1 (* Title: HOL/MicroJava/BV/Err.thy |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow |
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4 Copyright 2000 TUM |
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5 |
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6 The error type |
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7 *) |
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8 |
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9 header {* \isaheader{The Error Type} *} |
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10 |
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11 theory Err |
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12 imports Semilat |
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13 begin |
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14 |
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15 datatype 'a err = Err | OK 'a |
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16 |
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17 types 'a ebinop = "'a \<Rightarrow> 'a \<Rightarrow> 'a err" |
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18 'a esl = "'a set * 'a ord * 'a ebinop" |
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19 |
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20 consts |
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21 ok_val :: "'a err \<Rightarrow> 'a" |
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22 primrec |
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23 "ok_val (OK x) = x" |
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24 |
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25 constdefs |
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26 lift :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)" |
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27 "lift f e == case e of Err \<Rightarrow> Err | OK x \<Rightarrow> f x" |
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28 |
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29 lift2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c err) \<Rightarrow> 'a err \<Rightarrow> 'b err \<Rightarrow> 'c err" |
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30 "lift2 f e1 e2 == |
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31 case e1 of Err \<Rightarrow> Err |
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32 | OK x \<Rightarrow> (case e2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)" |
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33 |
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34 le :: "'a ord \<Rightarrow> 'a err ord" |
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35 "le r e1 e2 == |
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36 case e2 of Err \<Rightarrow> True | |
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37 OK y \<Rightarrow> (case e1 of Err \<Rightarrow> False | OK x \<Rightarrow> x <=_r y)" |
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38 |
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39 sup :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a err \<Rightarrow> 'b err \<Rightarrow> 'c err)" |
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40 "sup f == lift2(%x y. OK(x +_f y))" |
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41 |
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42 err :: "'a set \<Rightarrow> 'a err set" |
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43 "err A == insert Err {x . ? y:A. x = OK y}" |
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44 |
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45 esl :: "'a sl \<Rightarrow> 'a esl" |
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46 "esl == %(A,r,f). (A,r, %x y. OK(f x y))" |
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47 |
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48 sl :: "'a esl \<Rightarrow> 'a err sl" |
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49 "sl == %(A,r,f). (err A, le r, lift2 f)" |
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50 |
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51 syntax |
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52 err_semilat :: "'a esl \<Rightarrow> bool" |
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53 translations |
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54 "err_semilat L" == "semilat(Err.sl L)" |
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55 |
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56 |
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57 consts |
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58 strict :: "('a \<Rightarrow> 'b err) \<Rightarrow> ('a err \<Rightarrow> 'b err)" |
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59 primrec |
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60 "strict f Err = Err" |
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61 "strict f (OK x) = f x" |
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62 |
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63 lemma strict_Some [simp]: |
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64 "(strict f x = OK y) = (\<exists> z. x = OK z \<and> f z = OK y)" |
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65 by (cases x, auto) |
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66 |
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67 lemma not_Err_eq: |
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68 "(x \<noteq> Err) = (\<exists>a. x = OK a)" |
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69 by (cases x) auto |
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70 |
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71 lemma not_OK_eq: |
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72 "(\<forall>y. x \<noteq> OK y) = (x = Err)" |
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73 by (cases x) auto |
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74 |
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75 lemma unfold_lesub_err: |
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76 "e1 <=_(le r) e2 == le r e1 e2" |
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77 by (simp add: lesub_def) |
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78 |
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79 lemma le_err_refl: |
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80 "!x. x <=_r x \<Longrightarrow> e <=_(Err.le r) e" |
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81 apply (unfold lesub_def Err.le_def) |
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82 apply (simp split: err.split) |
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83 done |
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84 |
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85 lemma le_err_trans [rule_format]: |
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86 "order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e3 \<longrightarrow> e1 <=_(le r) e3" |
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87 apply (unfold unfold_lesub_err le_def) |
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88 apply (simp split: err.split) |
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89 apply (blast intro: order_trans) |
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90 done |
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91 |
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92 lemma le_err_antisym [rule_format]: |
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93 "order r \<Longrightarrow> e1 <=_(le r) e2 \<longrightarrow> e2 <=_(le r) e1 \<longrightarrow> e1=e2" |
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94 apply (unfold unfold_lesub_err le_def) |
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95 apply (simp split: err.split) |
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96 apply (blast intro: order_antisym) |
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97 done |
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98 |
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99 lemma OK_le_err_OK: |
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100 "(OK x <=_(le r) OK y) = (x <=_r y)" |
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101 by (simp add: unfold_lesub_err le_def) |
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102 |
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103 lemma order_le_err [iff]: |
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104 "order(le r) = order r" |
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105 apply (rule iffI) |
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106 apply (subst Semilat.order_def) |
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107 apply (blast dest: order_antisym OK_le_err_OK [THEN iffD2] |
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108 intro: order_trans OK_le_err_OK [THEN iffD1]) |
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109 apply (subst Semilat.order_def) |
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110 apply (blast intro: le_err_refl le_err_trans le_err_antisym |
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111 dest: order_refl) |
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112 done |
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113 |
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114 lemma le_Err [iff]: "e <=_(le r) Err" |
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115 by (simp add: unfold_lesub_err le_def) |
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116 |
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117 lemma Err_le_conv [iff]: |
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118 "Err <=_(le r) e = (e = Err)" |
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119 by (simp add: unfold_lesub_err le_def split: err.split) |
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120 |
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121 lemma le_OK_conv [iff]: |
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122 "e <=_(le r) OK x = (? y. e = OK y & y <=_r x)" |
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123 by (simp add: unfold_lesub_err le_def split: err.split) |
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124 |
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125 lemma OK_le_conv: |
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126 "OK x <=_(le r) e = (e = Err | (? y. e = OK y & x <=_r y))" |
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127 by (simp add: unfold_lesub_err le_def split: err.split) |
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128 |
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129 lemma top_Err [iff]: "top (le r) Err"; |
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130 by (simp add: top_def) |
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131 |
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132 lemma OK_less_conv [rule_format, iff]: |
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133 "OK x <_(le r) e = (e=Err | (? y. e = OK y & x <_r y))" |
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134 by (simp add: lesssub_def lesub_def le_def split: err.split) |
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135 |
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136 lemma not_Err_less [rule_format, iff]: |
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137 "~(Err <_(le r) x)" |
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138 by (simp add: lesssub_def lesub_def le_def split: err.split) |
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139 |
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140 lemma semilat_errI [intro]: |
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141 assumes semilat: "semilat (A, r, f)" |
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142 shows "semilat(err A, Err.le r, lift2(%x y. OK(f x y)))" |
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143 apply(insert semilat) |
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144 apply (unfold semilat_Def closed_def plussub_def lesub_def |
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145 lift2_def Err.le_def err_def) |
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146 apply (simp split: err.split) |
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147 done |
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148 |
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149 lemma err_semilat_eslI_aux: |
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150 assumes semilat: "semilat (A, r, f)" |
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151 shows "err_semilat(esl(A,r,f))" |
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152 apply (unfold sl_def esl_def) |
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153 apply (simp add: semilat_errI[OF semilat]) |
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154 done |
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155 |
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156 lemma err_semilat_eslI [intro, simp]: |
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157 "\<And>L. semilat L \<Longrightarrow> err_semilat(esl L)" |
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158 by(simp add: err_semilat_eslI_aux split_tupled_all) |
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159 |
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160 lemma acc_err [simp, intro!]: "acc r \<Longrightarrow> acc(le r)" |
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161 apply (unfold acc_def lesub_def le_def lesssub_def) |
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162 apply (simp add: wfP_eq_minimal split: err.split) |
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163 apply clarify |
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164 apply (case_tac "Err : Q") |
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165 apply blast |
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166 apply (erule_tac x = "{a . OK a : Q}" in allE) |
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167 apply (case_tac "x") |
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168 apply fast |
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169 apply blast |
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170 done |
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171 |
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172 lemma Err_in_err [iff]: "Err : err A" |
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173 by (simp add: err_def) |
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174 |
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175 lemma Ok_in_err [iff]: "(OK x : err A) = (x:A)" |
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176 by (auto simp add: err_def) |
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177 |
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178 section {* lift *} |
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179 |
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180 lemma lift_in_errI: |
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181 "\<lbrakk> e : err S; !x:S. e = OK x \<longrightarrow> f x : err S \<rbrakk> \<Longrightarrow> lift f e : err S" |
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182 apply (unfold lift_def) |
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183 apply (simp split: err.split) |
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184 apply blast |
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185 done |
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186 |
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187 lemma Err_lift2 [simp]: |
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188 "Err +_(lift2 f) x = Err" |
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189 by (simp add: lift2_def plussub_def) |
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190 |
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191 lemma lift2_Err [simp]: |
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192 "x +_(lift2 f) Err = Err" |
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193 by (simp add: lift2_def plussub_def split: err.split) |
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194 |
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195 lemma OK_lift2_OK [simp]: |
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196 "OK x +_(lift2 f) OK y = x +_f y" |
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197 by (simp add: lift2_def plussub_def split: err.split) |
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198 |
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199 |
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200 section {* sup *} |
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201 |
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202 lemma Err_sup_Err [simp]: |
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203 "Err +_(Err.sup f) x = Err" |
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204 by (simp add: plussub_def Err.sup_def Err.lift2_def) |
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205 |
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206 lemma Err_sup_Err2 [simp]: |
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207 "x +_(Err.sup f) Err = Err" |
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208 by (simp add: plussub_def Err.sup_def Err.lift2_def split: err.split) |
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209 |
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210 lemma Err_sup_OK [simp]: |
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211 "OK x +_(Err.sup f) OK y = OK(x +_f y)" |
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212 by (simp add: plussub_def Err.sup_def Err.lift2_def) |
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213 |
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214 lemma Err_sup_eq_OK_conv [iff]: |
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215 "(Err.sup f ex ey = OK z) = (? x y. ex = OK x & ey = OK y & f x y = z)" |
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216 apply (unfold Err.sup_def lift2_def plussub_def) |
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217 apply (rule iffI) |
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218 apply (simp split: err.split_asm) |
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219 apply clarify |
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220 apply simp |
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221 done |
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222 |
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223 lemma Err_sup_eq_Err [iff]: |
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224 "(Err.sup f ex ey = Err) = (ex=Err | ey=Err)" |
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225 apply (unfold Err.sup_def lift2_def plussub_def) |
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226 apply (simp split: err.split) |
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227 done |
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228 |
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229 section {* semilat (err A) (le r) f *} |
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230 |
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231 lemma semilat_le_err_Err_plus [simp]: |
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232 "\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> Err +_f x = Err" |
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233 by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1] |
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234 Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1]) |
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235 |
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236 lemma semilat_le_err_plus_Err [simp]: |
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237 "\<lbrakk> x: err A; semilat(err A, le r, f) \<rbrakk> \<Longrightarrow> x +_f Err = Err" |
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238 by (blast intro: Semilat.le_iff_plus_unchanged [OF Semilat.intro, THEN iffD1] |
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239 Semilat.le_iff_plus_unchanged2 [OF Semilat.intro, THEN iffD1]) |
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240 |
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241 lemma semilat_le_err_OK1: |
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242 "\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk> |
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243 \<Longrightarrow> x <=_r z"; |
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244 apply (rule OK_le_err_OK [THEN iffD1]) |
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245 apply (erule subst) |
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246 apply (simp add: Semilat.ub1 [OF Semilat.intro]) |
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247 done |
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248 |
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249 lemma semilat_le_err_OK2: |
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250 "\<lbrakk> x:A; y:A; semilat(err A, le r, f); OK x +_f OK y = OK z \<rbrakk> |
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251 \<Longrightarrow> y <=_r z" |
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252 apply (rule OK_le_err_OK [THEN iffD1]) |
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253 apply (erule subst) |
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254 apply (simp add: Semilat.ub2 [OF Semilat.intro]) |
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255 done |
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256 |
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257 lemma eq_order_le: |
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258 "\<lbrakk> x=y; order r \<rbrakk> \<Longrightarrow> x <=_r y" |
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259 apply (unfold Semilat.order_def) |
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260 apply blast |
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261 done |
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262 |
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263 lemma OK_plus_OK_eq_Err_conv [simp]: |
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264 assumes "x:A" and "y:A" and "semilat(err A, le r, fe)" |
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265 shows "((OK x) +_fe (OK y) = Err) = (~(? z:A. x <=_r z & y <=_r z))" |
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266 proof - |
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267 have plus_le_conv3: "\<And>A x y z f r. |
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268 \<lbrakk> semilat (A,r,f); x +_f y <=_r z; x:A; y:A; z:A \<rbrakk> |
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269 \<Longrightarrow> x <=_r z \<and> y <=_r z" |
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270 by (rule Semilat.plus_le_conv [OF Semilat.intro, THEN iffD1]) |
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271 from prems show ?thesis |
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272 apply (rule_tac iffI) |
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273 apply clarify |
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274 apply (drule OK_le_err_OK [THEN iffD2]) |
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275 apply (drule OK_le_err_OK [THEN iffD2]) |
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276 apply (drule Semilat.lub [OF Semilat.intro, of _ _ _ "OK x" _ "OK y"]) |
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277 apply assumption |
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278 apply assumption |
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279 apply simp |
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280 apply simp |
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281 apply simp |
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282 apply simp |
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283 apply (case_tac "(OK x) +_fe (OK y)") |
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284 apply assumption |
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285 apply (rename_tac z) |
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286 apply (subgoal_tac "OK z: err A") |
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287 apply (drule eq_order_le) |
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288 apply (erule Semilat.orderI [OF Semilat.intro]) |
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289 apply (blast dest: plus_le_conv3) |
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290 apply (erule subst) |
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291 apply (blast intro: Semilat.closedI [OF Semilat.intro] closedD) |
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292 done |
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293 qed |
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294 |
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295 section {* semilat (err(Union AS)) *} |
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296 |
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297 (* FIXME? *) |
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298 lemma all_bex_swap_lemma [iff]: |
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299 "(!x. (? y:A. x = f y) \<longrightarrow> P x) = (!y:A. P(f y))" |
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300 by blast |
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301 |
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302 lemma closed_err_Union_lift2I: |
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303 "\<lbrakk> !A:AS. closed (err A) (lift2 f); AS ~= {}; |
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304 !A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. a +_f b = Err) \<rbrakk> |
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305 \<Longrightarrow> closed (err(Union AS)) (lift2 f)" |
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306 apply (unfold closed_def err_def) |
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307 apply simp |
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308 apply clarify |
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309 apply simp |
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310 apply fast |
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311 done |
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312 |
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313 text {* |
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314 If @{term "AS = {}"} the thm collapses to |
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315 @{prop "order r & closed {Err} f & Err +_f Err = Err"} |
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316 which may not hold |
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317 *} |
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318 lemma err_semilat_UnionI: |
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319 "\<lbrakk> !A:AS. err_semilat(A, r, f); AS ~= {}; |
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320 !A:AS.!B:AS. A~=B \<longrightarrow> (!a:A.!b:B. ~ a <=_r b & a +_f b = Err) \<rbrakk> |
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321 \<Longrightarrow> err_semilat(Union AS, r, f)" |
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322 apply (unfold semilat_def sl_def) |
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323 apply (simp add: closed_err_Union_lift2I) |
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324 apply (rule conjI) |
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325 apply blast |
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326 apply (simp add: err_def) |
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327 apply (rule conjI) |
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328 apply clarify |
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329 apply (rename_tac A a u B b) |
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330 apply (case_tac "A = B") |
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331 apply simp |
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332 apply simp |
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333 apply (rule conjI) |
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334 apply clarify |
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335 apply (rename_tac A a u B b) |
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336 apply (case_tac "A = B") |
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337 apply simp |
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338 apply simp |
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339 apply clarify |
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340 apply (rename_tac A ya yb B yd z C c a b) |
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341 apply (case_tac "A = B") |
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342 apply (case_tac "A = C") |
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343 apply simp |
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344 apply (rotate_tac -1) |
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345 apply simp |
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346 apply (rotate_tac -1) |
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347 apply (case_tac "B = C") |
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348 apply simp |
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349 apply (rotate_tac -1) |
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350 apply simp |
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351 done |
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352 |
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353 end |
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