1 (* Title: HOL/Library/Extended_Reals.thy |
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2 Author: Johannes Hölzl, TU München |
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3 Author: Robert Himmelmann, TU München |
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4 Author: Armin Heller, TU München |
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5 Author: Bogdan Grechuk, University of Edinburgh |
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6 *) |
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7 |
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8 header {* Extended real number line *} |
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9 |
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10 theory Extended_Reals |
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11 imports Complex_Main |
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12 begin |
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13 |
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14 text {* |
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15 |
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16 For more lemmas about the extended real numbers go to |
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17 @{text "src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"} |
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18 |
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19 *} |
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20 |
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21 lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot" |
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22 proof |
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23 assume "{x..} = UNIV" |
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24 show "x = bot" |
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25 proof (rule ccontr) |
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26 assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less) |
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27 then show False using `{x..} = UNIV` by simp |
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28 qed |
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29 qed auto |
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30 |
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31 lemma SUPR_pair: |
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32 "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))" |
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33 by (rule antisym) (auto intro!: SUP_leI le_SUPI2) |
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34 |
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35 lemma INFI_pair: |
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36 "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))" |
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37 by (rule antisym) (auto intro!: le_INFI INF_leI2) |
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38 |
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39 subsection {* Definition and basic properties *} |
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40 |
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41 datatype extreal = extreal real | PInfty | MInfty |
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42 |
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43 notation (xsymbols) |
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44 PInfty ("\<infinity>") |
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45 |
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46 notation (HTML output) |
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47 PInfty ("\<infinity>") |
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48 |
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49 declare [[coercion "extreal :: real \<Rightarrow> extreal"]] |
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50 |
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51 instantiation extreal :: uminus |
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52 begin |
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53 fun uminus_extreal where |
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54 "- (extreal r) = extreal (- r)" |
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55 | "- \<infinity> = MInfty" |
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56 | "- MInfty = \<infinity>" |
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57 instance .. |
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58 end |
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59 |
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60 lemma inj_extreal[simp]: "inj_on extreal A" |
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61 unfolding inj_on_def by auto |
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62 |
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63 lemma MInfty_neq_PInfty[simp]: |
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64 "\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all |
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65 |
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66 lemma MInfty_neq_extreal[simp]: |
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67 "extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all |
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68 |
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69 lemma MInfinity_cases[simp]: |
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70 "(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z" |
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71 by simp |
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72 |
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73 lemma extreal_uminus_uminus[simp]: |
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74 fixes a :: extreal shows "- (- a) = a" |
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75 by (cases a) simp_all |
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76 |
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77 lemma MInfty_eq[simp, code_post]: |
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78 "MInfty = - \<infinity>" by simp |
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79 |
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80 declare uminus_extreal.simps(2)[code_inline, simp del] |
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81 |
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82 lemma extreal_cases[case_names real PInf MInf, cases type: extreal]: |
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83 assumes "\<And>r. x = extreal r \<Longrightarrow> P" |
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84 assumes "x = \<infinity> \<Longrightarrow> P" |
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85 assumes "x = -\<infinity> \<Longrightarrow> P" |
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86 shows P |
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87 using assms by (cases x) auto |
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88 |
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89 lemmas extreal2_cases = extreal_cases[case_product extreal_cases] |
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90 lemmas extreal3_cases = extreal2_cases[case_product extreal_cases] |
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91 |
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92 lemma extreal_uminus_eq_iff[simp]: |
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93 fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b" |
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94 by (cases rule: extreal2_cases[of a b]) simp_all |
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95 |
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96 function of_extreal :: "extreal \<Rightarrow> real" where |
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97 "of_extreal (extreal r) = r" | |
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98 "of_extreal \<infinity> = 0" | |
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99 "of_extreal (-\<infinity>) = 0" |
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100 by (auto intro: extreal_cases) |
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101 termination proof qed (rule wf_empty) |
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102 |
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103 defs (overloaded) |
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104 real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal" |
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105 |
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106 lemma real_of_extreal[simp]: |
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107 "real (- x :: extreal) = - (real x)" |
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108 "real (extreal r) = r" |
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109 "real \<infinity> = 0" |
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110 by (cases x) (simp_all add: real_of_extreal_def) |
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111 |
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112 lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}" |
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113 proof safe |
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114 fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>" |
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115 then show "x = -\<infinity>" by (cases x) auto |
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116 qed auto |
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117 |
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118 lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)" |
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119 proof safe |
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120 fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto |
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121 qed auto |
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122 |
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123 instantiation extreal :: number |
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124 begin |
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125 definition [simp]: "number_of x = extreal (number_of x)" |
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126 instance proof qed |
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127 end |
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128 |
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129 instantiation extreal :: abs |
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130 begin |
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131 function abs_extreal where |
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132 "\<bar>extreal r\<bar> = extreal \<bar>r\<bar>" |
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133 | "\<bar>-\<infinity>\<bar> = \<infinity>" |
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134 | "\<bar>\<infinity>\<bar> = \<infinity>" |
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135 by (auto intro: extreal_cases) |
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136 termination proof qed (rule wf_empty) |
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137 instance .. |
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138 end |
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139 |
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140 lemma abs_eq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> = \<infinity> ; x = \<infinity> \<Longrightarrow> P ; x = -\<infinity> \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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141 by (cases x) auto |
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142 |
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143 lemma abs_neq_infinity_cases[elim!]: "\<lbrakk> \<bar>x\<bar> \<noteq> \<infinity> ; \<And>r. x = extreal r \<Longrightarrow> P \<rbrakk> \<Longrightarrow> P" |
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144 by (cases x) auto |
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145 |
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146 lemma abs_extreal_uminus[simp]: "\<bar>- x\<bar> = \<bar>x::extreal\<bar>" |
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147 by (cases x) auto |
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148 |
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149 subsubsection "Addition" |
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150 |
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151 instantiation extreal :: comm_monoid_add |
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152 begin |
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153 |
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154 definition "0 = extreal 0" |
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155 |
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156 function plus_extreal where |
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157 "extreal r + extreal p = extreal (r + p)" | |
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158 "\<infinity> + a = \<infinity>" | |
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159 "a + \<infinity> = \<infinity>" | |
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160 "extreal r + -\<infinity> = - \<infinity>" | |
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161 "-\<infinity> + extreal p = -\<infinity>" | |
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162 "-\<infinity> + -\<infinity> = -\<infinity>" |
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163 proof - |
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164 case (goal1 P x) |
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165 moreover then obtain a b where "x = (a, b)" by (cases x) auto |
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166 ultimately show P |
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167 by (cases rule: extreal2_cases[of a b]) auto |
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168 qed auto |
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169 termination proof qed (rule wf_empty) |
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170 |
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171 lemma Infty_neq_0[simp]: |
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172 "\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>" |
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173 "-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>" |
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174 by (simp_all add: zero_extreal_def) |
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175 |
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176 lemma extreal_eq_0[simp]: |
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177 "extreal r = 0 \<longleftrightarrow> r = 0" |
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178 "0 = extreal r \<longleftrightarrow> r = 0" |
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179 unfolding zero_extreal_def by simp_all |
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180 |
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181 instance |
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182 proof |
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183 fix a :: extreal show "0 + a = a" |
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184 by (cases a) (simp_all add: zero_extreal_def) |
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185 fix b :: extreal show "a + b = b + a" |
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186 by (cases rule: extreal2_cases[of a b]) simp_all |
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187 fix c :: extreal show "a + b + c = a + (b + c)" |
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188 by (cases rule: extreal3_cases[of a b c]) simp_all |
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189 qed |
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190 end |
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191 |
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192 lemma real_of_extreal_0[simp]: "real (0::extreal) = 0" |
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193 unfolding real_of_extreal_def zero_extreal_def by simp |
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194 |
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195 lemma abs_extreal_zero[simp]: "\<bar>0\<bar> = (0::extreal)" |
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196 unfolding zero_extreal_def abs_extreal.simps by simp |
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197 |
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198 lemma extreal_uminus_zero[simp]: |
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199 "- 0 = (0::extreal)" |
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200 by (simp add: zero_extreal_def) |
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201 |
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202 lemma extreal_uminus_zero_iff[simp]: |
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203 fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0" |
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204 by (cases a) simp_all |
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205 |
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206 lemma extreal_plus_eq_PInfty[simp]: |
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207 shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
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208 by (cases rule: extreal2_cases[of a b]) auto |
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209 |
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210 lemma extreal_plus_eq_MInfty[simp]: |
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211 shows "a + b = -\<infinity> \<longleftrightarrow> |
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212 (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>" |
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213 by (cases rule: extreal2_cases[of a b]) auto |
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214 |
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215 lemma extreal_add_cancel_left: |
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216 assumes "a \<noteq> -\<infinity>" |
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217 shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)" |
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218 using assms by (cases rule: extreal3_cases[of a b c]) auto |
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219 |
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220 lemma extreal_add_cancel_right: |
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221 assumes "a \<noteq> -\<infinity>" |
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222 shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)" |
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223 using assms by (cases rule: extreal3_cases[of a b c]) auto |
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224 |
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225 lemma extreal_real: |
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226 "extreal (real x) = (if \<bar>x\<bar> = \<infinity> then 0 else x)" |
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227 by (cases x) simp_all |
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228 |
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229 lemma real_of_extreal_add: |
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230 fixes a b :: extreal |
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231 shows "real (a + b) = (if (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real a + real b else 0)" |
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232 by (cases rule: extreal2_cases[of a b]) auto |
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233 |
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234 subsubsection "Linear order on @{typ extreal}" |
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235 |
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236 instantiation extreal :: linorder |
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237 begin |
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238 |
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239 function less_extreal where |
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240 "extreal x < extreal y \<longleftrightarrow> x < y" | |
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241 " \<infinity> < a \<longleftrightarrow> False" | |
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242 " a < -\<infinity> \<longleftrightarrow> False" | |
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243 "extreal x < \<infinity> \<longleftrightarrow> True" | |
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244 " -\<infinity> < extreal r \<longleftrightarrow> True" | |
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245 " -\<infinity> < \<infinity> \<longleftrightarrow> True" |
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246 proof - |
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247 case (goal1 P x) |
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248 moreover then obtain a b where "x = (a,b)" by (cases x) auto |
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249 ultimately show P by (cases rule: extreal2_cases[of a b]) auto |
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250 qed simp_all |
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251 termination by (relation "{}") simp |
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252 |
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253 definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y" |
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254 |
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255 lemma extreal_infty_less[simp]: |
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256 "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)" |
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257 "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)" |
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258 by (cases x, simp_all) (cases x, simp_all) |
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259 |
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260 lemma extreal_infty_less_eq[simp]: |
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261 "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>" |
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262 "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>" |
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263 by (auto simp add: less_eq_extreal_def) |
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264 |
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265 lemma extreal_less[simp]: |
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266 "extreal r < 0 \<longleftrightarrow> (r < 0)" |
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267 "0 < extreal r \<longleftrightarrow> (0 < r)" |
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268 "0 < \<infinity>" |
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269 "-\<infinity> < 0" |
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270 by (simp_all add: zero_extreal_def) |
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271 |
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272 lemma extreal_less_eq[simp]: |
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273 "x \<le> \<infinity>" |
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274 "-\<infinity> \<le> x" |
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275 "extreal r \<le> extreal p \<longleftrightarrow> r \<le> p" |
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276 "extreal r \<le> 0 \<longleftrightarrow> r \<le> 0" |
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277 "0 \<le> extreal r \<longleftrightarrow> 0 \<le> r" |
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278 by (auto simp add: less_eq_extreal_def zero_extreal_def) |
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279 |
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280 lemma extreal_infty_less_eq2: |
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281 "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>" |
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282 "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>" |
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283 by simp_all |
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284 |
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285 instance |
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286 proof |
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287 fix x :: extreal show "x \<le> x" |
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288 by (cases x) simp_all |
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289 fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
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290 by (cases rule: extreal2_cases[of x y]) auto |
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291 show "x \<le> y \<or> y \<le> x " |
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292 by (cases rule: extreal2_cases[of x y]) auto |
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293 { assume "x \<le> y" "y \<le> x" then show "x = y" |
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294 by (cases rule: extreal2_cases[of x y]) auto } |
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295 { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z" |
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296 by (cases rule: extreal3_cases[of x y z]) auto } |
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297 qed |
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298 end |
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299 |
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300 instance extreal :: ordered_ab_semigroup_add |
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301 proof |
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302 fix a b c :: extreal assume "a \<le> b" then show "c + a \<le> c + b" |
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303 by (cases rule: extreal3_cases[of a b c]) auto |
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304 qed |
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305 |
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306 lemma real_of_extreal_positive_mono: |
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307 "\<lbrakk>0 \<le> x; x \<le> y; y \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> real x \<le> real y" |
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308 by (cases rule: extreal2_cases[of x y]) auto |
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309 |
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310 lemma extreal_MInfty_lessI[intro, simp]: |
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311 "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a" |
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312 by (cases a) auto |
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313 |
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314 lemma extreal_less_PInfty[intro, simp]: |
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315 "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>" |
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316 by (cases a) auto |
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317 |
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318 lemma extreal_less_extreal_Ex: |
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319 fixes a b :: extreal |
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320 shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)" |
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321 by (cases x) auto |
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322 |
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323 lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>n::nat. x < extreal (real n))" |
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324 proof (cases x) |
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325 case (real r) then show ?thesis |
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326 using reals_Archimedean2[of r] by simp |
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327 qed simp_all |
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328 |
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329 lemma extreal_add_mono: |
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330 fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d" |
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331 using assms |
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332 apply (cases a) |
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333 apply (cases rule: extreal3_cases[of b c d], auto) |
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334 apply (cases rule: extreal3_cases[of b c d], auto) |
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335 done |
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336 |
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337 lemma extreal_minus_le_minus[simp]: |
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338 fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a" |
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339 by (cases rule: extreal2_cases[of a b]) auto |
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340 |
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341 lemma extreal_minus_less_minus[simp]: |
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342 fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a" |
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343 by (cases rule: extreal2_cases[of a b]) auto |
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344 |
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345 lemma extreal_le_real_iff: |
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346 "x \<le> real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0))" |
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347 by (cases y) auto |
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348 |
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349 lemma real_le_extreal_iff: |
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350 "real y \<le> x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x))" |
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351 by (cases y) auto |
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352 |
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353 lemma extreal_less_real_iff: |
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354 "x < real y \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0))" |
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355 by (cases y) auto |
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356 |
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357 lemma real_less_extreal_iff: |
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358 "real y < x \<longleftrightarrow> ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))" |
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359 by (cases y) auto |
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360 |
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361 lemma real_of_extreal_pos: |
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362 fixes x :: extreal shows "0 \<le> x \<Longrightarrow> 0 \<le> real x" by (cases x) auto |
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363 |
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364 lemmas real_of_extreal_ord_simps = |
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365 extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff |
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366 |
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367 lemma abs_extreal_ge0[simp]: "0 \<le> x \<Longrightarrow> \<bar>x :: extreal\<bar> = x" |
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368 by (cases x) auto |
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369 |
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370 lemma abs_extreal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: extreal\<bar> = -x" |
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371 by (cases x) auto |
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372 |
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373 lemma abs_extreal_pos[simp]: "0 \<le> \<bar>x :: extreal\<bar>" |
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374 by (cases x) auto |
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375 |
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376 lemma real_of_extreal_le_0[simp]: "real (X :: extreal) \<le> 0 \<longleftrightarrow> (X \<le> 0 \<or> X = \<infinity>)" |
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377 by (cases X) auto |
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378 |
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379 lemma abs_real_of_extreal[simp]: "\<bar>real (X :: extreal)\<bar> = real \<bar>X\<bar>" |
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380 by (cases X) auto |
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381 |
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382 lemma zero_less_real_of_extreal: "0 < real X \<longleftrightarrow> (0 < X \<and> X \<noteq> \<infinity>)" |
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383 by (cases X) auto |
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384 |
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385 lemma extreal_0_le_uminus_iff[simp]: |
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386 fixes a :: extreal shows "0 \<le> -a \<longleftrightarrow> a \<le> 0" |
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387 by (cases rule: extreal2_cases[of a]) auto |
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388 |
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389 lemma extreal_uminus_le_0_iff[simp]: |
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390 fixes a :: extreal shows "-a \<le> 0 \<longleftrightarrow> 0 \<le> a" |
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391 by (cases rule: extreal2_cases[of a]) auto |
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392 |
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393 lemma extreal_dense: |
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394 fixes x y :: extreal assumes "x < y" |
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395 shows "EX z. x < z & z < y" |
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396 proof - |
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397 { assume a: "x = (-\<infinity>)" |
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398 { assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) } |
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399 moreover |
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400 { assume "y ~= \<infinity>" |
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401 with `x < y` obtain r where r: "y = extreal r" by (cases y) auto |
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402 hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"]) |
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403 } ultimately have ?thesis by auto |
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404 } |
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405 moreover |
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406 { assume "x ~= (-\<infinity>)" |
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407 with `x < y` obtain p where p: "x = extreal p" by (cases x) auto |
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408 { assume "y = \<infinity>" hence ?thesis using `x < y` p |
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409 by (auto intro!: exI[of _ "extreal (p + 1)"]) } |
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410 moreover |
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411 { assume "y ~= \<infinity>" |
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412 with `x < y` obtain r where r: "y = extreal r" by (cases y) auto |
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413 with p `x < y` have "p < r" by auto |
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414 with dense obtain z where "p < z" "z < r" by auto |
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415 hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"]) |
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416 } ultimately have ?thesis by auto |
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417 } ultimately show ?thesis by auto |
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418 qed |
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419 |
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420 lemma extreal_dense2: |
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421 fixes x y :: extreal assumes "x < y" |
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422 shows "EX z. x < extreal z & extreal z < y" |
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423 by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3)) |
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424 |
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425 lemma extreal_add_strict_mono: |
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426 fixes a b c d :: extreal |
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427 assumes "a = b" "0 \<le> a" "a \<noteq> \<infinity>" "c < d" |
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428 shows "a + c < b + d" |
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429 using assms by (cases rule: extreal3_cases[case_product extreal_cases, of a b c d]) auto |
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430 |
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431 lemma extreal_less_add: "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> a + c < a + b" |
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432 by (cases rule: extreal2_cases[of b c]) auto |
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433 |
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434 lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto |
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435 |
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436 lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)" |
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437 by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus) |
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438 |
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439 lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)" |
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440 by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus) |
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441 |
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442 lemmas extreal_uminus_reorder = |
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443 extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder |
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444 |
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445 lemma extreal_bot: |
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446 fixes x :: extreal assumes "\<And>B. x \<le> extreal B" shows "x = - \<infinity>" |
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447 proof (cases x) |
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448 case (real r) with assms[of "r - 1"] show ?thesis by auto |
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449 next case PInf with assms[of 0] show ?thesis by auto |
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450 next case MInf then show ?thesis by simp |
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451 qed |
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452 |
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453 lemma extreal_top: |
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454 fixes x :: extreal assumes "\<And>B. x \<ge> extreal B" shows "x = \<infinity>" |
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455 proof (cases x) |
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456 case (real r) with assms[of "r + 1"] show ?thesis by auto |
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457 next case MInf with assms[of 0] show ?thesis by auto |
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458 next case PInf then show ?thesis by simp |
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459 qed |
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460 |
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461 lemma |
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462 shows extreal_max[simp]: "extreal (max x y) = max (extreal x) (extreal y)" |
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463 and extreal_min[simp]: "extreal (min x y) = min (extreal x) (extreal y)" |
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464 by (simp_all add: min_def max_def) |
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465 |
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466 lemma extreal_max_0: "max 0 (extreal r) = extreal (max 0 r)" |
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467 by (auto simp: zero_extreal_def) |
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468 |
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469 lemma |
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470 fixes f :: "nat \<Rightarrow> extreal" |
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471 shows incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f" |
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472 and decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f" |
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473 unfolding decseq_def incseq_def by auto |
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474 |
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475 lemma incseq_extreal: "incseq f \<Longrightarrow> incseq (\<lambda>x. extreal (f x))" |
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476 unfolding incseq_def by auto |
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477 |
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478 lemma extreal_add_nonneg_nonneg: |
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479 fixes a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b" |
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480 using add_mono[of 0 a 0 b] by simp |
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481 |
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482 lemma image_eqD: "f ` A = B \<Longrightarrow> (\<forall>x\<in>A. f x \<in> B)" |
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483 by auto |
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484 |
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485 lemma incseq_setsumI: |
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486 fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
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487 assumes "\<And>i. 0 \<le> f i" |
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488 shows "incseq (\<lambda>i. setsum f {..< i})" |
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489 proof (intro incseq_SucI) |
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490 fix n have "setsum f {..< n} + 0 \<le> setsum f {..<n} + f n" |
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491 using assms by (rule add_left_mono) |
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492 then show "setsum f {..< n} \<le> setsum f {..< Suc n}" |
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493 by auto |
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494 qed |
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495 |
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496 lemma incseq_setsumI2: |
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497 fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{comm_monoid_add, ordered_ab_semigroup_add}" |
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498 assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" |
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499 shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)" |
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500 using assms unfolding incseq_def by (auto intro: setsum_mono) |
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501 |
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502 subsubsection "Multiplication" |
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503 |
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504 instantiation extreal :: "{comm_monoid_mult, sgn}" |
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505 begin |
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506 |
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507 definition "1 = extreal 1" |
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508 |
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509 function sgn_extreal where |
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510 "sgn (extreal r) = extreal (sgn r)" |
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511 | "sgn \<infinity> = 1" |
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512 | "sgn (-\<infinity>) = -1" |
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513 by (auto intro: extreal_cases) |
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514 termination proof qed (rule wf_empty) |
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515 |
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516 function times_extreal where |
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517 "extreal r * extreal p = extreal (r * p)" | |
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518 "extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
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519 "\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" | |
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520 "extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
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521 "-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" | |
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522 "\<infinity> * \<infinity> = \<infinity>" | |
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523 "-\<infinity> * \<infinity> = -\<infinity>" | |
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524 "\<infinity> * -\<infinity> = -\<infinity>" | |
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525 "-\<infinity> * -\<infinity> = \<infinity>" |
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526 proof - |
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527 case (goal1 P x) |
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528 moreover then obtain a b where "x = (a, b)" by (cases x) auto |
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529 ultimately show P by (cases rule: extreal2_cases[of a b]) auto |
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530 qed simp_all |
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531 termination by (relation "{}") simp |
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532 |
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533 instance |
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534 proof |
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535 fix a :: extreal show "1 * a = a" |
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536 by (cases a) (simp_all add: one_extreal_def) |
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537 fix b :: extreal show "a * b = b * a" |
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538 by (cases rule: extreal2_cases[of a b]) simp_all |
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539 fix c :: extreal show "a * b * c = a * (b * c)" |
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540 by (cases rule: extreal3_cases[of a b c]) |
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541 (simp_all add: zero_extreal_def zero_less_mult_iff) |
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542 qed |
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543 end |
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544 |
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545 lemma real_of_extreal_le_1: |
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546 fixes a :: extreal shows "a \<le> 1 \<Longrightarrow> real a \<le> 1" |
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547 by (cases a) (auto simp: one_extreal_def) |
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548 |
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549 lemma abs_extreal_one[simp]: "\<bar>1\<bar> = (1::extreal)" |
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550 unfolding one_extreal_def by simp |
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551 |
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552 lemma extreal_mult_zero[simp]: |
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553 fixes a :: extreal shows "a * 0 = 0" |
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554 by (cases a) (simp_all add: zero_extreal_def) |
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555 |
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556 lemma extreal_zero_mult[simp]: |
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557 fixes a :: extreal shows "0 * a = 0" |
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558 by (cases a) (simp_all add: zero_extreal_def) |
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559 |
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560 lemma extreal_m1_less_0[simp]: |
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561 "-(1::extreal) < 0" |
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562 by (simp add: zero_extreal_def one_extreal_def) |
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563 |
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564 lemma extreal_zero_m1[simp]: |
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565 "1 \<noteq> (0::extreal)" |
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566 by (simp add: zero_extreal_def one_extreal_def) |
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567 |
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568 lemma extreal_times_0[simp]: |
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569 fixes x :: extreal shows "0 * x = 0" |
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570 by (cases x) (auto simp: zero_extreal_def) |
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571 |
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572 lemma extreal_times[simp]: |
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573 "1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1" |
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574 "1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1" |
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575 by (auto simp add: times_extreal_def one_extreal_def) |
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576 |
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577 lemma extreal_plus_1[simp]: |
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578 "1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)" |
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579 "1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>" |
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580 unfolding one_extreal_def by auto |
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581 |
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582 lemma extreal_zero_times[simp]: |
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583 fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0" |
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584 by (cases rule: extreal2_cases[of a b]) auto |
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585 |
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586 lemma extreal_mult_eq_PInfty[simp]: |
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587 shows "a * b = \<infinity> \<longleftrightarrow> |
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588 (a = \<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = -\<infinity>)" |
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589 by (cases rule: extreal2_cases[of a b]) auto |
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590 |
|
591 lemma extreal_mult_eq_MInfty[simp]: |
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592 shows "a * b = -\<infinity> \<longleftrightarrow> |
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593 (a = \<infinity> \<and> b < 0) \<or> (a < 0 \<and> b = \<infinity>) \<or> (a = -\<infinity> \<and> b > 0) \<or> (a > 0 \<and> b = -\<infinity>)" |
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594 by (cases rule: extreal2_cases[of a b]) auto |
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595 |
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596 lemma extreal_0_less_1[simp]: "0 < (1::extreal)" |
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597 by (simp_all add: zero_extreal_def one_extreal_def) |
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598 |
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599 lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)" |
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600 by (simp_all add: zero_extreal_def one_extreal_def) |
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601 |
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602 lemma extreal_mult_minus_left[simp]: |
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603 fixes a b :: extreal shows "-a * b = - (a * b)" |
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604 by (cases rule: extreal2_cases[of a b]) auto |
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605 |
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606 lemma extreal_mult_minus_right[simp]: |
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607 fixes a b :: extreal shows "a * -b = - (a * b)" |
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608 by (cases rule: extreal2_cases[of a b]) auto |
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609 |
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610 lemma extreal_mult_infty[simp]: |
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611 "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
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612 by (cases a) auto |
|
613 |
|
614 lemma extreal_infty_mult[simp]: |
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615 "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)" |
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616 by (cases a) auto |
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617 |
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618 lemma extreal_mult_strict_right_mono: |
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619 assumes "a < b" and "0 < c" "c < \<infinity>" |
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620 shows "a * c < b * c" |
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621 using assms |
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622 by (cases rule: extreal3_cases[of a b c]) |
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623 (auto simp: zero_le_mult_iff extreal_less_PInfty) |
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624 |
|
625 lemma extreal_mult_strict_left_mono: |
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626 "\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b" |
|
627 using extreal_mult_strict_right_mono by (simp add: mult_commute[of c]) |
|
628 |
|
629 lemma extreal_mult_right_mono: |
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630 fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c" |
|
631 using assms |
|
632 apply (cases "c = 0") apply simp |
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633 by (cases rule: extreal3_cases[of a b c]) |
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634 (auto simp: zero_le_mult_iff extreal_less_PInfty) |
|
635 |
|
636 lemma extreal_mult_left_mono: |
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637 fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b" |
|
638 using extreal_mult_right_mono by (simp add: mult_commute[of c]) |
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639 |
|
640 lemma zero_less_one_extreal[simp]: "0 \<le> (1::extreal)" |
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641 by (simp add: one_extreal_def zero_extreal_def) |
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642 |
|
643 lemma extreal_0_le_mult[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * (b :: extreal)" |
|
644 by (cases rule: extreal2_cases[of a b]) (auto simp: mult_nonneg_nonneg) |
|
645 |
|
646 lemma extreal_right_distrib: |
|
647 fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> r * (a + b) = r * a + r * b" |
|
648 by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps) |
|
649 |
|
650 lemma extreal_left_distrib: |
|
651 fixes r a b :: extreal shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (a + b) * r = a * r + b * r" |
|
652 by (cases rule: extreal3_cases[of r a b]) (simp_all add: field_simps) |
|
653 |
|
654 lemma extreal_mult_le_0_iff: |
|
655 fixes a b :: extreal |
|
656 shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)" |
|
657 by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff) |
|
658 |
|
659 lemma extreal_zero_le_0_iff: |
|
660 fixes a b :: extreal |
|
661 shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)" |
|
662 by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff) |
|
663 |
|
664 lemma extreal_mult_less_0_iff: |
|
665 fixes a b :: extreal |
|
666 shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)" |
|
667 by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff) |
|
668 |
|
669 lemma extreal_zero_less_0_iff: |
|
670 fixes a b :: extreal |
|
671 shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)" |
|
672 by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff) |
|
673 |
|
674 lemma extreal_distrib: |
|
675 fixes a b c :: extreal |
|
676 assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "\<bar>c\<bar> \<noteq> \<infinity>" |
|
677 shows "(a + b) * c = a * c + b * c" |
|
678 using assms |
|
679 by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps) |
|
680 |
|
681 lemma extreal_le_epsilon: |
|
682 fixes x y :: extreal |
|
683 assumes "ALL e. 0 < e --> x <= y + e" |
|
684 shows "x <= y" |
|
685 proof- |
|
686 { assume a: "EX r. y = extreal r" |
|
687 from this obtain r where r_def: "y = extreal r" by auto |
|
688 { assume "x=(-\<infinity>)" hence ?thesis by auto } |
|
689 moreover |
|
690 { assume "~(x=(-\<infinity>))" |
|
691 from this obtain p where p_def: "x = extreal p" |
|
692 using a assms[rule_format, of 1] by (cases x) auto |
|
693 { fix e have "0 < e --> p <= r + e" |
|
694 using assms[rule_format, of "extreal e"] p_def r_def by auto } |
|
695 hence "p <= r" apply (subst field_le_epsilon) by auto |
|
696 hence ?thesis using r_def p_def by auto |
|
697 } ultimately have ?thesis by blast |
|
698 } |
|
699 moreover |
|
700 { assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis |
|
701 using assms[rule_format, of 1] by (cases x) auto |
|
702 } ultimately show ?thesis by (cases y) auto |
|
703 qed |
|
704 |
|
705 |
|
706 lemma extreal_le_epsilon2: |
|
707 fixes x y :: extreal |
|
708 assumes "ALL e. 0 < e --> x <= y + extreal e" |
|
709 shows "x <= y" |
|
710 proof- |
|
711 { fix e :: extreal assume "e>0" |
|
712 { assume "e=\<infinity>" hence "x<=y+e" by auto } |
|
713 moreover |
|
714 { assume "e~=\<infinity>" |
|
715 from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto |
|
716 hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto |
|
717 } ultimately have "x<=y+e" by blast |
|
718 } from this show ?thesis using extreal_le_epsilon by auto |
|
719 qed |
|
720 |
|
721 lemma extreal_le_real: |
|
722 fixes x y :: extreal |
|
723 assumes "ALL z. x <= extreal z --> y <= extreal z" |
|
724 shows "y <= x" |
|
725 by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1) |
|
726 extreal_less_eq(2) order_refl uminus_extreal.simps(2)) |
|
727 |
|
728 lemma extreal_le_extreal: |
|
729 fixes x y :: extreal |
|
730 assumes "\<And>B. B < x \<Longrightarrow> B <= y" |
|
731 shows "x <= y" |
|
732 by (metis assms extreal_dense leD linorder_le_less_linear) |
|
733 |
|
734 lemma extreal_ge_extreal: |
|
735 fixes x y :: extreal |
|
736 assumes "ALL B. B>x --> B >= y" |
|
737 shows "x >= y" |
|
738 by (metis assms extreal_dense leD linorder_le_less_linear) |
|
739 |
|
740 lemma setprod_extreal_0: |
|
741 fixes f :: "'a \<Rightarrow> extreal" |
|
742 shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> (finite A \<and> (\<exists>i\<in>A. f i = 0))" |
|
743 proof cases |
|
744 assume "finite A" |
|
745 then show ?thesis by (induct A) auto |
|
746 qed auto |
|
747 |
|
748 lemma setprod_extreal_pos: |
|
749 fixes f :: "'a \<Rightarrow> extreal" assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "0 \<le> (\<Prod>i\<in>I. f i)" |
|
750 proof cases |
|
751 assume "finite I" from this pos show ?thesis by induct auto |
|
752 qed simp |
|
753 |
|
754 lemma setprod_PInf: |
|
755 assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" |
|
756 shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)" |
|
757 proof cases |
|
758 assume "finite I" from this assms show ?thesis |
|
759 proof (induct I) |
|
760 case (insert i I) |
|
761 then have pos: "0 \<le> f i" "0 \<le> setprod f I" by (auto intro!: setprod_extreal_pos) |
|
762 from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> setprod f I * f i = \<infinity>" by auto |
|
763 also have "\<dots> \<longleftrightarrow> (setprod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> setprod f I \<noteq> 0" |
|
764 using setprod_extreal_pos[of I f] pos |
|
765 by (cases rule: extreal2_cases[of "f i" "setprod f I"]) auto |
|
766 also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)" |
|
767 using insert by (auto simp: setprod_extreal_0) |
|
768 finally show ?case . |
|
769 qed simp |
|
770 qed simp |
|
771 |
|
772 lemma setprod_extreal: "(\<Prod>i\<in>A. extreal (f i)) = extreal (setprod f A)" |
|
773 proof cases |
|
774 assume "finite A" then show ?thesis |
|
775 by induct (auto simp: one_extreal_def) |
|
776 qed (simp add: one_extreal_def) |
|
777 |
|
778 subsubsection {* Power *} |
|
779 |
|
780 instantiation extreal :: power |
|
781 begin |
|
782 primrec power_extreal where |
|
783 "power_extreal x 0 = 1" | |
|
784 "power_extreal x (Suc n) = x * x ^ n" |
|
785 instance .. |
|
786 end |
|
787 |
|
788 lemma extreal_power[simp]: "(extreal x) ^ n = extreal (x^n)" |
|
789 by (induct n) (auto simp: one_extreal_def) |
|
790 |
|
791 lemma extreal_power_PInf[simp]: "\<infinity> ^ n = (if n = 0 then 1 else \<infinity>)" |
|
792 by (induct n) (auto simp: one_extreal_def) |
|
793 |
|
794 lemma extreal_power_uminus[simp]: |
|
795 fixes x :: extreal |
|
796 shows "(- x) ^ n = (if even n then x ^ n else - (x^n))" |
|
797 by (induct n) (auto simp: one_extreal_def) |
|
798 |
|
799 lemma extreal_power_number_of[simp]: |
|
800 "(number_of num :: extreal) ^ n = extreal (number_of num ^ n)" |
|
801 by (induct n) (auto simp: one_extreal_def) |
|
802 |
|
803 lemma zero_le_power_extreal[simp]: |
|
804 fixes a :: extreal assumes "0 \<le> a" |
|
805 shows "0 \<le> a ^ n" |
|
806 using assms by (induct n) (auto simp: extreal_zero_le_0_iff) |
|
807 |
|
808 subsubsection {* Subtraction *} |
|
809 |
|
810 lemma extreal_minus_minus_image[simp]: |
|
811 fixes S :: "extreal set" |
|
812 shows "uminus ` uminus ` S = S" |
|
813 by (auto simp: image_iff) |
|
814 |
|
815 lemma extreal_uminus_lessThan[simp]: |
|
816 fixes a :: extreal shows "uminus ` {..<a} = {-a<..}" |
|
817 proof (safe intro!: image_eqI) |
|
818 fix x assume "-a < x" |
|
819 then have "- x < - (- a)" by (simp del: extreal_uminus_uminus) |
|
820 then show "- x < a" by simp |
|
821 qed auto |
|
822 |
|
823 lemma extreal_uminus_greaterThan[simp]: |
|
824 "uminus ` {(a::extreal)<..} = {..<-a}" |
|
825 by (metis extreal_uminus_lessThan extreal_uminus_uminus |
|
826 extreal_minus_minus_image) |
|
827 |
|
828 instantiation extreal :: minus |
|
829 begin |
|
830 definition "x - y = x + -(y::extreal)" |
|
831 instance .. |
|
832 end |
|
833 |
|
834 lemma extreal_minus[simp]: |
|
835 "extreal r - extreal p = extreal (r - p)" |
|
836 "-\<infinity> - extreal r = -\<infinity>" |
|
837 "extreal r - \<infinity> = -\<infinity>" |
|
838 "\<infinity> - x = \<infinity>" |
|
839 "-\<infinity> - \<infinity> = -\<infinity>" |
|
840 "x - -y = x + y" |
|
841 "x - 0 = x" |
|
842 "0 - x = -x" |
|
843 by (simp_all add: minus_extreal_def) |
|
844 |
|
845 lemma extreal_x_minus_x[simp]: |
|
846 "x - x = (if \<bar>x\<bar> = \<infinity> then \<infinity> else 0)" |
|
847 by (cases x) simp_all |
|
848 |
|
849 lemma extreal_eq_minus_iff: |
|
850 fixes x y z :: extreal |
|
851 shows "x = z - y \<longleftrightarrow> |
|
852 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and> |
|
853 (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and> |
|
854 (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and> |
|
855 (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)" |
|
856 by (cases rule: extreal3_cases[of x y z]) auto |
|
857 |
|
858 lemma extreal_eq_minus: |
|
859 fixes x y z :: extreal |
|
860 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z" |
|
861 by (auto simp: extreal_eq_minus_iff) |
|
862 |
|
863 lemma extreal_less_minus_iff: |
|
864 fixes x y z :: extreal |
|
865 shows "x < z - y \<longleftrightarrow> |
|
866 (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and> |
|
867 (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and> |
|
868 (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)" |
|
869 by (cases rule: extreal3_cases[of x y z]) auto |
|
870 |
|
871 lemma extreal_less_minus: |
|
872 fixes x y z :: extreal |
|
873 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z" |
|
874 by (auto simp: extreal_less_minus_iff) |
|
875 |
|
876 lemma extreal_le_minus_iff: |
|
877 fixes x y z :: extreal |
|
878 shows "x \<le> z - y \<longleftrightarrow> |
|
879 (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> |
|
880 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)" |
|
881 by (cases rule: extreal3_cases[of x y z]) auto |
|
882 |
|
883 lemma extreal_le_minus: |
|
884 fixes x y z :: extreal |
|
885 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z" |
|
886 by (auto simp: extreal_le_minus_iff) |
|
887 |
|
888 lemma extreal_minus_less_iff: |
|
889 fixes x y z :: extreal |
|
890 shows "x - y < z \<longleftrightarrow> |
|
891 y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> |
|
892 (y \<noteq> \<infinity> \<longrightarrow> x < z + y)" |
|
893 by (cases rule: extreal3_cases[of x y z]) auto |
|
894 |
|
895 lemma extreal_minus_less: |
|
896 fixes x y z :: extreal |
|
897 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y" |
|
898 by (auto simp: extreal_minus_less_iff) |
|
899 |
|
900 lemma extreal_minus_le_iff: |
|
901 fixes x y z :: extreal |
|
902 shows "x - y \<le> z \<longleftrightarrow> |
|
903 (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
904 (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and> |
|
905 (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)" |
|
906 by (cases rule: extreal3_cases[of x y z]) auto |
|
907 |
|
908 lemma extreal_minus_le: |
|
909 fixes x y z :: extreal |
|
910 shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y" |
|
911 by (auto simp: extreal_minus_le_iff) |
|
912 |
|
913 lemma extreal_minus_eq_minus_iff: |
|
914 fixes a b c :: extreal |
|
915 shows "a - b = a - c \<longleftrightarrow> |
|
916 b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)" |
|
917 by (cases rule: extreal3_cases[of a b c]) auto |
|
918 |
|
919 lemma extreal_add_le_add_iff: |
|
920 "c + a \<le> c + b \<longleftrightarrow> |
|
921 a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)" |
|
922 by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps) |
|
923 |
|
924 lemma extreal_mult_le_mult_iff: |
|
925 "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
|
926 by (cases rule: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left) |
|
927 |
|
928 lemma extreal_minus_mono: |
|
929 fixes A B C D :: extreal assumes "A \<le> B" "D \<le> C" |
|
930 shows "A - C \<le> B - D" |
|
931 using assms |
|
932 by (cases rule: extreal3_cases[case_product extreal_cases, of A B C D]) simp_all |
|
933 |
|
934 lemma real_of_extreal_minus: |
|
935 "real (a - b) = (if \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> then 0 else real a - real b)" |
|
936 by (cases rule: extreal2_cases[of a b]) auto |
|
937 |
|
938 lemma extreal_diff_positive: |
|
939 fixes a b :: extreal shows "a \<le> b \<Longrightarrow> 0 \<le> b - a" |
|
940 by (cases rule: extreal2_cases[of a b]) auto |
|
941 |
|
942 lemma extreal_between: |
|
943 fixes x e :: extreal |
|
944 assumes "\<bar>x\<bar> \<noteq> \<infinity>" "0 < e" |
|
945 shows "x - e < x" "x < x + e" |
|
946 using assms apply (cases x, cases e) apply auto |
|
947 using assms by (cases x, cases e) auto |
|
948 |
|
949 subsubsection {* Division *} |
|
950 |
|
951 instantiation extreal :: inverse |
|
952 begin |
|
953 |
|
954 function inverse_extreal where |
|
955 "inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" | |
|
956 "inverse \<infinity> = 0" | |
|
957 "inverse (-\<infinity>) = 0" |
|
958 by (auto intro: extreal_cases) |
|
959 termination by (relation "{}") simp |
|
960 |
|
961 definition "x / y = x * inverse (y :: extreal)" |
|
962 |
|
963 instance proof qed |
|
964 end |
|
965 |
|
966 lemma real_of_extreal_inverse[simp]: |
|
967 fixes a :: extreal |
|
968 shows "real (inverse a) = 1 / real a" |
|
969 by (cases a) (auto simp: inverse_eq_divide) |
|
970 |
|
971 lemma extreal_inverse[simp]: |
|
972 "inverse 0 = \<infinity>" |
|
973 "inverse (1::extreal) = 1" |
|
974 by (simp_all add: one_extreal_def zero_extreal_def) |
|
975 |
|
976 lemma extreal_divide[simp]: |
|
977 "extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))" |
|
978 unfolding divide_extreal_def by (auto simp: divide_real_def) |
|
979 |
|
980 lemma extreal_divide_same[simp]: |
|
981 "x / x = (if \<bar>x\<bar> = \<infinity> \<or> x = 0 then 0 else 1)" |
|
982 by (cases x) |
|
983 (simp_all add: divide_real_def divide_extreal_def one_extreal_def) |
|
984 |
|
985 lemma extreal_inv_inv[simp]: |
|
986 "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)" |
|
987 by (cases x) auto |
|
988 |
|
989 lemma extreal_inverse_minus[simp]: |
|
990 "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)" |
|
991 by (cases x) simp_all |
|
992 |
|
993 lemma extreal_uminus_divide[simp]: |
|
994 fixes x y :: extreal shows "- x / y = - (x / y)" |
|
995 unfolding divide_extreal_def by simp |
|
996 |
|
997 lemma extreal_divide_Infty[simp]: |
|
998 "x / \<infinity> = 0" "x / -\<infinity> = 0" |
|
999 unfolding divide_extreal_def by simp_all |
|
1000 |
|
1001 lemma extreal_divide_one[simp]: |
|
1002 "x / 1 = (x::extreal)" |
|
1003 unfolding divide_extreal_def by simp |
|
1004 |
|
1005 lemma extreal_divide_extreal[simp]: |
|
1006 "\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)" |
|
1007 unfolding divide_extreal_def by simp |
|
1008 |
|
1009 lemma zero_le_divide_extreal[simp]: |
|
1010 fixes a :: extreal assumes "0 \<le> a" "0 \<le> b" |
|
1011 shows "0 \<le> a / b" |
|
1012 using assms by (cases rule: extreal2_cases[of a b]) (auto simp: zero_le_divide_iff) |
|
1013 |
|
1014 lemma extreal_le_divide_pos: |
|
1015 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z" |
|
1016 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1017 |
|
1018 lemma extreal_divide_le_pos: |
|
1019 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y" |
|
1020 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1021 |
|
1022 lemma extreal_le_divide_neg: |
|
1023 "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y" |
|
1024 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1025 |
|
1026 lemma extreal_divide_le_neg: |
|
1027 "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z" |
|
1028 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
1029 |
|
1030 lemma extreal_inverse_antimono_strict: |
|
1031 fixes x y :: extreal |
|
1032 shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x" |
|
1033 by (cases rule: extreal2_cases[of x y]) auto |
|
1034 |
|
1035 lemma extreal_inverse_antimono: |
|
1036 fixes x y :: extreal |
|
1037 shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x" |
|
1038 by (cases rule: extreal2_cases[of x y]) auto |
|
1039 |
|
1040 lemma inverse_inverse_Pinfty_iff[simp]: |
|
1041 "inverse x = \<infinity> \<longleftrightarrow> x = 0" |
|
1042 by (cases x) auto |
|
1043 |
|
1044 lemma extreal_inverse_eq_0: |
|
1045 "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>" |
|
1046 by (cases x) auto |
|
1047 |
|
1048 lemma extreal_0_gt_inverse: |
|
1049 fixes x :: extreal shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x" |
|
1050 by (cases x) auto |
|
1051 |
|
1052 lemma extreal_mult_less_right: |
|
1053 assumes "b * a < c * a" "0 < a" "a < \<infinity>" |
|
1054 shows "b < c" |
|
1055 using assms |
|
1056 by (cases rule: extreal3_cases[of a b c]) |
|
1057 (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff) |
|
1058 |
|
1059 lemma extreal_power_divide: |
|
1060 "y \<noteq> 0 \<Longrightarrow> (x / y :: extreal) ^ n = x^n / y^n" |
|
1061 by (cases rule: extreal2_cases[of x y]) |
|
1062 (auto simp: one_extreal_def zero_extreal_def power_divide not_le |
|
1063 power_less_zero_eq zero_le_power_iff) |
|
1064 |
|
1065 lemma extreal_le_mult_one_interval: |
|
1066 fixes x y :: extreal |
|
1067 assumes y: "y \<noteq> -\<infinity>" |
|
1068 assumes z: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y" |
|
1069 shows "x \<le> y" |
|
1070 proof (cases x) |
|
1071 case PInf with z[of "1 / 2"] show "x \<le> y" by (simp add: one_extreal_def) |
|
1072 next |
|
1073 case (real r) note r = this |
|
1074 show "x \<le> y" |
|
1075 proof (cases y) |
|
1076 case (real p) note p = this |
|
1077 have "r \<le> p" |
|
1078 proof (rule field_le_mult_one_interval) |
|
1079 fix z :: real assume "0 < z" and "z < 1" |
|
1080 with z[of "extreal z"] |
|
1081 show "z * r \<le> p" using p r by (auto simp: zero_le_mult_iff one_extreal_def) |
|
1082 qed |
|
1083 then show "x \<le> y" using p r by simp |
|
1084 qed (insert y, simp_all) |
|
1085 qed simp |
|
1086 |
|
1087 subsection "Complete lattice" |
|
1088 |
|
1089 instantiation extreal :: lattice |
|
1090 begin |
|
1091 definition [simp]: "sup x y = (max x y :: extreal)" |
|
1092 definition [simp]: "inf x y = (min x y :: extreal)" |
|
1093 instance proof qed simp_all |
|
1094 end |
|
1095 |
|
1096 instantiation extreal :: complete_lattice |
|
1097 begin |
|
1098 |
|
1099 definition "bot = -\<infinity>" |
|
1100 definition "top = \<infinity>" |
|
1101 |
|
1102 definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)" |
|
1103 definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)" |
|
1104 |
|
1105 lemma extreal_complete_Sup: |
|
1106 fixes S :: "extreal set" assumes "S \<noteq> {}" |
|
1107 shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)" |
|
1108 proof cases |
|
1109 assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x" |
|
1110 then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto |
|
1111 then have "\<infinity> \<notin> S" by force |
|
1112 show ?thesis |
|
1113 proof cases |
|
1114 assume "S = {-\<infinity>}" |
|
1115 then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"]) |
|
1116 next |
|
1117 assume "S \<noteq> {-\<infinity>}" |
|
1118 with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto |
|
1119 with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y" |
|
1120 by (auto simp: real_of_extreal_ord_simps) |
|
1121 with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}` |
|
1122 obtain s where s: |
|
1123 "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z" |
|
1124 by auto |
|
1125 show ?thesis |
|
1126 proof (safe intro!: exI[of _ "extreal s"]) |
|
1127 fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s" |
|
1128 proof (cases z) |
|
1129 case (real r) |
|
1130 then show ?thesis |
|
1131 using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto |
|
1132 qed auto |
|
1133 next |
|
1134 fix z assume *: "\<forall>y\<in>S. y \<le> z" |
|
1135 with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z" |
|
1136 proof (cases z) |
|
1137 case (real u) |
|
1138 with * have "s \<le> u" |
|
1139 by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps) |
|
1140 then show ?thesis using real by simp |
|
1141 qed auto |
|
1142 qed |
|
1143 qed |
|
1144 next |
|
1145 assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)" |
|
1146 show ?thesis |
|
1147 proof (safe intro!: exI[of _ \<infinity>]) |
|
1148 fix y assume **: "\<forall>z\<in>S. z \<le> y" |
|
1149 with * show "\<infinity> \<le> y" |
|
1150 proof (cases y) |
|
1151 case MInf with * ** show ?thesis by (force simp: not_le) |
|
1152 qed auto |
|
1153 qed simp |
|
1154 qed |
|
1155 |
|
1156 lemma extreal_complete_Inf: |
|
1157 fixes S :: "extreal set" assumes "S ~= {}" |
|
1158 shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)" |
|
1159 proof- |
|
1160 def S1 == "uminus ` S" |
|
1161 hence "S1 ~= {}" using assms by auto |
|
1162 from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)" |
|
1163 using extreal_complete_Sup[of S1] by auto |
|
1164 { fix z assume "ALL y:S. z <= y" |
|
1165 hence "ALL y:S1. y <= -z" unfolding S1_def by auto |
|
1166 hence "x <= -z" using x_def by auto |
|
1167 hence "z <= -x" |
|
1168 apply (subst extreal_uminus_uminus[symmetric]) |
|
1169 unfolding extreal_minus_le_minus . } |
|
1170 moreover have "(ALL y:S. -x <= y)" |
|
1171 using x_def unfolding S1_def |
|
1172 apply simp |
|
1173 apply (subst (3) extreal_uminus_uminus[symmetric]) |
|
1174 unfolding extreal_minus_le_minus by simp |
|
1175 ultimately show ?thesis by auto |
|
1176 qed |
|
1177 |
|
1178 lemma extreal_complete_uminus_eq: |
|
1179 fixes S :: "extreal set" |
|
1180 shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z) |
|
1181 \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
1182 by simp (metis extreal_minus_le_minus extreal_uminus_uminus) |
|
1183 |
|
1184 lemma extreal_Sup_uminus_image_eq: |
|
1185 fixes S :: "extreal set" |
|
1186 shows "Sup (uminus ` S) = - Inf S" |
|
1187 proof cases |
|
1188 assume "S = {}" |
|
1189 moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)" |
|
1190 by (rule the_equality) (auto intro!: extreal_bot) |
|
1191 moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)" |
|
1192 by (rule some_equality) (auto intro!: extreal_top) |
|
1193 ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def |
|
1194 Least_def Greatest_def GreatestM_def by simp |
|
1195 next |
|
1196 assume "S \<noteq> {}" |
|
1197 with extreal_complete_Sup[of "uminus`S"] |
|
1198 obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)" |
|
1199 unfolding extreal_complete_uminus_eq by auto |
|
1200 show "Sup (uminus ` S) = - Inf S" |
|
1201 unfolding Inf_extreal_def Greatest_def GreatestM_def |
|
1202 proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"]) |
|
1203 show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)" |
|
1204 using x . |
|
1205 fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')" |
|
1206 then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)" |
|
1207 unfolding extreal_complete_uminus_eq by simp |
|
1208 then show "Sup (uminus ` S) = -x'" |
|
1209 unfolding Sup_extreal_def extreal_uminus_eq_iff |
|
1210 by (intro Least_equality) auto |
|
1211 qed |
|
1212 qed |
|
1213 |
|
1214 instance |
|
1215 proof |
|
1216 { fix x :: extreal and A |
|
1217 show "bot <= x" by (cases x) (simp_all add: bot_extreal_def) |
|
1218 show "x <= top" by (simp add: top_extreal_def) } |
|
1219 |
|
1220 { fix x :: extreal and A assume "x : A" |
|
1221 with extreal_complete_Sup[of A] |
|
1222 obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
|
1223 hence "x <= s" using `x : A` by auto |
|
1224 also have "... = Sup A" using s unfolding Sup_extreal_def |
|
1225 by (auto intro!: Least_equality[symmetric]) |
|
1226 finally show "x <= Sup A" . } |
|
1227 note le_Sup = this |
|
1228 |
|
1229 { fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)" |
|
1230 show "Sup A <= x" |
|
1231 proof (cases "A = {}") |
|
1232 case True |
|
1233 hence "Sup A = -\<infinity>" unfolding Sup_extreal_def |
|
1234 by (auto intro!: Least_equality) |
|
1235 thus "Sup A <= x" by simp |
|
1236 next |
|
1237 case False |
|
1238 with extreal_complete_Sup[of A] |
|
1239 obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto |
|
1240 hence "Sup A = s" |
|
1241 unfolding Sup_extreal_def by (auto intro!: Least_equality) |
|
1242 also have "s <= x" using * s by auto |
|
1243 finally show "Sup A <= x" . |
|
1244 qed } |
|
1245 note Sup_le = this |
|
1246 |
|
1247 { fix x :: extreal and A assume "x \<in> A" |
|
1248 with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x" |
|
1249 unfolding extreal_Sup_uminus_image_eq by simp } |
|
1250 |
|
1251 { fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)" |
|
1252 with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A" |
|
1253 unfolding extreal_Sup_uminus_image_eq by force } |
|
1254 qed |
|
1255 end |
|
1256 |
|
1257 lemma extreal_SUPR_uminus: |
|
1258 fixes f :: "'a => extreal" |
|
1259 shows "(SUP i : R. -(f i)) = -(INF i : R. f i)" |
|
1260 unfolding SUPR_def INFI_def |
|
1261 using extreal_Sup_uminus_image_eq[of "f`R"] |
|
1262 by (simp add: image_image) |
|
1263 |
|
1264 lemma extreal_INFI_uminus: |
|
1265 fixes f :: "'a => extreal" |
|
1266 shows "(INF i : R. -(f i)) = -(SUP i : R. f i)" |
|
1267 using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp |
|
1268 |
|
1269 lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)" |
|
1270 using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image) |
|
1271 |
|
1272 lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)" |
|
1273 by (auto intro!: inj_onI) |
|
1274 |
|
1275 lemma extreal_image_uminus_shift: |
|
1276 fixes X Y :: "extreal set" shows "uminus ` X = Y \<longleftrightarrow> X = uminus ` Y" |
|
1277 proof |
|
1278 assume "uminus ` X = Y" |
|
1279 then have "uminus ` uminus ` X = uminus ` Y" |
|
1280 by (simp add: inj_image_eq_iff) |
|
1281 then show "X = uminus ` Y" by (simp add: image_image) |
|
1282 qed (simp add: image_image) |
|
1283 |
|
1284 lemma Inf_extreal_iff: |
|
1285 fixes z :: extreal |
|
1286 shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y" |
|
1287 by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear |
|
1288 order_less_le_trans) |
|
1289 |
|
1290 lemma Sup_eq_MInfty: |
|
1291 fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}" |
|
1292 proof |
|
1293 assume a: "Sup S = -\<infinity>" |
|
1294 with complete_lattice_class.Sup_upper[of _ S] |
|
1295 show "S={} \<or> S={-\<infinity>}" by auto |
|
1296 next |
|
1297 assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>" |
|
1298 unfolding Sup_extreal_def by (auto intro!: Least_equality) |
|
1299 qed |
|
1300 |
|
1301 lemma Inf_eq_PInfty: |
|
1302 fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}" |
|
1303 using Sup_eq_MInfty[of "uminus`S"] |
|
1304 unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp |
|
1305 |
|
1306 lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>" |
|
1307 unfolding Inf_extreal_def |
|
1308 by (auto intro!: Greatest_equality) |
|
1309 |
|
1310 lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>" |
|
1311 unfolding Sup_extreal_def |
|
1312 by (auto intro!: Least_equality) |
|
1313 |
|
1314 lemma extreal_SUPI: |
|
1315 fixes x :: extreal |
|
1316 assumes "!!i. i : A ==> f i <= x" |
|
1317 assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y" |
|
1318 shows "(SUP i:A. f i) = x" |
|
1319 unfolding SUPR_def Sup_extreal_def |
|
1320 using assms by (auto intro!: Least_equality) |
|
1321 |
|
1322 lemma extreal_INFI: |
|
1323 fixes x :: extreal |
|
1324 assumes "!!i. i : A ==> f i >= x" |
|
1325 assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y" |
|
1326 shows "(INF i:A. f i) = x" |
|
1327 unfolding INFI_def Inf_extreal_def |
|
1328 using assms by (auto intro!: Greatest_equality) |
|
1329 |
|
1330 lemma Sup_extreal_close: |
|
1331 fixes e :: extreal |
|
1332 assumes "0 < e" and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}" |
|
1333 shows "\<exists>x\<in>S. Sup S - e < x" |
|
1334 using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1]) |
|
1335 |
|
1336 lemma Inf_extreal_close: |
|
1337 fixes e :: extreal assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>" "0 < e" |
|
1338 shows "\<exists>x\<in>X. x < Inf X + e" |
|
1339 proof (rule Inf_less_iff[THEN iffD1]) |
|
1340 show "Inf X < Inf X + e" using assms |
|
1341 by (cases e) auto |
|
1342 qed |
|
1343 |
|
1344 lemma Sup_eq_top_iff: |
|
1345 fixes A :: "'a::{complete_lattice, linorder} set" |
|
1346 shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)" |
|
1347 proof |
|
1348 assume *: "Sup A = top" |
|
1349 show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric] |
|
1350 proof (intro allI impI) |
|
1351 fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i" |
|
1352 unfolding less_Sup_iff by auto |
|
1353 qed |
|
1354 next |
|
1355 assume *: "\<forall>x<top. \<exists>i\<in>A. x < i" |
|
1356 show "Sup A = top" |
|
1357 proof (rule ccontr) |
|
1358 assume "Sup A \<noteq> top" |
|
1359 with top_greatest[of "Sup A"] |
|
1360 have "Sup A < top" unfolding le_less by auto |
|
1361 then have "Sup A < Sup A" |
|
1362 using * unfolding less_Sup_iff by auto |
|
1363 then show False by auto |
|
1364 qed |
|
1365 qed |
|
1366 |
|
1367 lemma SUP_eq_top_iff: |
|
1368 fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}" |
|
1369 shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)" |
|
1370 unfolding SUPR_def Sup_eq_top_iff by auto |
|
1371 |
|
1372 lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>" |
|
1373 proof - |
|
1374 { fix x assume "x \<noteq> \<infinity>" |
|
1375 then have "\<exists>k::nat. x < extreal (real k)" |
|
1376 proof (cases x) |
|
1377 case MInf then show ?thesis by (intro exI[of _ 0]) auto |
|
1378 next |
|
1379 case (real r) |
|
1380 moreover obtain k :: nat where "r < real k" |
|
1381 using ex_less_of_nat by (auto simp: real_eq_of_nat) |
|
1382 ultimately show ?thesis by auto |
|
1383 qed simp } |
|
1384 then show ?thesis |
|
1385 using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"] |
|
1386 by (auto simp: top_extreal_def) |
|
1387 qed |
|
1388 |
|
1389 lemma extreal_le_Sup: |
|
1390 fixes x :: extreal |
|
1391 shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))" |
|
1392 (is "?lhs <-> ?rhs") |
|
1393 proof- |
|
1394 { assume "?rhs" |
|
1395 { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le) |
|
1396 from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto |
|
1397 from this obtain i where "i : A & y <= f i" using `?rhs` by auto |
|
1398 hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto |
|
1399 hence False using y_def by auto |
|
1400 } hence "?lhs" by auto |
|
1401 } |
|
1402 moreover |
|
1403 { assume "?lhs" hence "?rhs" |
|
1404 by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff |
|
1405 inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
|
1406 } ultimately show ?thesis by auto |
|
1407 qed |
|
1408 |
|
1409 lemma extreal_Inf_le: |
|
1410 fixes x :: extreal |
|
1411 shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))" |
|
1412 (is "?lhs <-> ?rhs") |
|
1413 proof- |
|
1414 { assume "?rhs" |
|
1415 { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le) |
|
1416 from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto |
|
1417 from this obtain i where "i : A & f i <= y" using `?rhs` by auto |
|
1418 hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto |
|
1419 hence False using y_def by auto |
|
1420 } hence "?lhs" by auto |
|
1421 } |
|
1422 moreover |
|
1423 { assume "?lhs" hence "?rhs" |
|
1424 by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff |
|
1425 inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8)) |
|
1426 } ultimately show ?thesis by auto |
|
1427 qed |
|
1428 |
|
1429 lemma Inf_less: |
|
1430 fixes x :: extreal |
|
1431 assumes "(INF i:A. f i) < x" |
|
1432 shows "EX i. i : A & f i <= x" |
|
1433 proof(rule ccontr) |
|
1434 assume "~ (EX i. i : A & f i <= x)" |
|
1435 hence "ALL i:A. f i > x" by auto |
|
1436 hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto |
|
1437 thus False using assms by auto |
|
1438 qed |
|
1439 |
|
1440 lemma same_INF: |
|
1441 assumes "ALL e:A. f e = g e" |
|
1442 shows "(INF e:A. f e) = (INF e:A. g e)" |
|
1443 proof- |
|
1444 have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1445 thus ?thesis unfolding INFI_def by auto |
|
1446 qed |
|
1447 |
|
1448 lemma same_SUP: |
|
1449 assumes "ALL e:A. f e = g e" |
|
1450 shows "(SUP e:A. f e) = (SUP e:A. g e)" |
|
1451 proof- |
|
1452 have "f ` A = g ` A" unfolding image_def using assms by auto |
|
1453 thus ?thesis unfolding SUPR_def by auto |
|
1454 qed |
|
1455 |
|
1456 lemma SUPR_eq: |
|
1457 assumes "\<forall>i\<in>A. \<exists>j\<in>B. f i \<le> g j" |
|
1458 assumes "\<forall>j\<in>B. \<exists>i\<in>A. g j \<le> f i" |
|
1459 shows "(SUP i:A. f i) = (SUP j:B. g j)" |
|
1460 proof (intro antisym) |
|
1461 show "(SUP i:A. f i) \<le> (SUP j:B. g j)" |
|
1462 using assms by (metis SUP_leI le_SUPI2) |
|
1463 show "(SUP i:B. g i) \<le> (SUP j:A. f j)" |
|
1464 using assms by (metis SUP_leI le_SUPI2) |
|
1465 qed |
|
1466 |
|
1467 lemma SUP_extreal_le_addI: |
|
1468 assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>" |
|
1469 shows "SUPR UNIV f + y \<le> z" |
|
1470 proof (cases y) |
|
1471 case (real r) |
|
1472 then have "\<And>i. f i \<le> z - y" using assms by (simp add: extreal_le_minus_iff) |
|
1473 then have "SUPR UNIV f \<le> z - y" by (rule SUP_leI) |
|
1474 then show ?thesis using real by (simp add: extreal_le_minus_iff) |
|
1475 qed (insert assms, auto) |
|
1476 |
|
1477 lemma SUPR_extreal_add: |
|
1478 fixes f g :: "nat \<Rightarrow> extreal" |
|
1479 assumes "incseq f" "incseq g" and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>" |
|
1480 shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
|
1481 proof (rule extreal_SUPI) |
|
1482 fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> f i + g i \<le> y" |
|
1483 have f: "SUPR UNIV f \<noteq> -\<infinity>" using pos |
|
1484 unfolding SUPR_def Sup_eq_MInfty by (auto dest: image_eqD) |
|
1485 { fix j |
|
1486 { fix i |
|
1487 have "f i + g j \<le> f i + g (max i j)" |
|
1488 using `incseq g`[THEN incseqD] by (rule add_left_mono) auto |
|
1489 also have "\<dots> \<le> f (max i j) + g (max i j)" |
|
1490 using `incseq f`[THEN incseqD] by (rule add_right_mono) auto |
|
1491 also have "\<dots> \<le> y" using * by auto |
|
1492 finally have "f i + g j \<le> y" . } |
|
1493 then have "SUPR UNIV f + g j \<le> y" |
|
1494 using assms(4)[of j] by (intro SUP_extreal_le_addI) auto |
|
1495 then have "g j + SUPR UNIV f \<le> y" by (simp add: ac_simps) } |
|
1496 then have "SUPR UNIV g + SUPR UNIV f \<le> y" |
|
1497 using f by (rule SUP_extreal_le_addI) |
|
1498 then show "SUPR UNIV f + SUPR UNIV g \<le> y" by (simp add: ac_simps) |
|
1499 qed (auto intro!: add_mono le_SUPI) |
|
1500 |
|
1501 lemma SUPR_extreal_add_pos: |
|
1502 fixes f g :: "nat \<Rightarrow> extreal" |
|
1503 assumes inc: "incseq f" "incseq g" and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i" |
|
1504 shows "(SUP i. f i + g i) = SUPR UNIV f + SUPR UNIV g" |
|
1505 proof (intro SUPR_extreal_add inc) |
|
1506 fix i show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>" using pos[of i] by auto |
|
1507 qed |
|
1508 |
|
1509 lemma SUPR_extreal_setsum: |
|
1510 fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> extreal" |
|
1511 assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i" |
|
1512 shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPR UNIV (f n))" |
|
1513 proof cases |
|
1514 assume "finite A" then show ?thesis using assms |
|
1515 by induct (auto simp: incseq_setsumI2 setsum_nonneg SUPR_extreal_add_pos) |
|
1516 qed simp |
|
1517 |
|
1518 lemma SUPR_extreal_cmult: |
|
1519 fixes f :: "nat \<Rightarrow> extreal" assumes "\<And>i. 0 \<le> f i" "0 \<le> c" |
|
1520 shows "(SUP i. c * f i) = c * SUPR UNIV f" |
|
1521 proof (rule extreal_SUPI) |
|
1522 fix i have "f i \<le> SUPR UNIV f" by (rule le_SUPI) auto |
|
1523 then show "c * f i \<le> c * SUPR UNIV f" |
|
1524 using `0 \<le> c` by (rule extreal_mult_left_mono) |
|
1525 next |
|
1526 fix y assume *: "\<And>i. i \<in> UNIV \<Longrightarrow> c * f i \<le> y" |
|
1527 show "c * SUPR UNIV f \<le> y" |
|
1528 proof cases |
|
1529 assume c: "0 < c \<and> c \<noteq> \<infinity>" |
|
1530 with * have "SUPR UNIV f \<le> y / c" |
|
1531 by (intro SUP_leI) (auto simp: extreal_le_divide_pos) |
|
1532 with c show ?thesis |
|
1533 by (auto simp: extreal_le_divide_pos) |
|
1534 next |
|
1535 { assume "c = \<infinity>" have ?thesis |
|
1536 proof cases |
|
1537 assume "\<forall>i. f i = 0" |
|
1538 moreover then have "range f = {0}" by auto |
|
1539 ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def) |
|
1540 next |
|
1541 assume "\<not> (\<forall>i. f i = 0)" |
|
1542 then obtain i where "f i \<noteq> 0" by auto |
|
1543 with *[of i] `c = \<infinity>` `0 \<le> f i` show ?thesis by (auto split: split_if_asm) |
|
1544 qed } |
|
1545 moreover assume "\<not> (0 < c \<and> c \<noteq> \<infinity>)" |
|
1546 ultimately show ?thesis using * `0 \<le> c` by auto |
|
1547 qed |
|
1548 qed |
|
1549 |
|
1550 lemma SUP_PInfty: |
|
1551 fixes f :: "'a \<Rightarrow> extreal" |
|
1552 assumes "\<And>n::nat. \<exists>i\<in>A. extreal (real n) \<le> f i" |
|
1553 shows "(SUP i:A. f i) = \<infinity>" |
|
1554 unfolding SUPR_def Sup_eq_top_iff[where 'a=extreal, unfolded top_extreal_def] |
|
1555 apply simp |
|
1556 proof safe |
|
1557 fix x assume "x \<noteq> \<infinity>" |
|
1558 show "\<exists>i\<in>A. x < f i" |
|
1559 proof (cases x) |
|
1560 case PInf with `x \<noteq> \<infinity>` show ?thesis by simp |
|
1561 next |
|
1562 case MInf with assms[of "0"] show ?thesis by force |
|
1563 next |
|
1564 case (real r) |
|
1565 with less_PInf_Ex_of_nat[of x] obtain n :: nat where "x < extreal (real n)" by auto |
|
1566 moreover from assms[of n] guess i .. |
|
1567 ultimately show ?thesis |
|
1568 by (auto intro!: bexI[of _ i]) |
|
1569 qed |
|
1570 qed |
|
1571 |
|
1572 lemma Sup_countable_SUPR: |
|
1573 assumes "A \<noteq> {}" |
|
1574 shows "\<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> A \<and> Sup A = SUPR UNIV f" |
|
1575 proof (cases "Sup A") |
|
1576 case (real r) |
|
1577 have "\<forall>n::nat. \<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)" |
|
1578 proof |
|
1579 fix n ::nat have "\<exists>x\<in>A. Sup A - 1 / extreal (real n) < x" |
|
1580 using assms real by (intro Sup_extreal_close) (auto simp: one_extreal_def) |
|
1581 then guess x .. |
|
1582 then show "\<exists>x. x \<in> A \<and> Sup A < x + 1 / extreal (real n)" |
|
1583 by (auto intro!: exI[of _ x] simp: extreal_minus_less_iff) |
|
1584 qed |
|
1585 from choice[OF this] guess f .. note f = this |
|
1586 have "SUPR UNIV f = Sup A" |
|
1587 proof (rule extreal_SUPI) |
|
1588 fix i show "f i \<le> Sup A" using f |
|
1589 by (auto intro!: complete_lattice_class.Sup_upper) |
|
1590 next |
|
1591 fix y assume bound: "\<And>i. i \<in> UNIV \<Longrightarrow> f i \<le> y" |
|
1592 show "Sup A \<le> y" |
|
1593 proof (rule extreal_le_epsilon, intro allI impI) |
|
1594 fix e :: extreal assume "0 < e" |
|
1595 show "Sup A \<le> y + e" |
|
1596 proof (cases e) |
|
1597 case (real r) |
|
1598 hence "0 < r" using `0 < e` by auto |
|
1599 then obtain n ::nat where *: "1 / real n < r" "0 < n" |
|
1600 using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide) |
|
1601 have "Sup A \<le> f n + 1 / extreal (real n)" using f[THEN spec, of n] by auto |
|
1602 also have "1 / extreal (real n) \<le> e" using real * by (auto simp: one_extreal_def ) |
|
1603 with bound have "f n + 1 / extreal (real n) \<le> y + e" by (rule add_mono) simp |
|
1604 finally show "Sup A \<le> y + e" . |
|
1605 qed (insert `0 < e`, auto) |
|
1606 qed |
|
1607 qed |
|
1608 with f show ?thesis by (auto intro!: exI[of _ f]) |
|
1609 next |
|
1610 case PInf |
|
1611 from `A \<noteq> {}` obtain x where "x \<in> A" by auto |
|
1612 show ?thesis |
|
1613 proof cases |
|
1614 assume "\<infinity> \<in> A" |
|
1615 moreover then have "\<infinity> \<le> Sup A" by (intro complete_lattice_class.Sup_upper) |
|
1616 ultimately show ?thesis by (auto intro!: exI[of _ "\<lambda>x. \<infinity>"]) |
|
1617 next |
|
1618 assume "\<infinity> \<notin> A" |
|
1619 have "\<exists>x\<in>A. 0 \<le> x" |
|
1620 by (metis Infty_neq_0 PInf complete_lattice_class.Sup_least extreal_infty_less_eq2 linorder_linear) |
|
1621 then obtain x where "x \<in> A" "0 \<le> x" by auto |
|
1622 have "\<forall>n::nat. \<exists>f. f \<in> A \<and> x + extreal (real n) \<le> f" |
|
1623 proof (rule ccontr) |
|
1624 assume "\<not> ?thesis" |
|
1625 then have "\<exists>n::nat. Sup A \<le> x + extreal (real n)" |
|
1626 by (simp add: Sup_le_iff not_le less_imp_le Ball_def) (metis less_imp_le) |
|
1627 then show False using `x \<in> A` `\<infinity> \<notin> A` PInf |
|
1628 by(cases x) auto |
|
1629 qed |
|
1630 from choice[OF this] guess f .. note f = this |
|
1631 have "SUPR UNIV f = \<infinity>" |
|
1632 proof (rule SUP_PInfty) |
|
1633 fix n :: nat show "\<exists>i\<in>UNIV. extreal (real n) \<le> f i" |
|
1634 using f[THEN spec, of n] `0 \<le> x` |
|
1635 by (cases rule: extreal2_cases[of "f n" x]) (auto intro!: exI[of _ n]) |
|
1636 qed |
|
1637 then show ?thesis using f PInf by (auto intro!: exI[of _ f]) |
|
1638 qed |
|
1639 next |
|
1640 case MInf |
|
1641 with `A \<noteq> {}` have "A = {-\<infinity>}" by (auto simp: Sup_eq_MInfty) |
|
1642 then show ?thesis using MInf by (auto intro!: exI[of _ "\<lambda>x. -\<infinity>"]) |
|
1643 qed |
|
1644 |
|
1645 lemma SUPR_countable_SUPR: |
|
1646 "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> extreal. range f \<subseteq> g`A \<and> SUPR A g = SUPR UNIV f" |
|
1647 using Sup_countable_SUPR[of "g`A"] by (auto simp: SUPR_def) |
|
1648 |
|
1649 |
|
1650 lemma Sup_extreal_cadd: |
|
1651 fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
|
1652 shows "Sup ((\<lambda>x. a + x) ` A) = a + Sup A" |
|
1653 proof (rule antisym) |
|
1654 have *: "\<And>a::extreal. \<And>A. Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" |
|
1655 by (auto intro!: add_mono complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
|
1656 then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" . |
|
1657 show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)" |
|
1658 proof (cases a) |
|
1659 case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant) |
|
1660 next |
|
1661 case (real r) |
|
1662 then have **: "op + (- a) ` op + a ` A = A" |
|
1663 by (auto simp: image_iff ac_simps zero_extreal_def[symmetric]) |
|
1664 from *[of "-a" "(\<lambda>x. a + x) ` A"] real show ?thesis unfolding ** |
|
1665 by (cases rule: extreal2_cases[of "Sup A" "Sup (op + a ` A)"]) auto |
|
1666 qed (insert `a \<noteq> -\<infinity>`, auto) |
|
1667 qed |
|
1668 |
|
1669 lemma Sup_extreal_cminus: |
|
1670 fixes A :: "extreal set" assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
|
1671 shows "Sup ((\<lambda>x. a - x) ` A) = a - Inf A" |
|
1672 using Sup_extreal_cadd[of "uminus ` A" a] assms |
|
1673 by (simp add: comp_def image_image minus_extreal_def |
|
1674 extreal_Sup_uminus_image_eq) |
|
1675 |
|
1676 lemma SUPR_extreal_cminus: |
|
1677 fixes A assumes "A \<noteq> {}" and "a \<noteq> -\<infinity>" |
|
1678 shows "(SUP x:A. a - f x) = a - (INF x:A. f x)" |
|
1679 using Sup_extreal_cminus[of "f`A" a] assms |
|
1680 unfolding SUPR_def INFI_def image_image by auto |
|
1681 |
|
1682 lemma Inf_extreal_cminus: |
|
1683 fixes A :: "extreal set" assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
|
1684 shows "Inf ((\<lambda>x. a - x) ` A) = a - Sup A" |
|
1685 proof - |
|
1686 { fix x have "-a - -x = -(a - x)" using assms by (cases x) auto } |
|
1687 moreover then have "(\<lambda>x. -a - x)`uminus`A = uminus ` (\<lambda>x. a - x) ` A" |
|
1688 by (auto simp: image_image) |
|
1689 ultimately show ?thesis |
|
1690 using Sup_extreal_cminus[of "uminus ` A" "-a"] assms |
|
1691 by (auto simp add: extreal_Sup_uminus_image_eq extreal_Inf_uminus_image_eq) |
|
1692 qed |
|
1693 |
|
1694 lemma INFI_extreal_cminus: |
|
1695 fixes A assumes "A \<noteq> {}" and "\<bar>a\<bar> \<noteq> \<infinity>" |
|
1696 shows "(INF x:A. a - f x) = a - (SUP x:A. f x)" |
|
1697 using Inf_extreal_cminus[of "f`A" a] assms |
|
1698 unfolding SUPR_def INFI_def image_image |
|
1699 by auto |
|
1700 |
|
1701 lemma uminus_extreal_add_uminus_uminus: |
|
1702 fixes a b :: extreal shows "a \<noteq> \<infinity> \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> - (- a + - b) = a + b" |
|
1703 by (cases rule: extreal2_cases[of a b]) auto |
|
1704 |
|
1705 lemma INFI_extreal_add: |
|
1706 assumes "decseq f" "decseq g" and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>" |
|
1707 shows "(INF i. f i + g i) = INFI UNIV f + INFI UNIV g" |
|
1708 proof - |
|
1709 have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>" |
|
1710 using assms unfolding INF_less_iff by auto |
|
1711 { fix i from fin[of i] have "- ((- f i) + (- g i)) = f i + g i" |
|
1712 by (rule uminus_extreal_add_uminus_uminus) } |
|
1713 then have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))" |
|
1714 by simp |
|
1715 also have "\<dots> = INFI UNIV f + INFI UNIV g" |
|
1716 unfolding extreal_INFI_uminus |
|
1717 using assms INF_less |
|
1718 by (subst SUPR_extreal_add) |
|
1719 (auto simp: extreal_SUPR_uminus intro!: uminus_extreal_add_uminus_uminus) |
|
1720 finally show ?thesis . |
|
1721 qed |
|
1722 |
|
1723 subsection "Limits on @{typ extreal}" |
|
1724 |
|
1725 subsubsection "Topological space" |
|
1726 |
|
1727 instantiation extreal :: topological_space |
|
1728 begin |
|
1729 |
|
1730 definition "open A \<longleftrightarrow> open (extreal -` A) |
|
1731 \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A)) |
|
1732 \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))" |
|
1733 |
|
1734 lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {extreal x<..} \<subseteq> A)" |
|
1735 unfolding open_extreal_def by auto |
|
1736 |
|
1737 lemma open_MInfty: "open A \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A)" |
|
1738 unfolding open_extreal_def by auto |
|
1739 |
|
1740 lemma open_PInfty2: assumes "open A" "\<infinity> \<in> A" obtains x where "{extreal x<..} \<subseteq> A" |
|
1741 using open_PInfty[OF assms] by auto |
|
1742 |
|
1743 lemma open_MInfty2: assumes "open A" "-\<infinity> \<in> A" obtains x where "{..<extreal x} \<subseteq> A" |
|
1744 using open_MInfty[OF assms] by auto |
|
1745 |
|
1746 lemma extreal_openE: assumes "open A" obtains x y where |
|
1747 "open (extreal -` A)" |
|
1748 "\<infinity> \<in> A \<Longrightarrow> {extreal x<..} \<subseteq> A" |
|
1749 "-\<infinity> \<in> A \<Longrightarrow> {..<extreal y} \<subseteq> A" |
|
1750 using assms open_extreal_def by auto |
|
1751 |
|
1752 instance |
|
1753 proof |
|
1754 let ?U = "UNIV::extreal set" |
|
1755 show "open ?U" unfolding open_extreal_def |
|
1756 by (auto intro!: exI[of _ 0]) |
|
1757 next |
|
1758 fix S T::"extreal set" assume "open S" and "open T" |
|
1759 from `open S`[THEN extreal_openE] guess xS yS . |
|
1760 moreover from `open T`[THEN extreal_openE] guess xT yT . |
|
1761 ultimately have |
|
1762 "open (extreal -` (S \<inter> T))" |
|
1763 "\<infinity> \<in> S \<inter> T \<Longrightarrow> {extreal (max xS xT) <..} \<subseteq> S \<inter> T" |
|
1764 "-\<infinity> \<in> S \<inter> T \<Longrightarrow> {..< extreal (min yS yT)} \<subseteq> S \<inter> T" |
|
1765 by auto |
|
1766 then show "open (S Int T)" unfolding open_extreal_def by blast |
|
1767 next |
|
1768 fix K :: "extreal set set" assume "\<forall>S\<in>K. open S" |
|
1769 then have *: "\<forall>S. \<exists>x y. S \<in> K \<longrightarrow> open (extreal -` S) \<and> |
|
1770 (\<infinity> \<in> S \<longrightarrow> {extreal x <..} \<subseteq> S) \<and> (-\<infinity> \<in> S \<longrightarrow> {..< extreal y} \<subseteq> S)" |
|
1771 by (auto simp: open_extreal_def) |
|
1772 then show "open (Union K)" unfolding open_extreal_def |
|
1773 proof (intro conjI impI) |
|
1774 show "open (extreal -` \<Union>K)" |
|
1775 using *[THEN choice] by (auto simp: vimage_Union) |
|
1776 qed ((metis UnionE Union_upper subset_trans *)+) |
|
1777 qed |
|
1778 end |
|
1779 |
|
1780 lemma open_extreal: "open S \<Longrightarrow> open (extreal ` S)" |
|
1781 by (auto simp: inj_vimage_image_eq open_extreal_def) |
|
1782 |
|
1783 lemma open_extreal_vimage: "open S \<Longrightarrow> open (extreal -` S)" |
|
1784 unfolding open_extreal_def by auto |
|
1785 |
|
1786 lemma open_extreal_lessThan[intro, simp]: "open {..< a :: extreal}" |
|
1787 proof - |
|
1788 have "\<And>x. extreal -` {..<extreal x} = {..< x}" |
|
1789 "extreal -` {..< \<infinity>} = UNIV" "extreal -` {..< -\<infinity>} = {}" by auto |
|
1790 then show ?thesis by (cases a) (auto simp: open_extreal_def) |
|
1791 qed |
|
1792 |
|
1793 lemma open_extreal_greaterThan[intro, simp]: |
|
1794 "open {a :: extreal <..}" |
|
1795 proof - |
|
1796 have "\<And>x. extreal -` {extreal x<..} = {x<..}" |
|
1797 "extreal -` {\<infinity><..} = {}" "extreal -` {-\<infinity><..} = UNIV" by auto |
|
1798 then show ?thesis by (cases a) (auto simp: open_extreal_def) |
|
1799 qed |
|
1800 |
|
1801 lemma extreal_open_greaterThanLessThan[intro, simp]: "open {a::extreal <..< b}" |
|
1802 unfolding greaterThanLessThan_def by auto |
|
1803 |
|
1804 lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}" |
|
1805 proof - |
|
1806 have "- {a ..} = {..< a}" by auto |
|
1807 then show "closed {a ..}" |
|
1808 unfolding closed_def using open_extreal_lessThan by auto |
|
1809 qed |
|
1810 |
|
1811 lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}" |
|
1812 proof - |
|
1813 have "- {.. b} = {b <..}" by auto |
|
1814 then show "closed {.. b}" |
|
1815 unfolding closed_def using open_extreal_greaterThan by auto |
|
1816 qed |
|
1817 |
|
1818 lemma closed_extreal_atLeastAtMost[simp, intro]: |
|
1819 shows "closed {a :: extreal .. b}" |
|
1820 unfolding atLeastAtMost_def by auto |
|
1821 |
|
1822 lemma closed_extreal_singleton: |
|
1823 "closed {a :: extreal}" |
|
1824 by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost) |
|
1825 |
|
1826 lemma extreal_open_cont_interval: |
|
1827 assumes "open S" "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>" |
|
1828 obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S" |
|
1829 proof- |
|
1830 from `open S` have "open (extreal -` S)" by (rule extreal_openE) |
|
1831 then obtain e where "0 < e" and e: "\<And>y. dist y (real x) < e \<Longrightarrow> extreal y \<in> S" |
|
1832 using assms unfolding open_dist by force |
|
1833 show thesis |
|
1834 proof (intro that subsetI) |
|
1835 show "0 < extreal e" using `0 < e` by auto |
|
1836 fix y assume "y \<in> {x - extreal e<..<x + extreal e}" |
|
1837 with assms obtain t where "y = extreal t" "dist t (real x) < e" |
|
1838 apply (cases y) by (auto simp: dist_real_def) |
|
1839 then show "y \<in> S" using e[of t] by auto |
|
1840 qed |
|
1841 qed |
|
1842 |
|
1843 lemma extreal_open_cont_interval2: |
|
1844 assumes "open S" "x \<in> S" and x: "\<bar>x\<bar> \<noteq> \<infinity>" |
|
1845 obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S" |
|
1846 proof- |
|
1847 guess e using extreal_open_cont_interval[OF assms] . |
|
1848 with that[of "x-e" "x+e"] extreal_between[OF x, of e] |
|
1849 show thesis by auto |
|
1850 qed |
|
1851 |
|
1852 instance extreal :: t2_space |
|
1853 proof |
|
1854 fix x y :: extreal assume "x ~= y" |
|
1855 let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
1856 |
|
1857 { fix x y :: extreal assume "x < y" |
|
1858 from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto |
|
1859 have "?P x y" |
|
1860 apply (rule exI[of _ "{..<z}"]) |
|
1861 apply (rule exI[of _ "{z<..}"]) |
|
1862 using z by auto } |
|
1863 note * = this |
|
1864 |
|
1865 from `x ~= y` |
|
1866 show "EX U V. open U & open V & x : U & y : V & U Int V = {}" |
|
1867 proof (cases rule: linorder_cases) |
|
1868 assume "x = y" with `x ~= y` show ?thesis by simp |
|
1869 next assume "x < y" from *[OF this] show ?thesis by auto |
|
1870 next assume "y < x" from *[OF this] show ?thesis by auto |
|
1871 qed |
|
1872 qed |
|
1873 |
|
1874 subsubsection {* Convergent sequences *} |
|
1875 |
|
1876 lemma lim_extreal[simp]: |
|
1877 "((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r") |
|
1878 proof (intro iffI topological_tendstoI) |
|
1879 fix S assume "?l" "open S" "x \<in> S" |
|
1880 then show "eventually (\<lambda>x. f x \<in> S) net" |
|
1881 using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`] |
|
1882 by (simp add: inj_image_mem_iff) |
|
1883 next |
|
1884 fix S assume "?r" "open S" "extreal x \<in> S" |
|
1885 show "eventually (\<lambda>x. extreal (f x) \<in> S) net" |
|
1886 using `?r`[THEN topological_tendstoD, OF open_extreal_vimage, OF `open S`] |
|
1887 using `extreal x \<in> S` by auto |
|
1888 qed |
|
1889 |
|
1890 lemma lim_real_of_extreal[simp]: |
|
1891 assumes lim: "(f ---> extreal x) net" |
|
1892 shows "((\<lambda>x. real (f x)) ---> x) net" |
|
1893 proof (intro topological_tendstoI) |
|
1894 fix S assume "open S" "x \<in> S" |
|
1895 then have S: "open S" "extreal x \<in> extreal ` S" |
|
1896 by (simp_all add: inj_image_mem_iff) |
|
1897 have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto |
|
1898 from this lim[THEN topological_tendstoD, OF open_extreal, OF S] |
|
1899 show "eventually (\<lambda>x. real (f x) \<in> S) net" |
|
1900 by (rule eventually_mono) |
|
1901 qed |
|
1902 |
|
1903 lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r") |
|
1904 proof assume ?r show ?l apply(rule topological_tendstoI) |
|
1905 unfolding eventually_sequentially |
|
1906 proof- fix S assume "open S" "\<infinity> : S" |
|
1907 from open_PInfty[OF this] guess B .. note B=this |
|
1908 from `?r`[rule_format,of "B+1"] guess N .. note N=this |
|
1909 show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1910 proof safe case goal1 |
|
1911 have "extreal B < extreal (B + 1)" by auto |
|
1912 also have "... <= f n" using goal1 N by auto |
|
1913 finally show ?case using B by fastsimp |
|
1914 qed |
|
1915 qed |
|
1916 next assume ?l show ?r |
|
1917 proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto |
|
1918 from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
|
1919 guess N .. note N=this |
|
1920 show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto |
|
1921 qed |
|
1922 qed |
|
1923 |
|
1924 |
|
1925 lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r") |
|
1926 proof assume ?r show ?l apply(rule topological_tendstoI) |
|
1927 unfolding eventually_sequentially |
|
1928 proof- fix S assume "open S" "(-\<infinity>) : S" |
|
1929 from open_MInfty[OF this] guess B .. note B=this |
|
1930 from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this |
|
1931 show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI) |
|
1932 proof safe case goal1 |
|
1933 have "extreal (B - 1) >= f n" using goal1 N by auto |
|
1934 also have "... < extreal B" by auto |
|
1935 finally show ?case using B by fastsimp |
|
1936 qed |
|
1937 qed |
|
1938 next assume ?l show ?r |
|
1939 proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto |
|
1940 from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially] |
|
1941 guess N .. note N=this |
|
1942 show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto |
|
1943 qed |
|
1944 qed |
|
1945 |
|
1946 |
|
1947 lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>" |
|
1948 proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>" |
|
1949 from lim[unfolded this Lim_PInfty,rule_format,of "?B"] |
|
1950 guess N .. note N=this[rule_format,OF le_refl] |
|
1951 hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans) |
|
1952 hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto |
|
1953 thus False by auto |
|
1954 qed |
|
1955 |
|
1956 |
|
1957 lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)" |
|
1958 proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)" |
|
1959 from lim[unfolded this Lim_MInfty,rule_format,of "?B"] |
|
1960 guess N .. note N=this[rule_format,OF le_refl] |
|
1961 hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast |
|
1962 thus False by auto |
|
1963 qed |
|
1964 |
|
1965 |
|
1966 lemma tendsto_explicit: |
|
1967 "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))" |
|
1968 unfolding tendsto_def eventually_sequentially by auto |
|
1969 |
|
1970 |
|
1971 lemma tendsto_obtains_N: |
|
1972 assumes "f ----> f0" |
|
1973 assumes "open S" "f0 : S" |
|
1974 obtains N where "ALL n>=N. f n : S" |
|
1975 using tendsto_explicit[of f f0] assms by auto |
|
1976 |
|
1977 |
|
1978 lemma tail_same_limit: |
|
1979 fixes X Y N |
|
1980 assumes "X ----> L" "ALL n>=N. X n = Y n" |
|
1981 shows "Y ----> L" |
|
1982 proof- |
|
1983 { fix S assume "open S" and "L:S" |
|
1984 from this obtain N1 where "ALL n>=N1. X n : S" |
|
1985 using assms unfolding tendsto_def eventually_sequentially by auto |
|
1986 hence "ALL n>=max N N1. Y n : S" using assms by auto |
|
1987 hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto |
|
1988 } |
|
1989 thus ?thesis using tendsto_explicit by auto |
|
1990 qed |
|
1991 |
|
1992 |
|
1993 lemma Lim_bounded_PInfty2: |
|
1994 assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B" |
|
1995 shows "l ~= \<infinity>" |
|
1996 proof- |
|
1997 def g == "(%n. if n>=N then f n else extreal B)" |
|
1998 hence "g ----> l" using tail_same_limit[of f l N g] lim by auto |
|
1999 moreover have "!!n. g n <= extreal B" using g_def assms by auto |
|
2000 ultimately show ?thesis using Lim_bounded_PInfty by auto |
|
2001 qed |
|
2002 |
|
2003 lemma Lim_bounded_extreal: |
|
2004 assumes lim:"f ----> (l :: extreal)" |
|
2005 and "ALL n>=M. f n <= C" |
|
2006 shows "l<=C" |
|
2007 proof- |
|
2008 { assume "l=(-\<infinity>)" hence ?thesis by auto } |
|
2009 moreover |
|
2010 { assume "~(l=(-\<infinity>))" |
|
2011 { assume "C=\<infinity>" hence ?thesis by auto } |
|
2012 moreover |
|
2013 { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto |
|
2014 hence "l=(-\<infinity>)" using assms |
|
2015 tendsto_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto |
|
2016 hence ?thesis by auto } |
|
2017 moreover |
|
2018 { assume "EX B. C = extreal B" |
|
2019 from this obtain B where B_def: "C=extreal B" by auto |
|
2020 hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto |
|
2021 from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto |
|
2022 from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}" |
|
2023 apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto |
|
2024 { fix n assume "n>=N" |
|
2025 hence "EX r. extreal r = f n" using N_def by (cases "f n") auto |
|
2026 } from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis |
|
2027 hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto |
|
2028 hence *: "(%n. g n) ----> m" using m_def by auto |
|
2029 { fix n assume "n>=max N M" |
|
2030 hence "extreal (g n) <= extreal B" using assms g_def B_def by auto |
|
2031 hence "g n <= B" by auto |
|
2032 } hence "EX N. ALL n>=N. g n <= B" by blast |
|
2033 hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto |
|
2034 hence ?thesis using m_def B_def by auto |
|
2035 } ultimately have ?thesis by (cases C) auto |
|
2036 } ultimately show ?thesis by blast |
|
2037 qed |
|
2038 |
|
2039 lemma real_of_extreal_mult[simp]: |
|
2040 fixes a b :: extreal shows "real (a * b) = real a * real b" |
|
2041 by (cases rule: extreal2_cases[of a b]) auto |
|
2042 |
|
2043 lemma real_of_extreal_eq_0: |
|
2044 "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0" |
|
2045 by (cases x) auto |
|
2046 |
|
2047 lemma tendsto_extreal_realD: |
|
2048 fixes f :: "'a \<Rightarrow> extreal" |
|
2049 assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net" |
|
2050 shows "(f ---> x) net" |
|
2051 proof (intro topological_tendstoI) |
|
2052 fix S assume S: "open S" "x \<in> S" |
|
2053 with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto |
|
2054 from tendsto[THEN topological_tendstoD, OF this] |
|
2055 show "eventually (\<lambda>x. f x \<in> S) net" |
|
2056 by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0) |
|
2057 qed |
|
2058 |
|
2059 lemma tendsto_extreal_realI: |
|
2060 fixes f :: "'a \<Rightarrow> extreal" |
|
2061 assumes x: "\<bar>x\<bar> \<noteq> \<infinity>" and tendsto: "(f ---> x) net" |
|
2062 shows "((\<lambda>x. extreal (real (f x))) ---> x) net" |
|
2063 proof (intro topological_tendstoI) |
|
2064 fix S assume "open S" "x \<in> S" |
|
2065 with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto |
|
2066 from tendsto[THEN topological_tendstoD, OF this] |
|
2067 show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net" |
|
2068 by (elim eventually_elim1) (auto simp: extreal_real) |
|
2069 qed |
|
2070 |
|
2071 lemma extreal_mult_cancel_left: |
|
2072 fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow> |
|
2073 ((\<bar>a\<bar> = \<infinity> \<and> 0 < b * c) \<or> a = 0 \<or> b = c)" |
|
2074 by (cases rule: extreal3_cases[of a b c]) |
|
2075 (simp_all add: zero_less_mult_iff) |
|
2076 |
|
2077 lemma extreal_inj_affinity: |
|
2078 assumes "\<bar>m\<bar> \<noteq> \<infinity>" "m \<noteq> 0" "\<bar>t\<bar> \<noteq> \<infinity>" |
|
2079 shows "inj_on (\<lambda>x. m * x + t) A" |
|
2080 using assms |
|
2081 by (cases rule: extreal2_cases[of m t]) |
|
2082 (auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left) |
|
2083 |
|
2084 lemma extreal_PInfty_eq_plus[simp]: |
|
2085 shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>" |
|
2086 by (cases rule: extreal2_cases[of a b]) auto |
|
2087 |
|
2088 lemma extreal_MInfty_eq_plus[simp]: |
|
2089 shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)" |
|
2090 by (cases rule: extreal2_cases[of a b]) auto |
|
2091 |
|
2092 lemma extreal_less_divide_pos: |
|
2093 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z" |
|
2094 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
2095 |
|
2096 lemma extreal_divide_less_pos: |
|
2097 "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z" |
|
2098 by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps) |
|
2099 |
|
2100 lemma extreal_divide_eq: |
|
2101 "b \<noteq> 0 \<Longrightarrow> \<bar>b\<bar> \<noteq> \<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c" |
|
2102 by (cases rule: extreal3_cases[of a b c]) |
|
2103 (simp_all add: field_simps) |
|
2104 |
|
2105 lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>" |
|
2106 by (cases a) auto |
|
2107 |
|
2108 lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x" |
|
2109 by (cases x) auto |
|
2110 |
|
2111 lemma extreal_LimI_finite: |
|
2112 assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
2113 assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r" |
|
2114 shows "u ----> x" |
|
2115 proof (rule topological_tendstoI, unfold eventually_sequentially) |
|
2116 obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto |
|
2117 fix S assume "open S" "x : S" |
|
2118 then have "open (extreal -` S)" unfolding open_extreal_def by auto |
|
2119 with `x \<in> S` obtain r where "0 < r" and dist: "!!y. dist y rx < r ==> extreal y \<in> S" |
|
2120 unfolding open_real_def rx_def by auto |
|
2121 then obtain n where |
|
2122 upper: "!!N. n <= N ==> u N < x + extreal r" and |
|
2123 lower: "!!N. n <= N ==> x < u N + extreal r" using assms(2)[of "extreal r"] by auto |
|
2124 show "EX N. ALL n>=N. u n : S" |
|
2125 proof (safe intro!: exI[of _ n]) |
|
2126 fix N assume "n <= N" |
|
2127 from upper[OF this] lower[OF this] assms `0 < r` |
|
2128 have "u N ~: {\<infinity>,(-\<infinity>)}" by auto |
|
2129 from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto |
|
2130 hence "rx < ra + r" and "ra < rx + r" |
|
2131 using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto |
|
2132 hence "dist (real (u N)) rx < r" |
|
2133 using rx_def ra_def |
|
2134 by (auto simp: dist_real_def abs_diff_less_iff field_simps) |
|
2135 from dist[OF this] show "u N : S" using `u N ~: {\<infinity>, -\<infinity>}` |
|
2136 by (auto simp: extreal_real split: split_if_asm) |
|
2137 qed |
|
2138 qed |
|
2139 |
|
2140 lemma extreal_LimI_finite_iff: |
|
2141 assumes "\<bar>x\<bar> \<noteq> \<infinity>" |
|
2142 shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))" |
|
2143 (is "?lhs <-> ?rhs") |
|
2144 proof |
|
2145 assume lim: "u ----> x" |
|
2146 { fix r assume "(r::extreal)>0" |
|
2147 from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}" |
|
2148 apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"]) |
|
2149 using lim extreal_between[of x r] assms `r>0` by auto |
|
2150 hence "EX N. ALL n>=N. u n < x + r & x < u n + r" |
|
2151 using extreal_minus_less[of r x] by (cases r) auto |
|
2152 } then show "?rhs" by auto |
|
2153 next |
|
2154 assume ?rhs then show "u ----> x" |
|
2155 using extreal_LimI_finite[of x] assms by auto |
|
2156 qed |
|
2157 |
|
2158 |
|
2159 subsubsection {* @{text Liminf} and @{text Limsup} *} |
|
2160 |
|
2161 definition |
|
2162 "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)" |
|
2163 |
|
2164 definition |
|
2165 "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)" |
|
2166 |
|
2167 lemma Liminf_Sup: |
|
2168 fixes f :: "'a => 'b::{complete_lattice, linorder}" |
|
2169 shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}" |
|
2170 by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def) |
|
2171 |
|
2172 lemma Limsup_Inf: |
|
2173 fixes f :: "'a => 'b::{complete_lattice, linorder}" |
|
2174 shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}" |
|
2175 by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def) |
|
2176 |
|
2177 lemma extreal_SupI: |
|
2178 fixes x :: extreal |
|
2179 assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" |
|
2180 assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y" |
|
2181 shows "Sup A = x" |
|
2182 unfolding Sup_extreal_def |
|
2183 using assms by (auto intro!: Least_equality) |
|
2184 |
|
2185 lemma extreal_InfI: |
|
2186 fixes x :: extreal |
|
2187 assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i" |
|
2188 assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x" |
|
2189 shows "Inf A = x" |
|
2190 unfolding Inf_extreal_def |
|
2191 using assms by (auto intro!: Greatest_equality) |
|
2192 |
|
2193 lemma Limsup_const: |
|
2194 fixes c :: "'a::{complete_lattice, linorder}" |
|
2195 assumes ntriv: "\<not> trivial_limit net" |
|
2196 shows "Limsup net (\<lambda>x. c) = c" |
|
2197 unfolding Limsup_Inf |
|
2198 proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower) |
|
2199 fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net" |
|
2200 show "c \<le> x" |
|
2201 proof (rule ccontr) |
|
2202 assume "\<not> c \<le> x" then have "x < c" by auto |
|
2203 then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2204 qed |
|
2205 qed auto |
|
2206 |
|
2207 lemma Liminf_const: |
|
2208 fixes c :: "'a::{complete_lattice, linorder}" |
|
2209 assumes ntriv: "\<not> trivial_limit net" |
|
2210 shows "Liminf net (\<lambda>x. c) = c" |
|
2211 unfolding Liminf_Sup |
|
2212 proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper) |
|
2213 fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net" |
|
2214 show "x \<le> c" |
|
2215 proof (rule ccontr) |
|
2216 assume "\<not> x \<le> c" then have "c < x" by auto |
|
2217 then show False using ntriv * by (auto simp: trivial_limit_def) |
|
2218 qed |
|
2219 qed auto |
|
2220 |
|
2221 lemma mono_set: |
|
2222 fixes S :: "('a::order) set" |
|
2223 shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)" |
|
2224 by (auto simp: mono_def mem_def) |
|
2225 |
|
2226 lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto |
|
2227 lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto |
|
2228 lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto |
|
2229 lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto |
|
2230 |
|
2231 lemma mono_set_iff: |
|
2232 fixes S :: "'a::{linorder,complete_lattice} set" |
|
2233 defines "a \<equiv> Inf S" |
|
2234 shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c") |
|
2235 proof |
|
2236 assume "mono S" |
|
2237 then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set) |
|
2238 show ?c |
|
2239 proof cases |
|
2240 assume "a \<in> S" |
|
2241 show ?c |
|
2242 using mono[OF _ `a \<in> S`] |
|
2243 by (auto intro: complete_lattice_class.Inf_lower simp: a_def) |
|
2244 next |
|
2245 assume "a \<notin> S" |
|
2246 have "S = {a <..}" |
|
2247 proof safe |
|
2248 fix x assume "x \<in> S" |
|
2249 then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower) |
|
2250 then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto |
|
2251 next |
|
2252 fix x assume "a < x" |
|
2253 then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff .. |
|
2254 with mono[of y x] show "x \<in> S" by auto |
|
2255 qed |
|
2256 then show ?c .. |
|
2257 qed |
|
2258 qed auto |
|
2259 |
|
2260 lemma lim_imp_Liminf: |
|
2261 fixes f :: "'a \<Rightarrow> extreal" |
|
2262 assumes ntriv: "\<not> trivial_limit net" |
|
2263 assumes lim: "(f ---> f0) net" |
|
2264 shows "Liminf net f = f0" |
|
2265 unfolding Liminf_Sup |
|
2266 proof (safe intro!: extreal_SupI) |
|
2267 fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net" |
|
2268 show "y \<le> f0" |
|
2269 proof (rule extreal_le_extreal) |
|
2270 fix B assume "B < y" |
|
2271 { assume "f0 < B" |
|
2272 then have "eventually (\<lambda>x. f x < B \<and> B < f x) net" |
|
2273 using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`] |
|
2274 by (auto intro: eventually_conj) |
|
2275 also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2276 finally have False using ntriv[unfolded trivial_limit_def] by auto |
|
2277 } then show "B \<le> f0" by (metis linorder_le_less_linear) |
|
2278 qed |
|
2279 next |
|
2280 fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y" |
|
2281 show "f0 \<le> y" |
|
2282 proof (safe intro!: *[rule_format]) |
|
2283 fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net" |
|
2284 using lim[THEN topological_tendstoD, of "{y <..}"] by auto |
|
2285 qed |
|
2286 qed |
|
2287 |
|
2288 lemma extreal_Liminf_le_Limsup: |
|
2289 fixes f :: "'a \<Rightarrow> extreal" |
|
2290 assumes ntriv: "\<not> trivial_limit net" |
|
2291 shows "Liminf net f \<le> Limsup net f" |
|
2292 unfolding Limsup_Inf Liminf_Sup |
|
2293 proof (safe intro!: complete_lattice_class.Inf_greatest complete_lattice_class.Sup_least) |
|
2294 fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net" |
|
2295 show "u \<le> v" |
|
2296 proof (rule ccontr) |
|
2297 assume "\<not> u \<le> v" |
|
2298 then obtain t where "t < u" "v < t" |
|
2299 using extreal_dense[of v u] by (auto simp: not_le) |
|
2300 then have "eventually (\<lambda>x. t < f x \<and> f x < t) net" |
|
2301 using * by (auto intro: eventually_conj) |
|
2302 also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff) |
|
2303 finally show False using ntriv by (auto simp: trivial_limit_def) |
|
2304 qed |
|
2305 qed |
|
2306 |
|
2307 lemma Liminf_mono: |
|
2308 fixes f g :: "'a => extreal" |
|
2309 assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
|
2310 shows "Liminf net f \<le> Liminf net g" |
|
2311 unfolding Liminf_Sup |
|
2312 proof (safe intro!: Sup_mono bexI) |
|
2313 fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a" |
|
2314 then have "eventually (\<lambda>x. y < f x) net" by auto |
|
2315 then show "eventually (\<lambda>x. y < g x) net" |
|
2316 by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2317 qed simp |
|
2318 |
|
2319 lemma Liminf_eq: |
|
2320 fixes f g :: "'a \<Rightarrow> extreal" |
|
2321 assumes "eventually (\<lambda>x. f x = g x) net" |
|
2322 shows "Liminf net f = Liminf net g" |
|
2323 by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto |
|
2324 |
|
2325 lemma Liminf_mono_all: |
|
2326 fixes f g :: "'a \<Rightarrow> extreal" |
|
2327 assumes "\<And>x. f x \<le> g x" |
|
2328 shows "Liminf net f \<le> Liminf net g" |
|
2329 using assms by (intro Liminf_mono always_eventually) auto |
|
2330 |
|
2331 lemma Limsup_mono: |
|
2332 fixes f g :: "'a \<Rightarrow> extreal" |
|
2333 assumes ev: "eventually (\<lambda>x. f x \<le> g x) net" |
|
2334 shows "Limsup net f \<le> Limsup net g" |
|
2335 unfolding Limsup_Inf |
|
2336 proof (safe intro!: Inf_mono bexI) |
|
2337 fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y" |
|
2338 then have "eventually (\<lambda>x. g x < y) net" by auto |
|
2339 then show "eventually (\<lambda>x. f x < y) net" |
|
2340 by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto) |
|
2341 qed simp |
|
2342 |
|
2343 lemma Limsup_mono_all: |
|
2344 fixes f g :: "'a \<Rightarrow> extreal" |
|
2345 assumes "\<And>x. f x \<le> g x" |
|
2346 shows "Limsup net f \<le> Limsup net g" |
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2347 using assms by (intro Limsup_mono always_eventually) auto |
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2348 |
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2349 lemma Limsup_eq: |
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2350 fixes f g :: "'a \<Rightarrow> extreal" |
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2351 assumes "eventually (\<lambda>x. f x = g x) net" |
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2352 shows "Limsup net f = Limsup net g" |
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2353 by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto |
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2354 |
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2355 abbreviation "liminf \<equiv> Liminf sequentially" |
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2356 |
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2357 abbreviation "limsup \<equiv> Limsup sequentially" |
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2358 |
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2359 lemma (in complete_lattice) less_INFD: |
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2360 assumes "y < INFI A f"" i \<in> A" shows "y < f i" |
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2361 proof - |
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2362 note `y < INFI A f` |
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2363 also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI) |
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2364 finally show "y < f i" . |
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2365 qed |
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2366 |
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2367 lemma liminf_SUPR_INFI: |
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2368 fixes f :: "nat \<Rightarrow> extreal" |
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2369 shows "liminf f = (SUP n. INF m:{n..}. f m)" |
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2370 unfolding Liminf_Sup eventually_sequentially |
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2371 proof (safe intro!: antisym complete_lattice_class.Sup_least) |
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2372 fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)" |
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2373 proof (rule extreal_le_extreal) |
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2374 fix y assume "y < x" |
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2375 with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto |
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2376 then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff) |
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2377 also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto |
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2378 finally show "y \<le> (SUP n. INF m:{n..}. f m)" . |
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2379 qed |
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2380 next |
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2381 show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}" |
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2382 proof (unfold SUPR_def, safe intro!: Sup_mono bexI) |
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2383 fix y n assume "y < INFI {n..} f" |
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2384 from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto |
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2385 qed (rule order_refl) |
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2386 qed |
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2387 |
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2388 lemma tail_same_limsup: |
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2389 fixes X Y :: "nat => extreal" |
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2390 assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
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2391 shows "limsup X = limsup Y" |
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2392 using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto |
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2393 |
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2394 lemma tail_same_liminf: |
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2395 fixes X Y :: "nat => extreal" |
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2396 assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n" |
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2397 shows "liminf X = liminf Y" |
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2398 using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto |
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2399 |
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2400 lemma liminf_mono: |
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2401 fixes X Y :: "nat \<Rightarrow> extreal" |
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2402 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
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2403 shows "liminf X \<le> liminf Y" |
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2404 using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto |
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2405 |
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2406 lemma limsup_mono: |
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2407 fixes X Y :: "nat => extreal" |
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2408 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n" |
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2409 shows "limsup X \<le> limsup Y" |
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2410 using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto |
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2411 |
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2412 declare trivial_limit_sequentially[simp] |
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2413 |
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2414 lemma |
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2415 fixes X :: "nat \<Rightarrow> extreal" |
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2416 shows extreal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X" |
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2417 and extreal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X" |
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2418 unfolding incseq_def decseq_def by auto |
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2419 |
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2420 lemma liminf_bounded: |
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2421 fixes X Y :: "nat \<Rightarrow> extreal" |
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2422 assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n" |
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2423 shows "C \<le> liminf X" |
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2424 using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp |
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2425 |
|
2426 lemma limsup_bounded: |
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2427 fixes X Y :: "nat => extreal" |
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2428 assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C" |
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2429 shows "limsup X \<le> C" |
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2430 using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp |
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2431 |
|
2432 lemma liminf_bounded_iff: |
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2433 fixes x :: "nat \<Rightarrow> extreal" |
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2434 shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs") |
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2435 proof safe |
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2436 fix B assume "B < C" "C \<le> liminf x" |
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2437 then have "B < liminf x" by auto |
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2438 then obtain N where "B < (INF m:{N..}. x m)" |
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2439 unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto |
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2440 from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto |
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2441 next |
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2442 assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n" |
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2443 { fix B assume "B<C" |
|
2444 then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto |
|
2445 hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto |
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2446 also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp |
|
2447 finally have "B \<le> liminf x" . |
|
2448 } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear) |
|
2449 qed |
|
2450 |
|
2451 lemma liminf_subseq_mono: |
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2452 fixes X :: "nat \<Rightarrow> extreal" |
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2453 assumes "subseq r" |
|
2454 shows "liminf X \<le> liminf (X \<circ> r) " |
|
2455 proof- |
|
2456 have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)" |
|
2457 proof (safe intro!: INF_mono) |
|
2458 fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m" |
|
2459 using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto |
|
2460 qed |
|
2461 then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def) |
|
2462 qed |
|
2463 |
|
2464 lemma extreal_real': assumes "\<bar>x\<bar> \<noteq> \<infinity>" shows "extreal (real x) = x" |
|
2465 using assms by auto |
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2466 |
|
2467 lemma extreal_le_extreal_bounded: |
|
2468 fixes x y z :: extreal |
|
2469 assumes "z \<le> y" |
|
2470 assumes *: "\<And>B. z < B \<Longrightarrow> B < x \<Longrightarrow> B \<le> y" |
|
2471 shows "x \<le> y" |
|
2472 proof (rule extreal_le_extreal) |
|
2473 fix B assume "B < x" |
|
2474 show "B \<le> y" |
|
2475 proof cases |
|
2476 assume "z < B" from *[OF this `B < x`] show "B \<le> y" . |
|
2477 next |
|
2478 assume "\<not> z < B" with `z \<le> y` show "B \<le> y" by auto |
|
2479 qed |
|
2480 qed |
|
2481 |
|
2482 lemma fixes x y :: extreal |
|
2483 shows Sup_atMost[simp]: "Sup {.. y} = y" |
|
2484 and Sup_lessThan[simp]: "Sup {..< y} = y" |
|
2485 and Sup_atLeastAtMost[simp]: "x \<le> y \<Longrightarrow> Sup { x .. y} = y" |
|
2486 and Sup_greaterThanAtMost[simp]: "x < y \<Longrightarrow> Sup { x <.. y} = y" |
|
2487 and Sup_atLeastLessThan[simp]: "x < y \<Longrightarrow> Sup { x ..< y} = y" |
|
2488 by (auto simp: Sup_extreal_def intro!: Least_equality |
|
2489 intro: extreal_le_extreal extreal_le_extreal_bounded[of x]) |
|
2490 |
|
2491 lemma Sup_greaterThanlessThan[simp]: |
|
2492 fixes x y :: extreal assumes "x < y" shows "Sup { x <..< y} = y" |
|
2493 unfolding Sup_extreal_def |
|
2494 proof (intro Least_equality extreal_le_extreal_bounded[of _ _ y]) |
|
2495 fix z assume z: "\<forall>u\<in>{x<..<y}. u \<le> z" |
|
2496 from extreal_dense[OF `x < y`] guess w .. note w = this |
|
2497 with z[THEN bspec, of w] show "x \<le> z" by auto |
|
2498 qed auto |
|
2499 |
|
2500 lemma real_extreal_id: "real o extreal = id" |
|
2501 proof- |
|
2502 { fix x have "(real o extreal) x = id x" by auto } |
|
2503 from this show ?thesis using ext by blast |
|
2504 qed |
|
2505 |
|
2506 lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})" |
|
2507 by (metis range_extreal open_extreal open_UNIV) |
|
2508 |
|
2509 lemma extreal_le_distrib: |
|
2510 fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b" |
|
2511 by (cases rule: extreal3_cases[of a b c]) |
|
2512 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
2513 |
|
2514 lemma extreal_pos_distrib: |
|
2515 fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b" |
|
2516 using assms by (cases rule: extreal3_cases[of a b c]) |
|
2517 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff) |
|
2518 |
|
2519 lemma extreal_pos_le_distrib: |
|
2520 fixes a b c :: extreal |
|
2521 assumes "c>=0" |
|
2522 shows "c * (a + b) <= c * a + c * b" |
|
2523 using assms by (cases rule: extreal3_cases[of a b c]) |
|
2524 (auto simp add: field_simps) |
|
2525 |
|
2526 lemma extreal_max_mono: |
|
2527 "[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d" |
|
2528 by (metis sup_extreal_def sup_mono) |
|
2529 |
|
2530 |
|
2531 lemma extreal_max_least: |
|
2532 "[| (a::extreal) <= x; c <= x |] ==> max a c <= x" |
|
2533 by (metis sup_extreal_def sup_least) |
|
2534 |
|
2535 end |
|