1 | |
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2 | include "compiler.ma". |
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3 | |
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4 | include "common/SmallstepExec.ma". |
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5 | include "Clight/Cexec.ma". |
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6 | include "ASM/Interpret2.ma". |
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7 | |
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8 | include "Clight/labelSimulation.ma". |
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9 | |
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10 | theorem correct : |
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11 | ∀input_program. |
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12 | |
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13 | not_wrong … (exec_inf … clight_fullexec input_program) → |
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14 | |
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15 | ∀object_code,costlabel_map,labelled,cost_map. |
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16 | compile input_program = OK ? 〈〈object_code,costlabel_map〉,labelled,cost_map〉 → |
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17 | |
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18 | sim_with_labels (exec_inf … clight_fullexec input_program) (exec_inf … clight_fullexec labelled) |
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19 | ∧ |
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20 | True (* TODO *). |
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21 | |
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22 | #input_program |
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23 | #NOT_WRONG |
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24 | #object_code #costlabel_map #labelled #cost_map |
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25 | #COMPILE |
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26 | cases (bind_inversion ????? COMPILE) -COMPILE * * #init_cost #labelled' #rtlabs_program * #FRONTEND #COMPILE |
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27 | cases (bind_inversion ????? COMPILE) -COMPILE * #object_code' #costlabel_map' * #ASSEMBLER #COMPILE |
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28 | whd in COMPILE:(??%%); destruct |
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29 | cases (bind_inversion ????? FRONTEND) -FRONTEND #cminor_program * #CMINOR #FRONTEND |
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30 | whd in FRONTEND:(??%%); destruct |
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31 | |
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32 | % |
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33 | [ @labelling_sim @NOT_WRONG |
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34 | | @I |
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35 | ] qed. |
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36 | |
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37 | axiom Clight_classify : state → status_class. |
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38 | axiom Clight_labelled : state → bool. |
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39 | definition Clight_stack_T ≝ ∀s:state. match Clight_classify s with [ cl_call ⇒ True | cl_return ⇒ True | _ ⇒ False ] → nat. |
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40 | |
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41 | (* [will_return depth stack max time s trace] says that there is a prefix of [trace], |
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42 | whose successor state is [s], which exits [depth] number of functions, starting with [stack] |
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43 | amount of stack space, using a maximum of [max] amount of stack space and [time]. *) |
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44 | |
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45 | inductive will_return (time_cost:state → nat) (stack_cost:Clight_stack_T) : nat → nat → nat → nat → state → execution state io_out io_in → Prop ≝ |
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46 | | wr_step : ∀s,tr,depth,stack,mx,tm,s',trace. |
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47 | Clight_classify s = cl_other ∨ Clight_classify s = cl_jump → |
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48 | will_return ?? depth stack mx tm s' trace → |
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49 | will_return ?? depth stack mx (tm + time_cost s) s' (e_step … tr s trace) |
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50 | | wr_call : ∀s,tr,depth,stack,mx,tm,s',trace. |
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51 | ∀CL:Clight_classify s = cl_call. |
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52 | will_return ?? (S depth) (stack + stack_cost s ?) mx tm s' trace → |
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53 | will_return ?? depth stack mx (tm + time_cost s) s' (e_step … tr s trace) |
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54 | | wr_ret : ∀s,tr,depth,stack,mx,tm,s',trace. |
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55 | ∀CL:Clight_classify s = cl_return. |
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56 | will_return ?? depth (stack - stack_cost s ?) mx tm s' trace → |
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57 | will_return ?? (S depth) stack (max mx stack) (tm + time_cost s) s' (e_step … tr s trace) |
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58 | (* will_return ?? depth stack mx tm s' trace → |
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59 | let prev_stack ≝ (stack + stack_cost s ?) in |
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60 | will_return ?? (S depth) prev_stack (max mx prev_stack) (tm + time_cost s) s' (e_step … tr s trace)*) |
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61 | (* Note that we require the ability to make a step after the return (this |
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62 | corresponds to somewhere that will be guaranteed to be a label at the |
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63 | end of the compilation chain). *) |
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64 | | wr_base : ∀base_stack,s,tr,trace. |
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65 | will_return ?? O base_stack base_stack O s (e_step … tr s trace) |
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66 | . >CL @I qed. |
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67 | |
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68 | (* [nth_state_of_with_stack state stack_cost stack_bound exec n] returns [Some s] iff after |
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69 | [n] steps of [exec] we have reached [s] without exceeding the [stack_bound] |
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70 | according to the [stack_cost] function. *) |
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71 | axiom nth_state_of_with_stack : ∀state. (state → nat) → nat → execution state io_out io_in → nat → option state. |
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72 | axiom nth_state_of : ∀state. execution state io_out io_in → nat → option state. |
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73 | |
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74 | definition execution_prefix : Type[0] ≝ list (trace × state). |
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75 | axiom split_trace : execution state io_out io_in → nat → option (execution_prefix × (execution state io_out io_in)). |
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76 | axiom stack_after : execution_prefix → nat. |
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77 | axiom will_return' : Clight_stack_T → nat → nat → execution_prefix → option nat. |
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78 | |
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79 | definition measurable : clight_program → nat → nat → Clight_stack_T → nat → Prop ≝ |
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80 | λp,m,n,stack_cost,max_allowed_stack. ∀prefix,suffix,interesting,remainder,max_stack. |
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81 | let cl_trace ≝ exec_inf … clight_fullexec p in |
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82 | split_trace cl_trace m = Some ? 〈prefix,suffix〉 ∧ |
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83 | split_trace suffix n = Some ? 〈interesting,remainder〉 ∧ |
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84 | will_return' stack_cost O (stack_after prefix) interesting = Some ? max_stack ∧ |
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85 | max_stack < max_allowed_stack. |
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86 | |
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87 | (* From measurable on Clight, we will end up with an RTLabs flat trace where |
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88 | we know that there are some m' and n' such that the prefix in Clight matches |
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89 | the prefix in RTLabs given by m', the next n steps in Clight are equivalent |
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90 | to the n' steps in RTLabs, and we have a suitable "will_return" for RTLabs |
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91 | for those n' steps so that we can build a corresponding structured trace. |
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92 | |
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93 | "Equivalent" here means, in particular, that the observables will be the same, |
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94 | and those observables will include the stack space costs. |
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95 | *) |
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96 | |
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97 | axiom observables : clight_program → nat → nat → option ((list trace) × (list trace)). |
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98 | axiom observables_8051 : object_code → nat → nat → option ((list trace) × (list trace)). |
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99 | axiom clight_clock_after : clight_program → nat → option nat. |
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100 | axiom initial_8051_status : ∀oc. Status oc. |
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101 | |
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102 | definition simulates ≝ |
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103 | λstack_cost, stack_bound, labelled, object_code. |
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104 | let initial_status ≝ initial_8051_status (load_code_memory object_code) in |
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105 | ∀m1,m2. measurable labelled m1 m2 stack_cost stack_bound → |
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106 | ∀c1,c2. clight_clock_after labelled m1 = Some ? c1 → clight_clock_after labelled m2 = Some ? c2 → |
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107 | ∃n1,n2. observables labelled m1 m2 = observables_8051 object_code n1 n2 ∧ |
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108 | c2 - c1 = clock … (execute n2 ? initial_status) - clock … (execute n1 ? initial_status). |
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109 | |
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110 | axiom compile' : clight_program → res (object_code × costlabel_map × clight_program |
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111 | × ((Σl:costlabel.in_clight_program l)→ℕ) × Clight_stack_T × nat). |
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112 | |
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113 | theorem correct' : |
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114 | ∀input_program. |
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115 | |
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116 | not_wrong … (exec_inf … clight_fullexec input_program) → |
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117 | |
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118 | ∀object_code,costlabel_map,labelled,cost_map,stack_cost,stack_bound. |
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119 | compile' input_program = OK ? 〈〈〈〈object_code,costlabel_map〉,labelled〉,cost_map〉,stack_cost,stack_bound〉 → |
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120 | |
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121 | sim_with_labels (exec_inf … clight_fullexec input_program) (exec_inf … clight_fullexec labelled) |
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122 | ∧ |
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123 | |
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124 | simulates stack_cost stack_bound labelled object_code. |
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125 | |
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126 | |
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127 | |
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128 | (* start of old simulates |
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129 | |
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130 | let cl_trace ≝ exec_inf … clight_fullexec labelled in |
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131 | let asm_trace ≝ exec_inf … ASM_fullexec object_code in |
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132 | not_wrong ? cl_trace → |
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133 | ∀n,s. nth_state_of_with_stack ? stack_cost stack_bound cl_trace n = Some ? s → |
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134 | 𝚺m,s'. nth_state_of ? asm_trace m = Some ? s' ∧ s ≃ s' |
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135 | |
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136 | *) |
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137 | |
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138 | (* TODO |
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139 | |
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140 | |
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141 | ∀input_program. |
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142 | ! 〈object_code,costlabel_map,labelled,cost_map〉 ← compile input_program |
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143 | |
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144 | exec_inf … clight_fullexec input_program ≃l exec_inf … clight_fullexec labelled |
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145 | |
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146 | ∧ |
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147 | |
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148 | exec_inf … clight_fullexec labelled ≈ exec_inf … ASM_fullexec object_code |
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149 | (* Should we be lifting labels in some way here? *) |
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150 | |
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151 | ∧ |
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152 | |
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153 | ∀i,f : clight_status. |
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154 | Clight_labelled i → |
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155 | Clight_labelled f → |
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156 | ∀mx,time. |
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157 | let trace ≝ exec_inf_aux … clight_fullexec labelled i in |
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158 | will_return O O mx time f trace → |
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159 | mx < max_allowed_stack → |
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160 | ∃!i',f'. i ≃ i' ∧ f ≃ f' ∧ i' 8051~> f' ∧ |
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161 | time = clock f' - clock i'. |
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162 | |
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163 | |
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164 | ∀s,flat. |
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165 | let ge ≝ (globalenvs … labelled) in |
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166 | subtrace_of (exec_inf … RTLabs_fullexec labelled) flat → |
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167 | RTLabs_cost s = true → |
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168 | ∀WR : will_return ge 0 s flat. |
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169 | let structured_trace_rtlabs ≝ make_label_return' ge 0 s flat ??? WR in |
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170 | let labels_rtlabs ≝ flat_label_trace … flat WR in |
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171 | ∃!initial,final,structured_trace_asm. |
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172 | structured_trace_rtlabs ≈ structured_trace_asm ∧ |
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173 | clock … code_memory … final = clock … code_memory … initial + |
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174 | (Σ_{i < |labels_rtlabs|} (cost_map (match nth i labels_rtlabs with [ Some k ⇒ k | None ⇒ 0 ])). |
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175 | |
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176 | |
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177 | |
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178 | What is ≃l? Must show that "labelled" does everything that |
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179 | "input_program" does, without getting lost in some |
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180 | non-terminating loop part way. |
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181 | |
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182 | *) |
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183 | |
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