1 | include "common/Executions.ma". |
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2 | include "common/StructuredTraces.ma". (* just for status_class *) |
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3 | |
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4 | (* A small-step executable semantics, together with some kind of global |
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5 | environment, notion of cost labelled state, classification function and |
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6 | stack costs. *) |
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7 | |
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8 | record classified_system : Type[2] ≝ { |
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9 | cs_exec :> fullexec io_out io_in; |
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10 | cs_global : global … cs_exec; |
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11 | cs_labelled : state … cs_exec cs_global → bool; |
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12 | cs_classify : state … cs_exec cs_global → status_class; |
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13 | cs_callee : ∀s. match cs_classify … s with [ cl_call ⇒ True | _ ⇒ False ] → ident |
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14 | }. |
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15 | |
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16 | definition cs_state : classified_system → Type[0] ≝ |
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17 | λC. state … C (cs_global … C). |
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18 | |
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19 | (* Ideally we would also allow measurable traces to be from one cost label to |
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20 | another (in the same function call), but due to time constraints we'll stick |
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21 | to ending measurable traces at the end of the function only. |
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22 | *) |
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23 | |
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24 | definition trace_is_label_to_return : ∀C. list (cs_state … C × trace) → Prop ≝ |
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25 | λC,x. ∃tr1,s1,x',tr2,s2. x = 〈s1,tr1〉::(x'@[〈s2,tr2〉]) ∧ bool_to_Prop (cs_labelled C s1) ∧ cs_classify C s2 = cl_return. |
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26 | |
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27 | (* For intensional_event; should separate out definition *) |
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28 | include "common/StatusSimulation.ma". |
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29 | |
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30 | definition intensional_event_of_event : event → list intensional_event ≝ |
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31 | λev. match ev with |
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32 | [ EVcost l ⇒ [ IEVcost l ] |
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33 | | EVextcall _ _ _ ⇒ [ ] (* No equivalent, but there shouldn't be any for now *) |
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34 | ]. |
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35 | |
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36 | definition intensional_events_of_events : trace → list intensional_event ≝ |
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37 | λtr. flatten ? (map ?? intensional_event_of_event tr). |
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38 | |
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39 | let rec intensional_trace_of_trace C (callstack:list ident) (trace:list (cs_state … C × trace)) on trace : list ident × (list intensional_event) ≝ |
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40 | match trace with |
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41 | [ nil ⇒ 〈callstack, [ ]〉 |
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42 | | cons str tl ⇒ |
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43 | let 〈s,tr〉 ≝ str in |
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44 | let 〈callstack, call_event〉 ≝ |
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45 | match cs_classify C s return λx. (match x with [ cl_call ⇒ True | _ ⇒ False ] → ident) → list ident × (list intensional_event) with |
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46 | [ cl_call ⇒ λcallee. let id ≝ callee I in 〈id::callstack, [IEVcall id]〉 |
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47 | | cl_return ⇒ λ_. match callstack with [ nil ⇒ 〈[ ], [ ]〉 | cons id tl ⇒ 〈tl, [IEVret id]〉 ] |
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48 | | _ ⇒ λ_. 〈callstack, [ ]〉 |
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49 | ] (cs_callee C s) in |
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50 | let other_events ≝ intensional_events_of_events tr in |
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51 | let 〈callstack,rem〉 ≝ intensional_trace_of_trace C callstack tl in |
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52 | 〈callstack, call_event@other_events@rem〉 |
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53 | ]. |
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54 | |
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55 | definition normal_state : ∀C:classified_system. cs_state … C → bool ≝ |
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56 | λC,s. match cs_classify C s with [ cl_other ⇒ true | cl_jump ⇒ true | _ ⇒ false ]. |
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57 | |
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58 | lemma normal_state_inv : ∀C,s. |
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59 | normal_state C s → |
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60 | cs_classify C s = cl_other ∨ cs_classify C s = cl_jump. |
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61 | #C #s whd in ⊢ (?% → ?); cases (cs_classify C s) /2/ * |
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62 | qed. |
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63 | |
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64 | lemma int_trace_of_normal : ∀C,callstack,s,tr,trace. |
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65 | normal_state C s → |
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66 | intensional_trace_of_trace C callstack (〈s,tr〉::trace) = |
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67 | (let 〈stk',tl〉 ≝ intensional_trace_of_trace C callstack trace in |
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68 | 〈stk', (intensional_events_of_events tr)@tl〉). |
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69 | #C #callstack #s #tr #trace #NORMAL |
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70 | whd in ⊢ (??%?); |
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71 | generalize in match (cs_callee C s); |
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72 | cases (normal_state_inv … NORMAL) |
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73 | #CL >CL normalize nodelta #_ |
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74 | cases (intensional_trace_of_trace C callstack trace) |
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75 | #callstack' #tl normalize nodelta |
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76 | % |
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77 | qed. |
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78 | |
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79 | lemma flatten_append : ∀A,l1,l2. |
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80 | flatten A (l1@l2) = (flatten A l1)@(flatten A l2). |
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81 | #A #l1 #l2 |
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82 | elim l1 |
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83 | [ % |
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84 | | #h #t #IH whd in ⊢ (??%(??%?)); |
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85 | change with (flatten ??) in match (foldr ?????); >IH |
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86 | change with (flatten ??) in match (foldr ?????); |
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87 | >associative_append % |
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88 | ] qed. |
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89 | |
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90 | |
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91 | lemma int_events_append : ∀tr1,tr2. |
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92 | intensional_events_of_events (tr1@tr2) = (intensional_events_of_events tr1)@(intensional_events_of_events tr2). |
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93 | #tr1 #tr2 |
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94 | change with (flatten ??) in ⊢ (??%(??%%)); |
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95 | <map_append >flatten_append % |
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96 | qed. |
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97 | |
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98 | |
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99 | lemma int_trace_irr : ∀C,callstack,s,trace. |
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100 | normal_state C s → |
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101 | intensional_trace_of_trace C callstack (〈s,E0〉::trace) = intensional_trace_of_trace C callstack trace. |
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102 | #C #callstate #s #trace #NORMAL >(int_trace_of_normal … NORMAL) |
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103 | cases (intensional_trace_of_trace ???) // |
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104 | qed. |
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105 | |
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106 | lemma int_trace_append : ∀C,callstack,s,t1,t2,trace. |
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107 | normal_state C s → |
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108 | intensional_trace_of_trace C callstack (〈s,t1@t2〉::trace) = intensional_trace_of_trace C callstack (〈s,t1〉::〈s,t2〉::trace). |
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109 | #C #callstack #s #t1 #t2 #trace #NORMAL |
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110 | >(int_trace_of_normal … NORMAL) |
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111 | >(int_trace_of_normal … NORMAL) |
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112 | >(int_trace_of_normal … NORMAL) |
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113 | cases (intensional_trace_of_trace ???) #callstack' #trace' |
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114 | normalize nodelta |
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115 | >int_events_append |
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116 | >associative_append % |
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117 | qed. |
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118 | |
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119 | lemma build_eq_trace : ∀C,C',callstack,s,trace,rem,rem'. |
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120 | normal_state C s → |
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121 | all … (λstr. normal_state C' (\fst str)) trace → |
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122 | intensional_trace_of_trace C callstack rem = intensional_trace_of_trace C' callstack rem' → |
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123 | intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). |
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124 | #C #C' #callstack #s #trace #rem #rem' #NORMAL #NORMAL' |
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125 | >(int_trace_of_normal … NORMAL) |
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126 | cases (intensional_trace_of_trace C callstack rem) #callstack' #trace' |
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127 | #REM whd in ⊢ (??%?); |
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128 | elim trace in NORMAL' ⊢ %; |
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129 | [ #_ @REM |
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130 | | * #s' #tr' #tl #IH #NORMAL' |
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131 | cases (andb_Prop_true … NORMAL') #NORMALs #NORMALtl |
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132 | >int_trace_of_normal |
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133 | [ <(IH NORMALtl) whd in match (gather_trace ??); whd in ⊢ (???%); |
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134 | >int_events_append >associative_append % |
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135 | | @NORMALs |
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136 | ] |
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137 | ] qed. |
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138 | |
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139 | lemma int_trace_append' : ∀C,t1,t2,callstack. |
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140 | intensional_trace_of_trace C callstack (t1@t2) = |
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141 | (let 〈cs',t1'〉 ≝ intensional_trace_of_trace C callstack t1 in |
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142 | let 〈cs'',t2'〉 ≝ intensional_trace_of_trace C cs' t2 in |
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143 | 〈cs'', t1'@t2'〉). |
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144 | #C #t1 #t2 elim t1 |
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145 | [ #callstack whd in match ([ ]@t2); whd in ⊢ (???%); |
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146 | cases (intensional_trace_of_trace ???) #cs' #trace' % |
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147 | | * #s #tr #tl #IH |
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148 | #callstack |
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149 | whd in match (intensional_trace_of_trace ???); |
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150 | whd in match (intensional_trace_of_trace ???); |
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151 | generalize in match (cs_callee C s); |
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152 | cases (cs_classify C s) |
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153 | normalize nodelta #callee |
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154 | [ cases callstack [2: #cshd #cdtl] normalize nodelta ] |
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155 | >IH cases (intensional_trace_of_trace C ? tl) #cs' #tl' |
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156 | normalize nodelta |
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157 | cases (intensional_trace_of_trace C ? t2) #cs'' #tl'' |
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158 | normalize nodelta >associative_append >associative_append |
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159 | % |
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160 | ] qed. |
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161 | |
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162 | lemma int_trace_normal_cs : ∀C,callstack,trace. |
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163 | all ? (λstr. normal_state C (\fst str)) trace → |
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164 | callstack = \fst (intensional_trace_of_trace C callstack trace). |
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165 | #C #callstack #trace elim trace |
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166 | [ // |
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167 | | * #s #tr #tl #IH #N cases (andb_Prop_true … N) #N1 #Ntl |
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168 | >(int_trace_of_normal … N1) |
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169 | >(IH Ntl) in ⊢ (??%?); |
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170 | cases (intensional_trace_of_trace ???) /2/ |
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171 | ] qed. |
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172 | |
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173 | lemma int_trace_append_normal : ∀C,t1,t2,callstack. |
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174 | all ? (λstr. normal_state C (\fst str)) t1 → |
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175 | intensional_trace_of_trace C callstack (t1@t2) = |
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176 | (let t1' ≝ \snd (intensional_trace_of_trace C callstack t1) in |
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177 | let 〈cs'',t2'〉 ≝ intensional_trace_of_trace C callstack t2 in |
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178 | 〈cs'', t1'@t2'〉). |
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179 | #C #t1 #t2 #callstack #NORMAL lapply (int_trace_append' C t1 t2 callstack) |
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180 | lapply (int_trace_normal_cs C callstack t1 NORMAL) |
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181 | cases (intensional_trace_of_trace ?? t1) #cs #tl #E destruct // |
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182 | qed. |
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183 | |
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184 | lemma build_return_trace : ∀C,C',callstack,s,s',tr,trace',rem,rem'. |
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185 | cs_classify C s = cl_return → |
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186 | cs_classify C' s' = cl_return → |
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187 | all ? (λstr. normal_state C' (\fst str)) trace' → |
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188 | intensional_trace_of_trace C (tail … callstack) rem = intensional_trace_of_trace C' (tail … callstack) rem' → |
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189 | let trace ≝ 〈s',tr〉::trace' in |
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190 | intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). |
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191 | #C #C' #callstack #s #s' #tr #trace #rem #rem' #CL #CL' #NORMAL #E |
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192 | whd |
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193 | whd in ⊢ (??%%); normalize nodelta |
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194 | generalize in match (cs_callee C s); generalize in match (cs_callee C' s'); |
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195 | >CL >CL' normalize nodelta #_ #_ |
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196 | cases callstack in E ⊢ %; [2: #stkhd #stktl] |
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197 | normalize nodelta |
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198 | cases (intensional_trace_of_trace ?? rem) #cs_rem #ev_rem normalize nodelta |
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199 | >(int_trace_append_normal … NORMAL) normalize nodelta |
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200 | cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E |
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201 | destruct @eq_f @eq_f |
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202 | whd in match (gather_trace ??); >int_events_append |
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203 | >associative_append @eq_f |
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204 | elim trace in NORMAL ⊢ %; |
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205 | [ 1,3: #_ % |
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206 | | 2,4: |
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207 | * #s1 #tr1 #tl #IH |
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208 | #N cases (andb_Prop_true … N) #N1 #Ntl |
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209 | whd in match (gather_trace ??); >int_events_append |
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210 | >associative_append >(IH Ntl) |
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211 | >(int_trace_of_normal … N1) |
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212 | cases (intensional_trace_of_trace ?? tl) |
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213 | #cs' #tl' >associative_append % |
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214 | ] qed. |
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215 | |
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216 | lemma generalize_callee : ∀C,s,H. ∀P: ? → ? → Prop. |
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217 | (∀f. P f (f H)) → |
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218 | P (cs_callee C s) (cs_callee C s H). |
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219 | #C #s #H #P #f @f |
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220 | qed. |
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221 | |
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222 | lemma build_call_trace : ∀C,C',callstack,s,s',tr,trace',rem,rem',H,H'. |
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223 | cs_classify C s = cl_call → |
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224 | cs_classify C' s' = cl_call → |
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225 | all ? (λstr. normal_state C' (\fst str)) trace' → |
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226 | intensional_trace_of_trace C (cs_callee C s H::callstack) rem = intensional_trace_of_trace C' (cs_callee C s H::callstack) rem' → |
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227 | cs_callee C s H = cs_callee C' s' H' → |
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228 | let trace ≝ 〈s',tr〉::trace' in |
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229 | intensional_trace_of_trace C callstack (〈s,gather_trace … trace〉::rem) = intensional_trace_of_trace C' callstack (trace@rem'). |
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230 | #C #C' #callstack #s #s' #tr #trace #rem #rem' #H #H' #CL #CL' #NORMAL |
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231 | whd in ⊢ (? → ? → %); |
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232 | whd in ⊢ (? → ? → ??%%); normalize nodelta |
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233 | @generalize_callee |
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234 | @generalize_callee |
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235 | >CL in H ⊢ %; * >CL' in H' ⊢ %; * normalize nodelta #calleef #calleef' #E #CALLEE <CALLEE |
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236 | cases (intensional_trace_of_trace ?? rem) in E ⊢ %; #cs_rem #ev_rem normalize nodelta |
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237 | >(int_trace_append_normal … NORMAL) normalize nodelta |
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238 | cases (intensional_trace_of_trace ?? rem') #cs_rem' #ev_rem' normalize nodelta #E |
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239 | destruct @eq_f @eq_f |
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240 | whd in match (gather_trace ??); >int_events_append |
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241 | >associative_append @eq_f |
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242 | elim trace in NORMAL ⊢ %; |
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243 | [ 1,3: #_ % |
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244 | | 2,4: |
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245 | * #s1 #tr1 #tl #IH |
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246 | #N cases (andb_Prop_true … N) #N1 #Ntl |
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247 | whd in match (gather_trace ??); >int_events_append |
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248 | >associative_append >(IH Ntl) |
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249 | >(int_trace_of_normal … N1) |
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250 | cases (intensional_trace_of_trace ?? tl) |
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251 | #cs' #tl' >associative_append % |
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252 | ] qed. |
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253 | |
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254 | |
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255 | definition measure_stack : (ident → nat) → nat → list intensional_event → nat × nat ≝ |
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256 | λcosts,start. |
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257 | foldl ?? (λx. λev. |
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258 | match ev with |
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259 | [ IEVcall id ⇒ |
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260 | let 〈current_stack,max_stack〉 ≝ x in |
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261 | let new_stack ≝ current_stack + costs id in |
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262 | 〈new_stack, max new_stack max_stack〉 |
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263 | | IEVret id ⇒ |
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264 | let 〈current_stack,max_stack〉 ≝ x in |
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265 | 〈current_stack - costs id, max_stack〉 |
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266 | | _ ⇒ x |
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267 | ]) 〈start,start〉. |
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268 | |
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269 | definition max_stack : (ident → nat) → nat → list intensional_event → nat ≝ |
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270 | λcosts, start, trace. \snd (measure_stack costs start trace). |
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271 | |
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272 | lemma foldl_inv : ∀A,B. ∀P:A → Prop. ∀f. |
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273 | (∀acc,el. P acc → P (f acc el)) → |
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274 | ∀l,acc. P acc → |
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275 | P (foldl A B f acc l). |
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276 | #A #B #P #f #IH #l elim l |
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277 | [ // |
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278 | | #h #t #IH' #acc #H @IH' @IH @H |
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279 | ] qed. |
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280 | (* |
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281 | lemma max_stack_step : ∀C,a,m,a',m',tr,s. |
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282 | measure_stack_aux C 〈a,m〉 〈s,tr〉 = 〈a',m'〉 → |
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283 | m' = max m a'. |
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284 | #C #a #m #a' #m' #tr #s |
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285 | whd in match (measure_stack_aux ???); |
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286 | generalize in match (cs_stack C s); cases (cs_classify C s) #f |
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287 | normalize nodelta #E destruct // |
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288 | qed. |
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289 | |
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290 | lemma max_stack_step' : ∀C,a,a1',a2',m1,m1',m2,m2',str. |
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291 | measure_stack_aux C 〈a,m1〉 str = 〈a1',m1'〉 → |
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292 | measure_stack_aux C 〈a,m2〉 str = 〈a2',m2'〉 → |
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293 | a1' = a2'. |
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294 | #C #a #a1' #a2' #m1 #m2 #m1' #m2' * #s #tr |
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295 | whd in match (measure_stack_aux ???); whd in match (measure_stack_aux ???); |
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296 | generalize in match (cs_stack C s); cases (cs_classify C s) #f |
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297 | normalize nodelta |
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298 | #E1 #E2 destruct |
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299 | % |
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300 | qed. |
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301 | |
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302 | |
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303 | lemma max_stack_steps : ∀C,trace,a,m. |
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304 | \snd (foldl … (measure_stack_aux C) 〈a,m〉 trace) = |
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305 | max m (\snd (foldl … (measure_stack_aux C) 〈a,0〉 trace)). |
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306 | #C #trace elim trace |
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307 | [ #a #m >max_n_O % |
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308 | | * #tr #s #tl #IH #a #m |
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309 | whd in match (foldl ?????); |
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310 | letin x ≝ (measure_stack_aux ???) lapply (refl ? x) cases x in ⊢ (???% → %); #a' #m' #M |
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311 | whd in match (foldl ??? 〈a,O〉 ?); |
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312 | letin y ≝ (measure_stack_aux ? 〈a,O〉 ?) lapply (refl ? y) cases y in ⊢ (???% → %); #a'' #m'' #M' |
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313 | >(IH a'') >IH |
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314 | >(max_stack_step … M) |
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315 | >(max_stack_step … M') >max_O_n >(max_stack_step' … M M') >associative_max % |
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316 | ] qed. |
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317 | |
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318 | lemma max_stack_append : ∀C,c1,ex1,ex2. |
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319 | max (max_stack C c1 ex1) (max_stack C (stack_after C c1 ex1) ex2) = max_stack C c1 (ex1@ex2). |
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320 | #C #c1 #ex1 #ex2 |
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321 | whd in match (max_stack ???); whd in match (stack_after ???); |
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322 | whd in match (max_stack ?? (ex1@ex2)); |
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323 | whd in match (measure_stack ???); whd in match (measure_stack ?? (ex1@ex2)); |
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324 | generalize in match O; generalize in match c1; elim ex1 |
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325 | [ #c #m whd in ⊢ (??(?%?)%); >max_stack_steps % |
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326 | | * #tr #s #tl #IH #c #m |
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327 | whd in match (foldl ?????); whd in ⊢ (???(???%)); |
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328 | lapply (refl ? (measure_stack_aux C 〈c1,m〉 〈tr,s〉)) |
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329 | cases (measure_stack_aux ???) in ⊢ (???% → %); #a' #m' #M |
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330 | >IH % |
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331 | ] qed. |
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332 | *) |
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333 | |
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334 | (* Check that the trace ends with the return from the starting function and one |
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335 | further state. *) |
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336 | |
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337 | let rec will_return_aux C (depth:nat) |
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338 | (trace:list (cs_state … C × trace)) on trace : bool ≝ |
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339 | match trace with |
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340 | [ nil ⇒ false |
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341 | | cons h tl ⇒ |
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342 | let 〈s,tr〉 ≝ h in |
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343 | match cs_classify C s with |
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344 | [ cl_call ⇒ will_return_aux C (S depth) tl |
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345 | | cl_return ⇒ |
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346 | match depth with |
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347 | (* The state after the return will appear in the structured trace, but |
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348 | not here because it is the second part of the pair returned by |
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349 | exec_steps. *) |
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350 | [ O ⇒ match tl with [ nil ⇒ true | _ ⇒ false ] |
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351 | | S d ⇒ will_return_aux C d tl |
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352 | ] |
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353 | | _ ⇒ will_return_aux C depth tl |
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354 | ] |
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355 | ]. |
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356 | definition will_return' : ∀C. list (cs_state … C × trace) → bool ≝ λC. will_return_aux C O. |
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357 | |
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358 | (* Like classified_system, but we don't fix the global environment so that we |
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359 | talk about the program separately. *) |
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360 | |
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361 | record preclassified_system : Type[2] ≝ { |
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362 | pcs_exec :> fullexec io_out io_in; |
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363 | pcs_labelled : ∀g. state … pcs_exec g → bool; |
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364 | pcs_classify : ∀g. state … pcs_exec g → status_class; |
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365 | pcs_callee : ∀g,s. match pcs_classify g s with [ cl_call ⇒ True | _ ⇒ False ] → ident |
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366 | }. |
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367 | |
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368 | definition pcs_to_cs : ∀C:preclassified_system. global … C → classified_system ≝ |
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369 | λC,g. |
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370 | mk_classified_system (pcs_exec C) g (pcs_labelled C ?) (pcs_classify C ?) (pcs_callee C ?). |
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371 | |
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372 | (* FIXME: this definition is unreasonable because it presumes we can easily |
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373 | identify the change in stack usage from return states, but that information |
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374 | is rather implicit (we only really record the function called in the |
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375 | extended RTLabs states when building the structured traces). *) |
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376 | |
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377 | definition measurable : ∀C:preclassified_system. |
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378 | ∀p:program ?? C. nat → nat → |
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379 | (ident → nat) (* stack cost *) → |
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380 | nat → Prop ≝ |
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381 | λC,p,m,n,stack_cost,max_allowed_stack. ∃s0,prefix,s1,interesting,s2. |
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382 | let g ≝ make_global … (pcs_exec … C) p in |
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383 | let C' ≝ pcs_to_cs C g in |
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384 | make_initial_state … p = OK ? s0 ∧ |
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385 | exec_steps m ?? (cs_exec … C') g s0 = OK ? 〈prefix,s1〉 ∧ |
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386 | exec_steps n ?? (cs_exec … C') g s1 = OK ? 〈interesting,s2〉 ∧ |
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387 | trace_is_label_to_return C' interesting ∧ |
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388 | bool_to_Prop (will_return' C' interesting) ∧ |
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389 | le (max_stack stack_cost 0 (\snd (intensional_trace_of_trace C' [ ] (prefix@interesting)))) max_allowed_stack. |
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390 | |
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391 | (* TODO: probably ought to be elsewhere; use exec_steps instead |
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392 | |
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393 | Note that this does not include stack space |
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394 | *) |
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395 | |
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396 | definition observables : ∀C:preclassified_system. program ?? C → nat → nat → res ((list intensional_event) × (list intensional_event)) ≝ |
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397 | λC,p,m,n. |
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398 | let g ≝ make_global … (pcs_exec … C) p in |
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399 | let C' ≝ pcs_to_cs C g in |
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400 | ! s0 ← make_initial_state … p; |
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401 | ! 〈prefix,s1〉 ← exec_steps m ?? (cs_exec … C') g s0; |
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402 | ! 〈interesting,s2〉 ← exec_steps n ?? (cs_exec … C') g s1; |
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403 | let 〈cs,prefix'〉 ≝ intensional_trace_of_trace C' [ ] prefix in |
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404 | return 〈prefix', \snd (intensional_trace_of_trace C' cs interesting)〉. |
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405 | |
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