1 | include "utilities/extralib.ma". |
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2 | include "common/IOMonad.ma". |
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3 | include "common/Integers.ma". |
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4 | include "common/Events.ma". |
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5 | |
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6 | record trans_system (outty:Type[0]) (inty:outty → Type[0]) : Type[2] ≝ |
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7 | { global : Type[1] |
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8 | ; state : global → Type[0] |
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9 | ; is_final : ∀g. state g → option int |
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10 | ; step : ∀g. state g → IO outty inty (trace×(state g)) |
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11 | }. |
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12 | |
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13 | let rec repeat (n:nat) (outty:Type[0]) (inty:outty → Type[0]) (exec:trans_system outty inty) |
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14 | (g:global ?? exec) (s:state ?? exec g) |
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15 | : IO outty inty (trace × (state ?? exec g)) ≝ |
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16 | match n with |
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17 | [ O ⇒ Value ??? 〈E0, s〉 |
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18 | | S n' ⇒ ! 〈t1,s1〉 ← step ?? exec g s; |
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19 | ! 〈tn,sn〉 ← repeat n' ?? exec g s1; |
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20 | Value ??? 〈t1⧺tn,sn〉 |
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21 | ]. |
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22 | |
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23 | (* We take care here to check that we're not at the final state. It is not |
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24 | necessarily the case that a success step guarantees this (e.g., because |
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25 | there may be no way to stop a processor, so an infinite loop is used |
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26 | instead). *) |
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27 | inductive plus (O) (I) (TS:trans_system O I) (ge:global … TS) : state … TS ge → trace → state … TS ge → Prop ≝ |
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28 | | plus_one : ∀s1,tr,s2. |
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29 | is_final … TS ge s1 = None ? → |
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30 | step … TS ge s1 = OK … 〈tr,s2〉 → |
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31 | plus … ge s1 tr s2 |
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32 | | plus_succ : ∀s1,tr,s2,tr',s3. |
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33 | is_final … TS ge s1 = None ? → |
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34 | step … TS ge s1 = OK … 〈tr,s2〉 → |
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35 | plus … ge s2 tr' s3 → |
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36 | plus … ge s1 (tr⧺tr') s3. |
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37 | |
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38 | lemma plus_not_final : ∀O,I,FE. ∀gl,s,tr,s'. |
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39 | plus O I FE gl s tr s' → |
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40 | is_final … FE gl s = None ?. |
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41 | #O #I #FE #gl #s0 #tr0 #s0' * // |
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42 | qed. |
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43 | |
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44 | let rec trace_map (A,B:Type[0]) (f:A → res (trace × B)) |
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45 | (l:list A) on l : res (trace × (list B)) ≝ |
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46 | match l with |
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47 | [ nil ⇒ OK ? 〈E0, [ ]〉 |
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48 | | cons h t ⇒ |
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49 | do 〈tr,h'〉 ← f h; |
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50 | do 〈tr',t'〉 ← trace_map … f t; |
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51 | OK ? 〈tr ⧺ tr',h'::t'〉 |
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52 | ]. |
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53 | |
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54 | (* A third version of making several steps (!) |
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55 | |
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56 | The idea here is to have a computational definition of reducing serveral |
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57 | steps then showing some property about the resulting trace and state. By |
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58 | careful use of let rec we can ensure that reduction stops in a sensible |
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59 | way whenever the number of steps or the step currently being executed is |
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60 | not (reducible to) a value. |
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61 | |
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62 | An invariant is also asserted on every state other than the initial one. |
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63 | |
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64 | For example, we state a property to prove by something like |
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65 | |
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66 | ∃n. after_n_steps n … clight_exec ge s inv (λtr,s'. <some property>) |
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67 | |
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68 | then start reducing by giving the number of steps and reducing the |
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69 | definition |
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70 | |
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71 | %{3} whd |
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72 | |
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73 | and then whenever the reduction gets stuck, the currently reducing step is |
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74 | the third subterm, which we can reduce and unblock with (for example) |
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75 | rewriting |
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76 | |
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77 | whd in ⊢ (??%?); >EXe' |
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78 | |
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79 | and at the end reduce with whd to get the property as the goal. |
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80 | |
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81 | There are a few inversion-like results to get back the information contained in |
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82 | the proof below, and other that provides all the steps in an inductive form |
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83 | in Executions.ma. |
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84 | |
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85 | *) |
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86 | |
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87 | record await_value_stuff : Type[2] ≝ { |
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88 | avs_O : Type[0]; |
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89 | avs_I : avs_O → Type[0]; |
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90 | avs_exec : trans_system avs_O avs_I; |
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91 | avs_g : global ?? avs_exec; |
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92 | avs_inv : state ?? avs_exec avs_g → bool |
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93 | }. |
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94 | |
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95 | let rec await_value (avs:await_value_stuff) |
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96 | (v:IO (avs_O avs) (avs_I avs) (trace × (state ?? (avs_exec avs) (avs_g avs)))) |
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97 | (P:trace → state ?? (avs_exec avs) (avs_g avs) → Prop) on v : Prop ≝ |
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98 | match v with |
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99 | [ Value ts ⇒ P (\fst ts) (\snd ts) |
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100 | | _ ⇒ False |
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101 | ]. |
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102 | |
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103 | let rec assert (b:bool) (P:Prop) on b ≝ |
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104 | if b then P else False. |
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105 | |
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106 | let rec after_aux (avs:await_value_stuff) |
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107 | (n:nat) (s:state ?? (avs_exec avs) (avs_g avs)) (tr:trace) |
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108 | (P:trace → state ?? (avs_exec avs) (avs_g avs) → Prop) |
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109 | on n : Prop ≝ |
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110 | match n with |
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111 | [ O ⇒ P tr s |
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112 | | S n' ⇒ |
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113 | match is_final … s with |
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114 | [ None ⇒ |
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115 | await_value avs (step ?? (avs_exec avs) (avs_g avs) s) (λtr',s. |
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116 | assert (avs_inv avs s) (after_aux avs n' s (tr ⧺ tr') P)) |
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117 | | _ ⇒ False |
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118 | ] |
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119 | ]. |
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120 | |
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121 | definition after_n_steps : nat → |
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122 | ∀O,I. ∀exec:trans_system O I. ∀g:global ?? exec. |
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123 | state ?? exec g → |
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124 | (state ?? exec g → bool) → |
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125 | (trace → state ?? exec g → Prop) → Prop ≝ |
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126 | λn,O,I,exec,g,s,inv,P. after_aux (mk_await_value_stuff O I exec g inv) n s E0 P. |
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127 | |
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128 | lemma after_aux_covariant : ∀avs,P,Q,tr'. |
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129 | (∀tr,s. P (tr'⧺tr) s → Q tr s) → |
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130 | ∀n,s,tr. |
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131 | after_aux avs n s (tr'⧺tr) P → |
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132 | after_aux avs n s tr Q. |
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133 | #avs #P #Q #tr' #H #n elim n |
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134 | [ normalize /2/ |
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135 | | #n' #IH #s #tr |
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136 | whd in ⊢ (% → %); cases (is_final … s) [ 2: #x * ] |
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137 | whd in ⊢ (% → %); |
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138 | cases (step … s) [1,3: normalize /2/ ] |
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139 | * #tr'' #s'' |
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140 | whd in ⊢ (% → %); |
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141 | cases (avs_inv avs s'') [ 2: * ] |
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142 | whd in ⊢ (% → %); |
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143 | >Eapp_assoc @IH |
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144 | ] qed. |
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145 | |
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146 | lemma after_n_covariant : ∀n,O,I,exec,g,P,inv,Q. |
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147 | (∀tr,s. P tr s → Q tr s) → |
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148 | ∀s. |
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149 | after_n_steps n O I exec g s inv P → |
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150 | after_n_steps n O I exec g s inv Q. |
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151 | normalize /3/ |
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152 | qed. |
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153 | |
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154 | (* for joining a couple of these together *) |
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155 | |
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156 | lemma after_n_m : ∀n,m,O,I,exec,g,inv,P,Q,s,s'. |
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157 | after_n_steps m O I exec g s' inv Q → |
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158 | after_n_steps n O I exec g s inv (λtr1,s1. s' = s1 ∧ (∀tr'',s''. Q tr'' s'' → P (tr1⧺tr'') s'')) → |
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159 | after_n_steps (n+m) O I exec g s inv P. |
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160 | #n #m #O #I #exec #g #inv #P #Q whd in ⊢ (? → ? → ? → % → %); generalize in match E0; elim n |
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161 | [ #tr #s #s' #H whd in ⊢ (% → %); * #E destruct #H2 |
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162 | whd in H; <(E0_right tr) generalize in match E0 in H ⊢ %; |
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163 | generalize in match s; -s |
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164 | elim m |
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165 | [ #s #tr' whd in ⊢ (% → %); @H2 |
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166 | | #m' #IH #s #tr' whd in ⊢ (% → %); |
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167 | cases (is_final … s) [2: #r * ] |
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168 | whd in ⊢ (% → %); |
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169 | cases (step O I exec g s) [1,3: normalize //] * #tr'' #s'' whd in ⊢ (% → %); |
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170 | cases (inv s'') [2: * ] |
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171 | >Eapp_assoc @IH ] |
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172 | | #n' #IH #tr #s #s' #H whd in ⊢ (% → %); |
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173 | cases (is_final … s) [2: #r * ] |
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174 | whd in ⊢ (% → %); |
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175 | cases (step O I exec g s) [1,3: normalize // ] |
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176 | * #tr1 #s1 whd in ⊢ (% → %); |
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177 | cases (inv s1) [2:*] |
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178 | @IH @H |
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179 | ] qed. |
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180 | |
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181 | (* Inversion lemmas. *) |
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182 | |
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183 | lemma after_1_of_n_steps : ∀n,O,I,exec,g,inv,P,s. |
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184 | after_n_steps (S n) O I exec g s inv P → |
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185 | ∃tr,s'. step ?? exec g s = Value … 〈tr,s'〉 ∧ |
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186 | bool_to_Prop (inv s') ∧ |
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187 | after_n_steps n O I exec g s' inv (λtr'',s''. P (tr⧺tr'') s''). |
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188 | #n #O #I #exec #g #inv #P #s |
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189 | whd in ⊢ (% → ?); |
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190 | cases (is_final … s) |
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191 | [ whd in ⊢ (% → ?); |
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192 | cases (step … s) |
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193 | [ #o #k * |
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194 | | * #tr #s' whd in ⊢ (% → ?); >E0_left <(E0_right tr) #AFTER |
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195 | %{tr} %{s'} cases (inv s') in AFTER ⊢ %; #AFTER % /2/ cases AFTER |
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196 | | #m * |
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197 | ] |
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198 | | #r * |
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199 | ] qed. |
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200 | |
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201 | lemma after_1_step : ∀O,I,exec,g,inv,P,s. |
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202 | after_n_steps 1 O I exec g s inv P → |
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203 | ∃tr,s'. step ?? exec g s = Value … 〈tr,s'〉 ∧ bool_to_Prop (inv s') ∧ P tr s'. |
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204 | #O #I #exec #g #inv #P #s #AFTER |
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205 | cases (after_1_of_n_steps … AFTER) |
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206 | #tr * #s' * * #STEP #INV #FIN %{tr} %{s'} % [%] // whd in FIN; >E0_right in FIN; // |
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207 | qed. |
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208 | |
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209 | (* A typical way to use the following: |
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210 | |
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211 | With a hypothesis |
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212 | EX : after_n_steps n ... (State ...) ... |
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213 | reduce it [whd in EX;] to |
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214 | EX : await_value ... |
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215 | then perform inversion |
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216 | [cases (await_value_bind_inv … EX) -EX * #x * #E1 #EX] |
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217 | x : T |
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218 | E1 : f = return x |
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219 | EX : await_value ... |
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220 | *) |
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221 | |
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222 | lemma await_value_bind_inv : ∀avs,T,f,g,P. |
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223 | await_value avs (m_bind … f g) P → |
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224 | ∃x:T. (f = return x) ∧ await_value avs (g x) P. |
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225 | #avs #T * |
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226 | [ #o #k #g #P * |
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227 | | #x #g #P #AWAIT %{x} % // |
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228 | | #m #g #P * |
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229 | ] qed. |
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230 | |
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231 | (* A (possibly non-terminating) execution. *) |
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232 | coinductive execution (state:Type[0]) (output:Type[0]) (input:output → Type[0]) : Type[0] ≝ |
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233 | | e_stop : trace → int → state → execution state output input |
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234 | | e_step : trace → state → execution state output input → execution state output input |
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235 | | e_wrong : errmsg → execution state output input |
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236 | | e_interact : ∀o:output. (input o → execution state output input) → execution state output input. |
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237 | |
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238 | (* This definition is slightly awkward because of the need to provide resumptions. |
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239 | It records the last trace/state passed in, then recursively processes the next |
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240 | state. *) |
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241 | |
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242 | let corec exec_inf_aux (output:Type[0]) (input:output → Type[0]) |
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243 | (exec:trans_system output input) (g:global ?? exec) |
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244 | (s:IO output input (trace×(state ?? exec g))) |
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245 | : execution ??? ≝ |
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246 | match s with |
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247 | [ Wrong m ⇒ e_wrong ??? m |
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248 | | Value v ⇒ let 〈t,s'〉 ≝ v in |
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249 | match is_final ?? exec g s' with |
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250 | [ Some r ⇒ e_stop ??? t r s' |
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251 | | None ⇒ e_step ??? t s' (exec_inf_aux ??? g (step ?? exec g s')) ] |
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252 | | Interact out k' ⇒ e_interact ??? out (λv. exec_inf_aux ??? g (k' v)) |
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253 | ]. |
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254 | |
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255 | lemma execution_cases: ∀o,i,s.∀e:execution s o i. |
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256 | e = match e with [ e_stop tr r m ⇒ e_stop ??? tr r m |
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257 | | e_step tr s e ⇒ e_step ??? tr s e |
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258 | | e_wrong m ⇒ e_wrong ??? m | e_interact o k ⇒ e_interact ??? o k ]. |
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259 | #o #i #s #e cases e; [1: #T #I #M % | 2: #T #S #E % | 3: #E % |
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260 | | 4: #O #I % ] qed. (* XXX: assertion failed: superposition.ml when using auto |
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261 | here, used reflexivity instead *) |
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262 | |
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263 | axiom exec_inf_aux_unfold: ∀o,i,exec,g,s. exec_inf_aux o i exec g s = |
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264 | match s with |
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265 | [ Wrong m ⇒ e_wrong ??? m |
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266 | | Value v ⇒ let 〈t,s'〉 ≝ v in |
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267 | match is_final ?? exec g s' with |
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268 | [ Some r ⇒ e_stop ??? t r s' |
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269 | | None ⇒ e_step ??? t s' (exec_inf_aux ??? g (step ?? exec g s')) ] |
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270 | | Interact out k' ⇒ e_interact ??? out (λv. exec_inf_aux ??? g (k' v)) |
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271 | ]. |
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272 | (* |
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273 | #exec #ge #s >(execution_cases ? (exec_inf_aux …)) cases s |
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274 | [ #o #k |
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275 | | #x cases x #tr #s' (* XXX Can't unfold exec_inf_aux here *) |
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276 | | ] |
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277 | whd in ⊢ (??%%); //; |
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278 | qed. |
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279 | *) |
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280 | |
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281 | lemma exec_inf_stops_at_final : ∀O,I,TS,ge,s,s',tr,r. |
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282 | exec_inf_aux O I TS ge (step … ge s) = e_stop … tr r s' → |
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283 | step … ge s = Value … 〈tr, s'〉 ∧ |
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284 | is_final … s' = Some ? r. |
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285 | #O #I #TS #ge #s #s' #tr #r |
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286 | >exec_inf_aux_unfold |
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287 | cases (step … ge s) |
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288 | [ 1,3: normalize #H1 #H2 try #H3 destruct |
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289 | | * #tr' #s'' |
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290 | whd in ⊢ (??%? → ?); |
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291 | lapply (refl ? (is_final … s'')) |
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292 | cases (is_final … s'') in ⊢ (???% → %); |
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293 | [ #_ whd in ⊢ (??%? → ?); #E destruct |
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294 | | #r' #E1 #E2 whd in E2:(??%?); destruct % // |
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295 | ] |
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296 | ] qed. |
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297 | |
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298 | lemma extract_step : ∀O,I,TS,ge,s,s',tr,e. |
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299 | exec_inf_aux O I TS ge (step … ge s) = e_step … tr s' e → |
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300 | step … ge s = Value … 〈tr,s'〉 ∧ e = exec_inf_aux O I TS ge (step … ge s'). |
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301 | #O #I #TS #ge #s #s' #tr #e |
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302 | >exec_inf_aux_unfold |
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303 | cases (step … s) |
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304 | [ 1,3: normalize #H1 #H2 try #H3 destruct |
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305 | | * #tr' #s'' |
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306 | whd in ⊢ (??%? → ?); |
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307 | cases (is_final … s'') |
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308 | [ #E normalize in E; destruct /2/ |
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309 | | #r #E normalize in E; destruct |
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310 | ] |
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311 | ] qed. |
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312 | |
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313 | lemma extract_interact : ∀O,I,TS,ge,s,o,k. |
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314 | exec_inf_aux O I TS ge (step … ge s) = e_interact … o k → |
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315 | ∃k'. step … ge s = Interact … o k' ∧ k = (λv. exec_inf_aux ??? ge (k' v)). |
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316 | #O #I #TS #ge #s #o #k >exec_inf_aux_unfold |
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317 | cases (step … s) |
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318 | [ #o' #k' normalize #E destruct %{k'} /2/ |
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319 | | * #x #y normalize cases (is_final ?????) normalize |
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320 | #X try #Y destruct |
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321 | | normalize #m #E destruct |
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322 | ] qed. |
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323 | |
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324 | lemma exec_e_step_not_final : ∀O,I,TS,ge,s,s',tr,e. |
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325 | exec_inf_aux O I TS ge (step … ge s) = e_step … tr s' e → |
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326 | is_final … s' = None ?. |
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327 | #O #I #TS #ge #s #s' #tr #e |
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328 | >exec_inf_aux_unfold |
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329 | cases (step … s) |
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330 | [ 1,3: normalize #H1 #H2 try #H3 destruct |
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331 | | * #tr' #s'' |
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332 | whd in ⊢ (??%? → ?); |
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333 | lapply (refl ? (is_final … s'')) |
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334 | cases (is_final … s'') in ⊢ (???% → %); |
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335 | [ #F whd in ⊢ (??%? → ?); #E destruct @F |
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336 | | #r' #_ #E whd in E:(??%?); destruct |
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337 | ] |
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338 | ] qed. |
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339 | |
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340 | |
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341 | |
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342 | record fullexec (outty:Type[0]) (inty:outty → Type[0]) : Type[2] ≝ |
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343 | { program : Type[0] |
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344 | ; es1 :> trans_system outty inty |
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345 | ; make_global : program → global ?? es1 |
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346 | ; make_initial_state : ∀p:program. res (state ?? es1 (make_global p)) |
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347 | }. |
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348 | |
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349 | definition exec_inf : ∀o,i.∀fx:fullexec o i. ∀p:program ?? fx. execution (state ?? fx (make_global … fx p)) o i ≝ |
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350 | λo,i,fx,p. |
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351 | match make_initial_state ?? fx p with |
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352 | [ OK s ⇒ exec_inf_aux ?? fx (make_global … fx p) (Value … 〈E0,s〉) |
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353 | | Error m ⇒ e_wrong ??? m |
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354 | ]. |
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355 | |
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356 | |
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357 | |
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358 | |
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359 | definition execution_prefix : Type[0] → Type[0] ≝ λstate. list (trace × state). |
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360 | |
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361 | let rec split_trace O I (state:Type[0]) (x:execution state O I) (n:nat) on n : option (execution_prefix state × (execution state O I)) ≝ |
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362 | match n with |
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363 | [ O ⇒ Some ? 〈[ ], x〉 |
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364 | | S n' ⇒ |
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365 | match x with |
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366 | [ e_step tr s x' ⇒ |
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367 | ! 〈pre,x''〉 ← split_trace ?? state x' n'; |
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368 | Some ? 〈〈tr,s〉::pre,x''〉 |
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369 | (* Necessary for a measurable trace at the end of the program *) |
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370 | | e_stop tr r s ⇒ |
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371 | match n' with |
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372 | [ O ⇒ Some ? 〈[〈tr,s〉], x〉 |
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373 | | _ ⇒ None ? |
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374 | ] |
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375 | | _ ⇒ None ? |
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376 | ] |
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377 | ]. |
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378 | |
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379 | |
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