a + c * b" |
|
43920 | 2505 |
using assms by (cases rule: ereal3_cases[of a b c]) |
41973 | 2506 |
(auto simp add: field_simps) |
2507 |
||
43920 | 2508 |
lemma ereal_max_mono: |
2509 |
"[| (a::ereal) <= b; c <= d |] ==> max a c <= max b d" |
|
2510 |
by (metis sup_ereal_def sup_mono) |
|
41973 | 2511 |
|
2512 |
||
43920 | 2513 |
lemma ereal_max_least: |
2514 |
"[| (a::ereal) <= x; c <= x |] ==> max a c <= x" |
|
2515 |
by (metis sup_ereal_def sup_least) |
|
41973 | 2516 |
|
43933 | 2517 |
subsubsection {* Tests for code generator *} |
2518 |
||
2519 |
(* A small list of simple arithmetic expressions *) |
|
2520 |
||
2521 |
value [code] "- \<infinity> :: ereal" |
|
2522 |
value [code] "\<bar>-\<infinity>\<bar> :: ereal" |
|
2523 |
value [code] "4 + 5 / 4 - ereal 2 :: ereal" |
|
2524 |
value [code] "ereal 3 < \<infinity>" |
|
2525 |
value [code] "real (\<infinity>::ereal) = 0" |
|
2526 |
||
41973 | 2527 |
end |