1 (* Title: ZF/EquivClass.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1994 University of Cambridge |
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5 |
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6 *) |
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7 |
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8 header{*Equivalence Relations*} |
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9 |
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10 theory EquivClass imports Trancl Perm begin |
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11 |
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12 constdefs |
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13 |
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14 quotient :: "[i,i]=>i" (infixl "'/'/" 90) (*set of equiv classes*) |
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15 "A//r == {r``{x} . x:A}" |
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16 |
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17 congruent :: "[i,i=>i]=>o" |
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18 "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)" |
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19 |
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20 congruent2 :: "[i,i,[i,i]=>i]=>o" |
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21 "congruent2(r1,r2,b) == ALL y1 z1 y2 z2. |
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22 <y1,z1>:r1 --> <y2,z2>:r2 --> b(y1,y2) = b(z1,z2)" |
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23 |
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24 syntax |
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25 RESPECTS ::"[i=>i, i] => o" (infixr "respects" 80) |
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26 RESPECTS2 ::"[i=>i, i] => o" (infixr "respects2 " 80) |
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27 --{*Abbreviation for the common case where the relations are identical*} |
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28 |
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29 translations |
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30 "f respects r" == "congruent(r,f)" |
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31 "f respects2 r" => "congruent2(r,r,f)" |
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32 |
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33 subsection{*Suppes, Theorem 70: |
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34 @{term r} is an equiv relation iff @{term "converse(r) O r = r"}*} |
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35 |
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36 (** first half: equiv(A,r) ==> converse(r) O r = r **) |
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37 |
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38 lemma sym_trans_comp_subset: |
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39 "[| sym(r); trans(r) |] ==> converse(r) O r <= r" |
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40 by (unfold trans_def sym_def, blast) |
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41 |
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42 lemma refl_comp_subset: |
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43 "[| refl(A,r); r <= A*A |] ==> r <= converse(r) O r" |
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44 by (unfold refl_def, blast) |
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45 |
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46 lemma equiv_comp_eq: |
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47 "equiv(A,r) ==> converse(r) O r = r" |
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48 apply (unfold equiv_def) |
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49 apply (blast del: subsetI intro!: sym_trans_comp_subset refl_comp_subset) |
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50 done |
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51 |
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52 (*second half*) |
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53 lemma comp_equivI: |
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54 "[| converse(r) O r = r; domain(r) = A |] ==> equiv(A,r)" |
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55 apply (unfold equiv_def refl_def sym_def trans_def) |
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56 apply (erule equalityE) |
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57 apply (subgoal_tac "ALL x y. <x,y> : r --> <y,x> : r", blast+) |
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58 done |
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59 |
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60 (** Equivalence classes **) |
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61 |
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62 (*Lemma for the next result*) |
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63 lemma equiv_class_subset: |
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64 "[| sym(r); trans(r); <a,b>: r |] ==> r``{a} <= r``{b}" |
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65 by (unfold trans_def sym_def, blast) |
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66 |
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67 lemma equiv_class_eq: |
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68 "[| equiv(A,r); <a,b>: r |] ==> r``{a} = r``{b}" |
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69 apply (unfold equiv_def) |
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70 apply (safe del: subsetI intro!: equalityI equiv_class_subset) |
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71 apply (unfold sym_def, blast) |
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72 done |
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73 |
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74 lemma equiv_class_self: |
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75 "[| equiv(A,r); a: A |] ==> a: r``{a}" |
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76 by (unfold equiv_def refl_def, blast) |
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77 |
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78 (*Lemma for the next result*) |
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79 lemma subset_equiv_class: |
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80 "[| equiv(A,r); r``{b} <= r``{a}; b: A |] ==> <a,b>: r" |
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81 by (unfold equiv_def refl_def, blast) |
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82 |
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83 lemma eq_equiv_class: "[| r``{a} = r``{b}; equiv(A,r); b: A |] ==> <a,b>: r" |
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84 by (assumption | rule equalityD2 subset_equiv_class)+ |
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85 |
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86 (*thus r``{a} = r``{b} as well*) |
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87 lemma equiv_class_nondisjoint: |
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88 "[| equiv(A,r); x: (r``{a} Int r``{b}) |] ==> <a,b>: r" |
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89 by (unfold equiv_def trans_def sym_def, blast) |
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90 |
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91 lemma equiv_type: "equiv(A,r) ==> r <= A*A" |
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92 by (unfold equiv_def, blast) |
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93 |
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94 lemma equiv_class_eq_iff: |
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95 "equiv(A,r) ==> <x,y>: r <-> r``{x} = r``{y} & x:A & y:A" |
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96 by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type) |
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97 |
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98 lemma eq_equiv_class_iff: |
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99 "[| equiv(A,r); x: A; y: A |] ==> r``{x} = r``{y} <-> <x,y>: r" |
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100 by (blast intro: eq_equiv_class equiv_class_eq dest: equiv_type) |
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101 |
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102 (*** Quotients ***) |
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103 |
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104 (** Introduction/elimination rules -- needed? **) |
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105 |
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106 lemma quotientI [TC]: "x:A ==> r``{x}: A//r" |
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107 apply (unfold quotient_def) |
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108 apply (erule RepFunI) |
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109 done |
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110 |
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111 lemma quotientE: |
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112 "[| X: A//r; !!x. [| X = r``{x}; x:A |] ==> P |] ==> P" |
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113 by (unfold quotient_def, blast) |
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114 |
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115 lemma Union_quotient: |
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116 "equiv(A,r) ==> Union(A//r) = A" |
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117 by (unfold equiv_def refl_def quotient_def, blast) |
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118 |
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119 lemma quotient_disj: |
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120 "[| equiv(A,r); X: A//r; Y: A//r |] ==> X=Y | (X Int Y <= 0)" |
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121 apply (unfold quotient_def) |
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122 apply (safe intro!: equiv_class_eq, assumption) |
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123 apply (unfold equiv_def trans_def sym_def, blast) |
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124 done |
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125 |
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126 subsection{*Defining Unary Operations upon Equivalence Classes*} |
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127 |
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128 (** Could have a locale with the premises equiv(A,r) and congruent(r,b) |
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129 **) |
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130 |
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131 (*Conversion rule*) |
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132 lemma UN_equiv_class: |
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133 "[| equiv(A,r); b respects r; a: A |] ==> (UN x:r``{a}. b(x)) = b(a)" |
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134 apply (subgoal_tac "\<forall>x \<in> r``{a}. b(x) = b(a)") |
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135 apply simp |
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136 apply (blast intro: equiv_class_self) |
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137 apply (unfold equiv_def sym_def congruent_def, blast) |
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138 done |
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139 |
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140 (*type checking of UN x:r``{a}. b(x) *) |
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141 lemma UN_equiv_class_type: |
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142 "[| equiv(A,r); b respects r; X: A//r; !!x. x : A ==> b(x) : B |] |
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143 ==> (UN x:X. b(x)) : B" |
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144 apply (unfold quotient_def, safe) |
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145 apply (simp (no_asm_simp) add: UN_equiv_class) |
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146 done |
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147 |
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148 (*Sufficient conditions for injectiveness. Could weaken premises! |
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149 major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B |
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150 *) |
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151 lemma UN_equiv_class_inject: |
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152 "[| equiv(A,r); b respects r; |
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153 (UN x:X. b(x))=(UN y:Y. b(y)); X: A//r; Y: A//r; |
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154 !!x y. [| x:A; y:A; b(x)=b(y) |] ==> <x,y>:r |] |
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155 ==> X=Y" |
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156 apply (unfold quotient_def, safe) |
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157 apply (rule equiv_class_eq, assumption) |
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158 apply (simp add: UN_equiv_class [of A r b]) |
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159 done |
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160 |
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161 |
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162 subsection{*Defining Binary Operations upon Equivalence Classes*} |
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163 |
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164 lemma congruent2_implies_congruent: |
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165 "[| equiv(A,r1); congruent2(r1,r2,b); a: A |] ==> congruent(r2,b(a))" |
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166 by (unfold congruent_def congruent2_def equiv_def refl_def, blast) |
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167 |
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168 lemma congruent2_implies_congruent_UN: |
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169 "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a: A2 |] ==> |
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170 congruent(r1, %x1. \<Union>x2 \<in> r2``{a}. b(x1,x2))" |
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171 apply (unfold congruent_def, safe) |
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172 apply (frule equiv_type [THEN subsetD], assumption) |
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173 apply clarify |
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174 apply (simp add: UN_equiv_class congruent2_implies_congruent) |
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175 apply (unfold congruent2_def equiv_def refl_def, blast) |
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176 done |
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177 |
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178 lemma UN_equiv_class2: |
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179 "[| equiv(A1,r1); equiv(A2,r2); congruent2(r1,r2,b); a1: A1; a2: A2 |] |
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180 ==> (\<Union>x1 \<in> r1``{a1}. \<Union>x2 \<in> r2``{a2}. b(x1,x2)) = b(a1,a2)" |
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181 by (simp add: UN_equiv_class congruent2_implies_congruent |
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182 congruent2_implies_congruent_UN) |
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183 |
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184 (*type checking*) |
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185 lemma UN_equiv_class_type2: |
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186 "[| equiv(A,r); b respects2 r; |
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187 X1: A//r; X2: A//r; |
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188 !!x1 x2. [| x1: A; x2: A |] ==> b(x1,x2) : B |
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189 |] ==> (UN x1:X1. UN x2:X2. b(x1,x2)) : B" |
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190 apply (unfold quotient_def, safe) |
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191 apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN |
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192 congruent2_implies_congruent quotientI) |
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193 done |
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194 |
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195 |
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196 (*Suggested by John Harrison -- the two subproofs may be MUCH simpler |
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197 than the direct proof*) |
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198 lemma congruent2I: |
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199 "[| equiv(A1,r1); equiv(A2,r2); |
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200 !! y z w. [| w \<in> A2; <y,z> \<in> r1 |] ==> b(y,w) = b(z,w); |
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201 !! y z w. [| w \<in> A1; <y,z> \<in> r2 |] ==> b(w,y) = b(w,z) |
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202 |] ==> congruent2(r1,r2,b)" |
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203 apply (unfold congruent2_def equiv_def refl_def, safe) |
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204 apply (blast intro: trans) |
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205 done |
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206 |
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207 lemma congruent2_commuteI: |
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208 assumes equivA: "equiv(A,r)" |
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209 and commute: "!! y z. [| y: A; z: A |] ==> b(y,z) = b(z,y)" |
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210 and congt: "!! y z w. [| w: A; <y,z>: r |] ==> b(w,y) = b(w,z)" |
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211 shows "b respects2 r" |
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212 apply (insert equivA [THEN equiv_type, THEN subsetD]) |
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213 apply (rule congruent2I [OF equivA equivA]) |
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214 apply (rule commute [THEN trans]) |
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215 apply (rule_tac [3] commute [THEN trans, symmetric]) |
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216 apply (rule_tac [5] sym) |
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217 apply (blast intro: congt)+ |
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218 done |
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219 |
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220 (*Obsolete?*) |
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221 lemma congruent_commuteI: |
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222 "[| equiv(A,r); Z: A//r; |
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223 !!w. [| w: A |] ==> congruent(r, %z. b(w,z)); |
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224 !!x y. [| x: A; y: A |] ==> b(y,x) = b(x,y) |
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225 |] ==> congruent(r, %w. UN z: Z. b(w,z))" |
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226 apply (simp (no_asm) add: congruent_def) |
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227 apply (safe elim!: quotientE) |
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228 apply (frule equiv_type [THEN subsetD], assumption) |
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229 apply (simp add: UN_equiv_class [of A r]) |
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230 apply (simp add: congruent_def) |
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231 done |
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232 |
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233 ML |
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234 {* |
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235 val sym_trans_comp_subset = thm "sym_trans_comp_subset"; |
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236 val refl_comp_subset = thm "refl_comp_subset"; |
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237 val equiv_comp_eq = thm "equiv_comp_eq"; |
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238 val comp_equivI = thm "comp_equivI"; |
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239 val equiv_class_subset = thm "equiv_class_subset"; |
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240 val equiv_class_eq = thm "equiv_class_eq"; |
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241 val equiv_class_self = thm "equiv_class_self"; |
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242 val subset_equiv_class = thm "subset_equiv_class"; |
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243 val eq_equiv_class = thm "eq_equiv_class"; |
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244 val equiv_class_nondisjoint = thm "equiv_class_nondisjoint"; |
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245 val equiv_type = thm "equiv_type"; |
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246 val equiv_class_eq_iff = thm "equiv_class_eq_iff"; |
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247 val eq_equiv_class_iff = thm "eq_equiv_class_iff"; |
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248 val quotientI = thm "quotientI"; |
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249 val quotientE = thm "quotientE"; |
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250 val Union_quotient = thm "Union_quotient"; |
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251 val quotient_disj = thm "quotient_disj"; |
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252 val UN_equiv_class = thm "UN_equiv_class"; |
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253 val UN_equiv_class_type = thm "UN_equiv_class_type"; |
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254 val UN_equiv_class_inject = thm "UN_equiv_class_inject"; |
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255 val congruent2_implies_congruent = thm "congruent2_implies_congruent"; |
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256 val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN"; |
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257 val congruent_commuteI = thm "congruent_commuteI"; |
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258 val UN_equiv_class2 = thm "UN_equiv_class2"; |
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259 val UN_equiv_class_type2 = thm "UN_equiv_class_type2"; |
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260 val congruent2I = thm "congruent2I"; |
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261 val congruent2_commuteI = thm "congruent2_commuteI"; |
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262 val congruent_commuteI = thm "congruent_commuteI"; |
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263 *} |
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264 |
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265 end |
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