1 (* Title: CCL/wfd.ML |
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2 ID: $Id$ |
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3 |
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4 For wfd.thy. |
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5 |
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6 Based on |
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7 Titles: ZF/wf.ML and HOL/ex/lex-prod |
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8 Authors: Lawrence C Paulson and Tobias Nipkow |
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9 Copyright 1992 University of Cambridge |
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10 |
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11 *) |
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12 |
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13 open Wfd; |
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14 |
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15 (***********) |
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16 |
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17 val [major,prem] = goalw Wfd.thy [Wfd_def] |
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18 "[| Wfd(R); \ |
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19 \ !!x.[| ALL y. <y,x>: R --> P(y) |] ==> P(x) |] ==> \ |
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20 \ P(a)"; |
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21 by (rtac (major RS spec RS mp RS spec RS CollectD) 1); |
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22 by (fast_tac (set_cs addSIs [prem RS CollectI]) 1); |
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23 val wfd_induct = result(); |
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24 |
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25 val [p1,p2,p3] = goal Wfd.thy |
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26 "[| !!x y.<x,y> : R ==> Q(x); \ |
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27 \ ALL x. (ALL y. <y,x> : R --> y : P) --> x : P; \ |
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28 \ !!x.Q(x) ==> x:P |] ==> a:P"; |
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29 br (p2 RS spec RS mp) 1; |
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30 by (fast_tac (set_cs addSIs [p1 RS p3]) 1); |
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31 val wfd_strengthen_lemma = result(); |
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32 |
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33 fun wfd_strengthen_tac s i = res_inst_tac [("Q",s)] wfd_strengthen_lemma i THEN |
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34 assume_tac (i+1); |
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35 |
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36 val wfd::prems = goal Wfd.thy "[| Wfd(r); <a,x>:r; <x,a>:r |] ==> P"; |
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37 by (subgoal_tac "ALL x. <a,x>:r --> <x,a>:r --> P" 1); |
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38 by (fast_tac (FOL_cs addIs prems) 1); |
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39 br (wfd RS wfd_induct) 1; |
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40 by (ALLGOALS (fast_tac (ccl_cs addSIs prems))); |
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41 val wf_anti_sym = result(); |
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42 |
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43 val prems = goal Wfd.thy "[| Wfd(r); <a,a>: r |] ==> P"; |
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44 by (rtac wf_anti_sym 1); |
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45 by (REPEAT (resolve_tac prems 1)); |
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46 val wf_anti_refl = result(); |
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47 |
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48 (*** Irreflexive transitive closure ***) |
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49 |
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50 val [prem] = goal Wfd.thy "Wfd(R) ==> Wfd(R^+)"; |
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51 by (rewtac Wfd_def); |
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52 by (REPEAT (ares_tac [allI,ballI,impI] 1)); |
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53 (*must retain the universal formula for later use!*) |
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54 by (rtac allE 1 THEN assume_tac 1); |
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55 by (etac mp 1); |
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56 br (prem RS wfd_induct) 1; |
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57 by (rtac (impI RS allI) 1); |
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58 by (etac tranclE 1); |
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59 by (fast_tac ccl_cs 1); |
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60 be (spec RS mp RS spec RS mp) 1; |
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61 by (REPEAT (atac 1)); |
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62 val trancl_wf = result(); |
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63 |
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64 (*** Lexicographic Ordering ***) |
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65 |
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66 goalw Wfd.thy [lex_def] |
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67 "p : ra**rb <-> (EX a a' b b'.p = <<a,b>,<a',b'>> & (<a,a'> : ra | a=a' & <b,b'> : rb))"; |
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68 by (fast_tac ccl_cs 1); |
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69 val lexXH = result(); |
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70 |
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71 val prems = goal Wfd.thy |
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72 "<a,a'> : ra ==> <<a,b>,<a',b'>> : ra**rb"; |
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73 by (fast_tac (ccl_cs addSIs (prems @ [lexXH RS iffD2])) 1); |
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74 val lexI1 = result(); |
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75 |
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76 val prems = goal Wfd.thy |
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77 "<b,b'> : rb ==> <<a,b>,<a,b'>> : ra**rb"; |
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78 by (fast_tac (ccl_cs addSIs (prems @ [lexXH RS iffD2])) 1); |
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79 val lexI2 = result(); |
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80 |
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81 val major::prems = goal Wfd.thy |
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82 "[| p : ra**rb; \ |
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83 \ !!a a' b b'.[| <a,a'> : ra; p=<<a,b>,<a',b'>> |] ==> R; \ |
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84 \ !!a b b'.[| <b,b'> : rb; p = <<a,b>,<a,b'>> |] ==> R |] ==> \ |
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85 \ R"; |
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86 br (major RS (lexXH RS iffD1) RS exE) 1; |
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87 by (REPEAT_SOME (eresolve_tac ([exE,conjE,disjE]@prems))); |
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88 by (ALLGOALS (fast_tac ccl_cs)); |
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89 val lexE = result(); |
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90 |
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91 val [major,minor] = goal Wfd.thy |
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92 "[| p : r**s; !!a a' b b'. p = <<a,b>,<a',b'>> ==> P |] ==>P"; |
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93 br (major RS lexE) 1; |
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94 by (ALLGOALS (fast_tac (set_cs addSEs [minor]))); |
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95 val lex_pair = result(); |
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96 |
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97 val [wfa,wfb] = goal Wfd.thy |
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98 "[| Wfd(R); Wfd(S) |] ==> Wfd(R**S)"; |
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99 bw Wfd_def; |
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100 by (safe_tac ccl_cs); |
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101 by (wfd_strengthen_tac "%x.EX a b.x=<a,b>" 1); |
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102 by (fast_tac (term_cs addSEs [lex_pair]) 1); |
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103 by (subgoal_tac "ALL a b.<a,b>:P" 1); |
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104 by (fast_tac ccl_cs 1); |
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105 br (wfa RS wfd_induct RS allI) 1; |
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106 br (wfb RS wfd_induct RS allI) 1;back(); |
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107 by (fast_tac (type_cs addSEs [lexE]) 1); |
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108 val lex_wf = result(); |
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109 |
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110 (*** Mapping ***) |
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111 |
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112 goalw Wfd.thy [wmap_def] |
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113 "p : wmap(f,r) <-> (EX x y. p=<x,y> & <f(x),f(y)> : r)"; |
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114 by (fast_tac ccl_cs 1); |
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115 val wmapXH = result(); |
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116 |
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117 val prems = goal Wfd.thy |
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118 "<f(a),f(b)> : r ==> <a,b> : wmap(f,r)"; |
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119 by (fast_tac (ccl_cs addSIs (prems @ [wmapXH RS iffD2])) 1); |
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120 val wmapI = result(); |
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121 |
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122 val major::prems = goal Wfd.thy |
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123 "[| p : wmap(f,r); !!a b.[| <f(a),f(b)> : r; p=<a,b> |] ==> R |] ==> R"; |
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124 br (major RS (wmapXH RS iffD1) RS exE) 1; |
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125 by (REPEAT_SOME (eresolve_tac ([exE,conjE,disjE]@prems))); |
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126 by (ALLGOALS (fast_tac ccl_cs)); |
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127 val wmapE = result(); |
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128 |
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129 val [wf] = goal Wfd.thy |
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130 "Wfd(r) ==> Wfd(wmap(f,r))"; |
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131 bw Wfd_def; |
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132 by (safe_tac ccl_cs); |
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133 by (subgoal_tac "ALL b.ALL a.f(a)=b-->a:P" 1); |
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134 by (fast_tac ccl_cs 1); |
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135 br (wf RS wfd_induct RS allI) 1; |
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136 by (safe_tac ccl_cs); |
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137 be (spec RS mp) 1; |
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138 by (safe_tac (ccl_cs addSEs [wmapE])); |
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139 be (spec RS mp RS spec RS mp) 1; |
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140 ba 1; |
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141 br refl 1; |
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142 val wmap_wf = result(); |
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143 |
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144 (* Projections *) |
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145 |
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146 val prems = goal Wfd.thy "<xa,ya> : r ==> <<xa,xb>,<ya,yb>> : wmap(fst,r)"; |
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147 br wmapI 1; |
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148 by (simp_tac (term_ss addsimps prems) 1); |
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149 val wfstI = result(); |
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150 |
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151 val prems = goal Wfd.thy "<xb,yb> : r ==> <<xa,xb>,<ya,yb>> : wmap(snd,r)"; |
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152 br wmapI 1; |
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153 by (simp_tac (term_ss addsimps prems) 1); |
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154 val wsndI = result(); |
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155 |
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156 val prems = goal Wfd.thy "<xc,yc> : r ==> <<xa,<xb,xc>>,<ya,<yb,yc>>> : wmap(thd,r)"; |
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157 br wmapI 1; |
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158 by (simp_tac (term_ss addsimps prems) 1); |
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159 val wthdI = result(); |
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160 |
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161 (*** Ground well-founded relations ***) |
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162 |
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163 val prems = goalw Wfd.thy [wf_def] |
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164 "[| Wfd(r); a : r |] ==> a : wf(r)"; |
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165 by (fast_tac (set_cs addSIs prems) 1); |
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166 val wfI = result(); |
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167 |
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168 val prems = goalw Wfd.thy [Wfd_def] "Wfd({})"; |
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169 by (fast_tac (set_cs addEs [EmptyXH RS iffD1 RS FalseE]) 1); |
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170 val Empty_wf = result(); |
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171 |
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172 val prems = goalw Wfd.thy [wf_def] "Wfd(wf(R))"; |
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173 by (res_inst_tac [("Q","Wfd(R)")] (excluded_middle RS disjE) 1); |
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174 by (ALLGOALS (asm_simp_tac CCL_ss)); |
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175 br Empty_wf 1; |
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176 val wf_wf = result(); |
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177 |
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178 goalw Wfd.thy [NatPR_def] "p : NatPR <-> (EX x:Nat.p=<x,succ(x)>)"; |
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179 by (fast_tac set_cs 1); |
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180 val NatPRXH = result(); |
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181 |
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182 goalw Wfd.thy [ListPR_def] "p : ListPR(A) <-> (EX h:A.EX t:List(A).p=<t,h$t>)"; |
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183 by (fast_tac set_cs 1); |
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184 val ListPRXH = result(); |
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185 |
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186 val NatPRI = refl RS (bexI RS (NatPRXH RS iffD2)); |
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187 val ListPRI = refl RS (bexI RS (bexI RS (ListPRXH RS iffD2))); |
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188 |
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189 goalw Wfd.thy [Wfd_def] "Wfd(NatPR)"; |
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190 by (safe_tac set_cs); |
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191 by (wfd_strengthen_tac "%x.x:Nat" 1); |
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192 by (fast_tac (type_cs addSEs [XH_to_E NatPRXH]) 1); |
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193 be Nat_ind 1; |
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194 by (ALLGOALS (fast_tac (type_cs addEs [XH_to_E NatPRXH]))); |
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195 val NatPR_wf = result(); |
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196 |
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197 goalw Wfd.thy [Wfd_def] "Wfd(ListPR(A))"; |
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198 by (safe_tac set_cs); |
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199 by (wfd_strengthen_tac "%x.x:List(A)" 1); |
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200 by (fast_tac (type_cs addSEs [XH_to_E ListPRXH]) 1); |
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201 be List_ind 1; |
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202 by (ALLGOALS (fast_tac (type_cs addEs [XH_to_E ListPRXH]))); |
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203 val ListPR_wf = result(); |
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204 |
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