1 (* Title: HOL/Hoare/HoareAbort.thy |
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2 Author: Leonor Prensa Nieto & Tobias Nipkow |
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3 Copyright 2003 TUM |
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4 |
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5 Like Hoare.thy, but with an Abort statement for modelling run time errors. |
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6 *) |
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7 |
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8 theory HoareAbort |
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9 imports Main |
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10 uses ("hoare_tac.ML") |
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11 begin |
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12 |
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13 types |
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14 'a bexp = "'a set" |
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15 'a assn = "'a set" |
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16 |
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17 datatype |
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18 'a com = Basic "'a \<Rightarrow> 'a" |
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19 | Abort |
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20 | Seq "'a com" "'a com" ("(_;/ _)" [61,60] 60) |
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21 | Cond "'a bexp" "'a com" "'a com" ("(1IF _/ THEN _ / ELSE _/ FI)" [0,0,0] 61) |
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22 | While "'a bexp" "'a assn" "'a com" ("(1WHILE _/ INV {_} //DO _ /OD)" [0,0,0] 61) |
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23 |
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24 abbreviation annskip ("SKIP") where "SKIP == Basic id" |
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25 |
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26 types 'a sem = "'a option => 'a option => bool" |
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27 |
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28 consts iter :: "nat => 'a bexp => 'a sem => 'a sem" |
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29 primrec |
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30 "iter 0 b S = (\<lambda>s s'. s \<notin> Some ` b \<and> s=s')" |
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31 "iter (Suc n) b S = |
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32 (\<lambda>s s'. s \<in> Some ` b \<and> (\<exists>s''. S s s'' \<and> iter n b S s'' s'))" |
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33 |
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34 consts Sem :: "'a com => 'a sem" |
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35 primrec |
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36 "Sem(Basic f) s s' = (case s of None \<Rightarrow> s' = None | Some t \<Rightarrow> s' = Some(f t))" |
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37 "Sem Abort s s' = (s' = None)" |
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38 "Sem(c1;c2) s s' = (\<exists>s''. Sem c1 s s'' \<and> Sem c2 s'' s')" |
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39 "Sem(IF b THEN c1 ELSE c2 FI) s s' = |
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40 (case s of None \<Rightarrow> s' = None |
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41 | Some t \<Rightarrow> ((t \<in> b \<longrightarrow> Sem c1 s s') \<and> (t \<notin> b \<longrightarrow> Sem c2 s s')))" |
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42 "Sem(While b x c) s s' = |
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43 (if s = None then s' = None else \<exists>n. iter n b (Sem c) s s')" |
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44 |
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45 constdefs Valid :: "'a bexp \<Rightarrow> 'a com \<Rightarrow> 'a bexp \<Rightarrow> bool" |
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46 "Valid p c q == \<forall>s s'. Sem c s s' \<longrightarrow> s : Some ` p \<longrightarrow> s' : Some ` q" |
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47 |
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48 |
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49 |
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50 (** parse translations **) |
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51 |
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52 syntax |
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53 "_assign" :: "id => 'b => 'a com" ("(2_ :=/ _)" [70,65] 61) |
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54 |
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55 syntax |
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56 "_hoare_vars" :: "[idts, 'a assn,'a com,'a assn] => bool" |
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57 ("VARS _// {_} // _ // {_}" [0,0,55,0] 50) |
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58 syntax ("" output) |
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59 "_hoare" :: "['a assn,'a com,'a assn] => bool" |
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60 ("{_} // _ // {_}" [0,55,0] 50) |
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61 ML {* |
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62 |
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63 local |
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64 fun free a = Free(a,dummyT) |
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65 fun abs((a,T),body) = |
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66 let val a = absfree(a, dummyT, body) |
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67 in if T = Bound 0 then a else Const(Syntax.constrainAbsC,dummyT) $ a $ T end |
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68 in |
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69 |
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70 fun mk_abstuple [x] body = abs (x, body) |
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71 | mk_abstuple (x::xs) body = |
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72 Syntax.const @{const_syntax split} $ abs (x, mk_abstuple xs body); |
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73 |
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74 fun mk_fbody a e [x as (b,_)] = if a=b then e else free b |
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75 | mk_fbody a e ((b,_)::xs) = |
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76 Syntax.const @{const_syntax Pair} $ (if a=b then e else free b) $ mk_fbody a e xs; |
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77 |
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78 fun mk_fexp a e xs = mk_abstuple xs (mk_fbody a e xs) |
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79 end |
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80 *} |
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81 |
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82 (* bexp_tr & assn_tr *) |
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83 (*all meta-variables for bexp except for TRUE are translated as if they |
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84 were boolean expressions*) |
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85 ML{* |
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86 fun bexp_tr (Const ("TRUE", _)) xs = Syntax.const "TRUE" (* FIXME !? *) |
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87 | bexp_tr b xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs b; |
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88 |
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89 fun assn_tr r xs = Syntax.const @{const_syntax Collect} $ mk_abstuple xs r; |
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90 *} |
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91 (* com_tr *) |
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92 ML{* |
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93 fun com_tr (Const (@{syntax_const "_assign"},_) $ Free (a,_) $ e) xs = |
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94 Syntax.const @{const_syntax Basic} $ mk_fexp a e xs |
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95 | com_tr (Const (@{const_syntax Basic},_) $ f) xs = Syntax.const @{const_syntax Basic} $ f |
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96 | com_tr (Const (@{const_syntax Seq},_) $ c1 $ c2) xs = |
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97 Syntax.const @{const_syntax Seq} $ com_tr c1 xs $ com_tr c2 xs |
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98 | com_tr (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) xs = |
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99 Syntax.const @{const_syntax Cond} $ bexp_tr b xs $ com_tr c1 xs $ com_tr c2 xs |
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100 | com_tr (Const (@{const_syntax While},_) $ b $ I $ c) xs = |
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101 Syntax.const @{const_syntax While} $ bexp_tr b xs $ assn_tr I xs $ com_tr c xs |
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102 | com_tr t _ = t (* if t is just a Free/Var *) |
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103 *} |
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104 |
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105 (* triple_tr *) (* FIXME does not handle "_idtdummy" *) |
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106 ML{* |
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107 local |
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108 |
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109 fun var_tr (Free (a, _)) = (a, Bound 0) (* Bound 0 = dummy term *) |
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110 | var_tr (Const (@{syntax_const "_constrain"}, _) $ Free (a, _) $ T) = (a, T); |
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111 |
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112 fun vars_tr (Const (@{syntax_const "_idts"}, _) $ idt $ vars) = var_tr idt :: vars_tr vars |
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113 | vars_tr t = [var_tr t] |
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114 |
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115 in |
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116 fun hoare_vars_tr [vars, pre, prg, post] = |
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117 let val xs = vars_tr vars |
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118 in Syntax.const @{const_syntax Valid} $ |
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119 assn_tr pre xs $ com_tr prg xs $ assn_tr post xs |
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120 end |
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121 | hoare_vars_tr ts = raise TERM ("hoare_vars_tr", ts); |
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122 end |
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123 *} |
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124 |
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125 parse_translation {* [(@{syntax_const "_hoare_vars"}, hoare_vars_tr)] *} |
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126 |
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127 |
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128 (*****************************************************************************) |
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129 |
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130 (*** print translations ***) |
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131 ML{* |
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132 fun dest_abstuple (Const (@{const_syntax split},_) $ (Abs(v,_, body))) = |
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133 subst_bound (Syntax.free v, dest_abstuple body) |
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134 | dest_abstuple (Abs(v,_, body)) = subst_bound (Syntax.free v, body) |
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135 | dest_abstuple trm = trm; |
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136 |
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137 fun abs2list (Const (@{const_syntax split},_) $ (Abs(x,T,t))) = Free (x, T)::abs2list t |
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138 | abs2list (Abs(x,T,t)) = [Free (x, T)] |
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139 | abs2list _ = []; |
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140 |
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141 fun mk_ts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = mk_ts t |
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142 | mk_ts (Abs(x,_,t)) = mk_ts t |
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143 | mk_ts (Const (@{const_syntax Pair},_) $ a $ b) = a::(mk_ts b) |
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144 | mk_ts t = [t]; |
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145 |
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146 fun mk_vts (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = |
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147 ((Syntax.free x)::(abs2list t), mk_ts t) |
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148 | mk_vts (Abs(x,_,t)) = ([Syntax.free x], [t]) |
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149 | mk_vts t = raise Match; |
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150 |
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151 fun find_ch [] i xs = (false, (Syntax.free "not_ch", Syntax.free "not_ch")) |
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152 | find_ch ((v,t)::vts) i xs = |
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153 if t = Bound i then find_ch vts (i-1) xs |
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154 else (true, (v, subst_bounds (xs,t))); |
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155 |
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156 fun is_f (Const (@{const_syntax split},_) $ (Abs(x,_,t))) = true |
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157 | is_f (Abs(x,_,t)) = true |
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158 | is_f t = false; |
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159 *} |
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160 |
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161 (* assn_tr' & bexp_tr'*) |
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162 ML{* |
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163 fun assn_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T |
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164 | assn_tr' (Const (@{const_syntax inter},_) $ (Const (@{const_syntax Collect},_) $ T1) $ |
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165 (Const (@{const_syntax Collect},_) $ T2)) = |
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166 Syntax.const @{const_syntax inter} $ dest_abstuple T1 $ dest_abstuple T2 |
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167 | assn_tr' t = t; |
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168 |
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169 fun bexp_tr' (Const (@{const_syntax Collect},_) $ T) = dest_abstuple T |
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170 | bexp_tr' t = t; |
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171 *} |
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172 |
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173 (*com_tr' *) |
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174 ML{* |
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175 fun mk_assign f = |
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176 let val (vs, ts) = mk_vts f; |
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177 val (ch, which) = find_ch (vs~~ts) ((length vs)-1) (rev vs) |
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178 in |
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179 if ch then Syntax.const @{syntax_const "_assign"} $ fst which $ snd which |
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180 else Syntax.const @{const_syntax annskip} |
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181 end; |
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182 |
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183 fun com_tr' (Const (@{const_syntax Basic},_) $ f) = |
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184 if is_f f then mk_assign f else Syntax.const @{const_syntax Basic} $ f |
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185 | com_tr' (Const (@{const_syntax Seq},_) $ c1 $ c2) = |
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186 Syntax.const @{const_syntax Seq} $ com_tr' c1 $ com_tr' c2 |
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187 | com_tr' (Const (@{const_syntax Cond},_) $ b $ c1 $ c2) = |
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188 Syntax.const @{const_syntax Cond} $ bexp_tr' b $ com_tr' c1 $ com_tr' c2 |
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189 | com_tr' (Const (@{const_syntax While},_) $ b $ I $ c) = |
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190 Syntax.const @{const_syntax While} $ bexp_tr' b $ assn_tr' I $ com_tr' c |
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191 | com_tr' t = t; |
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192 |
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193 fun spec_tr' [p, c, q] = |
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194 Syntax.const @{syntax_const "_hoare"} $ assn_tr' p $ com_tr' c $ assn_tr' q |
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195 *} |
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196 |
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197 print_translation {* [(@{const_syntax Valid}, spec_tr')] *} |
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198 |
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199 (*** The proof rules ***) |
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200 |
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201 lemma SkipRule: "p \<subseteq> q \<Longrightarrow> Valid p (Basic id) q" |
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202 by (auto simp:Valid_def) |
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203 |
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204 lemma BasicRule: "p \<subseteq> {s. f s \<in> q} \<Longrightarrow> Valid p (Basic f) q" |
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205 by (auto simp:Valid_def) |
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206 |
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207 lemma SeqRule: "Valid P c1 Q \<Longrightarrow> Valid Q c2 R \<Longrightarrow> Valid P (c1;c2) R" |
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208 by (auto simp:Valid_def) |
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209 |
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210 lemma CondRule: |
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211 "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')} |
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212 \<Longrightarrow> Valid w c1 q \<Longrightarrow> Valid w' c2 q \<Longrightarrow> Valid p (Cond b c1 c2) q" |
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213 by (fastsimp simp:Valid_def image_def) |
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214 |
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215 lemma iter_aux: |
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216 "! s s'. Sem c s s' \<longrightarrow> s \<in> Some ` (I \<inter> b) \<longrightarrow> s' \<in> Some ` I \<Longrightarrow> |
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217 (\<And>s s'. s \<in> Some ` I \<Longrightarrow> iter n b (Sem c) s s' \<Longrightarrow> s' \<in> Some ` (I \<inter> -b))"; |
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218 apply(unfold image_def) |
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219 apply(induct n) |
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220 apply clarsimp |
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221 apply(simp (no_asm_use)) |
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222 apply blast |
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223 done |
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224 |
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225 lemma WhileRule: |
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226 "p \<subseteq> i \<Longrightarrow> Valid (i \<inter> b) c i \<Longrightarrow> i \<inter> (-b) \<subseteq> q \<Longrightarrow> Valid p (While b i c) q" |
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227 apply(simp add:Valid_def) |
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228 apply(simp (no_asm) add:image_def) |
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229 apply clarify |
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230 apply(drule iter_aux) |
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231 prefer 2 apply assumption |
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232 apply blast |
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233 apply blast |
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234 done |
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235 |
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236 lemma AbortRule: "p \<subseteq> {s. False} \<Longrightarrow> Valid p Abort q" |
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237 by(auto simp:Valid_def) |
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238 |
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239 |
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240 subsection {* Derivation of the proof rules and, most importantly, the VCG tactic *} |
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241 |
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242 lemma Compl_Collect: "-(Collect b) = {x. ~(b x)}" |
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243 by blast |
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244 |
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245 use "hoare_tac.ML" |
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246 |
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247 method_setup vcg = {* |
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248 Scan.succeed (fn ctxt => SIMPLE_METHOD' (hoare_tac ctxt (K all_tac))) *} |
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249 "verification condition generator" |
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250 |
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251 method_setup vcg_simp = {* |
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252 Scan.succeed (fn ctxt => |
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253 SIMPLE_METHOD' (hoare_tac ctxt (asm_full_simp_tac (simpset_of ctxt)))) *} |
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254 "verification condition generator plus simplification" |
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255 |
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256 (* Special syntax for guarded statements and guarded array updates: *) |
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257 |
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258 syntax |
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259 guarded_com :: "bool \<Rightarrow> 'a com \<Rightarrow> 'a com" ("(2_ \<rightarrow>/ _)" 71) |
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260 array_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a com" ("(2_[_] :=/ _)" [70, 65] 61) |
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261 translations |
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262 "P \<rightarrow> c" == "IF P THEN c ELSE CONST Abort FI" |
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263 "a[i] := v" => "(i < CONST length a) \<rightarrow> (a := CONST list_update a i v)" |
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264 (* reverse translation not possible because of duplicate "a" *) |
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265 |
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266 text{* Note: there is no special syntax for guarded array access. Thus |
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267 you must write @{text"j < length a \<rightarrow> a[i] := a!j"}. *} |
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268 |
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269 end |
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