4 Copyright 1994 University of Cambridge |
4 Copyright 1994 University of Cambridge |
5 |
5 |
6 Ordinals in Zermelo-Fraenkel Set Theory |
6 Ordinals in Zermelo-Fraenkel Set Theory |
7 *) |
7 *) |
8 |
8 |
9 Ordinal = WF + Bool + equalities + |
9 theory Ordinal = WF + Bool + equalities: |
10 consts |
10 |
11 Memrel :: i=>i |
11 constdefs |
12 Transset,Ord :: i=>o |
12 |
13 "<" :: [i,i] => o (infixl 50) (*less than on ordinals*) |
13 Memrel :: "i=>i" |
14 Limit :: i=>o |
14 "Memrel(A) == {z: A*A . EX x y. z=<x,y> & x:y }" |
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15 |
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16 Transset :: "i=>o" |
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17 "Transset(i) == ALL x:i. x<=i" |
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18 |
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19 Ord :: "i=>o" |
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20 "Ord(i) == Transset(i) & (ALL x:i. Transset(x))" |
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21 |
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22 lt :: "[i,i] => o" (infixl "<" 50) (*less-than on ordinals*) |
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23 "i<j == i:j & Ord(j)" |
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24 |
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25 Limit :: "i=>o" |
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26 "Limit(i) == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)" |
15 |
27 |
16 syntax |
28 syntax |
17 "le" :: [i,i] => o (infixl 50) (*less than or equals*) |
29 "le" :: "[i,i] => o" (infixl 50) (*less-than or equals*) |
18 |
30 |
19 translations |
31 translations |
20 "x le y" == "x < succ(y)" |
32 "x le y" == "x < succ(y)" |
21 |
33 |
22 syntax (xsymbols) |
34 syntax (xsymbols) |
23 "op le" :: [i,i] => o (infixl "\\<le>" 50) (*less than or equals*) |
35 "op le" :: "[i,i] => o" (infixl "\<le>" 50) (*less-than or equals*) |
24 |
36 |
25 defs |
37 |
26 Memrel_def "Memrel(A) == {z: A*A . EX x y. z=<x,y> & x:y }" |
38 (*** Rules for Transset ***) |
27 Transset_def "Transset(i) == ALL x:i. x<=i" |
39 |
28 Ord_def "Ord(i) == Transset(i) & (ALL x:i. Transset(x))" |
40 (** Three neat characterisations of Transset **) |
29 lt_def "i<j == i:j & Ord(j)" |
41 |
30 Limit_def "Limit(i) == Ord(i) & 0<i & (ALL y. y<i --> succ(y)<i)" |
42 lemma Transset_iff_Pow: "Transset(A) <-> A<=Pow(A)" |
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43 by (unfold Transset_def, blast) |
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44 |
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45 lemma Transset_iff_Union_succ: "Transset(A) <-> Union(succ(A)) = A" |
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46 apply (unfold Transset_def) |
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47 apply (blast elim!: equalityE) |
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48 done |
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49 |
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50 lemma Transset_iff_Union_subset: "Transset(A) <-> Union(A) <= A" |
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51 by (unfold Transset_def, blast) |
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52 |
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53 (** Consequences of downwards closure **) |
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54 |
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55 lemma Transset_doubleton_D: |
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56 "[| Transset(C); {a,b}: C |] ==> a:C & b: C" |
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57 by (unfold Transset_def, blast) |
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58 |
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59 lemma Transset_Pair_D: |
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60 "[| Transset(C); <a,b>: C |] ==> a:C & b: C" |
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61 apply (simp add: Pair_def) |
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62 apply (blast dest: Transset_doubleton_D) |
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63 done |
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64 |
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65 lemma Transset_includes_domain: |
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66 "[| Transset(C); A*B <= C; b: B |] ==> A <= C" |
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67 by (blast dest: Transset_Pair_D) |
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68 |
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69 lemma Transset_includes_range: |
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70 "[| Transset(C); A*B <= C; a: A |] ==> B <= C" |
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71 by (blast dest: Transset_Pair_D) |
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72 |
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73 (** Closure properties **) |
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74 |
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75 lemma Transset_0: "Transset(0)" |
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76 by (unfold Transset_def, blast) |
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77 |
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78 lemma Transset_Un: |
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79 "[| Transset(i); Transset(j) |] ==> Transset(i Un j)" |
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80 by (unfold Transset_def, blast) |
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81 |
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82 lemma Transset_Int: |
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83 "[| Transset(i); Transset(j) |] ==> Transset(i Int j)" |
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84 by (unfold Transset_def, blast) |
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85 |
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86 lemma Transset_succ: "Transset(i) ==> Transset(succ(i))" |
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87 by (unfold Transset_def, blast) |
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88 |
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89 lemma Transset_Pow: "Transset(i) ==> Transset(Pow(i))" |
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90 by (unfold Transset_def, blast) |
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91 |
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92 lemma Transset_Union: "Transset(A) ==> Transset(Union(A))" |
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93 by (unfold Transset_def, blast) |
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94 |
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95 lemma Transset_Union_family: |
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96 "[| !!i. i:A ==> Transset(i) |] ==> Transset(Union(A))" |
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97 by (unfold Transset_def, blast) |
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98 |
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99 lemma Transset_Inter_family: |
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100 "[| j:A; !!i. i:A ==> Transset(i) |] ==> Transset(Inter(A))" |
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101 by (unfold Transset_def, blast) |
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102 |
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103 (*** Natural Deduction rules for Ord ***) |
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104 |
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105 lemma OrdI: |
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106 "[| Transset(i); !!x. x:i ==> Transset(x) |] ==> Ord(i)" |
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107 by (simp add: Ord_def) |
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108 |
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109 lemma Ord_is_Transset: "Ord(i) ==> Transset(i)" |
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110 by (simp add: Ord_def) |
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111 |
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112 lemma Ord_contains_Transset: |
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113 "[| Ord(i); j:i |] ==> Transset(j) " |
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114 by (unfold Ord_def, blast) |
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115 |
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116 (*** Lemmas for ordinals ***) |
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117 |
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118 lemma Ord_in_Ord: "[| Ord(i); j:i |] ==> Ord(j)" |
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119 by (unfold Ord_def Transset_def, blast) |
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120 |
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121 (* Ord(succ(j)) ==> Ord(j) *) |
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122 lemmas Ord_succD = Ord_in_Ord [OF _ succI1] |
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123 |
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124 lemma Ord_subset_Ord: "[| Ord(i); Transset(j); j<=i |] ==> Ord(j)" |
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125 by (simp add: Ord_def Transset_def, blast) |
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126 |
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127 lemma OrdmemD: "[| j:i; Ord(i) |] ==> j<=i" |
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128 by (unfold Ord_def Transset_def, blast) |
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129 |
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130 lemma Ord_trans: "[| i:j; j:k; Ord(k) |] ==> i:k" |
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131 by (blast dest: OrdmemD) |
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132 |
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133 lemma Ord_succ_subsetI: "[| i:j; Ord(j) |] ==> succ(i) <= j" |
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134 by (blast dest: OrdmemD) |
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135 |
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136 |
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137 (*** The construction of ordinals: 0, succ, Union ***) |
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138 |
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139 lemma Ord_0 [iff,TC]: "Ord(0)" |
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140 by (blast intro: OrdI Transset_0) |
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141 |
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142 lemma Ord_succ [TC]: "Ord(i) ==> Ord(succ(i))" |
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143 by (blast intro: OrdI Transset_succ Ord_is_Transset Ord_contains_Transset) |
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144 |
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145 lemmas Ord_1 = Ord_0 [THEN Ord_succ] |
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146 |
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147 lemma Ord_succ_iff [iff]: "Ord(succ(i)) <-> Ord(i)" |
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148 by (blast intro: Ord_succ dest!: Ord_succD) |
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149 |
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150 lemma Ord_Un [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Un j)" |
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151 apply (unfold Ord_def) |
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152 apply (blast intro!: Transset_Un) |
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153 done |
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154 |
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155 lemma Ord_Int [TC]: "[| Ord(i); Ord(j) |] ==> Ord(i Int j)" |
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156 apply (unfold Ord_def) |
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157 apply (blast intro!: Transset_Int) |
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158 done |
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159 |
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160 |
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161 lemma Ord_Inter: |
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162 "[| j:A; !!i. i:A ==> Ord(i) |] ==> Ord(Inter(A))" |
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163 apply (rule Transset_Inter_family [THEN OrdI], assumption) |
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164 apply (blast intro: Ord_is_Transset) |
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165 apply (blast intro: Ord_contains_Transset) |
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166 done |
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167 |
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168 lemma Ord_INT: |
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169 "[| j:A; !!x. x:A ==> Ord(B(x)) |] ==> Ord(INT x:A. B(x))" |
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170 by (rule RepFunI [THEN Ord_Inter], assumption, blast) |
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171 |
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172 (*There is no set of all ordinals, for then it would contain itself*) |
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173 lemma ON_class: "~ (ALL i. i:X <-> Ord(i))" |
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174 apply (rule notI) |
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175 apply (frule_tac x = "X" in spec) |
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176 apply (safe elim!: mem_irrefl) |
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177 apply (erule swap, rule OrdI [OF _ Ord_is_Transset]) |
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178 apply (simp add: Transset_def) |
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179 apply (blast intro: Ord_in_Ord)+ |
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180 done |
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181 |
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182 (*** < is 'less than' for ordinals ***) |
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183 |
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184 lemma ltI: "[| i:j; Ord(j) |] ==> i<j" |
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185 by (unfold lt_def, blast) |
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186 |
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187 lemma ltE: |
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188 "[| i<j; [| i:j; Ord(i); Ord(j) |] ==> P |] ==> P" |
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189 apply (unfold lt_def) |
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190 apply (blast intro: Ord_in_Ord) |
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191 done |
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192 |
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193 lemma ltD: "i<j ==> i:j" |
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194 by (erule ltE, assumption) |
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195 |
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196 lemma not_lt0 [simp]: "~ i<0" |
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197 by (unfold lt_def, blast) |
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198 |
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199 lemma lt_Ord: "j<i ==> Ord(j)" |
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200 by (erule ltE, assumption) |
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201 |
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202 lemma lt_Ord2: "j<i ==> Ord(i)" |
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203 by (erule ltE, assumption) |
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204 |
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205 (* "ja le j ==> Ord(j)" *) |
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206 lemmas le_Ord2 = lt_Ord2 [THEN Ord_succD] |
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207 |
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208 (* i<0 ==> R *) |
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209 lemmas lt0E = not_lt0 [THEN notE, elim!] |
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210 |
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211 lemma lt_trans: "[| i<j; j<k |] ==> i<k" |
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212 by (blast intro!: ltI elim!: ltE intro: Ord_trans) |
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213 |
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214 lemma lt_not_sym: "i<j ==> ~ (j<i)" |
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215 apply (unfold lt_def) |
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216 apply (blast elim: mem_asym) |
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217 done |
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218 |
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219 (* [| i<j; ~P ==> j<i |] ==> P *) |
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220 lemmas lt_asym = lt_not_sym [THEN swap] |
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221 |
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222 lemma lt_irrefl [elim!]: "i<i ==> P" |
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223 by (blast intro: lt_asym) |
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224 |
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225 lemma lt_not_refl: "~ i<i" |
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226 apply (rule notI) |
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227 apply (erule lt_irrefl) |
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228 done |
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229 |
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230 |
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231 (** le is less than or equals; recall i le j abbrevs i<succ(j) !! **) |
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232 |
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233 lemma le_iff: "i le j <-> i<j | (i=j & Ord(j))" |
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234 by (unfold lt_def, blast) |
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235 |
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236 (*Equivalently, i<j ==> i < succ(j)*) |
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237 lemma leI: "i<j ==> i le j" |
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238 by (simp (no_asm_simp) add: le_iff) |
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239 |
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240 lemma le_eqI: "[| i=j; Ord(j) |] ==> i le j" |
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241 by (simp (no_asm_simp) add: le_iff) |
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242 |
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243 lemmas le_refl = refl [THEN le_eqI] |
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244 |
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245 lemma le_refl_iff [iff]: "i le i <-> Ord(i)" |
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246 by (simp (no_asm_simp) add: lt_not_refl le_iff) |
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247 |
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248 lemma leCI: "(~ (i=j & Ord(j)) ==> i<j) ==> i le j" |
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249 by (simp add: le_iff, blast) |
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250 |
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251 lemma leE: |
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252 "[| i le j; i<j ==> P; [| i=j; Ord(j) |] ==> P |] ==> P" |
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253 by (simp add: le_iff, blast) |
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254 |
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255 lemma le_anti_sym: "[| i le j; j le i |] ==> i=j" |
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256 apply (simp add: le_iff) |
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257 apply (blast elim: lt_asym) |
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258 done |
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259 |
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260 lemma le0_iff [simp]: "i le 0 <-> i=0" |
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261 by (blast elim!: leE) |
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262 |
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263 lemmas le0D = le0_iff [THEN iffD1, dest!] |
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264 |
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265 (*** Natural Deduction rules for Memrel ***) |
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266 |
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267 (*The lemmas MemrelI/E give better speed than [iff] here*) |
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268 lemma Memrel_iff [simp]: "<a,b> : Memrel(A) <-> a:b & a:A & b:A" |
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269 by (unfold Memrel_def, blast) |
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270 |
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271 lemma MemrelI [intro!]: "[| a: b; a: A; b: A |] ==> <a,b> : Memrel(A)" |
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272 by auto |
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273 |
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274 lemma MemrelE [elim!]: |
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275 "[| <a,b> : Memrel(A); |
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276 [| a: A; b: A; a:b |] ==> P |] |
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277 ==> P" |
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278 by auto |
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279 |
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280 lemma Memrel_type: "Memrel(A) <= A*A" |
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281 by (unfold Memrel_def, blast) |
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282 |
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283 lemma Memrel_mono: "A<=B ==> Memrel(A) <= Memrel(B)" |
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284 by (unfold Memrel_def, blast) |
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285 |
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286 lemma Memrel_0 [simp]: "Memrel(0) = 0" |
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287 by (unfold Memrel_def, blast) |
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288 |
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289 lemma Memrel_1 [simp]: "Memrel(1) = 0" |
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290 by (unfold Memrel_def, blast) |
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291 |
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292 (*The membership relation (as a set) is well-founded. |
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293 Proof idea: show A<=B by applying the foundation axiom to A-B *) |
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294 lemma wf_Memrel: "wf(Memrel(A))" |
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295 apply (unfold wf_def) |
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296 apply (rule foundation [THEN disjE, THEN allI], erule disjI1, blast) |
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297 done |
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298 |
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299 (*Transset(i) does not suffice, though ALL j:i.Transset(j) does*) |
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300 lemma trans_Memrel: |
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301 "Ord(i) ==> trans(Memrel(i))" |
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302 by (unfold Ord_def Transset_def trans_def, blast) |
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303 |
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304 (*If Transset(A) then Memrel(A) internalizes the membership relation below A*) |
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305 lemma Transset_Memrel_iff: |
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306 "Transset(A) ==> <a,b> : Memrel(A) <-> a:b & b:A" |
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307 by (unfold Transset_def, blast) |
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308 |
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309 |
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310 (*** Transfinite induction ***) |
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311 |
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312 (*Epsilon induction over a transitive set*) |
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313 lemma Transset_induct: |
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314 "[| i: k; Transset(k); |
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315 !!x.[| x: k; ALL y:x. P(y) |] ==> P(x) |] |
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316 ==> P(i)" |
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317 apply (simp add: Transset_def) |
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318 apply (erule wf_Memrel [THEN wf_induct2], blast) |
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319 apply blast |
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320 done |
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321 |
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322 (*Induction over an ordinal*) |
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323 lemmas Ord_induct = Transset_induct [OF _ Ord_is_Transset] |
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324 |
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325 (*Induction over the class of ordinals -- a useful corollary of Ord_induct*) |
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326 |
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327 lemma trans_induct: |
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328 "[| Ord(i); |
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329 !!x.[| Ord(x); ALL y:x. P(y) |] ==> P(x) |] |
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330 ==> P(i)" |
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331 apply (rule Ord_succ [THEN succI1 [THEN Ord_induct]], assumption) |
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332 apply (blast intro: Ord_succ [THEN Ord_in_Ord]) |
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333 done |
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334 |
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335 |
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336 (*** Fundamental properties of the epsilon ordering (< on ordinals) ***) |
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337 |
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338 |
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339 (** Proving that < is a linear ordering on the ordinals **) |
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340 |
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341 lemma Ord_linear [rule_format]: |
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342 "Ord(i) ==> (ALL j. Ord(j) --> i:j | i=j | j:i)" |
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343 apply (erule trans_induct) |
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344 apply (rule impI [THEN allI]) |
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345 apply (erule_tac i=j in trans_induct) |
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346 apply (blast dest: Ord_trans) |
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347 done |
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348 |
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349 (*The trichotomy law for ordinals!*) |
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350 lemma Ord_linear_lt: |
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351 "[| Ord(i); Ord(j); i<j ==> P; i=j ==> P; j<i ==> P |] ==> P" |
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352 apply (simp add: lt_def) |
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353 apply (rule_tac i1=i and j1=j in Ord_linear [THEN disjE], blast+) |
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354 done |
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355 |
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356 lemma Ord_linear2: |
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357 "[| Ord(i); Ord(j); i<j ==> P; j le i ==> P |] ==> P" |
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358 apply (rule_tac i = "i" and j = "j" in Ord_linear_lt) |
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359 apply (blast intro: leI le_eqI sym ) + |
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360 done |
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361 |
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362 lemma Ord_linear_le: |
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363 "[| Ord(i); Ord(j); i le j ==> P; j le i ==> P |] ==> P" |
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364 apply (rule_tac i = "i" and j = "j" in Ord_linear_lt) |
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365 apply (blast intro: leI le_eqI ) + |
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366 done |
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367 |
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368 lemma le_imp_not_lt: "j le i ==> ~ i<j" |
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369 by (blast elim!: leE elim: lt_asym) |
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370 |
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371 lemma not_lt_imp_le: "[| ~ i<j; Ord(i); Ord(j) |] ==> j le i" |
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372 by (rule_tac i = "i" and j = "j" in Ord_linear2, auto) |
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373 |
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374 (** Some rewrite rules for <, le **) |
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375 |
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376 lemma Ord_mem_iff_lt: "Ord(j) ==> i:j <-> i<j" |
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377 by (unfold lt_def, blast) |
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378 |
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379 lemma not_lt_iff_le: "[| Ord(i); Ord(j) |] ==> ~ i<j <-> j le i" |
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380 by (blast dest: le_imp_not_lt not_lt_imp_le) |
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381 |
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382 lemma not_le_iff_lt: "[| Ord(i); Ord(j) |] ==> ~ i le j <-> j<i" |
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383 by (simp (no_asm_simp) add: not_lt_iff_le [THEN iff_sym]) |
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384 |
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385 (*This is identical to 0<succ(i) *) |
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386 lemma Ord_0_le: "Ord(i) ==> 0 le i" |
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387 by (erule not_lt_iff_le [THEN iffD1], auto) |
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388 |
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389 lemma Ord_0_lt: "[| Ord(i); i~=0 |] ==> 0<i" |
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390 apply (erule not_le_iff_lt [THEN iffD1]) |
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391 apply (rule Ord_0, blast) |
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392 done |
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393 |
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394 lemma Ord_0_lt_iff: "Ord(i) ==> i~=0 <-> 0<i" |
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395 by (blast intro: Ord_0_lt) |
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396 |
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397 |
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398 (*** Results about less-than or equals ***) |
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399 |
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400 (** For ordinals, j<=i (subset) implies j le i (less-than or equals) **) |
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401 |
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402 lemma zero_le_succ_iff [iff]: "0 le succ(x) <-> Ord(x)" |
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403 by (blast intro: Ord_0_le elim: ltE) |
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404 |
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405 lemma subset_imp_le: "[| j<=i; Ord(i); Ord(j) |] ==> j le i" |
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406 apply (rule not_lt_iff_le [THEN iffD1], assumption) |
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407 apply assumption |
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408 apply (blast elim: ltE mem_irrefl) |
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409 done |
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410 |
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411 lemma le_imp_subset: "i le j ==> i<=j" |
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412 by (blast dest: OrdmemD elim: ltE leE) |
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413 |
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414 lemma le_subset_iff: "j le i <-> j<=i & Ord(i) & Ord(j)" |
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415 by (blast dest: subset_imp_le le_imp_subset elim: ltE) |
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416 |
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417 lemma le_succ_iff: "i le succ(j) <-> i le j | i=succ(j) & Ord(i)" |
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418 apply (simp (no_asm) add: le_iff) |
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419 apply blast |
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420 done |
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421 |
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422 (*Just a variant of subset_imp_le*) |
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423 lemma all_lt_imp_le: "[| Ord(i); Ord(j); !!x. x<j ==> x<i |] ==> j le i" |
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424 by (blast intro: not_lt_imp_le dest: lt_irrefl) |
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425 |
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426 (** Transitive laws **) |
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427 |
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428 lemma lt_trans1: "[| i le j; j<k |] ==> i<k" |
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429 by (blast elim!: leE intro: lt_trans) |
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430 |
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431 lemma lt_trans2: "[| i<j; j le k |] ==> i<k" |
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432 by (blast elim!: leE intro: lt_trans) |
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433 |
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434 lemma le_trans: "[| i le j; j le k |] ==> i le k" |
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435 by (blast intro: lt_trans1) |
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436 |
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437 lemma succ_leI: "i<j ==> succ(i) le j" |
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438 apply (rule not_lt_iff_le [THEN iffD1]) |
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439 apply (blast elim: ltE leE lt_asym)+ |
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440 done |
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441 |
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442 (*Identical to succ(i) < succ(j) ==> i<j *) |
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443 lemma succ_leE: "succ(i) le j ==> i<j" |
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444 apply (rule not_le_iff_lt [THEN iffD1]) |
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445 apply (blast elim: ltE leE lt_asym)+ |
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446 done |
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447 |
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448 lemma succ_le_iff [iff]: "succ(i) le j <-> i<j" |
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449 by (blast intro: succ_leI succ_leE) |
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450 |
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451 lemma succ_le_imp_le: "succ(i) le succ(j) ==> i le j" |
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452 by (blast dest!: succ_leE) |
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453 |
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454 lemma lt_subset_trans: "[| i <= j; j<k; Ord(i) |] ==> i<k" |
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455 apply (rule subset_imp_le [THEN lt_trans1]) |
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456 apply (blast intro: elim: ltE) + |
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457 done |
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458 |
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459 (** Union and Intersection **) |
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460 |
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461 lemma Un_upper1_le: "[| Ord(i); Ord(j) |] ==> i le i Un j" |
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462 by (rule Un_upper1 [THEN subset_imp_le], auto) |
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463 |
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464 lemma Un_upper2_le: "[| Ord(i); Ord(j) |] ==> j le i Un j" |
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465 by (rule Un_upper2 [THEN subset_imp_le], auto) |
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466 |
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467 (*Replacing k by succ(k') yields the similar rule for le!*) |
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468 lemma Un_least_lt: "[| i<k; j<k |] ==> i Un j < k" |
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469 apply (rule_tac i = "i" and j = "j" in Ord_linear_le) |
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470 apply (auto simp add: Un_commute le_subset_iff subset_Un_iff lt_Ord) |
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471 done |
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472 |
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473 lemma Un_least_lt_iff: "[| Ord(i); Ord(j) |] ==> i Un j < k <-> i<k & j<k" |
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474 apply (safe intro!: Un_least_lt) |
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475 apply (rule_tac [2] Un_upper2_le [THEN lt_trans1]) |
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476 apply (rule Un_upper1_le [THEN lt_trans1], auto) |
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477 done |
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478 |
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479 lemma Un_least_mem_iff: |
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480 "[| Ord(i); Ord(j); Ord(k) |] ==> i Un j : k <-> i:k & j:k" |
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481 apply (insert Un_least_lt_iff [of i j k]) |
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482 apply (simp add: lt_def) |
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483 done |
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484 |
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485 (*Replacing k by succ(k') yields the similar rule for le!*) |
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486 lemma Int_greatest_lt: "[| i<k; j<k |] ==> i Int j < k" |
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487 apply (rule_tac i = "i" and j = "j" in Ord_linear_le) |
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488 apply (auto simp add: Int_commute le_subset_iff subset_Int_iff lt_Ord) |
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489 done |
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490 |
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491 (*FIXME: the Intersection duals are missing!*) |
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492 |
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493 (*** Results about limits ***) |
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494 |
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495 lemma Ord_Union: "[| !!i. i:A ==> Ord(i) |] ==> Ord(Union(A))" |
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496 apply (rule Ord_is_Transset [THEN Transset_Union_family, THEN OrdI]) |
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497 apply (blast intro: Ord_contains_Transset)+ |
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498 done |
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499 |
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500 lemma Ord_UN: "[| !!x. x:A ==> Ord(B(x)) |] ==> Ord(UN x:A. B(x))" |
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501 by (rule Ord_Union, blast) |
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502 |
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503 (* No < version; consider (UN i:nat.i)=nat *) |
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504 lemma UN_least_le: |
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505 "[| Ord(i); !!x. x:A ==> b(x) le i |] ==> (UN x:A. b(x)) le i" |
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506 apply (rule le_imp_subset [THEN UN_least, THEN subset_imp_le]) |
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507 apply (blast intro: Ord_UN elim: ltE)+ |
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508 done |
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509 |
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510 lemma UN_succ_least_lt: |
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511 "[| j<i; !!x. x:A ==> b(x)<j |] ==> (UN x:A. succ(b(x))) < i" |
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512 apply (rule ltE, assumption) |
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513 apply (rule UN_least_le [THEN lt_trans2]) |
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514 apply (blast intro: succ_leI)+ |
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515 done |
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516 |
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517 lemma UN_upper_le: |
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518 "[| a: A; i le b(a); Ord(UN x:A. b(x)) |] ==> i le (UN x:A. b(x))" |
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519 apply (frule ltD) |
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520 apply (rule le_imp_subset [THEN subset_trans, THEN subset_imp_le]) |
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521 apply (blast intro: lt_Ord UN_upper)+ |
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522 done |
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523 |
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524 lemma le_implies_UN_le_UN: |
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525 "[| !!x. x:A ==> c(x) le d(x) |] ==> (UN x:A. c(x)) le (UN x:A. d(x))" |
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526 apply (rule UN_least_le) |
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527 apply (rule_tac [2] UN_upper_le) |
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528 apply (blast intro: Ord_UN le_Ord2)+ |
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529 done |
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530 |
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531 lemma Ord_equality: "Ord(i) ==> (UN y:i. succ(y)) = i" |
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532 by (blast intro: Ord_trans) |
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533 |
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534 (*Holds for all transitive sets, not just ordinals*) |
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535 lemma Ord_Union_subset: "Ord(i) ==> Union(i) <= i" |
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536 by (blast intro: Ord_trans) |
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537 |
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538 |
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539 (*** Limit ordinals -- general properties ***) |
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540 |
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541 lemma Limit_Union_eq: "Limit(i) ==> Union(i) = i" |
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542 apply (unfold Limit_def) |
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543 apply (fast intro!: ltI elim!: ltE elim: Ord_trans) |
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544 done |
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545 |
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546 lemma Limit_is_Ord: "Limit(i) ==> Ord(i)" |
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547 apply (unfold Limit_def) |
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548 apply (erule conjunct1) |
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549 done |
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550 |
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551 lemma Limit_has_0: "Limit(i) ==> 0 < i" |
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552 apply (unfold Limit_def) |
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553 apply (erule conjunct2 [THEN conjunct1]) |
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554 done |
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555 |
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556 lemma Limit_has_succ: "[| Limit(i); j<i |] ==> succ(j) < i" |
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557 by (unfold Limit_def, blast) |
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558 |
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559 lemma non_succ_LimitI: |
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560 "[| 0<i; ALL y. succ(y) ~= i |] ==> Limit(i)" |
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561 apply (unfold Limit_def) |
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562 apply (safe del: subsetI) |
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563 apply (rule_tac [2] not_le_iff_lt [THEN iffD1]) |
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564 apply (simp_all add: lt_Ord lt_Ord2) |
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565 apply (blast elim: leE lt_asym) |
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566 done |
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567 |
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568 lemma succ_LimitE [elim!]: "Limit(succ(i)) ==> P" |
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569 apply (rule lt_irrefl) |
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570 apply (rule Limit_has_succ, assumption) |
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571 apply (erule Limit_is_Ord [THEN Ord_succD, THEN le_refl]) |
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572 done |
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573 |
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574 lemma not_succ_Limit [simp]: "~ Limit(succ(i))" |
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575 by blast |
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576 |
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577 lemma Limit_le_succD: "[| Limit(i); i le succ(j) |] ==> i le j" |
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578 by (blast elim!: leE) |
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579 |
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580 (** Traditional 3-way case analysis on ordinals **) |
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581 |
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582 lemma Ord_cases_disj: "Ord(i) ==> i=0 | (EX j. Ord(j) & i=succ(j)) | Limit(i)" |
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583 by (blast intro!: non_succ_LimitI Ord_0_lt) |
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584 |
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585 lemma Ord_cases: |
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586 "[| Ord(i); |
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587 i=0 ==> P; |
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588 !!j. [| Ord(j); i=succ(j) |] ==> P; |
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589 Limit(i) ==> P |
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590 |] ==> P" |
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591 by (drule Ord_cases_disj, blast) |
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592 |
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593 lemma trans_induct3: |
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594 "[| Ord(i); |
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595 P(0); |
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596 !!x. [| Ord(x); P(x) |] ==> P(succ(x)); |
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597 !!x. [| Limit(x); ALL y:x. P(y) |] ==> P(x) |
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598 |] ==> P(i)" |
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599 apply (erule trans_induct) |
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600 apply (erule Ord_cases, blast+) |
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601 done |
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602 |
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603 ML |
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604 {* |
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605 val Memrel_def = thm "Memrel_def"; |
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606 val Transset_def = thm "Transset_def"; |
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607 val Ord_def = thm "Ord_def"; |
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608 val lt_def = thm "lt_def"; |
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609 val Limit_def = thm "Limit_def"; |
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610 |
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611 val Transset_iff_Pow = thm "Transset_iff_Pow"; |
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612 val Transset_iff_Union_succ = thm "Transset_iff_Union_succ"; |
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613 val Transset_iff_Union_subset = thm "Transset_iff_Union_subset"; |
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614 val Transset_doubleton_D = thm "Transset_doubleton_D"; |
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615 val Transset_Pair_D = thm "Transset_Pair_D"; |
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616 val Transset_includes_domain = thm "Transset_includes_domain"; |
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617 val Transset_includes_range = thm "Transset_includes_range"; |
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618 val Transset_0 = thm "Transset_0"; |
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619 val Transset_Un = thm "Transset_Un"; |
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620 val Transset_Int = thm "Transset_Int"; |
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621 val Transset_succ = thm "Transset_succ"; |
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622 val Transset_Pow = thm "Transset_Pow"; |
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623 val Transset_Union = thm "Transset_Union"; |
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624 val Transset_Union_family = thm "Transset_Union_family"; |
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625 val Transset_Inter_family = thm "Transset_Inter_family"; |
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626 val OrdI = thm "OrdI"; |
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627 val Ord_is_Transset = thm "Ord_is_Transset"; |
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628 val Ord_contains_Transset = thm "Ord_contains_Transset"; |
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629 val Ord_in_Ord = thm "Ord_in_Ord"; |
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630 val Ord_succD = thm "Ord_succD"; |
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631 val Ord_subset_Ord = thm "Ord_subset_Ord"; |
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632 val OrdmemD = thm "OrdmemD"; |
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633 val Ord_trans = thm "Ord_trans"; |
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634 val Ord_succ_subsetI = thm "Ord_succ_subsetI"; |
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635 val Ord_0 = thm "Ord_0"; |
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636 val Ord_succ = thm "Ord_succ"; |
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637 val Ord_1 = thm "Ord_1"; |
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638 val Ord_succ_iff = thm "Ord_succ_iff"; |
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639 val Ord_Un = thm "Ord_Un"; |
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640 val Ord_Int = thm "Ord_Int"; |
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641 val Ord_Inter = thm "Ord_Inter"; |
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642 val Ord_INT = thm "Ord_INT"; |
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643 val ON_class = thm "ON_class"; |
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644 val ltI = thm "ltI"; |
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645 val ltE = thm "ltE"; |
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646 val ltD = thm "ltD"; |
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647 val not_lt0 = thm "not_lt0"; |
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648 val lt_Ord = thm "lt_Ord"; |
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649 val lt_Ord2 = thm "lt_Ord2"; |
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650 val le_Ord2 = thm "le_Ord2"; |
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651 val lt0E = thm "lt0E"; |
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652 val lt_trans = thm "lt_trans"; |
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653 val lt_not_sym = thm "lt_not_sym"; |
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654 val lt_asym = thm "lt_asym"; |
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655 val lt_irrefl = thm "lt_irrefl"; |
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656 val lt_not_refl = thm "lt_not_refl"; |
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657 val le_iff = thm "le_iff"; |
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658 val leI = thm "leI"; |
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659 val le_eqI = thm "le_eqI"; |
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660 val le_refl = thm "le_refl"; |
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661 val le_refl_iff = thm "le_refl_iff"; |
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662 val leCI = thm "leCI"; |
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663 val leE = thm "leE"; |
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664 val le_anti_sym = thm "le_anti_sym"; |
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665 val le0_iff = thm "le0_iff"; |
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666 val le0D = thm "le0D"; |
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667 val Memrel_iff = thm "Memrel_iff"; |
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668 val MemrelI = thm "MemrelI"; |
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669 val MemrelE = thm "MemrelE"; |
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670 val Memrel_type = thm "Memrel_type"; |
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671 val Memrel_mono = thm "Memrel_mono"; |
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672 val Memrel_0 = thm "Memrel_0"; |
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673 val Memrel_1 = thm "Memrel_1"; |
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674 val wf_Memrel = thm "wf_Memrel"; |
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675 val trans_Memrel = thm "trans_Memrel"; |
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676 val Transset_Memrel_iff = thm "Transset_Memrel_iff"; |
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677 val Transset_induct = thm "Transset_induct"; |
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678 val Ord_induct = thm "Ord_induct"; |
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679 val trans_induct = thm "trans_induct"; |
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680 val Ord_linear = thm "Ord_linear"; |
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681 val Ord_linear_lt = thm "Ord_linear_lt"; |
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682 val Ord_linear2 = thm "Ord_linear2"; |
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683 val Ord_linear_le = thm "Ord_linear_le"; |
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684 val le_imp_not_lt = thm "le_imp_not_lt"; |
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685 val not_lt_imp_le = thm "not_lt_imp_le"; |
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686 val Ord_mem_iff_lt = thm "Ord_mem_iff_lt"; |
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687 val not_lt_iff_le = thm "not_lt_iff_le"; |
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688 val not_le_iff_lt = thm "not_le_iff_lt"; |
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689 val Ord_0_le = thm "Ord_0_le"; |
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690 val Ord_0_lt = thm "Ord_0_lt"; |
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691 val Ord_0_lt_iff = thm "Ord_0_lt_iff"; |
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692 val zero_le_succ_iff = thm "zero_le_succ_iff"; |
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693 val subset_imp_le = thm "subset_imp_le"; |
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694 val le_imp_subset = thm "le_imp_subset"; |
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695 val le_subset_iff = thm "le_subset_iff"; |
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696 val le_succ_iff = thm "le_succ_iff"; |
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697 val all_lt_imp_le = thm "all_lt_imp_le"; |
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698 val lt_trans1 = thm "lt_trans1"; |
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699 val lt_trans2 = thm "lt_trans2"; |
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700 val le_trans = thm "le_trans"; |
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701 val succ_leI = thm "succ_leI"; |
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702 val succ_leE = thm "succ_leE"; |
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703 val succ_le_iff = thm "succ_le_iff"; |
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704 val succ_le_imp_le = thm "succ_le_imp_le"; |
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705 val lt_subset_trans = thm "lt_subset_trans"; |
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706 val Un_upper1_le = thm "Un_upper1_le"; |
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707 val Un_upper2_le = thm "Un_upper2_le"; |
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708 val Un_least_lt = thm "Un_least_lt"; |
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709 val Un_least_lt_iff = thm "Un_least_lt_iff"; |
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710 val Un_least_mem_iff = thm "Un_least_mem_iff"; |
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711 val Int_greatest_lt = thm "Int_greatest_lt"; |
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712 val Ord_Union = thm "Ord_Union"; |
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713 val Ord_UN = thm "Ord_UN"; |
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714 val UN_least_le = thm "UN_least_le"; |
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715 val UN_succ_least_lt = thm "UN_succ_least_lt"; |
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716 val UN_upper_le = thm "UN_upper_le"; |
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717 val le_implies_UN_le_UN = thm "le_implies_UN_le_UN"; |
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718 val Ord_equality = thm "Ord_equality"; |
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719 val Ord_Union_subset = thm "Ord_Union_subset"; |
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720 val Limit_Union_eq = thm "Limit_Union_eq"; |
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721 val Limit_is_Ord = thm "Limit_is_Ord"; |
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722 val Limit_has_0 = thm "Limit_has_0"; |
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723 val Limit_has_succ = thm "Limit_has_succ"; |
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724 val non_succ_LimitI = thm "non_succ_LimitI"; |
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725 val succ_LimitE = thm "succ_LimitE"; |
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726 val not_succ_Limit = thm "not_succ_Limit"; |
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727 val Limit_le_succD = thm "Limit_le_succD"; |
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728 val Ord_cases_disj = thm "Ord_cases_disj"; |
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729 val Ord_cases = thm "Ord_cases"; |
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730 val trans_induct3 = thm "trans_induct3"; |
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731 *} |
31 |
732 |
32 end |
733 end |