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24 <title>Constructing ZF Set Theory in Agda </title>
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25 </head>
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26 <body>
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27 <div class="main" id="mmm">
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28 <h1>Constructing ZF Set Theory in Agda </h1>
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29 <a href="#" right="0px" onclick="javascript:showElement('menu')">
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30 <span>Menu</span>
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31 </a>
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32 <a href="#" left="0px" onclick="javascript:showElement('menu')">
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33 <span>Menu</span>
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34 </a>
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35
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36 <p>
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37
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38 <author> Shinji KONO</author>
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39
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40 <hr/>
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279
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41 <h2><a name="content000">ZF in Agda</a></h2>
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42
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43 <pre>
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44 zf.agda axiom of ZF
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45 zfc.agda axiom of choice
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46 Ordinals.agda axiom of Ordinals
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47 ordinal.agda countable model of Ordinals
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48 OD.agda model of ZF based on Ordinal Definable Set with assumptions
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49 ODC.agda Law of exclude middle from axiom of choice assumptions
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50 LEMC.agda model of choice with assumption of the Law of exclude middle
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51 OPair.agda ordered pair on OD
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52 BAlgbra.agda Boolean algebra on OD (not yet done)
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53 filter.agda Filter on OD (not yet done)
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54 cardinal.agda Caedinal number on OD (not yet done)
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55 logic.agda some basics on logic
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56 nat.agda some basics on Nat
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57
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58 </pre>
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59
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60 <hr/>
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61 <h2><a name="content001">Programming Mathematics</a></h2>
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62
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63 <p>
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273
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64 Programming is processing data structure with λ terms.
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65 <p>
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66 We are going to handle Mathematics in intuitionistic logic with λ terms.
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67 <p>
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68 Mathematics is a functional programming which values are proofs.
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69 <p>
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70 Programming ZF Set Theory in Agda
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71 <p>
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72
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73 <hr/>
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279
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74 <h2><a name="content002">Target</a></h2>
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75
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76 <pre>
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77 Describe ZF axioms in Agda
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78 Construction a Model of ZF Set Theory in Agda
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79 Show necessary assumptions for the model
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80 Show validities of ZF axioms on the model
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81
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82 </pre>
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83 This shows consistency of Set Theory (with some assumptions),
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84 without circulating ZF Theory assumption.
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85 <p>
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86 <a href="https://github.com/shinji-kono/zf-in-agda">
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87 ZF in Agda https://github.com/shinji-kono/zf-in-agda
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88 </a>
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89 <p>
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90
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91 <hr/>
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279
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92 <h2><a name="content003">Why Set Theory</a></h2>
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93 If we can formulate Set theory, it suppose to work on any mathematical theory.
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94 <p>
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95 Set Theory is a difficult point for beginners especially axiom of choice.
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96 <p>
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97 It has some amount of difficulty and self circulating discussion.
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98 <p>
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99 I'm planning to do it in my old age, but I'm enough age now.
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100 <p>
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338
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101 if you familier with Agda, you can skip to <a href="#set-theory">
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102 there
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103 </a>
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273
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104 <p>
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105
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106 <hr/>
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279
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107 <h2><a name="content004">Agda and Intuitionistic Logic </a></h2>
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273
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108 Curry Howard Isomorphism
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109 <p>
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110
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111 <pre>
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112 Proposition : Proof ⇔ Type : Value
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113
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114 </pre>
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115 which means
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116 <p>
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117 constructing a typed lambda calculus which corresponds a logic
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118 <p>
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119 Typed lambda calculus which allows complex type as a value of a variable (System FC)
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120 <p>
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121 First class Type / Dependent Type
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122 <p>
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123 Agda is a such a programming language which has similar syntax of Haskell
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124 <p>
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125 Coq is specialized in proof assistance such as command and tactics .
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126 <p>
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127
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128 <hr/>
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129 <h2><a name="content005">Introduction of Agda </a></h2>
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130 A length of a list of type A.
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131 <p>
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132
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133 <pre>
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134 length : {A : Set } → List A → Nat
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135 length [] = zero
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136 length (_ ∷ t) = suc ( length t )
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137
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138 </pre>
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139 Simple functional programming language. Type declaration is mandatory.
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140 A colon means type, an equal means value. Indentation based.
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141 <p>
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142 Set is a base type (which may have a level ).
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143 <p>
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144 {} means implicit variable which can be omitted if Agda infers its value.
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145 <p>
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146
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147 <hr/>
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148 <h2><a name="content006">data ( Sum type )</a></h2>
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149 A data type which as exclusive multiple constructors. A similar one as
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150 union in C or case class in Scala.
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151 <p>
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152 It has a similar syntax as Haskell but it has a slight difference.
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153 <p>
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154
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155 <pre>
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156 data List (A : Set ) : Set where
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157 [] : List A
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158 _∷_ : A → List A → List A
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159
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160 </pre>
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161 _∷_ means infix operator. If use explicit _, it can be used in a normal function
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162 syntax.
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163 <p>
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164 Natural number can be defined as a usual way.
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165 <p>
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166
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167 <pre>
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168 data Nat : Set where
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169 zero : Nat
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170 suc : Nat → Nat
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171
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172 </pre>
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173
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174 <hr/>
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175 <h2><a name="content007"> A → B means "A implies B"</a></h2>
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176
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177 <p>
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178 In Agda, a type can be a value of a variable, which is usually called dependent type.
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179 Type has a name Set in Agda.
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180 <p>
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181
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182 <pre>
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183 ex3 : {A B : Set} → Set
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184 ex3 {A}{B} = A → B
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185
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186 </pre>
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187 ex3 is a type : A → B, which is a value of Set. It also means a formula : A implies B.
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188 <p>
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189
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190 <pre>
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191 A type is a formula, the value is the proof
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192
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193 </pre>
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194 A value of A → B can be interpreted as an inference from the formula A to the formula B, which
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195 can be a function from a proof of A to a proof of B.
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196 <p>
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197
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198 <hr/>
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199 <h2><a name="content008">introduction と elimination</a></h2>
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200 For a logical operator, there are two types of inference, an introduction and an elimination.
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201 <p>
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202
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203 <pre>
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204 intro creating symbol / constructor / introduction
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205 elim using symbolic / accessors / elimination
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206
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207 </pre>
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208 In Natural deduction, this can be written in proof schema.
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209 <p>
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210
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211 <pre>
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212 A
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213 :
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214 B A A → B
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215 ------------- →intro ------------------ →elim
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216 A → B B
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217
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218 </pre>
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219 In Agda, this is a pair of type and value as follows. Introduction of → uses λ.
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220 <p>
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221
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222 <pre>
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223 →intro : {A B : Set } → A → B → ( A → B )
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224 →intro _ b = λ x → b
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225 →elim : {A B : Set } → A → ( A → B ) → B
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226 →elim a f = f a
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227
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228 </pre>
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229 Important
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230 <p>
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231
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232 <pre>
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233 {A B : Set } → A → B → ( A → B )
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234
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235 </pre>
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236 is
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237 <p>
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238
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239 <pre>
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240 {A B : Set } → ( A → ( B → ( A → B ) ))
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241
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242 </pre>
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243 This makes currying of function easy.
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244 <p>
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245
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246 <hr/>
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247 <h2><a name="content009"> To prove A → B </a></h2>
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248 Make a left type as an argument. (intros in Coq)
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249 <p>
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250
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251 <pre>
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252 ex5 : {A B C : Set } → A → B → C → ?
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253 ex5 a b c = ?
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254
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255 </pre>
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256 ? is called a hole, which is unspecified part. Agda tell us which kind type is required for the Hole.
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257 <p>
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258 We are going to fill the holes, and if we have no warnings nor errors such as type conflict (Red),
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259 insufficient proof or instance (Yellow), Non-termination, the proof is completed.
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260 <p>
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261
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262 <hr/>
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263 <h2><a name="content010"> A ∧ B</a></h2>
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273
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264 Well known conjunction's introduction and elimination is as follow.
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265 <p>
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266
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267 <pre>
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268 A B A ∧ B A ∧ B
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269 ------------- ----------- proj1 ---------- proj2
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270 A ∧ B A B
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271
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272 </pre>
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273 We can introduce a corresponding structure in our functional programming language.
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274 <p>
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275
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276 <hr/>
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279
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277 <h2><a name="content011"> record</a></h2>
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273
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278
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279 <pre>
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280 record _∧_ A B : Set
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281 field
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282 proj1 : A
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283 proj2 : B
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284
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285 </pre>
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286 _∧_ means infix operator. _∧_ A B can be written as A ∧ B (Haskell uses (∧) )
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287 <p>
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288 This a type which constructed from type A and type B. You may think this as an object
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289 or struct.
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290 <p>
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291
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292 <pre>
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293 record { proj1 = x ; proj2 = y }
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294
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295 </pre>
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296 is a constructor of _∧_.
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297 <p>
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298
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299 <pre>
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300 ex3 : {A B : Set} → A → B → ( A ∧ B )
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301 ex3 a b = record { proj1 = a ; proj2 = b }
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302 ex1 : {A B : Set} → ( A ∧ B ) → A
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303 ex1 a∧b = proj1 a∧b
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304
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305 </pre>
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306 a∧b is a variable name. If we have no spaces in a string, it is a word even if we have symbols
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307 except parenthesis or colons. A symbol requires space separation such as a type defining colon.
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308 <p>
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309 Defining record can be recursively, but we don't use the recursion here.
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310 <p>
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311
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312 <hr/>
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313 <h2><a name="content012"> Mathematical structure</a></h2>
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314 We have types of elements and the relationship in a mathematical structure.
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315 <p>
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316
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317 <pre>
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318 logical relation has no ordering
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319 there is a natural ordering in arguments and a value in a function
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320
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321 </pre>
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322 So we have typical definition style of mathematical structure with records.
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323 <p>
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324
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325 <pre>
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326 record IsOrdinals {n : Level} (ord : Set n)
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327 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
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328 field
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329 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
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330 record Ordinals {n : Level} : Set (suc (suc n)) where
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331 field
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332 ord : Set n
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333 _o<_ : ord → ord → Set n
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334 isOrdinal : IsOrdinals ord _o<_
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335
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336 </pre>
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337 In IsOrdinals, axioms are written in flat way. In Ordinal, we may have
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338 inputs and outputs are put in the field including IsOrdinal.
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339 <p>
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340 Fields of Ordinal is existential objects in the mathematical structure.
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341 <p>
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342
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343 <hr/>
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344 <h2><a name="content013"> A Model and a theory</a></h2>
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345 Agda record is a type, so we can write it in the argument, but is it really exists?
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346 <p>
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347 If we have a value of the record, it simply exists, that is, we need to create all the existence
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348 in the record satisfies all the axioms (= field of IsOrdinal) should be valid.
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349 <p>
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350
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351 <pre>
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352 type of record = theory
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353 value of record = model
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354
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355 </pre>
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356 We call the value of the record as a model. If mathematical structure has a
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357 model, it exists. Pretty Obvious.
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358 <p>
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359
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360 <hr/>
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361 <h2><a name="content014"> postulate と module</a></h2>
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362 Agda proofs are separated by modules, which are large records.
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363 <p>
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364 postulates are assumptions. We can assume a type without proofs.
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365 <p>
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366
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367 <pre>
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368 postulate
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369 sup-o : ( Ordinal → Ordinal ) → Ordinal
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370 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
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371
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372 </pre>
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373 sup-o is an example of upper bound of a function and sup-o< assumes it actually satisfies all the value is less than upper bound.
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374 <p>
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375 Writing some type in a module argument is the same as postulating a type, but
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376 postulate can be written the middle of a proof.
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377 <p>
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378 postulate can be constructive.
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379 <p>
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338
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380 postulate can be inconsistent, which result everything has a proof. Actualy this assumption
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381 doesnot work for Ordinals, we discuss this later.
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273
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382 <p>
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383
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384 <hr/>
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385 <h2><a name="content015"> A ∨ B</a></h2>
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386
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387 <pre>
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388 data _∨_ (A B : Set) : Set where
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389 case1 : A → A ∨ B
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390 case2 : B → A ∨ B
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391
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392 </pre>
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393 As Haskell, case1/case2 are patterns.
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394 <p>
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395
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396 <pre>
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397 ex3 : {A B : Set} → ( A ∨ A ) → A
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398 ex3 = ?
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399
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400 </pre>
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401 In a case statement, Agda command C-C C-C generates possible cases in the head.
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402 <p>
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403
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404 <pre>
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405 ex3 : {A B : Set} → ( A ∨ A ) → A
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406 ex3 (case1 x) = ?
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407 ex3 (case2 x) = ?
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408
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409 </pre>
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410 Proof schema of ∨ is omit due to the complexity.
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411 <p>
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412
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413 <hr/>
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414 <h2><a name="content016"> Negation</a></h2>
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273
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415
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416 <pre>
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417 ⊥
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418 ------------- ⊥-elim
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419 A
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420
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421 </pre>
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422 Anything can be derived from bottom, in this case a Set A. There is no introduction rule
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423 in ⊥, which can be implemented as data which has no constructor.
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424 <p>
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425
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426 <pre>
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427 data ⊥ : Set where
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428
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429 </pre>
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430 ⊥-elim can be proved like this.
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431 <p>
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432
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433 <pre>
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434 ⊥-elim : {A : Set } -> ⊥ -> A
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435 ⊥-elim ()
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436
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437 </pre>
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438 () means no match argument nor value.
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439 <p>
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440 A negation can be defined using ⊥ like this.
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441 <p>
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442
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443 <pre>
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444 ¬_ : Set → Set
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445 ¬ A = A → ⊥
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446
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447 </pre>
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448
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449 <hr/>
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279
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450 <h2><a name="content017">Equality </a></h2>
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273
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451
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452 <p>
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453 All the value in Agda are terms. If we have the same normalized form, two terms are equal.
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454 If we have variables in the terms, we will perform an unification. unifiable terms are equal.
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455 We don't go further on the unification.
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456 <p>
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457
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458 <pre>
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459 { x : A } x ≡ y f x y
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460 --------------- ≡-intro --------------------- ≡-elim
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461 x ≡ x f x x
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462
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463 </pre>
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464 equality _≡_ can be defined as a data.
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465 <p>
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466
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467 <pre>
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468 data _≡_ {A : Set } : A → A → Set where
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469 refl : {x : A} → x ≡ x
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470
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471 </pre>
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472 The elimination of equality is a substitution in a term.
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473 <p>
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474
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475 <pre>
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476 subst : {A : Set } → { x y : A } → ( f : A → Set ) → x ≡ y → f x → f y
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477 subst {A} {x} {y} f refl fx = fx
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478 ex5 : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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479 ex5 {A} {x} {y} {z} x≡y y≡z = subst ( λ k → x ≡ k ) y≡z x≡y
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480
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481 </pre>
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482
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483 <hr/>
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279
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484 <h2><a name="content018">Equivalence relation</a></h2>
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273
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485
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486 <p>
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487
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488 <pre>
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489 refl' : {A : Set} {x : A } → x ≡ x
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490 refl' = ?
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491 sym : {A : Set} {x y : A } → x ≡ y → y ≡ x
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492 sym = ?
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493 trans : {A : Set} {x y z : A } → x ≡ y → y ≡ z → x ≡ z
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494 trans = ?
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495 cong : {A B : Set} {x y : A } { f : A → B } → x ≡ y → f x ≡ f y
|
|
496 cong = ?
|
|
497
|
|
498 </pre>
|
|
499
|
|
500 <hr/>
|
279
|
501 <h2><a name="content019">Ordering</a></h2>
|
273
|
502
|
|
503 <p>
|
|
504 Relation is a predicate on two value which has a same type.
|
|
505 <p>
|
|
506
|
|
507 <pre>
|
|
508 A → A → Set
|
|
509
|
|
510 </pre>
|
|
511 Defining order is the definition of this type with predicate or a data.
|
|
512 <p>
|
|
513
|
|
514 <pre>
|
|
515 data _≤_ : Rel ℕ 0ℓ where
|
|
516 z≤n : ∀ {n} → zero ≤ n
|
|
517 s≤s : ∀ {m n} (m≤n : m ≤ n) → suc m ≤ suc n
|
|
518
|
|
519 </pre>
|
|
520
|
|
521 <hr/>
|
279
|
522 <h2><a name="content020">Quantifier</a></h2>
|
273
|
523
|
|
524 <p>
|
|
525 Handling quantifier in an intuitionistic logic requires special cares.
|
|
526 <p>
|
|
527 In the input of a function, there are no restriction on it, that is, it has
|
|
528 a universal quantifier. (If we explicitly write ∀, Agda gives us a type inference on it)
|
|
529 <p>
|
|
530 There is no ∃ in agda, the one way is using negation like this.
|
|
531 <p>
|
|
532 ∃ (x : A ) → p x = ¬ ( ( x : A ) → ¬ ( p x ) )
|
|
533 <p>
|
|
534 On the another way, f : A can be used like this.
|
|
535 <p>
|
|
536
|
|
537 <pre>
|
|
538 p f
|
|
539
|
|
540 </pre>
|
|
541 If we use a function which can be defined globally which has stronger meaning the
|
|
542 usage of ∃ x in a logical expression.
|
|
543 <p>
|
|
544
|
|
545 <hr/>
|
279
|
546 <h2><a name="content021">Can we do math in this way?</a></h2>
|
273
|
547 Yes, we can. Actually we have Principia Mathematica by Russell and Whitehead (with out computer support).
|
|
548 <p>
|
|
549 In some sense, this story is a reprinting of the work, (but Principia Mathematica has a different formulation than ZF).
|
|
550 <p>
|
|
551
|
|
552 <pre>
|
|
553 define mathematical structure as a record
|
|
554 program inferences as if we have proofs in variables
|
|
555
|
|
556 </pre>
|
|
557
|
|
558 <hr/>
|
279
|
559 <h2><a name="content022">Things which Agda cannot prove</a></h2>
|
273
|
560
|
|
561 <p>
|
|
562 The infamous Internal Parametricity is a limitation of Agda, it cannot prove so called Free Theorem, which
|
|
563 leads uniqueness of a functor in Category Theory.
|
|
564 <p>
|
|
565 Functional extensionality cannot be proved.
|
|
566 <pre>
|
|
567 (∀ x → f x ≡ g x) → f ≡ g
|
|
568
|
|
569 </pre>
|
|
570 Agda has no law of exclude middle.
|
|
571 <p>
|
|
572
|
|
573 <pre>
|
|
574 a ∨ ( ¬ a )
|
|
575
|
|
576 </pre>
|
|
577 For example, (A → B) → ¬ B → ¬ A can be proved but, ( ¬ B → ¬ A ) → A → B cannot.
|
|
578 <p>
|
|
579 It also other problems such as termination, type inference or unification which we may overcome with
|
|
580 efforts or devices or may not.
|
|
581 <p>
|
|
582 If we cannot prove something, we can safely postulate it unless it leads a contradiction.
|
|
583 <pre>
|
|
584
|
|
585
|
|
586 </pre>
|
|
587
|
|
588 <hr/>
|
279
|
589 <h2><a name="content023">Classical story of ZF Set Theory</a></h2>
|
273
|
590
|
|
591 <p>
|
338
|
592 <a name="set-theory">
|
273
|
593 Assuming ZF, constructing a model of ZF is a flow of classical Set Theory, which leads
|
|
594 a relative consistency proof of the Set Theory.
|
|
595 Ordinal number is used in the flow.
|
|
596 <p>
|
|
597 In Agda, first we defines Ordinal numbers (Ordinals), then introduce Ordinal Definable Set (OD).
|
|
598 We need some non constructive assumptions in the construction. A model of Set theory is
|
|
599 constructed based on these assumptions.
|
|
600 <p>
|
|
601 <img src="fig/set-theory.svg">
|
|
602
|
|
603 <p>
|
|
604
|
|
605 <hr/>
|
279
|
606 <h2><a name="content024">Ordinals</a></h2>
|
273
|
607 Ordinals are our intuition of infinite things, which has ∅ and orders on the things.
|
|
608 It also has a successor osuc.
|
|
609 <p>
|
|
610
|
|
611 <pre>
|
|
612 record Ordinals {n : Level} : Set (suc (suc n)) where
|
|
613 field
|
|
614 ord : Set n
|
|
615 o∅ : ord
|
|
616 osuc : ord → ord
|
|
617 _o<_ : ord → ord → Set n
|
|
618 isOrdinal : IsOrdinals ord o∅ osuc _o<_
|
|
619
|
|
620 </pre>
|
|
621 It is different from natural numbers in way. The order of Ordinals is not defined in terms
|
|
622 of successor. It is given from outside, which make it possible to have higher order infinity.
|
|
623 <p>
|
|
624
|
|
625 <hr/>
|
279
|
626 <h2><a name="content025">Axiom of Ordinals</a></h2>
|
273
|
627 Properties of infinite things. We request a transfinite induction, which states that if
|
|
628 some properties are satisfied below all possible ordinals, the properties are true on all
|
|
629 ordinals.
|
|
630 <p>
|
|
631 Successor osuc has no ordinal between osuc and the base ordinal. There are some ordinals
|
|
632 which is not a successor of any ordinals. It is called limit ordinal.
|
|
633 <p>
|
|
634 Any two ordinal can be compared, that is less, equal or more, that is total order.
|
|
635 <p>
|
|
636
|
|
637 <pre>
|
|
638 record IsOrdinals {n : Level} (ord : Set n) (o∅ : ord )
|
|
639 (osuc : ord → ord )
|
|
640 (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
|
|
641 field
|
|
642 Otrans : {x y z : ord } → x o< y → y o< z → x o< z
|
|
643 OTri : Trichotomous {n} _≡_ _o<_
|
|
644 ¬x<0 : { x : ord } → ¬ ( x o< o∅ )
|
|
645 <-osuc : { x : ord } → x o< osuc x
|
|
646 osuc-≡< : { a x : ord } → x o< osuc a → (x ≡ a ) ∨ (x o< a)
|
|
647 TransFinite : { ψ : ord → Set (suc n) }
|
|
648 → ( (x : ord) → ( (y : ord ) → y o< x → ψ y ) → ψ x )
|
|
649 → ∀ (x : ord) → ψ x
|
|
650
|
|
651 </pre>
|
|
652
|
|
653 <hr/>
|
338
|
654 <h2><a name="content026">Concrete Ordinals or Countable Ordinals</a></h2>
|
273
|
655
|
|
656 <p>
|
|
657 We can define a list like structure with level, which is a kind of two dimensional infinite array.
|
|
658 <p>
|
|
659
|
|
660 <pre>
|
|
661 data OrdinalD {n : Level} : (lv : Nat) → Set n where
|
|
662 Φ : (lv : Nat) → OrdinalD lv
|
|
663 OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
|
|
664
|
|
665 </pre>
|
|
666 The order of the OrdinalD can be defined in this way.
|
|
667 <p>
|
|
668
|
|
669 <pre>
|
|
670 data _d<_ {n : Level} : {lx ly : Nat} → OrdinalD {n} lx → OrdinalD {n} ly → Set n where
|
|
671 Φ< : {lx : Nat} → {x : OrdinalD {n} lx} → Φ lx d< OSuc lx x
|
|
672 s< : {lx : Nat} → {x y : OrdinalD {n} lx} → x d< y → OSuc lx x d< OSuc lx y
|
|
673
|
|
674 </pre>
|
|
675 This is a simple data structure, it has no abstract assumptions, and it is countable many data
|
|
676 structure.
|
|
677 <p>
|
|
678
|
|
679 <pre>
|
|
680 Φ 0
|
|
681 OSuc 2 ( Osuc 2 ( Osuc 2 (Φ 2)))
|
|
682 Osuc 0 (Φ 0) d< Φ 1
|
|
683
|
|
684 </pre>
|
|
685
|
|
686 <hr/>
|
279
|
687 <h2><a name="content027">Model of Ordinals</a></h2>
|
273
|
688
|
|
689 <p>
|
|
690 It is easy to show OrdinalD and its order satisfies the axioms of Ordinals.
|
|
691 <p>
|
|
692 So our Ordinals has a mode. This means axiom of Ordinals are consistent.
|
|
693 <p>
|
|
694
|
|
695 <hr/>
|
279
|
696 <h2><a name="content028">Debugging axioms using Model</a></h2>
|
273
|
697 Whether axiom is correct or not can be checked by a validity on a mode.
|
|
698 <p>
|
|
699 If not, we may fix the axioms or the model, such as the definitions of the order.
|
|
700 <p>
|
|
701 We can also ask whether the inputs exist.
|
|
702 <p>
|
|
703
|
|
704 <hr/>
|
279
|
705 <h2><a name="content029">Countable Ordinals can contains uncountable set?</a></h2>
|
273
|
706 Yes, the ordinals contains any level of infinite Set in the axioms.
|
|
707 <p>
|
|
708 If we handle real-number in the model, only countable number of real-number is used.
|
|
709 <p>
|
|
710
|
|
711 <pre>
|
|
712 from the outside view point, it is countable
|
|
713 from the internal view point, it is uncountable
|
|
714
|
|
715 </pre>
|
|
716 The definition of countable/uncountable is the same, but the properties are different
|
|
717 depending on the context.
|
|
718 <p>
|
|
719 We don't show the definition of cardinal number here.
|
|
720 <p>
|
|
721
|
|
722 <hr/>
|
279
|
723 <h2><a name="content030">What is Set</a></h2>
|
273
|
724 The word Set in Agda is not a Set of ZF Set, but it is a type (why it is named Set?).
|
|
725 <p>
|
|
726 From naive point view, a set i a list, but in Agda, elements have all the same type.
|
|
727 A set in ZF may contain other Sets in ZF, which not easy to implement it as a list.
|
|
728 <p>
|
|
729 Finite set may be written in finite series of ∨, but ...
|
|
730 <p>
|
|
731
|
|
732 <hr/>
|
279
|
733 <h2><a name="content031">We don't ask the contents of Set. It can be anything.</a></h2>
|
273
|
734 From empty set φ, we can think a set contains a φ, and a pair of φ and the set, and so on,
|
|
735 and all of them, and again we repeat this.
|
|
736 <p>
|
|
737
|
|
738 <pre>
|
|
739 φ {φ} {φ,{φ}}, {φ,{φ},...}
|
|
740
|
|
741 </pre>
|
|
742 It is called V.
|
|
743 <p>
|
|
744 This operation can be performed within a ZF Set theory. Classical Set Theory assumes
|
|
745 ZF, so this kind of thing is allowed.
|
|
746 <p>
|
|
747 But in our case, we have no ZF theory, so we are going to use Ordinals.
|
|
748 <p>
|
338
|
749 The idea is to use an ordinal as a pointer to a record which defines a Set.
|
|
750 If the recored defines a series of Ordinals which is a pointer to the Set. This record looks like a Set.
|
|
751 <p>
|
273
|
752
|
|
753 <hr/>
|
279
|
754 <h2><a name="content032">Ordinal Definable Set</a></h2>
|
273
|
755 We can define a sbuset of Ordinals using predicates. What is a subset?
|
|
756 <p>
|
|
757
|
|
758 <pre>
|
|
759 a predicate has an Ordinal argument
|
|
760
|
|
761 </pre>
|
|
762 is an Ordinal Definable Set (OD). In Agda, OD is defined as follows.
|
|
763 <p>
|
|
764
|
|
765 <pre>
|
|
766 record OD : Set (suc n ) where
|
|
767 field
|
|
768 def : (x : Ordinal ) → Set n
|
|
769
|
|
770 </pre>
|
|
771 Ordinals itself is not a set in a ZF Set theory but a class. In OD,
|
|
772 <p>
|
|
773
|
|
774 <pre>
|
338
|
775 data One : Set n where
|
|
776 OneObj : One
|
|
777 record { def = λ x → One }
|
273
|
778
|
|
779 </pre>
|
338
|
780 means it accepets all Ordinals, i.e. this is Ordinals itself, so ODs are larger than ZF Set.
|
|
781 You can say OD is a class in ZF Set Theory term.
|
|
782 <p>
|
|
783
|
|
784 <hr/>
|
|
785 <h2><a name="content033">OD is not ZF Set</a></h2>
|
|
786 If we have 1 to 1 mapping between an OD and an Ordinal, OD contains several ODs and OD looks like
|
|
787 a Set. The idea is to use an ordinal as a pointer to OD.
|
|
788 Unfortunately this scheme does not work well. As we saw OD includes all Ordinals, which is a maximum of OD, but Ordinals has no maximum at all. So we have a contradction like
|
|
789 <p>
|
|
790
|
|
791 <pre>
|
|
792 ¬OD-order : ( od→ord : OD → Ordinal )
|
|
793 → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥
|
|
794 ¬OD-order od→ord ord→od c<→o< = ?
|
|
795
|
|
796 </pre>
|
|
797 Actualy we can prove this contrdction, so we need some restrctions on OD.
|
|
798 <p>
|
|
799 This is a kind of Russel paradox, that is if OD contains everything, what happens if it contains itself.
|
273
|
800 <p>
|
|
801
|
|
802 <hr/>
|
338
|
803 <h2><a name="content034"> HOD : Hereditarily Ordinal Definable</a></h2>
|
|
804 What we need is a bounded OD, the containment is limited by an ordinal.
|
|
805 <p>
|
|
806
|
|
807 <pre>
|
|
808 record HOD : Set (suc n) where
|
|
809 field
|
|
810 od : OD
|
|
811 odmax : Ordinal
|
|
812 <odmax : {y : Ordinal} → def od y → y o< odmax
|
|
813
|
|
814 </pre>
|
|
815 In classical Set Theory, HOD stands for Hereditarily Ordinal Definable, which means
|
273
|
816 <p>
|
|
817
|
|
818 <pre>
|
338
|
819 HOD = { x | TC x ⊆ OD }
|
273
|
820
|
|
821 </pre>
|
338
|
822 TC x is all transitive closure of x, that is elements of x and following all elements of them are all OD. But what is x? In this case, x is an Set which we don't have yet. In our case, HOD is a bounded OD.
|
|
823 <p>
|
|
824
|
|
825 <hr/>
|
|
826 <h2><a name="content035">1 to 1 mapping between an HOD and an Ordinal</a></h2>
|
|
827 HOD is a predicate on Ordinals and the solution is bounded by some ordinal. If we have a mapping
|
|
828 <p>
|
|
829
|
|
830 <pre>
|
|
831 od→ord : HOD → Ordinal
|
|
832 ord→od : Ordinal → HOD
|
|
833 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
|
|
834 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
835
|
|
836 </pre>
|
|
837 we can check an HOD is an element of the OD using def.
|
273
|
838 <p>
|
|
839 A ∋ x can be define as follows.
|
|
840 <p>
|
|
841
|
|
842 <pre>
|
338
|
843 _∋_ : ( A x : HOD ) → Set n
|
|
844 _∋_ A x = def (od A) ( od→ord x )
|
273
|
845
|
|
846 </pre>
|
|
847 In ψ : Ordinal → Set, if A is a record { def = λ x → ψ x } , then
|
|
848 <p>
|
|
849
|
|
850 <pre>
|
|
851 A x = def A ( od→ord x ) = ψ (od→ord x)
|
|
852
|
|
853 </pre>
|
|
854 They say the existing of the mappings can be proved in Classical Set Theory, but we
|
|
855 simply assumes these non constructively.
|
|
856 <p>
|
|
857 <img src="fig/ord-od-mapping.svg">
|
|
858
|
|
859 <p>
|
|
860
|
|
861 <hr/>
|
338
|
862 <h2><a name="content036">Order preserving in the mapping of OD and Ordinal</a></h2>
|
|
863 Ordinals have the order and HOD has a natural order based on inclusion ( def / ∋ ).
|
273
|
864 <p>
|
|
865
|
|
866 <pre>
|
338
|
867 def (od y) ( od→ord x )
|
273
|
868
|
|
869 </pre>
|
338
|
870 An elements of HOD should be defined before the HOD, that is, an ordinal corresponding an elements
|
273
|
871 have to be smaller than the corresponding ordinal of the containing OD.
|
338
|
872 We also assumes subset is always smaller. This is necessary to make a limit of Power Set.
|
273
|
873 <p>
|
|
874
|
|
875 <pre>
|
338
|
876 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
|
|
877 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
|
273
|
878
|
|
879 </pre>
|
338
|
880 If wa assumes reverse order preservation,
|
273
|
881 <p>
|
|
882
|
|
883 <pre>
|
|
884 o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x
|
|
885
|
|
886 </pre>
|
338
|
887 ∀ x ∋ ∅ becomes true, which manes all OD becomes Ordinals in the model.
|
273
|
888 <p>
|
|
889 <img src="fig/ODandOrdinals.svg">
|
|
890
|
|
891 <p>
|
|
892
|
|
893 <hr/>
|
279
|
894 <h2><a name="content037">Various Sets</a></h2>
|
273
|
895 In classical Set Theory, there is a hierarchy call L, which can be defined by a predicate.
|
|
896 <p>
|
|
897
|
|
898 <pre>
|
|
899 Ordinal / things satisfies axiom of Ordinal / extension of natural number
|
|
900 V / hierarchical construction of Set from φ
|
|
901 L / hierarchical predicate definable construction of Set from φ
|
338
|
902 HOD / Hereditarily Ordinal Definable
|
273
|
903 OD / equational formula on Ordinals
|
|
904 Agda Set / Type / it also has a level
|
|
905
|
|
906 </pre>
|
|
907
|
|
908 <hr/>
|
279
|
909 <h2><a name="content038">Fixes on ZF to intuitionistic logic</a></h2>
|
273
|
910
|
|
911 <p>
|
|
912 We use ODs as Sets in ZF, and defines record ZF, that is, we have to define
|
|
913 ZF axioms in Agda.
|
|
914 <p>
|
|
915 It may not valid in our model. We have to debug it.
|
|
916 <p>
|
|
917 Fixes are depends on axioms.
|
|
918 <p>
|
|
919 <img src="fig/axiom-type.svg">
|
|
920
|
|
921 <p>
|
|
922 <a href="fig/zf-record.html">
|
|
923 ZFのrecord </a>
|
|
924 <p>
|
|
925
|
|
926 <hr/>
|
279
|
927 <h2><a name="content039">Pure logical axioms</a></h2>
|
273
|
928
|
|
929 <pre>
|
279
|
930 empty, pair, select, ε-induction??infinity
|
273
|
931
|
|
932 </pre>
|
|
933 These are logical relations among OD.
|
|
934 <p>
|
|
935
|
|
936 <pre>
|
|
937 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
|
|
938 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
939 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
940 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
|
|
941 infinity∅ : ∅ ∈ infinite
|
|
942 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ( x , x ) ) ∈ infinite
|
|
943 ε-induction : { ψ : OD → Set (suc n)}
|
|
944 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
945 → (x : OD ) → ψ x
|
|
946
|
|
947 </pre>
|
|
948 finitely can be define by Agda data.
|
|
949 <p>
|
|
950
|
|
951 <pre>
|
|
952 data infinite-d : ( x : Ordinal ) → Set n where
|
|
953 iφ : infinite-d o∅
|
|
954 isuc : {x : Ordinal } → infinite-d x →
|
|
955 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
|
|
956
|
|
957 </pre>
|
|
958 Union (x , ( x , x )) should be an direct successor of x, but we cannot prove it in our model.
|
|
959 <p>
|
|
960
|
|
961 <hr/>
|
279
|
962 <h2><a name="content040">Axiom of Pair</a></h2>
|
273
|
963 In the Tanaka's book, axiom of pair is as follows.
|
|
964 <p>
|
|
965
|
|
966 <pre>
|
|
967 ∀ x ∀ y ∃ z ∀ t ( z ∋ t ↔ t ≈ x ∨ t ≈ y)
|
|
968
|
|
969 </pre>
|
|
970 We have fix ∃ z, a function (x , y) is defined, which is _,_ .
|
|
971 <p>
|
|
972
|
|
973 <pre>
|
|
974 _,_ : ( A B : ZFSet ) → ZFSet
|
|
975
|
|
976 </pre>
|
|
977 using this, we can define two directions in separates axioms, like this.
|
|
978 <p>
|
|
979
|
|
980 <pre>
|
|
981 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y )
|
|
982 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t
|
|
983
|
|
984 </pre>
|
|
985 This is already written in Agda, so we use these as axioms. All inputs have ∀.
|
|
986 <p>
|
|
987
|
|
988 <hr/>
|
279
|
989 <h2><a name="content041">pair in OD</a></h2>
|
273
|
990 OD is an equation on Ordinals, we can simply write axiom of pair in the OD.
|
|
991 <p>
|
|
992
|
|
993 <pre>
|
338
|
994 _,_ : HOD → HOD → HOD
|
|
995 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = ? ; <odmax = ? }
|
273
|
996
|
|
997 </pre>
|
338
|
998 It is easy to find out odmax from odmax of x and y.
|
|
999 <p>
|
273
|
1000 ≡ is an equality of λ terms, but please not that this is equality on Ordinals.
|
|
1001 <p>
|
|
1002
|
|
1003 <hr/>
|
279
|
1004 <h2><a name="content042">Validity of Axiom of Pair</a></h2>
|
273
|
1005 Assuming ZFSet is OD, we are going to prove pair→ .
|
|
1006 <p>
|
|
1007
|
|
1008 <pre>
|
|
1009 pair→ : ( x y t : OD ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y )
|
|
1010 pair→ x y t p = ?
|
|
1011
|
|
1012 </pre>
|
|
1013 In this program, type of p is ( x , y ) ∋ t , that is def ( x , y ) that is, (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) .
|
|
1014 <p>
|
|
1015 Since _∨_ is a data, it can be developed as (C-c C-c : agda2-make-case ).
|
|
1016 <p>
|
|
1017
|
|
1018 <pre>
|
|
1019 pair→ x y t (case1 t≡x ) = ?
|
|
1020 pair→ x y t (case2 t≡y ) = ?
|
|
1021
|
|
1022 </pre>
|
|
1023 The type of the ? is ( t == x ) ∨ ( t == y ), again it is data _∨_ .
|
|
1024 <p>
|
|
1025
|
|
1026 <pre>
|
|
1027 pair→ x y t (case1 t≡x ) = case1 ?
|
|
1028 pair→ x y t (case2 t≡y ) = case2 ?
|
|
1029
|
|
1030 </pre>
|
|
1031 The ? in case1 is t == x, so we have to create this from t≡x, which is a name of a variable
|
|
1032 which type is
|
|
1033 <p>
|
|
1034
|
|
1035 <pre>
|
|
1036 t≡x : od→ord t ≡ od→ord x
|
|
1037
|
|
1038 </pre>
|
|
1039 which is shown by an Agda command (C-C C-E : agda2-show-context ).
|
|
1040 <p>
|
|
1041 But we haven't defined == yet.
|
|
1042 <p>
|
|
1043
|
|
1044 <hr/>
|
279
|
1045 <h2><a name="content043">Equality of OD and Axiom of Extensionality </a></h2>
|
273
|
1046 OD is defined by a predicates, if we compares normal form of the predicates, even if
|
|
1047 it contains the same elements, it may be different, which is no good as an equality of
|
|
1048 Sets.
|
|
1049 <p>
|
|
1050 Axiom of Extensionality requires sets having the same elements are handled in the same way
|
|
1051 each other.
|
|
1052 <p>
|
|
1053
|
|
1054 <pre>
|
|
1055 ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
|
|
1056
|
|
1057 </pre>
|
|
1058 We can write this axiom in Agda as follows.
|
|
1059 <p>
|
|
1060
|
|
1061 <pre>
|
|
1062 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
|
|
1063
|
|
1064 </pre>
|
|
1065 So we use ( A ∋ z ) ⇔ (B ∋ z) as an equality (_==_) of our model. We have to show
|
|
1066 A ∈ w ⇔ B ∈ w from A == B.
|
|
1067 <p>
|
|
1068 x == y can be defined in this way.
|
|
1069 <p>
|
|
1070
|
|
1071 <pre>
|
|
1072 record _==_ ( a b : OD ) : Set n where
|
|
1073 field
|
|
1074 eq→ : ∀ { x : Ordinal } → def a x → def b x
|
|
1075 eq← : ∀ { x : Ordinal } → def b x → def a x
|
|
1076
|
|
1077 </pre>
|
338
|
1078 Actually, (z : HOD) → (A ∋ z) ⇔ (B ∋ z) implies od A == od B.
|
273
|
1079 <p>
|
|
1080
|
|
1081 <pre>
|
338
|
1082 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → od A == od B
|
273
|
1083 eq→ (extensionality0 {A} {B} eq ) {x} d = ?
|
|
1084 eq← (extensionality0 {A} {B} eq ) {x} d = ?
|
|
1085
|
|
1086 </pre>
|
|
1087 ? are def B x and def A x and these are generated from eq : (z : OD) → (A ∋ z) ⇔ (B ∋ z) .
|
|
1088 <p>
|
|
1089 Actual proof is rather complicated.
|
|
1090 <p>
|
|
1091
|
|
1092 <pre>
|
338
|
1093 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
|
|
1094 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
|
273
|
1095
|
|
1096 </pre>
|
|
1097 where
|
|
1098 <p>
|
|
1099
|
|
1100 <pre>
|
338
|
1101 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (def A (od y) → def (od B) y) → def (od A) x → def (od B) x
|
|
1102 odef-iso refl t = t
|
273
|
1103
|
|
1104 </pre>
|
|
1105
|
|
1106 <hr/>
|
279
|
1107 <h2><a name="content044">Validity of Axiom of Extensionality</a></h2>
|
273
|
1108
|
|
1109 <p>
|
338
|
1110 If we can derive (w ∋ A) ⇔ (w ∋ B) from od A == od B, the axiom becomes valid, but it seems impossible, so we assumes
|
273
|
1111 <p>
|
|
1112
|
|
1113 <pre>
|
338
|
1114 ==→o≡ : { x y : HOD } → (od x == od y) → x ≡ y
|
273
|
1115
|
|
1116 </pre>
|
|
1117 Using this, we have
|
|
1118 <p>
|
|
1119
|
|
1120 <pre>
|
338
|
1121 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
|
273
|
1122 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
|
338
|
1123 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
|
273
|
1124
|
|
1125 </pre>
|
|
1126
|
|
1127 <hr/>
|
279
|
1128 <h2><a name="content045">Non constructive assumptions so far</a></h2>
|
338
|
1129
|
273
|
1130 <p>
|
|
1131
|
|
1132 <pre>
|
338
|
1133 od→ord : HOD → Ordinal
|
|
1134 ord→od : Ordinal → HOD
|
|
1135 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
|
|
1136 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z)
|
|
1137 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x
|
|
1138 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
1139 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
|
|
1140 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
|
|
1141 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
|
273
|
1142
|
|
1143 </pre>
|
|
1144
|
|
1145 <hr/>
|
279
|
1146 <h2><a name="content046">Axiom which have negation form</a></h2>
|
273
|
1147
|
|
1148 <p>
|
|
1149
|
|
1150 <pre>
|
|
1151 Union, Selection
|
|
1152
|
|
1153 </pre>
|
|
1154 These axioms contains ∃ x as a logical relation, which can be described in ¬ ( ∀ x ( ¬ p )).
|
|
1155 <p>
|
|
1156 Axiom of replacement uses upper bound of function on Ordinals, which makes it non-constructive.
|
|
1157 <p>
|
|
1158 Power Set axiom requires double negation,
|
|
1159 <p>
|
|
1160
|
|
1161 <pre>
|
|
1162 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x )
|
|
1163 power← : ∀( A t : ZFSet ) → t ⊆_ A → Power A ∋ t
|
|
1164
|
|
1165 </pre>
|
|
1166 If we have an assumption of law of exclude middle, we can recover the original A ∋ x form.
|
|
1167 <p>
|
|
1168
|
|
1169 <hr/>
|
279
|
1170 <h2><a name="content047">Union </a></h2>
|
273
|
1171 The original form of the Axiom of Union is
|
|
1172 <p>
|
|
1173
|
|
1174 <pre>
|
|
1175 ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
|
|
1176
|
|
1177 </pre>
|
|
1178 Union requires the existence of b in a ⊇ ∃ b ∋ x . We will use negation form of ∃.
|
|
1179 <p>
|
|
1180
|
|
1181 <pre>
|
|
1182 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
|
|
1183 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
1184
|
|
1185 </pre>
|
|
1186 The definition of Union in OD is like this.
|
|
1187 <p>
|
|
1188
|
|
1189 <pre>
|
|
1190 Union : OD → OD
|
|
1191 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
|
|
1192
|
|
1193 </pre>
|
|
1194 Proof of validity is straight forward.
|
|
1195 <p>
|
|
1196
|
|
1197 <pre>
|
|
1198 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
|
|
1199 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
|
|
1200 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
|
|
1201 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
|
|
1202 union← X z UX∋z = FExists _ lemma UX∋z where
|
|
1203 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
|
|
1204 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
|
|
1205
|
|
1206 </pre>
|
|
1207 where
|
|
1208 <p>
|
|
1209
|
|
1210 <pre>
|
|
1211 FExists : {m l : Level} → ( ψ : Ordinal → Set m )
|
|
1212 → {p : Set l} ( P : { y : Ordinal } → ψ y → ¬ p )
|
|
1213 → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
|
|
1214 → ¬ p
|
|
1215 FExists {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p )
|
|
1216
|
|
1217 </pre>
|
|
1218 which checks existence using contra-position.
|
|
1219 <p>
|
|
1220
|
|
1221 <hr/>
|
279
|
1222 <h2><a name="content048">Axiom of replacement</a></h2>
|
273
|
1223 We can replace the elements of a set by a function and it becomes a set. From the book,
|
|
1224 <p>
|
|
1225
|
|
1226 <pre>
|
|
1227 ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
|
|
1228
|
|
1229 </pre>
|
|
1230 The existential quantifier can be related by a function,
|
|
1231 <p>
|
|
1232
|
|
1233 <pre>
|
|
1234 Replace : OD → (OD → OD ) → OD
|
|
1235
|
|
1236 </pre>
|
|
1237 The axioms becomes as follows.
|
|
1238 <p>
|
|
1239
|
|
1240 <pre>
|
|
1241 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
|
|
1242 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
|
|
1243
|
|
1244 </pre>
|
|
1245 In the axiom, the existence of the original elements is necessary. In order to do that we use OD which has
|
|
1246 negation form of existential quantifier in the definition.
|
|
1247 <p>
|
|
1248
|
|
1249 <pre>
|
|
1250 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD
|
|
1251 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
|
|
1252
|
|
1253 </pre>
|
|
1254 Besides this upper bounds is required.
|
|
1255 <p>
|
|
1256
|
|
1257 <pre>
|
|
1258 Replace : OD → (OD → OD ) → OD
|
|
1259 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
|
|
1260
|
|
1261 </pre>
|
|
1262 We omit the proof of the validity, but it is rather straight forward.
|
|
1263 <p>
|
|
1264
|
|
1265 <hr/>
|
279
|
1266 <h2><a name="content049">Validity of Power Set Axiom</a></h2>
|
273
|
1267 The original Power Set Axiom is this.
|
|
1268 <p>
|
|
1269
|
|
1270 <pre>
|
|
1271 ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
|
|
1272
|
|
1273 </pre>
|
|
1274 The existential quantifier is replaced by a function
|
|
1275 <p>
|
|
1276
|
|
1277 <pre>
|
|
1278 Power : ( A : OD ) → OD
|
|
1279
|
|
1280 </pre>
|
|
1281 t ⊆ X is a record like this.
|
|
1282 <p>
|
|
1283
|
|
1284 <pre>
|
|
1285 record _⊆_ ( A B : OD ) : Set (suc n) where
|
|
1286 field
|
|
1287 incl : { x : OD } → A ∋ x → B ∋ x
|
|
1288
|
|
1289 </pre>
|
|
1290 Axiom becomes likes this.
|
|
1291 <p>
|
|
1292
|
|
1293 <pre>
|
|
1294 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
1295 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
1296
|
|
1297 </pre>
|
|
1298 The validity of the axioms are slight complicated, we have to define set of all subset. We define
|
|
1299 subset in a different form.
|
|
1300 <p>
|
|
1301
|
|
1302 <pre>
|
|
1303 ZFSubset : (A x : OD ) → OD
|
|
1304 ZFSubset A x = record { def = λ y → def A y ∧ def x y }
|
|
1305
|
|
1306 </pre>
|
|
1307 We can prove,
|
|
1308 <p>
|
|
1309
|
|
1310 <pre>
|
|
1311 ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
|
|
1312
|
|
1313 </pre>
|
|
1314 We only have upper bound as an ordinal, but we have an obvious OD based on the order of Ordinals,
|
|
1315 which is an Ordinals with our Model.
|
|
1316 <p>
|
|
1317
|
|
1318 <pre>
|
|
1319 Ord : ( a : Ordinal ) → OD
|
|
1320 Ord a = record { def = λ y → y o< a }
|
|
1321 Def : (A : OD ) → OD
|
|
1322 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
|
|
1323
|
|
1324 </pre>
|
|
1325 This is slight larger than Power A, so we replace all elements x by A ∩ x (some of them may empty).
|
|
1326 <p>
|
|
1327
|
|
1328 <pre>
|
|
1329 Power : OD → OD
|
|
1330 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
|
|
1331
|
|
1332 </pre>
|
|
1333 Creating Power Set of Ordinals is rather easy, then we use replacement axiom on A ∩ x since we have this.
|
|
1334 <p>
|
|
1335
|
|
1336 <pre>
|
|
1337 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
|
|
1338
|
|
1339 </pre>
|
|
1340 In case of Ord a intro of Power Set axiom becomes valid.
|
|
1341 <p>
|
|
1342
|
|
1343 <pre>
|
|
1344 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
|
|
1345
|
|
1346 </pre>
|
|
1347 Using this, we can prove,
|
|
1348 <p>
|
|
1349
|
|
1350 <pre>
|
|
1351 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
|
|
1352 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
|
|
1353
|
|
1354 </pre>
|
|
1355
|
|
1356 <hr/>
|
279
|
1357 <h2><a name="content050">Axiom of regularity, Axiom of choice, ε-induction</a></h2>
|
273
|
1358
|
|
1359 <p>
|
|
1360 Axiom of regularity requires non self intersectable elements (which is called minimum), if we
|
|
1361 replace it by a function, it becomes a choice function. It makes axiom of choice valid.
|
|
1362 <p>
|
|
1363 This means we cannot prove axiom regularity form our model, and if we postulate this, axiom of
|
|
1364 choice also becomes valid.
|
|
1365 <p>
|
|
1366
|
|
1367 <pre>
|
|
1368 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
1369 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
1370 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
1371
|
|
1372 </pre>
|
|
1373 We can avoid this using ε-induction (a predicate is valid on all set if the predicate is true on some element of set).
|
|
1374 Assuming law of exclude middle, they say axiom of regularity will be proved, but we haven't check it yet.
|
|
1375 <p>
|
|
1376
|
|
1377 <pre>
|
|
1378 ε-induction : { ψ : OD → Set (suc n)}
|
|
1379 → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x )
|
|
1380 → (x : OD ) → ψ x
|
|
1381
|
|
1382 </pre>
|
|
1383 In our model, we assumes the mapping between Ordinals and OD, this is actually the TransFinite induction in Ordinals.
|
|
1384 <p>
|
|
1385 The axiom of choice in the book is complicated using any pair in a set, so we use use a form in the Wikipedia.
|
|
1386 <p>
|
|
1387
|
|
1388 <pre>
|
|
1389 ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
|
|
1390
|
|
1391 </pre>
|
|
1392 We can formulate like this.
|
|
1393 <p>
|
|
1394
|
|
1395 <pre>
|
|
1396 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
|
|
1397 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
|
|
1398
|
|
1399 </pre>
|
|
1400 It does not requires ∅ ∉ X .
|
|
1401 <p>
|
|
1402
|
|
1403 <hr/>
|
279
|
1404 <h2><a name="content051">Axiom of choice and Law of Excluded Middle</a></h2>
|
273
|
1405 In our model, since OD has a mapping to Ordinals, it has evident order, which means well ordering theorem is valid,
|
|
1406 but it don't have correct form of the axiom yet. They say well ordering axiom is equivalent to the axiom of choice,
|
|
1407 but it requires law of the exclude middle.
|
|
1408 <p>
|
|
1409 Actually, it is well known to prove law of the exclude middle from axiom of choice in intuitionistic logic, and we can
|
|
1410 perform the proof in our mode. Using the definition like this, predicates and ODs are related and we can ask the
|
|
1411 set is empty or not if we have an axiom of choice, so we have the law of the exclude middle p ∨ ( ¬ p ) .
|
|
1412 <p>
|
|
1413
|
|
1414 <pre>
|
|
1415 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
|
|
1416 ppp {p} {a} d = d
|
|
1417
|
|
1418 </pre>
|
|
1419 We can prove axiom of choice from law excluded middle since we have TransFinite induction. So Axiom of choice
|
|
1420 and Law of Excluded Middle is equivalent in our mode.
|
|
1421 <p>
|
|
1422
|
|
1423 <hr/>
|
279
|
1424 <h2><a name="content052">Relation-ship among ZF axiom</a></h2>
|
273
|
1425 <img src="fig/axiom-dependency.svg">
|
|
1426
|
|
1427 <p>
|
|
1428
|
|
1429 <hr/>
|
279
|
1430 <h2><a name="content053">Non constructive assumption in our model</a></h2>
|
273
|
1431 mapping between OD and Ordinals
|
|
1432 <p>
|
|
1433
|
|
1434 <pre>
|
|
1435 od→ord : OD → Ordinal
|
|
1436 ord→od : Ordinal → OD
|
|
1437 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x
|
|
1438 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
|
|
1439 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y
|
|
1440
|
|
1441 </pre>
|
|
1442 Equivalence on OD
|
|
1443 <p>
|
|
1444
|
|
1445 <pre>
|
|
1446 ==→o≡ : { x y : OD } → (x == y) → x ≡ y
|
|
1447
|
|
1448 </pre>
|
|
1449 Upper bound
|
|
1450 <p>
|
|
1451
|
|
1452 <pre>
|
|
1453 sup-o : ( Ordinal → Ordinal ) → Ordinal
|
|
1454 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ
|
|
1455
|
|
1456 </pre>
|
|
1457 Axiom of choice and strong axiom of regularity.
|
|
1458 <p>
|
|
1459
|
|
1460 <pre>
|
|
1461 minimal : (x : OD ) → ¬ (x == od∅ )→ OD
|
|
1462 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
|
|
1463 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) )
|
|
1464
|
|
1465 </pre>
|
|
1466
|
|
1467 <hr/>
|
279
|
1468 <h2><a name="content054">So it this correct?</a></h2>
|
273
|
1469
|
|
1470 <p>
|
|
1471 Our axiom are syntactically the same in the text book, but negations are slightly different.
|
|
1472 <p>
|
|
1473 If we assumes excluded middle, these are exactly same.
|
|
1474 <p>
|
|
1475 Even if we assumes excluded middle, intuitionistic logic itself remains consistent, but we cannot prove it.
|
|
1476 <p>
|
|
1477 Except the upper bound, axioms are simple logical relation.
|
|
1478 <p>
|
|
1479 Proof of existence of mapping between OD and Ordinals are not obvious. We don't know we prove it or not.
|
|
1480 <p>
|
|
1481 Existence of the Upper bounds is a pure assumption, if we have not limit on Ordinals, it may contradicts,
|
|
1482 but we don't have explicit upper limit on Ordinals.
|
|
1483 <p>
|
|
1484 Several inference on our model or our axioms are basically parallel to the set theory text book, so it looks like correct.
|
|
1485 <p>
|
|
1486
|
|
1487 <hr/>
|
279
|
1488 <h2><a name="content055">How to use Agda Set Theory</a></h2>
|
273
|
1489 Assuming record ZF, classical set theory can be developed. If necessary, axiom of choice can be
|
|
1490 postulated or assuming law of excluded middle.
|
|
1491 <p>
|
|
1492 Instead, simply assumes non constructive assumption, various theory can be developed. We haven't check
|
|
1493 these assumptions are proved in record ZF, so we are not sure, these development is a result of ZF Set theory.
|
|
1494 <p>
|
|
1495 ZF record itself is not necessary, for example, topology theory without ZF can be possible.
|
|
1496 <p>
|
|
1497
|
|
1498 <hr/>
|
279
|
1499 <h2><a name="content056">Topos and Set Theory</a></h2>
|
273
|
1500 Topos is a mathematical structure in Category Theory, which is a Cartesian closed category which has a
|
|
1501 sub-object classifier.
|
|
1502 <p>
|
|
1503 Topos itself is model of intuitionistic logic.
|
|
1504 <p>
|
|
1505 Transitive Sets are objects of Cartesian closed category.
|
|
1506 It is possible to introduce Power Set Functor on it
|
|
1507 <p>
|
|
1508 We can use replacement A ∩ x for each element in Transitive Set,
|
|
1509 in the similar way of our power set axiom. I
|
|
1510 <p>
|
|
1511 A model of ZF Set theory can be constructed on top of the Topos which is shown in Oisus.
|
|
1512 <p>
|
|
1513 Our Agda model is a proof theoretic version of it.
|
|
1514 <p>
|
|
1515
|
|
1516 <hr/>
|
279
|
1517 <h2><a name="content057">Cardinal number and Continuum hypothesis</a></h2>
|
273
|
1518 Axiom of choice is required to define cardinal number
|
|
1519 <p>
|
|
1520 definition of cardinal number is not yet done
|
|
1521 <p>
|
|
1522 definition of filter is not yet done
|
|
1523 <p>
|
|
1524 we may have a model without axiom of choice or without continuum hypothesis
|
|
1525 <p>
|
|
1526 Possible representation of continuum hypothesis is this.
|
|
1527 <p>
|
|
1528
|
|
1529 <pre>
|
|
1530 continuum-hyphotheis : (a : Ordinal) → Power (Ord a) ⊆ Ord (osuc a)
|
|
1531
|
|
1532 </pre>
|
|
1533
|
|
1534 <hr/>
|
279
|
1535 <h2><a name="content058">Filter</a></h2>
|
273
|
1536
|
|
1537 <p>
|
|
1538 filter is a dual of ideal on boolean algebra or lattice. Existence on natural number
|
|
1539 is depends on axiom of choice.
|
|
1540 <p>
|
|
1541
|
|
1542 <pre>
|
|
1543 record Filter ( L : OD ) : Set (suc n) where
|
|
1544 field
|
|
1545 filter : OD
|
|
1546 proper : ¬ ( filter ∋ od∅ )
|
|
1547 inL : filter ⊆ L
|
|
1548 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
|
|
1549 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
|
|
1550
|
|
1551 </pre>
|
|
1552 We may construct a model of non standard analysis or set theory.
|
|
1553 <p>
|
|
1554 This may be simpler than classical forcing theory ( not yet done).
|
|
1555 <p>
|
|
1556
|
|
1557 <hr/>
|
279
|
1558 <h2><a name="content059">Programming Mathematics</a></h2>
|
273
|
1559 Mathematics is a functional programming in Agda where proof is a value of a variable. The mathematical
|
|
1560 structure are
|
|
1561 <p>
|
|
1562
|
|
1563 <pre>
|
|
1564 record and data
|
|
1565
|
|
1566 </pre>
|
|
1567 Proof is check by type consistency not by the computation, but it may include some normalization.
|
|
1568 <p>
|
|
1569 Type inference and termination is not so clear in multi recursions.
|
|
1570 <p>
|
|
1571 Defining Agda record is a good way to understand mathematical theory, for examples,
|
|
1572 <p>
|
|
1573
|
|
1574 <pre>
|
|
1575 Category theory ( Yoneda lemma, Floyd Adjunction functor theorem, Applicative functor )
|
|
1576 Automaton ( Subset construction、Language containment)
|
|
1577
|
|
1578 </pre>
|
|
1579 are proved in Agda.
|
|
1580 <p>
|
|
1581
|
|
1582 <hr/>
|
279
|
1583 <h2><a name="content060">link</a></h2>
|
273
|
1584 Summer school of foundation of mathematics (in Japanese)<br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/">https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/</a>
|
|
1585 <p>
|
|
1586 Foundation of axiomatic set theory (in Japanese)<br> <a href="https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf">https://www.sci.shizuoka.ac.jp/~math/yorioka/ss2019/sakai0.pdf
|
|
1587 </a>
|
|
1588 <p>
|
|
1589 Agda
|
|
1590 <br> <a href="https://agda.readthedocs.io/en/v2.6.0.1/">https://agda.readthedocs.io/en/v2.6.0.1/</a>
|
|
1591 <p>
|
|
1592 ZF-in-Agda source
|
|
1593 <br> <a href="https://github.com/shinji-kono/zf-in-agda.git">https://github.com/shinji-kono/zf-in-agda.git
|
|
1594 </a>
|
|
1595 <p>
|
|
1596 Category theory in Agda source
|
|
1597 <br> <a href="https://github.com/shinji-kono/category-exercise-in-agda">https://github.com/shinji-kono/category-exercise-in-agda
|
|
1598 </a>
|
|
1599 <p>
|
|
1600 </div>
|
|
1601 <ol class="side" id="menu">
|
|
1602 Constructing ZF Set Theory in Agda
|
279
|
1603 <li><a href="#content000"> ZF in Agda</a>
|
|
1604 <li><a href="#content001"> Programming Mathematics</a>
|
|
1605 <li><a href="#content002"> Target</a>
|
|
1606 <li><a href="#content003"> Why Set Theory</a>
|
|
1607 <li><a href="#content004"> Agda and Intuitionistic Logic </a>
|
|
1608 <li><a href="#content005"> Introduction of Agda </a>
|
|
1609 <li><a href="#content006"> data ( Sum type )</a>
|
|
1610 <li><a href="#content007"> A → B means "A implies B"</a>
|
|
1611 <li><a href="#content008"> introduction と elimination</a>
|
|
1612 <li><a href="#content009"> To prove A → B </a>
|
|
1613 <li><a href="#content010"> A ∧ B</a>
|
|
1614 <li><a href="#content011"> record</a>
|
|
1615 <li><a href="#content012"> Mathematical structure</a>
|
|
1616 <li><a href="#content013"> A Model and a theory</a>
|
|
1617 <li><a href="#content014"> postulate と module</a>
|
|
1618 <li><a href="#content015"> A ∨ B</a>
|
|
1619 <li><a href="#content016"> Negation</a>
|
|
1620 <li><a href="#content017"> Equality </a>
|
|
1621 <li><a href="#content018"> Equivalence relation</a>
|
|
1622 <li><a href="#content019"> Ordering</a>
|
|
1623 <li><a href="#content020"> Quantifier</a>
|
|
1624 <li><a href="#content021"> Can we do math in this way?</a>
|
|
1625 <li><a href="#content022"> Things which Agda cannot prove</a>
|
|
1626 <li><a href="#content023"> Classical story of ZF Set Theory</a>
|
|
1627 <li><a href="#content024"> Ordinals</a>
|
|
1628 <li><a href="#content025"> Axiom of Ordinals</a>
|
338
|
1629 <li><a href="#content026"> Concrete Ordinals or Countable Ordinals</a>
|
279
|
1630 <li><a href="#content027"> Model of Ordinals</a>
|
|
1631 <li><a href="#content028"> Debugging axioms using Model</a>
|
|
1632 <li><a href="#content029"> Countable Ordinals can contains uncountable set?</a>
|
|
1633 <li><a href="#content030"> What is Set</a>
|
|
1634 <li><a href="#content031"> We don't ask the contents of Set. It can be anything.</a>
|
|
1635 <li><a href="#content032"> Ordinal Definable Set</a>
|
338
|
1636 <li><a href="#content033"> OD is not ZF Set</a>
|
|
1637 <li><a href="#content034"> HOD : Hereditarily Ordinal Definable</a>
|
|
1638 <li><a href="#content035"> 1 to 1 mapping between an HOD and an Ordinal</a>
|
|
1639 <li><a href="#content036"> Order preserving in the mapping of OD and Ordinal</a>
|
279
|
1640 <li><a href="#content037"> Various Sets</a>
|
|
1641 <li><a href="#content038"> Fixes on ZF to intuitionistic logic</a>
|
|
1642 <li><a href="#content039"> Pure logical axioms</a>
|
|
1643 <li><a href="#content040"> Axiom of Pair</a>
|
|
1644 <li><a href="#content041"> pair in OD</a>
|
|
1645 <li><a href="#content042"> Validity of Axiom of Pair</a>
|
|
1646 <li><a href="#content043"> Equality of OD and Axiom of Extensionality </a>
|
|
1647 <li><a href="#content044"> Validity of Axiom of Extensionality</a>
|
|
1648 <li><a href="#content045"> Non constructive assumptions so far</a>
|
|
1649 <li><a href="#content046"> Axiom which have negation form</a>
|
|
1650 <li><a href="#content047"> Union </a>
|
|
1651 <li><a href="#content048"> Axiom of replacement</a>
|
|
1652 <li><a href="#content049"> Validity of Power Set Axiom</a>
|
|
1653 <li><a href="#content050"> Axiom of regularity, Axiom of choice, ε-induction</a>
|
|
1654 <li><a href="#content051"> Axiom of choice and Law of Excluded Middle</a>
|
|
1655 <li><a href="#content052"> Relation-ship among ZF axiom</a>
|
|
1656 <li><a href="#content053"> Non constructive assumption in our model</a>
|
|
1657 <li><a href="#content054"> So it this correct?</a>
|
|
1658 <li><a href="#content055"> How to use Agda Set Theory</a>
|
|
1659 <li><a href="#content056"> Topos and Set Theory</a>
|
|
1660 <li><a href="#content057"> Cardinal number and Continuum hypothesis</a>
|
|
1661 <li><a href="#content058"> Filter</a>
|
|
1662 <li><a href="#content059"> Programming Mathematics</a>
|
|
1663 <li><a href="#content060"> link</a>
|
273
|
1664 </ol>
|
|
1665
|
338
|
1666 <hr/> Shinji KONO / Tue Jul 7 15:44:35 2020
|
273
|
1667 </body></html>
|