1 | |
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2 | include "ASM/ASMCosts.ma". |
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3 | include "ASM/WellLabeled.ma". |
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4 | include "ASM/Status.ma". |
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5 | include "common/StructuredTraces.ma". |
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6 | |
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7 | (* |
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8 | definition next_instruction_is_labelled ≝ |
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9 | λcost_labels: BitVectorTrie Byte 16. |
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10 | λs: Status. |
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11 | let pc ≝ program_counter … (execute_1 s) in |
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12 | match lookup_opt … pc cost_labels with |
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13 | [ None ⇒ False |
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14 | | _ ⇒ True |
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15 | ]. |
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16 | |
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17 | definition current_instruction_cost ≝ |
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18 | λs: Status. \snd (fetch (code_memory … s) (program_counter … s)). |
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19 | |
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20 | definition is_call_p ≝ |
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21 | λs. |
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22 | match s with |
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23 | [ ACALL _ ⇒ True |
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24 | | LCALL _ ⇒ True |
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25 | | JMP ⇒ True (* XXX: is function pointer call *) |
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26 | | _ ⇒ False |
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27 | ]. |
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28 | |
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29 | definition next_instruction ≝ |
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30 | λs: Status. |
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31 | let 〈instruction, ignore, ignore〉 ≝ fetch (code_memory … s) (program_counter … s) in |
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32 | instruction. |
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33 | |
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34 | inductive trace_ends_with_ret: Type[0] ≝ |
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35 | | ends_with_ret: trace_ends_with_ret |
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36 | | doesnt_end_with_ret: trace_ends_with_ret. |
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37 | |
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38 | definition next_instruction_is_labelled ≝ |
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39 | λcost_labels: BitVectorTrie Byte 16. |
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40 | λs: Status. |
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41 | let pc ≝ program_counter … (execute 1 s) in |
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42 | match lookup_opt … pc cost_labels with |
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43 | [ None ⇒ False |
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44 | | _ ⇒ True |
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45 | ]. |
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46 | |
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47 | definition next_instruction_properly_relates_program_counters ≝ |
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48 | λbefore: Status. |
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49 | λafter : Status. |
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50 | let size ≝ current_instruction_cost before in |
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51 | let pc_before ≝ program_counter … before in |
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52 | let pc_after ≝ program_counter … after in |
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53 | let sum ≝ \snd (half_add … pc_before (bitvector_of_nat … size)) in |
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54 | sum = pc_after. |
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55 | |
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56 | definition current_instruction ≝ |
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57 | λs: Status. |
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58 | let pc ≝ program_counter … s in |
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59 | \fst (\fst (fetch … (code_memory … s) pc)). |
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60 | |
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61 | definition current_instruction_is_labelled ≝ |
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62 | λcost_labels: BitVectorTrie Byte 16. |
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63 | λs: Status. |
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64 | let pc ≝ program_counter … s in |
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65 | match lookup_opt … pc cost_labels with |
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66 | [ None ⇒ False |
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67 | | _ ⇒ True |
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68 | ]. |
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69 | |
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70 | inductive trace_label_return (cost_labels: BitVectorTrie Byte 16): Status → Status → Type[0] ≝ |
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71 | | tlr_base: |
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72 | ∀status_before: Status. |
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73 | ∀status_after: Status. |
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74 | trace_label_label cost_labels ends_with_ret status_before status_after → |
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75 | trace_label_return cost_labels status_before status_after |
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76 | | tlr_step: |
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77 | ∀status_initial: Status. |
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78 | ∀status_labelled: Status. |
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79 | ∀status_final: Status. |
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80 | trace_label_label cost_labels doesnt_end_with_ret status_initial status_labelled → |
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81 | trace_label_return cost_labels status_labelled status_final → |
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82 | trace_label_return cost_labels status_initial status_final |
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83 | with trace_label_label: trace_ends_with_ret → Status → Status → Type[0] ≝ |
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84 | | tll_base: |
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85 | ∀ends_flag: trace_ends_with_ret. |
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86 | ∀start_status: Status. |
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87 | ∀end_status: Status. |
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88 | trace_label_label_pre cost_labels ends_flag start_status end_status → |
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89 | current_instruction_is_labelled cost_labels start_status → |
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90 | trace_label_label cost_labels ends_flag start_status end_status |
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91 | with trace_label_label_pre: trace_ends_with_ret → Status → Status → Type[0] ≝ |
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92 | | tllp_base_not_return: |
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93 | ∀start_status: Status. |
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94 | let final_status ≝ execute 1 start_status in |
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95 | current_instruction start_status ≠ (RealInstruction … (RET [[relative]])) → |
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96 | ¬ (is_jump_p (current_instruction start_status)) → |
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97 | current_instruction_is_labelled cost_labels final_status → |
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98 | trace_label_label_pre cost_labels doesnt_end_with_ret start_status final_status |
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99 | | tllp_base_return: |
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100 | ∀start_status: Status. |
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101 | let final_status ≝ execute 1 start_status in |
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102 | current_instruction start_status = (RealInstruction … (RET [[relative]])) → |
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103 | trace_label_label_pre cost_labels ends_with_ret start_status final_status |
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104 | | tllp_base_jump: |
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105 | ∀start_status: Status. |
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106 | let final_status ≝ execute 1 start_status in |
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107 | is_jump_p (current_instruction start_status) → |
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108 | current_instruction_is_labelled cost_labels final_status → |
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109 | trace_label_label_pre cost_labels doesnt_end_with_ret start_status final_status |
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110 | | tllp_step_call: |
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111 | ∀end_flag: trace_ends_with_ret. |
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112 | ∀status_pre_fun_call: Status. |
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113 | ∀status_after_fun_call: Status. |
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114 | ∀status_final: Status. |
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115 | let status_start_fun_call ≝ execute 1 status_pre_fun_call in |
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116 | is_call_p (current_instruction status_pre_fun_call) → |
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117 | next_instruction_properly_relates_program_counters status_pre_fun_call status_after_fun_call → |
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118 | trace_label_return cost_labels status_start_fun_call status_after_fun_call → |
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119 | trace_label_label_pre cost_labels end_flag status_after_fun_call status_final → |
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120 | trace_label_label_pre cost_labels end_flag status_pre_fun_call status_final |
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121 | | tllp_step_default: |
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122 | ∀end_flag: trace_ends_with_ret. |
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123 | ∀status_pre: Status. |
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124 | ∀status_end: Status. |
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125 | let status_init ≝ execute 1 status_pre in |
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126 | trace_label_label_pre cost_labels end_flag status_init status_end → |
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127 | ¬ (is_call_p (current_instruction status_pre)) → |
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128 | ¬ (is_jump_p (current_instruction status_pre)) → |
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129 | (current_instruction status_pre) ≠ (RealInstruction … (RET [[relative]])) → |
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130 | ¬ (current_instruction_is_labelled cost_labels status_init) → |
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131 | trace_label_label_pre cost_labels end_flag status_pre status_end. |
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132 | *) |
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133 | |
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134 | (* XXX: not us |
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135 | definition compute_simple_path0 (s:Status) (is_labelled s) (well_balanced s) (labelled_p p): trace_lab_ret ≝ …. |
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136 | |
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137 | lemma simple_path_ok: |
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138 | ∀st: Status.∀H. |
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139 | let p ≝ compute_simple_path0 st H in |
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140 | ∀n. |
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141 | nth_path n p = execute n st ∧ |
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142 | (execute' (path_length p) st = 〈st',τ〉 → |
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143 | τ = cost_trace p) |
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144 | ]. |
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145 | *) |
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146 | |
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147 | let rec compute_max_trace_label_label_cost |
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148 | (cost_labels: BitVectorTrie costlabel 16) |
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149 | (trace_ends_flag: trace_ends_with_ret) |
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150 | (start_status: Status) (final_status: Status) |
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151 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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152 | start_status final_status) on the_trace: nat ≝ |
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153 | match the_trace with |
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154 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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155 | compute_max_trace_any_label_cost … given_trace |
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156 | ] |
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157 | and compute_max_trace_any_label_cost |
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158 | (cost_labels: BitVectorTrie costlabel 16) |
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159 | (trace_ends_flag: trace_ends_with_ret) |
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160 | (start_status: Status) (final_status: Status) |
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161 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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162 | on the_trace: nat ≝ |
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163 | match the_trace with |
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164 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
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165 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
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166 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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167 | _ _ _ call_trace final_trace ⇒ |
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168 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
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169 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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170 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
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171 | call_trace_cost + current_instruction_cost + final_trace_cost |
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172 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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173 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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174 | let tail_trace_cost ≝ |
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175 | compute_max_trace_any_label_cost cost_labels end_flag |
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176 | status_init status_end tail_trace |
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177 | in |
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178 | current_instruction_cost + tail_trace_cost |
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179 | ] |
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180 | and compute_max_trace_label_return_cost |
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181 | (cost_labels: BitVectorTrie costlabel 16) |
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182 | (start_status: Status) (final_status: Status) |
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183 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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184 | on the_trace: nat ≝ |
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185 | match the_trace with |
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186 | [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift |
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187 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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188 | let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in |
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189 | let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in |
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190 | labelled_cost + return_cost |
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191 | ]. |
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192 | |
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193 | (*Useless now? |
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194 | (* To be moved *) |
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195 | lemma pred_minus_1: |
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196 | ∀m, n: nat. |
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197 | ∀proof: n < m. |
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198 | pred (m - n) = m - n - 1. |
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199 | #m #n |
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200 | cases m |
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201 | [ #proof |
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202 | cases(lt_to_not_zero … proof) |
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203 | | #m' #proof |
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204 | normalize in ⊢ (???%); |
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205 | cases n |
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206 | [ normalize // |
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207 | | #n' normalize |
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208 | cases(m' - n') |
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209 | [ % |
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210 | | #Sm_n' |
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211 | normalize // |
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212 | ] |
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213 | ] |
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214 | ] |
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215 | qed. |
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216 | |
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217 | lemma succ_m_plus_one: |
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218 | ∀m: nat. |
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219 | S m = m + 1. |
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220 | // |
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221 | qed.*) |
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222 | |
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223 | include alias "arithmetics/nat.ma". |
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224 | |
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225 | (* |
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226 | lemma minus_m_n_minus_minus_plus_1: |
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227 | ∀m, n: nat. |
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228 | ∀proof: n < m. |
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229 | m - n = (m - n - 1) + 1. |
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230 | /3 by lt_plus_to_minus_r, plus_minus/ |
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231 | qed. |
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232 | *) |
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233 | |
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234 | lemma plus_minus_rearrangement_1: |
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235 | ∀m, n, o: nat. |
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236 | ∀proof_n_m: n ≤ m. |
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237 | ∀proof_m_o: m ≤ o. |
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238 | (m - n) + (o - m) = o - n. |
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239 | #m #n #o #H1 #H2 |
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240 | lapply (minus_to_plus … H1 (refl …)) #K1 >K1 |
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241 | lapply (minus_to_plus … H2 (refl …)) #K2 >K2 |
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242 | /2 by plus_minus/ |
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243 | qed. |
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244 | |
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245 | lemma plus_minus_rearrangement_2: |
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246 | ∀m, n, o: nat. n ≤ m → o ≤ n → |
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247 | (m - n) + (n - o) = m - o. |
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248 | /2 by plus_minus_rearrangement_1/ |
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249 | qed. |
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250 | |
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251 | lemma m_le_plus_n_m: |
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252 | ∀m, n: nat. |
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253 | m ≤ n + m. |
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254 | #m #n // |
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255 | qed. |
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256 | |
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257 | lemma n_plus_m_le_o_to_m_le_o: |
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258 | ∀m, n, o: nat. |
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259 | n + m ≤ o → m ≤ o. |
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260 | #m #n #o #assm /2 by le_plus_b/ |
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261 | qed. |
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262 | |
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263 | lemma m_minus_n_plus_o_m_minus_n_minus_o: |
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264 | ∀m, n, o: nat. |
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265 | m - (n + o) = m - n - o. |
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266 | #m #n #o /2 by / |
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267 | qed. |
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268 | |
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269 | let rec compute_max_trace_label_label_cost_is_ok |
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270 | (cost_labels: BitVectorTrie costlabel 16) |
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271 | (trace_ends_flag: trace_ends_with_ret) |
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272 | (start_status: Status) (final_status: Status) |
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273 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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274 | start_status final_status) on the_trace: |
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275 | clock … final_status = (compute_max_trace_label_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ? |
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276 | and compute_max_trace_any_label_cost_is_ok |
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277 | (cost_labels: BitVectorTrie costlabel 16) |
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278 | (trace_ends_flag: trace_ends_with_ret) |
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279 | (start_status: Status) (final_status: Status) |
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280 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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281 | on the_trace: |
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282 | clock … final_status = (compute_max_trace_any_label_cost cost_labels trace_ends_flag start_status final_status the_trace) + (clock … start_status) ≝ ? |
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283 | and compute_max_trace_label_return_cost_is_ok |
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284 | (cost_labels: BitVectorTrie costlabel 16) |
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285 | (start_status: Status) (final_status: Status) |
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286 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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287 | on the_trace: |
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288 | clock … final_status = (compute_max_trace_label_return_cost cost_labels start_status final_status the_trace) + clock … start_status ≝ ?. |
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289 | [1: |
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290 | cases the_trace |
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291 | #ends_flag #start_status #end_status #any_label_trace #is_costed |
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292 | normalize @compute_max_trace_any_label_cost_is_ok |
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293 | |2: |
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294 | cases the_trace |
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295 | [1,2: |
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296 | #start_status #final_status #is_next #is_not_return try (#is_costed) |
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297 | change with (current_instruction_cost start_status) in ⊢ (???(?%?)); |
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298 | cases(is_next) |
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299 | cases(execute_1 start_status) |
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300 | whd in match eject; normalize nodelta; |
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301 | #the_status #assm >assm % |
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302 | |3: |
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303 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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304 | #status_final #is_next #is_call #is_after_return #call_trace #final_trace |
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305 | change with ( |
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306 | let current_instruction_cost ≝ current_instruction_cost status_pre_fun_call in |
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307 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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308 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
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309 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?)); |
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310 | normalize nodelta; |
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311 | >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
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312 | status_final final_trace) |
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313 | >(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
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314 | status_after_fun_call call_trace) |
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315 | cases(is_next) in match (clock … status_start_fun_call); |
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316 | cases(execute_1 status_pre_fun_call); |
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317 | #the_status |
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318 | whd in match eject; normalize nodelta; |
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319 | #assm >assm |
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320 | <associative_plus in ⊢ (??%?); |
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321 | <commutative_plus in match ( |
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322 | compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace |
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323 | + compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace); |
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324 | >associative_plus in ⊢ (??%?); |
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325 | <commutative_plus in match ( |
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326 | compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace |
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327 | + (current_instruction_cost status_pre_fun_call |
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328 | + clock (BitVectorTrie Byte 16) status_pre_fun_call)); |
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329 | >associative_plus in ⊢ (??%?); |
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330 | <commutative_plus in match ( |
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331 | clock (BitVectorTrie Byte 16) status_pre_fun_call |
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332 | + compute_max_trace_any_label_cost cost_labels end_flag status_after_fun_call status_final final_trace); |
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333 | <(associative_plus ( |
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334 | (compute_max_trace_label_return_cost cost_labels status_start_fun_call status_after_fun_call call_trace))) |
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335 | <associative_plus in ⊢ (??%?); % |
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336 | |4: |
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337 | #end_flag #status_pre #status_init #status_end #is_next |
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338 | #trace_any_label #is_other #is_not_costed |
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339 | change with ( |
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340 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
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341 | let tail_trace_cost ≝ |
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342 | compute_max_trace_any_label_cost cost_labels end_flag |
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343 | status_init status_end trace_any_label |
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344 | in |
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345 | current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?)); |
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346 | normalize nodelta; |
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347 | >(compute_max_trace_any_label_cost_is_ok cost_labels end_flag |
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348 | status_init status_end trace_any_label) |
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349 | cases(is_next) in match (clock … status_init); |
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350 | cases(execute_1 status_pre) |
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351 | #the_status whd in match eject; normalize nodelta; |
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352 | #assm >assm <associative_plus in ⊢ (??%?); |
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353 | >commutative_plus in match ( |
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354 | compute_max_trace_any_label_cost cost_labels end_flag status_init status_end trace_any_label |
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355 | + current_instruction_cost status_pre); |
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356 | % |
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357 | ] |
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358 | |3: |
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359 | cases the_trace |
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360 | [1: |
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361 | #status_before #status_after #trace_to_lift |
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362 | normalize @compute_max_trace_label_label_cost_is_ok |
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363 | |2: |
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364 | #status_initial #status_labelled #status_final #labelled_trace #ret_trace |
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365 | normalize |
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366 | >(compute_max_trace_label_return_cost_is_ok cost_labels status_labelled status_final ret_trace); |
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367 | >(compute_max_trace_label_label_cost_is_ok cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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368 | <associative_plus in ⊢ (??%?); |
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369 | >commutative_plus in match ( |
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370 | compute_max_trace_label_return_cost cost_labels status_labelled status_final ret_trace |
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371 | + compute_max_trace_label_label_cost cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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372 | % |
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373 | ] |
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374 | ]. |
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375 | qed. |
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376 | |
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377 | let rec compute_max_trace_label_label_cost_is_ok |
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378 | (cost_labels: BitVectorTrie costlabel 16) |
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379 | (trace_ends_flag: trace_ends_with_ret) |
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380 | (start_status: Status) (final_status: Status) |
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381 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
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382 | start_status final_status) on the_trace: |
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383 | And (compute_max_trace_label_label_cost cost_labels trace_ends_flag start_status final_status the_trace = |
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384 | (clock … final_status) - (clock … start_status)) (clock … start_status ≤ clock … final_status) ≝ ? |
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385 | and compute_max_trace_any_label_cost_is_ok |
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386 | (cost_labels: BitVectorTrie costlabel 16) |
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387 | (trace_ends_flag: trace_ends_with_ret) |
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388 | (start_status: Status) (final_status: Status) |
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389 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
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390 | on the_trace: |
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391 | (compute_max_trace_any_label_cost cost_labels trace_ends_flag start_status final_status the_trace = |
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392 | (clock … final_status) - (clock … start_status)) ∧ (clock … start_status ≤ clock … final_status) ≝ ? |
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393 | and compute_max_trace_label_return_cost_is_ok |
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394 | (cost_labels: BitVectorTrie costlabel 16) |
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395 | (start_status: Status) (final_status: Status) |
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396 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
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397 | on the_trace: |
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398 | (compute_max_trace_label_return_cost cost_labels start_status final_status the_trace = |
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399 | (clock … final_status) - (clock … start_status)) ∧ (clock … start_status ≤ clock … final_status) ≝ ?. |
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400 | [1: |
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401 | cases the_trace |
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402 | #ends_flag #start_status #end_status #any_label_trace #is_costed |
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403 | normalize @compute_max_trace_any_label_cost_is_ok |
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404 | |2: |
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405 | cases the_trace |
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406 | [1: |
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407 | #start_status #final_status #is_next #is_not_return #is_costed |
---|
408 | change with (current_instruction_cost start_status) in ⊢ (?(??%?)?); |
---|
409 | cases(is_next) |
---|
410 | @conj |
---|
411 | cases(execute_1 start_status) |
---|
412 | #the_status #assm |
---|
413 | whd in match eject; normalize nodelta |
---|
414 | >assm |
---|
415 | [1: |
---|
416 | @minus_plus_m_m |
---|
417 | |2: |
---|
418 | @m_le_plus_n_m |
---|
419 | ] |
---|
420 | |2: |
---|
421 | #start_status #final_status #is_next #is_return |
---|
422 | change with (current_instruction_cost start_status) in ⊢ (?(??%?)?); |
---|
423 | cases(is_next) |
---|
424 | @conj |
---|
425 | cases(execute_1 start_status) |
---|
426 | #the_status #assm |
---|
427 | whd in match eject; normalize nodelta |
---|
428 | >assm |
---|
429 | [1: |
---|
430 | @minus_plus_m_m |
---|
431 | |2: |
---|
432 | @m_le_plus_n_m |
---|
433 | ] |
---|
434 | |3: |
---|
435 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
---|
436 | #status_final #is_next #is_call #is_after_return #call_trace #final_trace |
---|
437 | change with ( |
---|
438 | let current_instruction_cost ≝ current_instruction_cost status_pre_fun_call in |
---|
439 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
---|
440 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
---|
441 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (?(??%?)?); |
---|
442 | normalize nodelta; |
---|
443 | @conj |
---|
444 | [1: |
---|
445 | cases(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
---|
446 | status_after_fun_call call_trace) |
---|
447 | #trace_label_return_eq #trace_label_return_le |
---|
448 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
---|
449 | status_final final_trace) |
---|
450 | #trace_any_label_eq #trace_any_label_le |
---|
451 | >trace_any_label_eq >trace_label_return_eq |
---|
452 | cases(is_next) |
---|
453 | cases(execute_1 status_pre_fun_call) |
---|
454 | #the_status #assm |
---|
455 | whd in match eject; normalize nodelta; |
---|
456 | >assm >commutative_plus in match |
---|
457 | (current_instruction_cost status_pre_fun_call |
---|
458 | + clock (BitVectorTrie Byte 16) status_pre_fun_call); |
---|
459 | >m_minus_n_plus_o_m_minus_n_minus_o |
---|
460 | <(plus_minus_m_m |
---|
461 | (clock (BitVectorTrie Byte 16) status_after_fun_call |
---|
462 | - clock (BitVectorTrie Byte 16) status_pre_fun_call) |
---|
463 | (current_instruction_cost status_pre_fun_call)) |
---|
464 | [1: |
---|
465 | @plus_minus_rearrangement_1 |
---|
466 | lapply(transitive_le |
---|
467 | (clock … status_pre_fun_call) |
---|
468 | (clock … status_start_fun_call) |
---|
469 | (clock … status_after_fun_call) ? trace_label_return_le) |
---|
470 | [1,3: |
---|
471 | cases(is_next) |
---|
472 | cases(execute_1 status_pre_fun_call) |
---|
473 | whd in match eject; normalize nodelta; |
---|
474 | #the_new_status #assm >assm @m_le_plus_n_m |
---|
475 | |2,4: |
---|
476 | #assm assumption |
---|
477 | ] |
---|
478 | |2: |
---|
479 | cases(is_next) |
---|
480 | assumption |
---|
481 | |2: |
---|
482 | cases(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
---|
483 | status_after_fun_call call_trace) |
---|
484 | #trace_label_return_eq |
---|
485 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
---|
486 | status_final final_trace) |
---|
487 | #trace_any_label_eq cases(is_next) |
---|
488 | cases(execute_1 status_pre_fun_call) |
---|
489 | #the_status #assm |
---|
490 | whd in match eject; normalize nodelta; |
---|
491 | >assm #after_final_le #pre_after_le |
---|
492 | lapply(n_plus_m_le_o_to_m_le_o |
---|
493 | (clock … status_pre_fun_call) |
---|
494 | (current_instruction_cost status_pre_fun_call) |
---|
495 | (clock … status_after_fun_call) pre_after_le); |
---|
496 | #pre_after_le' |
---|
497 | @(transitive_le |
---|
498 | (clock … status_pre_fun_call) |
---|
499 | (clock … status_after_fun_call) |
---|
500 | (clock … status_final)) |
---|
501 | assumption |
---|
502 | ] |
---|
503 | |*: cases daemon (* XXX: for now |
---|
504 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
---|
505 | #status_final #is_next #is_call #is_after_return #call_trace #final_trace |
---|
506 | change with ( |
---|
507 | let current_instruction_cost ≝ current_instruction_cost status_pre_fun_call in |
---|
508 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
---|
509 | let final_trace_cost ≝ compute_max_trace_any_label_cost cost_labels end_flag … final_trace in |
---|
510 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (?(??%?)?); |
---|
511 | normalize nodelta; |
---|
512 | @conj |
---|
513 | [1: |
---|
514 | cases(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
---|
515 | status_after_fun_call call_trace) |
---|
516 | #trace_label_return_eq #trace_label_return_le |
---|
517 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
---|
518 | status_final final_trace) |
---|
519 | #trace_any_label_eq #trace_any_label_le |
---|
520 | >trace_label_return_eq >trace_any_label_eq |
---|
521 | cases(is_next) |
---|
522 | >(clock_execute_1 status_pre_fun_call) in match (clock … (execute_1 status_pre_fun_call)); |
---|
523 | >commutative_plus in match ( |
---|
524 | (current_instruction_cost status_pre_fun_call |
---|
525 | + clock (BitVectorTrie Byte 16) status_pre_fun_call)); |
---|
526 | <minus_plus |
---|
527 | <(plus_minus_m_m (clock (BitVectorTrie Byte 16) status_after_fun_call |
---|
528 | -clock (BitVectorTrie Byte 16) status_pre_fun_call) (current_instruction_cost status_pre_fun_call)) |
---|
529 | [1: |
---|
530 | >plus_minus_rearrangement_1 |
---|
531 | [1: |
---|
532 | % |
---|
533 | |2: |
---|
534 | assumption |
---|
535 | |3: |
---|
536 | generalize in match trace_label_return_le; |
---|
537 | cases(is_next) |
---|
538 | >(clock_execute_1 status_pre_fun_call) |
---|
539 | @plus_le_le |
---|
540 | ] |
---|
541 | |2: |
---|
542 | cases daemon (* XXX: todo *) |
---|
543 | ] |
---|
544 | |2: |
---|
545 | cases(compute_max_trace_label_return_cost_is_ok cost_labels status_start_fun_call |
---|
546 | status_after_fun_call call_trace) |
---|
547 | #trace_label_return_eq cases(is_next) |
---|
548 | >clock_execute_1 |
---|
549 | #trace_label_return_le |
---|
550 | lapply (plus_le_le … trace_label_return_le) |
---|
551 | #trace_label_return_le' |
---|
552 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag status_after_fun_call |
---|
553 | status_final final_trace) |
---|
554 | #trace_any_label_eq #trace_any_label_le |
---|
555 | @(transitive_le (clock … status_pre_fun_call) (clock … status_after_fun_call) (clock … status_final)) |
---|
556 | assumption |
---|
557 | ] |
---|
558 | |4: |
---|
559 | #end_flag #status_pre #status_init #status_end #is_next |
---|
560 | #trace_any_label #is_other #is_not_costed |
---|
561 | change with ( |
---|
562 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
---|
563 | let tail_trace_cost ≝ |
---|
564 | compute_max_trace_any_label_cost cost_labels end_flag |
---|
565 | status_init status_end trace_any_label |
---|
566 | in |
---|
567 | current_instruction_cost + tail_trace_cost) in ⊢ (?(??%?)?); |
---|
568 | normalize nodelta; |
---|
569 | @conj |
---|
570 | [1: |
---|
571 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag |
---|
572 | status_init status_end trace_any_label) |
---|
573 | #trace_any_label_eq #trace_any_label_le |
---|
574 | >trace_any_label_eq |
---|
575 | cases(is_next) |
---|
576 | >commutative_plus |
---|
577 | >clock_execute_1 |
---|
578 | >commutative_plus in match ((current_instruction_cost status_pre+clock (BitVectorTrie Byte 16) status_pre)); |
---|
579 | <minus_plus |
---|
580 | <plus_minus_m_m |
---|
581 | [1: % |
---|
582 | |2: cases daemon (* XXX: todo *) |
---|
583 | ] |
---|
584 | |2: |
---|
585 | cases(compute_max_trace_any_label_cost_is_ok cost_labels end_flag |
---|
586 | status_init status_end trace_any_label) |
---|
587 | #trace_any_label_eq |
---|
588 | cases(is_next) |
---|
589 | >clock_execute_1 |
---|
590 | @plus_le_le |
---|
591 | ]*) |
---|
592 | ] |
---|
593 | |3: |
---|
594 | cases the_trace |
---|
595 | [1: |
---|
596 | #status_before #status_after #trace_to_lift |
---|
597 | normalize @compute_max_trace_label_label_cost_is_ok |
---|
598 | |2: |
---|
599 | #status_initial #status_labelled #status_final #labelled_trace #ret_trace |
---|
600 | normalize |
---|
601 | cases(compute_max_trace_label_label_cost_is_ok cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
---|
602 | #label_label_trace_equality #label_label_trace_leq |
---|
603 | cases(compute_max_trace_label_return_cost_is_ok cost_labels status_labelled status_final ret_trace); |
---|
604 | #label_return_trace_equality #label_return_trace_leq |
---|
605 | >label_label_trace_equality >label_return_trace_equality |
---|
606 | @conj |
---|
607 | [2: @(transitive_le (clock (BitVectorTrie (Vector bool 8) 16) status_initial) |
---|
608 | (clock (BitVectorTrie Byte 16) status_labelled) |
---|
609 | (clock (BitVectorTrie (Vector bool 8) 16) status_final)) |
---|
610 | |1: @plus_minus_rearrangement_1 |
---|
611 | ] |
---|
612 | assumption |
---|
613 | ] |
---|
614 | ] |
---|
615 | qed. |
---|
616 | |
---|
617 | (* XXX: work below here: *) |
---|
618 | |
---|
619 | let rec compute_paid_trace_any_label |
---|
620 | (cost_labels: BitVectorTrie costlabel 16) |
---|
621 | (trace_ends_flag: trace_ends_with_ret) |
---|
622 | (start_status: Status) (final_status: Status) |
---|
623 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
624 | on the_trace: nat ≝ |
---|
625 | match the_trace with |
---|
626 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
---|
627 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
---|
628 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
---|
629 | _ _ _ call_trace final_trace ⇒ |
---|
630 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
---|
631 | let final_trace_cost ≝ compute_paid_trace_any_label cost_labels end_flag … final_trace in |
---|
632 | current_instruction_cost + final_trace_cost |
---|
633 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
---|
634 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
---|
635 | let tail_trace_cost ≝ |
---|
636 | compute_paid_trace_any_label cost_labels end_flag |
---|
637 | status_init status_end tail_trace |
---|
638 | in |
---|
639 | current_instruction_cost + tail_trace_cost |
---|
640 | ]. |
---|
641 | |
---|
642 | definition compute_paid_trace_label_label ≝ |
---|
643 | λcost_labels: BitVectorTrie costlabel 16. |
---|
644 | λtrace_ends_flag: trace_ends_with_ret. |
---|
645 | λstart_status: Status. |
---|
646 | λfinal_status: Status. |
---|
647 | λthe_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
648 | start_status final_status. |
---|
649 | match the_trace with |
---|
650 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
---|
651 | compute_paid_trace_any_label … given_trace |
---|
652 | ]. |
---|
653 | |
---|
654 | let rec compute_trace_label_label_cost_using_paid |
---|
655 | (cost_labels: BitVectorTrie costlabel 16) |
---|
656 | (trace_ends_flag: trace_ends_with_ret) |
---|
657 | (start_status: Status) (final_status: Status) |
---|
658 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
659 | start_status final_status) on the_trace: nat ≝ |
---|
660 | match the_trace with |
---|
661 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
---|
662 | compute_paid_trace_label_label cost_labels … the_trace + |
---|
663 | compute_trace_any_label_cost_using_paid … given_trace |
---|
664 | ] |
---|
665 | and compute_trace_any_label_cost_using_paid |
---|
666 | (cost_labels: BitVectorTrie costlabel 16) |
---|
667 | (trace_ends_flag: trace_ends_with_ret) |
---|
668 | (start_status: Status) (final_status: Status) |
---|
669 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
670 | on the_trace: nat ≝ |
---|
671 | match the_trace with |
---|
672 | [ tal_base_not_return the_status _ _ _ _ ⇒ 0 |
---|
673 | | tal_base_return the_status _ _ _ ⇒ 0 |
---|
674 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
---|
675 | _ _ _ call_trace final_trace ⇒ |
---|
676 | let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in |
---|
677 | let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cost_labels end_flag … final_trace in |
---|
678 | call_trace_cost + final_trace_cost |
---|
679 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
---|
680 | compute_trace_any_label_cost_using_paid cost_labels end_flag |
---|
681 | status_init status_end tail_trace |
---|
682 | ] |
---|
683 | and compute_trace_label_return_cost_using_paid |
---|
684 | (cost_labels: BitVectorTrie costlabel 16) |
---|
685 | (start_status: Status) (final_status: Status) |
---|
686 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
687 | on the_trace: nat ≝ |
---|
688 | match the_trace with |
---|
689 | [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift |
---|
690 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
---|
691 | let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in |
---|
692 | let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in |
---|
693 | labelled_cost + return_cost |
---|
694 | ]. |
---|
695 | |
---|
696 | let rec compute_trace_label_label_cost_using_paid_ok |
---|
697 | (cost_labels: BitVectorTrie costlabel 16) |
---|
698 | (trace_ends_flag: trace_ends_with_ret) |
---|
699 | (start_status: Status) (final_status: Status) |
---|
700 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
701 | start_status final_status) on the_trace: |
---|
702 | compute_trace_label_label_cost_using_paid cost_labels … the_trace = |
---|
703 | compute_max_trace_label_label_cost … the_trace ≝ ? |
---|
704 | and compute_trace_any_label_cost_using_paid_ok |
---|
705 | (cost_labels: BitVectorTrie costlabel 16) |
---|
706 | (trace_ends_flag: trace_ends_with_ret) |
---|
707 | (start_status: Status) (final_status: Status) |
---|
708 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) |
---|
709 | trace_ends_flag start_status final_status) on the_trace: |
---|
710 | compute_paid_trace_any_label cost_labels trace_ends_flag … the_trace |
---|
711 | +compute_trace_any_label_cost_using_paid cost_labels trace_ends_flag … the_trace |
---|
712 | =compute_max_trace_any_label_cost cost_labels trace_ends_flag … the_trace ≝ ? |
---|
713 | and compute_trace_label_return_cost_using_paid_ok |
---|
714 | (cost_labels: BitVectorTrie costlabel 16) |
---|
715 | (start_status: Status) (final_status: Status) |
---|
716 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) |
---|
717 | start_status final_status) on the_trace: |
---|
718 | compute_trace_label_return_cost_using_paid cost_labels … the_trace = |
---|
719 | compute_max_trace_label_return_cost cost_labels … the_trace ≝ ?. |
---|
720 | [ cases the_trace #endsf #ss #es #tr #H normalize |
---|
721 | @compute_trace_any_label_cost_using_paid_ok |
---|
722 | | cases the_trace |
---|
723 | [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O % |
---|
724 | | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O % |
---|
725 | | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #tr2 whd in ⊢ (??(?%%)%); |
---|
726 | <compute_trace_any_label_cost_using_paid_ok |
---|
727 | <compute_trace_label_return_cost_using_paid_ok |
---|
728 | -compute_trace_label_label_cost_using_paid_ok |
---|
729 | -compute_trace_label_return_cost_using_paid_ok |
---|
730 | -compute_trace_any_label_cost_using_paid_ok |
---|
731 | >commutative_plus in ⊢ (???(?%?)); |
---|
732 | >commutative_plus in ⊢ (??(??%)?); |
---|
733 | >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try % |
---|
734 | <associative_plus <commutative_plus % |
---|
735 | | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2 |
---|
736 | [ % | @compute_trace_any_label_cost_using_paid_ok ] |
---|
737 | ] |
---|
738 | | cases the_trace |
---|
739 | [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok |
---|
740 | | #si #sl #sf #tr1 #tr2 normalize @eq_f2 |
---|
741 | [ @compute_trace_label_label_cost_using_paid_ok |
---|
742 | | @compute_trace_label_return_cost_using_paid_ok ]]] |
---|
743 | qed. |
---|
744 | |
---|
745 | (* |
---|
746 | let rec compute_paid_trace_label_return |
---|
747 | (cost_labels: BitVectorTrie costlabel 16) |
---|
748 | (start_status: Status) (final_status: Status) |
---|
749 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
750 | on the_trace: nat ≝ |
---|
751 | match the_trace with |
---|
752 | [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label … trace_to_lift |
---|
753 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
---|
754 | let labelled_cost ≝ compute_paid_trace_label_label … labelled_trace in |
---|
755 | let return_cost ≝ compute_paid_trace_label_return … ret_trace in |
---|
756 | labelled_cost + return_cost |
---|
757 | ]. |
---|
758 | *) |
---|
759 | |
---|
760 | let rec compute_cost_trace_label_label |
---|
761 | (cost_labels: BitVectorTrie costlabel 16) |
---|
762 | (trace_ends_flag: trace_ends_with_ret) |
---|
763 | (start_status: Status) (final_status: Status) |
---|
764 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
765 | start_status final_status) on the_trace: |
---|
766 | list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
---|
767 | match the_trace with |
---|
768 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
---|
769 | let pc ≝ program_counter … initial in |
---|
770 | let label ≝ |
---|
771 | match lookup_opt … pc cost_labels return λx. match x with [None ⇒ False | Some _ ⇒ True] → costlabel with |
---|
772 | [ None ⇒ λabs. ⊥ |
---|
773 | | Some l ⇒ λ_. l ] labelled_proof in |
---|
774 | (dp … label ?)::compute_cost_trace_any_label … given_trace |
---|
775 | ] |
---|
776 | and compute_cost_trace_any_label |
---|
777 | (cost_labels: BitVectorTrie costlabel 16) |
---|
778 | (trace_ends_flag: trace_ends_with_ret) |
---|
779 | (start_status: Status) (final_status: Status) |
---|
780 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
781 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
---|
782 | match the_trace with |
---|
783 | [ tal_base_not_return the_status _ _ _ _ ⇒ [] |
---|
784 | | tal_base_return the_status _ _ _ ⇒ [] |
---|
785 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
---|
786 | _ _ _ call_trace final_trace ⇒ |
---|
787 | let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in |
---|
788 | let final_cost_trace ≝ compute_cost_trace_any_label cost_labels end_flag … final_trace in |
---|
789 | call_cost_trace @ final_cost_trace |
---|
790 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
---|
791 | compute_cost_trace_any_label cost_labels end_flag |
---|
792 | status_init status_end tail_trace |
---|
793 | ] |
---|
794 | and compute_cost_trace_label_return |
---|
795 | (cost_labels: BitVectorTrie costlabel 16) |
---|
796 | (start_status: Status) (final_status: Status) |
---|
797 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
798 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
---|
799 | match the_trace with |
---|
800 | [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label … trace_to_lift |
---|
801 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
---|
802 | let labelled_cost ≝ compute_cost_trace_label_label … labelled_trace in |
---|
803 | let return_cost ≝ compute_cost_trace_label_return … ret_trace in |
---|
804 | labelled_cost @ return_cost |
---|
805 | ]. |
---|
806 | [ %{pc} whd in match label; generalize in match labelled_proof; whd in ⊢ (% → ?); |
---|
807 | cases (lookup_opt costlabel … pc cost_labels) normalize |
---|
808 | [ #abs cases abs | // ] |
---|
809 | | // ] |
---|
810 | qed. |
---|
811 | |
---|
812 | (* ??????????????????????? *) |
---|
813 | axiom block_cost_static_dynamic_ok: |
---|
814 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
815 | ∀trace_ends_flag. |
---|
816 | ∀start_status: Status. |
---|
817 | ∀final_status: Status. |
---|
818 | ∀the_trace: |
---|
819 | trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
820 | start_status final_status. |
---|
821 | let mem ≝ code_memory … start_status in |
---|
822 | let pc ≝ program_counter … start_status in |
---|
823 | let program_size ≝ 2^16 in |
---|
824 | block_cost mem cost_labels pc program_size = |
---|
825 | compute_paid_trace_label_label cost_labels trace_ends_flag |
---|
826 | start_status final_status the_trace. |
---|
827 | |
---|
828 | (* |
---|
829 | (* To be moved elsewhere*) |
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830 | lemma le_S_n_m_to_le_n_m: ∀n,m. S n ≤ m → n ≤ m. |
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831 | #n #m #H change with (pred (S n) ≤ m) @(transitive_le ? (S n)) // |
---|
832 | qed. |
---|
833 | *) |
---|
834 | |
---|
835 | include "arithmetics/bigops.ma". |
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836 | |
---|
837 | (* This shoudl go in bigops! *) |
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838 | theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B. |
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839 | \big[op,nil]_{i<k1+k2|p i} (f i) = |
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840 | op \big[op,nil]_{i<k2|p (i+k1)} (f (i+k1)) \big[op,nil]_{i<k1|p i} (f i). |
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841 | #k1 #k2 #p #B #nil #op #f >bigop_sum >commutative_plus @same_bigop #i @leb_elim normalize |
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842 | [2,4: // |
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843 | | #H1 #H2 <plus_minus_m_m // |
---|
844 | | #H1 #H2 #H3 <plus_minus_m_m //] |
---|
845 | qed. |
---|
846 | |
---|
847 | (* This is taken by sigma_pi.ma that does not compile now *) |
---|
848 | definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
849 | (λa,b,c.sym_eq ??? (associative_plus a b c)). |
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850 | |
---|
851 | definition natACop ≝ mk_ACop nat 0 natAop commutative_plus. |
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852 | |
---|
853 | definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n))) |
---|
854 | distributive_times_plus. |
---|
855 | |
---|
856 | unification hint 0 ≔ ; |
---|
857 | S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
858 | (λa,b,c.sym_eq ??? (associative_plus a b c)) |
---|
859 | (* ---------------------------------------- *) ⊢ |
---|
860 | plus ≡ op ? ? S. |
---|
861 | |
---|
862 | unification hint 0 ≔ ; |
---|
863 | S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
---|
864 | (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus |
---|
865 | (* ---------------------------------------- *) ⊢ |
---|
866 | plus ≡ op ? ? S. |
---|
867 | |
---|
868 | unification hint 0 ≔ ; |
---|
869 | S ≟ natDop |
---|
870 | (* ---------------------------------------- *) ⊢ |
---|
871 | plus ≡ sum ? ? S. |
---|
872 | |
---|
873 | unification hint 0 ≔ ; |
---|
874 | S ≟ natDop |
---|
875 | (* ---------------------------------------- *) ⊢ |
---|
876 | times ≡ prod ? ? S. |
---|
877 | |
---|
878 | notation "Σ_{ ident i < n } f" |
---|
879 | with precedence 20 |
---|
880 | for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}. |
---|
881 | |
---|
882 | axiom code_memory_ro_label_label: |
---|
883 | ∀cost_labels. ∀ends_flag. |
---|
884 | ∀initial,final. trace_label_label (ASM_abstract_status cost_labels) ends_flag initial final → |
---|
885 | code_memory … initial = code_memory … final. |
---|
886 | |
---|
887 | (* |
---|
888 | axiom code_memory_ro_label_return: |
---|
889 | ∀cost_labels. |
---|
890 | ∀initial,final. trace_label_return (ASM_abstract_status cost_labels) initial final → |
---|
891 | code_memory … initial = code_memory … final. |
---|
892 | *) |
---|
893 | |
---|
894 | definition tech_cost_of_label0: |
---|
895 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
896 | ∀cost_map: identifier_map CostTag nat. |
---|
897 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
---|
898 | ∀ctrace:list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k). |
---|
899 | ∀i,p. present … cost_map (nth_safe ? i ctrace p). |
---|
900 | #cost_labels #cost_map #codom_dom #ctrace #i #p |
---|
901 | cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K |
---|
902 | lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres) |
---|
903 | % #abs destruct (abs) |
---|
904 | qed. |
---|
905 | |
---|
906 | lemma ltb_rect: |
---|
907 | ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P. |
---|
908 | #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2 |
---|
909 | [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ] |
---|
910 | qed. |
---|
911 | |
---|
912 | lemma same_ltb_rect: |
---|
913 | ∀P,n,m,H1,H2,n',m',H1',H2'. |
---|
914 | ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) → |
---|
915 | ltb_rect P n m H1 H2 = |
---|
916 | ltb_rect P n' m' H1' H2'. |
---|
917 | #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?); |
---|
918 | cut (∀xxx,yyy,xxx',yyy'. |
---|
919 | match ltb n m |
---|
920 | return λx:bool. |
---|
921 | eq bool (ltb n m) x |
---|
922 | → (lt n m → P) → (Not (lt n m) → P) → P |
---|
923 | with |
---|
924 | [ true ⇒ |
---|
925 | λE0:eq bool (ltb n m) true. |
---|
926 | λH10:lt n m → P. |
---|
927 | λH20:Not (lt n m) → P. H10 (xxx E0) |
---|
928 | | false ⇒ |
---|
929 | λE0:eq bool (ltb n m) false. |
---|
930 | λH10:lt n m → P. |
---|
931 | λH20:Not (lt n m) → P. H20 (yyy E0)] |
---|
932 | (refl … (ltb n m)) H1 H2 = |
---|
933 | match ltb n' m' |
---|
934 | return λx:bool. |
---|
935 | eq bool (ltb n' m') x |
---|
936 | → (lt n' m' → P) → (Not (lt n' m') → P) → P |
---|
937 | with |
---|
938 | [ true ⇒ |
---|
939 | λE0:eq bool (ltb n' m') true. |
---|
940 | λH10:lt n' m' → P. |
---|
941 | λH20:Not (lt n' m') → P. H10 (xxx' E0) |
---|
942 | | false ⇒ |
---|
943 | λE0:eq bool (ltb n' m') false. |
---|
944 | λH10:lt n' m' → P. |
---|
945 | λH20:Not (lt n' m') → P. H20 (yyy' E0)] |
---|
946 | (refl … (ltb n' m')) H1' H2' |
---|
947 | ) [2: #X @X] |
---|
948 | >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize |
---|
949 | [ @K1 | @K2 ] |
---|
950 | qed. |
---|
951 | |
---|
952 | |
---|
953 | definition tech_cost_of_label: |
---|
954 | ∀cost_labels: BitVectorTrie costlabel 16. |
---|
955 | ∀cost_map: identifier_map CostTag nat. |
---|
956 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
---|
957 | list (Σk:costlabel.∃pc. lookup_opt costlabel 16 pc cost_labels = Some ? k) → |
---|
958 | nat → nat |
---|
959 | ≝ λcost_labels,cost_map,codom_dom,ctrace,i. |
---|
960 | ltb_rect ? i (|ctrace|) |
---|
961 | (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?) |
---|
962 | (λ_.0). |
---|
963 | @tech_cost_of_label0 @codom_dom |
---|
964 | qed. |
---|
965 | |
---|
966 | lemma shift_nth_safe: |
---|
967 | ∀T,i,l2,l1,K1,K2. |
---|
968 | nth_safe T i l1 K1 = nth_safe T (i+|l2|) (l2@l1) K2. |
---|
969 | #T #i #l2 elim l2 normalize |
---|
970 | [ #l1 #K1 <plus_n_O #K2 % |
---|
971 | | #hd #tl #IH #l1 #K1 <plus_n_Sm #K2 change with (? < ?) in K1; change with (? < ?) in K2; |
---|
972 | whd in ⊢ (???%); @IH ] |
---|
973 | qed. |
---|
974 | |
---|
975 | lemma tech_cost_of_label_shift: |
---|
976 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
---|
977 | i < |l1| → |
---|
978 | tech_cost_of_label cost_labels cost_map codom_dom l2 i = |
---|
979 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|). |
---|
980 | #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H |
---|
981 | cut (ltb i (|l2|) = ltb (i+|l1|) (|l1@l2|)) |
---|
982 | [ change with (leb ?? = leb ??); @leb_elim |
---|
983 | [ #H1 change with (i < ?) in H1; >le_to_leb_true [//] change with (? < ?) |
---|
984 | >append_length >commutative_plus @monotonic_lt_plus_r // |
---|
985 | | #H1 change with (¬ i < ?) in H1; >not_le_to_leb_false [//] |
---|
986 | >append_length >commutative_plus % #H2 |
---|
987 | change with (? < ?) in H2; @(absurd ?? H1) @(lt_plus_to_lt_r … H2) ] |
---|
988 | | #E whd in match tech_cost_of_label; normalize nodelta |
---|
989 | @same_ltb_rect [@E] |
---|
990 | [ #K1 #K2 lapply (shift_nth_safe ? i l1 l2 K1 K2) #H check eq_f |
---|
991 | lapply (eq_f ?? (lookup_present CostTag nat cost_Map) ?? H) |
---|
992 | |
---|
993 | |
---|
994 | |
---|
995 | cut (decide_bool (ltb i (|l2|)) = decide_bool (ltb (i+(|l1|)) (|l1@l2|))) |
---|
996 | whd in match tech_cost_of_label; normalize nodelta |
---|
997 | |
---|
998 | |
---|
999 | let rec compute_max_trace_label_return_cost_ok_with_trace |
---|
1000 | (cost_labels: BitVectorTrie costlabel 16) |
---|
1001 | (cost_map: identifier_map CostTag nat) |
---|
1002 | (initial: Status) (final: Status) |
---|
1003 | (trace: trace_label_return (ASM_abstract_status cost_labels) initial final) |
---|
1004 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
---|
1005 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∃pc. lookup_opt … pc cost_labels = Some … k → |
---|
1006 | block_cost (code_memory … initial) cost_labels pc 2^16 = lookup_present … k_pres)) |
---|
1007 | on trace: |
---|
1008 | let ctrace ≝ compute_cost_trace_label_return … trace in |
---|
1009 | compute_max_trace_label_return_cost … trace = |
---|
1010 | Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i) |
---|
1011 | ≝ ?. |
---|
1012 | lapply (refl … initial) cases trace in ⊢ (??%? → %); |
---|
1013 | [ #sb #sa #tr #_ whd in ⊢ (let ctrace ≝ % in ??%?); |
---|
1014 | | #si #sl #sf #tr1 #tr2 #E whd in ⊢ (let ctrace ≝ % in ??%?); |
---|
1015 | change with (let ctrace ≝ ? in ? = bigop (|? @ ?|) ?????) |
---|
1016 | >append_length in ⊢ (let ctrace ≝ ? in ???(?%?????)); |
---|
1017 | change with (?=?) >bigop_sum_rev >commutative_plus @eq_f2 |
---|
1018 | [ >compute_max_trace_label_return_cost_ok_with_trace |
---|
1019 | -compute_max_trace_label_return_cost_ok_with_trace |
---|
1020 | [3: @cost_map |
---|
1021 | |2: @codom_dom |
---|
1022 | |4:lapply (code_memory_ro_label_label … tr1) >E #E2 <E2 @dom_codom] |
---|
1023 | @same_bigop [//] #i #H #_ -dom_codom |
---|
1024 | generalize in match (compute_cost_trace_label_return cost_labels sl sf tr2); #l1 |
---|
1025 | generalize in match (compute_cost_trace_label_label cost_labels doesnt_end_with_ret si sl tr1); |
---|
1026 | |
---|
1027 | i < |l1| → |
---|
1028 | tech_cost_of_label cost_labels cost_map codom_dom l1 i = |
---|
1029 | tech_cost_of_label cost_labels cost_map codom_dom (l2@l1) (i+|l1|) |
---|
1030 | |
---|
1031 | |
---|
1032 | whd in ⊢ (??%%); |
---|
1033 | | |
---|
1034 | ] |
---|
1035 | ] |
---|
1036 | |
---|
1037 | lemma |
---|
1038 | compute_max_trace_any_label_cost cost_label trace_ends_flag start_status |
---|
1039 | final_status the_trace = |
---|
1040 | |
---|
1041 | |
---|
1042 | let rec compute_paid_trace_label_label_cost |
---|
1043 | (cost_labels: BitVectorTrie Byte 16) |
---|
1044 | (trace_ends_flag: trace_ends_with_ret) |
---|
1045 | (start_status: Status) (final_status: Status) |
---|
1046 | (the_trace: trace_label_label (ASM_abstract_status cost_labels) trace_ends_flag |
---|
1047 | start_status final_status) on the_trace: nat ≝ |
---|
1048 | match the_trace with |
---|
1049 | [ tll_base _ _ _ given_trace _ ⇒ |
---|
1050 | compute_paid_trace_any_label_cost … given_trace |
---|
1051 | ] |
---|
1052 | and compute_paid_trace_any_label_cost |
---|
1053 | (cost_labels: BitVectorTrie Byte 16) |
---|
1054 | (trace_ends_flag: trace_ends_with_ret) |
---|
1055 | (start_status: Status) (final_status: Status) |
---|
1056 | (the_trace: trace_any_label (ASM_abstract_status cost_labels) trace_ends_flag start_status final_status) |
---|
1057 | on the_trace: nat ≝ |
---|
1058 | match the_trace with |
---|
1059 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost the_status |
---|
1060 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost the_status |
---|
1061 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
---|
1062 | _ _ _ call_trace final_trace ⇒ |
---|
1063 | let current_instruction_cost ≝ current_instruction_cost pre_fun_call in |
---|
1064 | let final_trace_cost ≝ compute_paid_trace_any_label_cost cost_labels end_flag … final_trace in |
---|
1065 | current_instruction_cost + final_trace_cost |
---|
1066 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
---|
1067 | let current_instruction_cost ≝ current_instruction_cost status_pre in |
---|
1068 | let tail_trace_cost ≝ |
---|
1069 | compute_paid_trace_any_label_cost cost_labels end_flag |
---|
1070 | status_init status_end tail_trace |
---|
1071 | in |
---|
1072 | current_instruction_cost + tail_trace_cost |
---|
1073 | ] |
---|
1074 | and compute_paid_trace_label_return_cost |
---|
1075 | (cost_labels: BitVectorTrie Byte 16) |
---|
1076 | (start_status: Status) (final_status: Status) |
---|
1077 | (the_trace: trace_label_return (ASM_abstract_status cost_labels) start_status final_status) |
---|
1078 | on the_trace: nat ≝ |
---|
1079 | match the_trace with |
---|
1080 | [ tlr_base before after trace_to_lift ⇒ compute_paid_trace_label_label_cost … trace_to_lift |
---|
1081 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
---|
1082 | let labelled_cost ≝ compute_paid_trace_label_label_cost … labelled_trace in |
---|
1083 | let return_cost ≝ compute_paid_trace_label_return_cost … ret_trace in |
---|
1084 | labelled_cost + return_cost |
---|
1085 | ]. |
---|
1086 | |
---|
1087 | let rec trace_lab_rec_cost' (p: trace_lab_ret) : nat. |
---|
1088 | | (call b bf tr af tl) as self ⇒ |
---|
1089 | trace_lab_lab_cost_nocall self + trace_lab_ret_cost' tr + |
---|
1090 | trace_lab_rec_cost' tl |
---|
1091 | |
---|
1092 | theorem main_lemma: |
---|
1093 | ∀p. trace_lab_rec_cost p = trace_lab_rec_cost' p. |
---|
1094 | |
---|
1095 | axiom lemma1: |
---|
1096 | ∀p: simple_path. |
---|
1097 | traverse_cost_internal (pc (hd p)) … = trace_lab_lab_cost_nocall p. |
---|
1098 | |
---|
1099 | axiom lemma2: |
---|
1100 | ∀s,l,cost_map. |
---|
1101 | is_labelled l s → |
---|
1102 | traverse_cost_internal s = cost_map l. |
---|
1103 | |
---|
1104 | axiom main_statement: |
---|
1105 | ∀s. |
---|
1106 | ∀cost_map. |
---|
1107 | let p ≝ compute_simple_path0 s in |
---|
1108 | ∑ (cost_trace p) cost_map = trace_lab_rec_cost' p. |
---|
1109 | |
---|
1110 | axiom main_statement: |
---|
1111 | ∀s. |
---|
1112 | ∀cost_map. |
---|
1113 | let p ≝ compute_simple_path0 s in |
---|
1114 | execute' (path_length p) s = 〈s',τ〉 → |
---|
1115 | Timer s' - Timer s = ∑ τ cost_map. |
---|