1 | include "ASM/ASMCostsSplit.ma". |
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2 | include "ASM/WellLabeled.ma". |
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3 | include "ASM/Status.ma". |
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4 | include "common/StructuredTraces.ma". |
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5 | include "arithmetics/bigops.ma". |
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6 | |
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7 | include alias "arithmetics/nat.ma". |
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8 | include alias "basics/logic.ma". |
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9 | |
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10 | let rec compute_max_trace_label_label_cost |
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11 | (cm: ?) |
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12 | (cost_labels: costlabel_map) |
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13 | (trace_ends_flag: trace_ends_with_ret) |
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14 | (start_status: Status cm) (final_status: Status cm) |
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15 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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16 | start_status final_status) on the_trace: nat ≝ |
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17 | match the_trace with |
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18 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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19 | compute_max_trace_any_label_cost … given_trace |
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20 | ] |
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21 | and compute_max_trace_any_label_cost |
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22 | (cm: ?) |
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23 | (cost_labels: costlabel_map) |
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24 | (trace_ends_flag: trace_ends_with_ret) |
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25 | (start_status: Status cm) (final_status: Status cm) |
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26 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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27 | on the_trace: nat ≝ |
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28 | match the_trace with |
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29 | [ tal_base_not_return the_status _ _ _ _ ⇒ current_instruction_cost cm the_status |
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30 | | tal_base_return the_status _ _ _ ⇒ current_instruction_cost cm the_status |
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31 | | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ |
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32 | let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in |
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33 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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34 | call_trace_cost + current_instruction_cost |
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35 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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36 | _ _ _ call_trace _ final_trace ⇒ |
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37 | let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in |
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38 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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39 | let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in |
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40 | call_trace_cost + current_instruction_cost + final_trace_cost |
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41 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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42 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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43 | let tail_trace_cost ≝ |
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44 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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45 | status_init status_end tail_trace |
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46 | in |
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47 | current_instruction_cost + tail_trace_cost |
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48 | ] |
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49 | and compute_max_trace_label_return_cost |
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50 | (cm: ?) |
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51 | (cost_labels: costlabel_map) |
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52 | (start_status: Status cm) (final_status: Status cm) |
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53 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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54 | on the_trace: nat ≝ |
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55 | match the_trace with |
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56 | [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift |
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57 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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58 | let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in |
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59 | let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in |
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60 | labelled_cost + return_cost |
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61 | ]. |
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62 | |
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63 | include alias "arithmetics/nat.ma". |
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64 | |
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65 | let rec compute_max_trace_label_label_cost_is_ok |
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66 | (cm: ?) |
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67 | (cost_labels: costlabel_map) |
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68 | (trace_ends_flag: trace_ends_with_ret) |
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69 | (start_status: Status cm) (final_status: Status cm) |
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70 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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71 | start_status final_status) on the_trace: |
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72 | clock … cm … final_status = (compute_max_trace_label_label_cost cm cost_labels trace_ends_flag start_status final_status the_trace) + (clock … cm … start_status) ≝ ? |
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73 | and compute_max_trace_any_label_cost_is_ok |
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74 | (cm: ?) |
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75 | (cost_labels: costlabel_map) |
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76 | (trace_ends_flag: trace_ends_with_ret) |
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77 | (start_status: Status cm) (final_status: Status cm) |
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78 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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79 | on the_trace: |
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80 | clock … cm … final_status = (compute_max_trace_any_label_cost cm cost_labels trace_ends_flag start_status final_status the_trace) + (clock … cm … start_status) ≝ ? |
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81 | and compute_max_trace_label_return_cost_is_ok |
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82 | (cm: ?) |
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83 | (cost_labels: costlabel_map) |
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84 | (start_status: Status cm) (final_status: Status cm) |
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85 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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86 | on the_trace: |
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87 | clock … cm … final_status = (compute_max_trace_label_return_cost cm cost_labels start_status final_status the_trace) + clock … cm … start_status ≝ ?. |
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88 | [1: |
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89 | cases the_trace |
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90 | #ends_flag #start_status #end_status #any_label_trace #is_costed |
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91 | normalize @compute_max_trace_any_label_cost_is_ok |
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92 | |2: |
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93 | cases the_trace |
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94 | [1,2: |
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95 | #start_status #final_status #is_next #is_not_return try (#is_costed) |
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96 | change with (current_instruction_cost cm start_status) in ⊢ (???(?%?)); |
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97 | cases(is_next) @execute_1_ok_clock |
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98 | |3: |
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99 | #status_pre_fun_call #status_start_fun_call #status_final #is_next |
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100 | #classifier_assm #after_return_assm #call_trace #costed_assm |
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101 | whd in match (compute_max_trace_any_label_cost … (tal_base_call …)); |
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102 | >(compute_max_trace_label_return_cost_is_ok … call_trace) |
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103 | >associative_plus @eq_f cases(is_next) |
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104 | @execute_1_ok_clock |
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105 | |4: |
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106 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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107 | #status_final #is_next #is_call #is_after_return #call_trace #not_costed #final_trace |
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108 | change with ( |
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109 | let current_instruction_cost ≝ current_instruction_cost cm status_pre_fun_call in |
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110 | let call_trace_cost ≝ compute_max_trace_label_return_cost cm … call_trace in |
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111 | let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in |
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112 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?)); |
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113 | normalize nodelta; |
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114 | >(compute_max_trace_any_label_cost_is_ok … cost_labels end_flag status_after_fun_call |
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115 | status_final final_trace) |
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116 | >(compute_max_trace_label_return_cost_is_ok … cost_labels status_start_fun_call |
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117 | status_after_fun_call call_trace) |
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118 | cases(is_next) in match (clock … cm status_start_fun_call); |
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119 | >(execute_1_ok_clock cm status_pre_fun_call) |
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120 | <associative_plus in ⊢ (??%?); |
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121 | <commutative_plus in match ( |
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122 | compute_max_trace_any_label_cost cm cost_labels end_flag status_after_fun_call status_final final_trace |
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123 | + compute_max_trace_label_return_cost cm cost_labels status_start_fun_call status_after_fun_call call_trace); |
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124 | >associative_plus in ⊢ (??%?); >associative_plus in ⊢ (???%); >associative_plus in ⊢ (???%); |
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125 | @eq_f >commutative_plus in ⊢ (??%?); >associative_plus in ⊢ (??%?); |
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126 | @eq_f @commutative_plus |
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127 | |5: |
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128 | #end_flag #status_pre #status_init #status_end #is_next |
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129 | #trace_any_label #is_other #is_not_costed |
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130 | change with ( |
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131 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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132 | let tail_trace_cost ≝ |
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133 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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134 | status_init status_end trace_any_label |
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135 | in |
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136 | current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?)); |
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137 | normalize nodelta; |
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138 | >(compute_max_trace_any_label_cost_is_ok cm cost_labels end_flag |
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139 | status_init status_end trace_any_label) |
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140 | cases(is_next) in match (clock … cm status_init); |
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141 | >(execute_1_ok_clock … status_pre) |
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142 | >commutative_plus >associative_plus >associative_plus @eq_f |
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143 | @commutative_plus |
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144 | ] |
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145 | |3: |
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146 | cases the_trace |
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147 | [1: |
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148 | #status_before #status_after #trace_to_lift |
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149 | normalize @compute_max_trace_label_label_cost_is_ok |
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150 | |2: |
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151 | #status_initial #status_labelled #status_final #labelled_trace #ret_trace |
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152 | normalize |
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153 | >(compute_max_trace_label_return_cost_is_ok cm cost_labels status_labelled status_final ret_trace); |
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154 | >(compute_max_trace_label_label_cost_is_ok cm cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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155 | <associative_plus in ⊢ (??%?); |
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156 | >commutative_plus in match ( |
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157 | compute_max_trace_label_return_cost cm cost_labels status_labelled status_final ret_trace |
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158 | + compute_max_trace_label_label_cost cm cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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159 | % |
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160 | ] |
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161 | ] |
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162 | qed. |
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163 | |
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164 | let rec compute_trace_label_label_cost_using_paid |
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165 | (cm: ?) |
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166 | (cost_labels: costlabel_map) |
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167 | (trace_ends_flag: trace_ends_with_ret) |
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168 | (start_status: Status cm) (final_status: Status cm) |
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169 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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170 | start_status final_status) on the_trace: nat ≝ |
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171 | match the_trace with |
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172 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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173 | compute_paid_trace_label_label cm cost_labels … the_trace + |
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174 | compute_trace_any_label_cost_using_paid … given_trace |
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175 | ] |
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176 | and compute_trace_any_label_cost_using_paid |
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177 | (cm: ?) |
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178 | (cost_labels: costlabel_map) |
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179 | (trace_ends_flag: trace_ends_with_ret) |
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180 | (start_status: Status cm) (final_status: Status cm) |
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181 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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182 | on the_trace: nat ≝ |
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183 | match the_trace with |
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184 | [ tal_base_not_return the_status _ _ _ _ ⇒ 0 |
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185 | | tal_base_return the_status _ _ _ ⇒ 0 |
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186 | | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ |
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187 | compute_trace_label_return_cost_using_paid … call_trace |
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188 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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189 | _ _ _ call_trace _ final_trace ⇒ |
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190 | let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in |
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191 | let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cm cost_labels end_flag … final_trace in |
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192 | call_trace_cost + final_trace_cost |
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193 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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194 | compute_trace_any_label_cost_using_paid cm cost_labels end_flag |
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195 | status_init status_end tail_trace |
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196 | ] |
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197 | and compute_trace_label_return_cost_using_paid |
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198 | (cm: ?) |
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199 | (cost_labels: costlabel_map) |
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200 | (start_status: Status cm) (final_status: Status cm) |
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201 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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202 | on the_trace: nat ≝ |
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203 | match the_trace with |
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204 | [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift |
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205 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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206 | let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in |
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207 | let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in |
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208 | labelled_cost + return_cost |
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209 | ]. |
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210 | |
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211 | let rec compute_trace_label_label_cost_using_paid_ok |
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212 | (cm: ?) |
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213 | (cost_labels: costlabel_map) |
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214 | (trace_ends_flag: trace_ends_with_ret) |
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215 | (start_status: Status cm) (final_status: Status cm) |
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216 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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217 | start_status final_status) on the_trace: |
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218 | compute_trace_label_label_cost_using_paid cm cost_labels … the_trace = |
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219 | compute_max_trace_label_label_cost … the_trace ≝ ? |
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220 | and compute_trace_any_label_cost_using_paid_ok |
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221 | (cm: ?) |
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222 | (cost_labels: costlabel_map) |
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223 | (trace_ends_flag: trace_ends_with_ret) |
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224 | (start_status: Status cm) (final_status: Status cm) |
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225 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) |
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226 | trace_ends_flag start_status final_status) on the_trace: |
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227 | compute_paid_trace_any_label cm cost_labels trace_ends_flag … the_trace |
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228 | +compute_trace_any_label_cost_using_paid cm cost_labels trace_ends_flag … the_trace |
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229 | =compute_max_trace_any_label_cost cm cost_labels trace_ends_flag … the_trace ≝ ? |
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230 | and compute_trace_label_return_cost_using_paid_ok |
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231 | (cm: ?) |
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232 | (cost_labels: costlabel_map) |
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233 | (start_status: Status cm) (final_status: Status cm) |
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234 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) |
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235 | start_status final_status) on the_trace: |
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236 | compute_trace_label_return_cost_using_paid cm cost_labels … the_trace = |
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237 | compute_max_trace_label_return_cost cm cost_labels … the_trace ≝ ?. |
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238 | [ cases the_trace #endsf #ss #es #tr #H normalize |
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239 | @compute_trace_any_label_cost_using_paid_ok |
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240 | | cases the_trace |
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241 | [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O % |
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242 | | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O % |
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243 | | |
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244 | #sp #ss #sf #H1 #H2 #tr1 #tr2 #H3 |
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245 | whd in ⊢ (???%); whd in ⊢ (??(??%)?); whd in ⊢ (??(?%?)?); |
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246 | >compute_trace_label_return_cost_using_paid_ok in ⊢ (??%?); |
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247 | >commutative_plus in ⊢ (??%?); @eq_f % |
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248 | | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #H4 #tr2 whd in ⊢ (??(?%%)%); |
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249 | <compute_trace_any_label_cost_using_paid_ok |
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250 | <compute_trace_label_return_cost_using_paid_ok |
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251 | -compute_trace_label_label_cost_using_paid_ok |
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252 | -compute_trace_label_return_cost_using_paid_ok |
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253 | -compute_trace_any_label_cost_using_paid_ok |
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254 | >commutative_plus in ⊢ (???(?%?)); |
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255 | >commutative_plus in ⊢ (??(??%)?); |
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256 | >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try % |
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257 | <associative_plus <commutative_plus % |
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258 | | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2 |
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259 | [ % | @compute_trace_any_label_cost_using_paid_ok ] |
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260 | ] |
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261 | | cases the_trace |
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262 | [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok |
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263 | | #si #sl #sf #tr1 #tr2 normalize @eq_f2 |
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264 | [ @compute_trace_label_label_cost_using_paid_ok |
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265 | | @compute_trace_label_return_cost_using_paid_ok ]]] |
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266 | qed. |
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267 | |
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268 | include alias "ASM/BitVectorTrie.ma". |
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269 | |
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270 | (* |
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271 | let rec compute_cost_trace_label_label |
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272 | (cm: BitVectorTrie Byte 16) |
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273 | (cost_labels: costlabel_map) |
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274 | (trace_ends_flag: trace_ends_with_ret) |
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275 | (start_status: Status cm) (final_status: Status cm) |
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276 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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277 | start_status final_status) on the_trace: |
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278 | list (Σk:costlabel. ∃pc: Word. lookup_opt … pc cost_labels = Some … k) ≝ |
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279 | match the_trace with |
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280 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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281 | let pc ≝ program_counter ? cm initial in |
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282 | let label ≝ |
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283 | match lookup_opt … pc cost_labels return λx: option ?. x ≠ None … → costlabel with |
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284 | [ None ⇒ λabs. ⊥ |
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285 | | Some l ⇒ λ_. l ] labelled_proof in |
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286 | (mk_Sig ?? label ?)::compute_cost_trace_any_label cm cost_labels ends_flag initial final … given_trace |
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287 | ] |
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288 | and compute_cost_trace_any_label |
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289 | (cm: BitVectorTrie Byte 16) |
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290 | (cost_labels: costlabel_map) |
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291 | (trace_ends_flag: trace_ends_with_ret) |
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292 | (start_status: Status cm) (final_status: Status cm) |
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293 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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294 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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295 | match the_trace with |
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296 | [ tal_base_not_return the_status _ _ _ _ ⇒ [ ] |
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297 | | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ |
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298 | compute_cost_trace_label_return … call_trace |
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299 | | tal_base_return the_status _ _ _ ⇒ [ ] |
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300 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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301 | _ _ _ call_trace _ final_trace ⇒ |
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302 | let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in |
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303 | let final_cost_trace ≝ compute_cost_trace_any_label cm cost_labels end_flag … final_trace in |
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304 | call_cost_trace @ final_cost_trace |
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305 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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306 | compute_cost_trace_any_label cm cost_labels end_flag |
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307 | status_init status_end tail_trace |
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308 | ] |
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309 | and compute_cost_trace_label_return |
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310 | (cm: BitVectorTrie Byte 16) |
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311 | (cost_labels: costlabel_map) |
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312 | (start_status: Status cm) (final_status: Status cm) |
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313 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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314 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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315 | match the_trace with |
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316 | [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label cm … trace_to_lift |
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317 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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318 | let labelled_cost ≝ compute_cost_trace_label_label cm … labelled_trace in |
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319 | let return_cost ≝ compute_cost_trace_label_return cm … ret_trace in |
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320 | labelled_cost @ return_cost |
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321 | ]. |
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322 | [1: |
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323 | %{pc} whd in match label; generalize in match labelled_proof; whd in ⊢ (% → ?); |
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324 | whd in match (as_costed ??); whd in match (as_label ??); normalize nodelta |
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325 | cases (lookup_opt costlabel … (program_counter … initial) cost_labels) |
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326 | normalize |
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327 | [ #abs cases abs #absurd @⊥ @absurd % | // ] |
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328 | | cases abs #absurd @absurd % ] |
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329 | qed. |
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330 | *) |
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331 | |
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332 | include alias "arithmetics/nat.ma". |
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333 | include alias "basics/logic.ma". |
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334 | |
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335 | include alias "arithmetics/bigops.ma". |
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336 | |
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337 | (* This shoudl go in bigops! *) |
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338 | theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B. |
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339 | \big[op,nil]_{i<k1+k2|p i} (f i) = |
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340 | 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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341 | #k1 #k2 #p #B #nil #op #f >bigop_sum |
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342 | >commutative_plus @same_bigop #i @leb_elim normalize |
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343 | [2,4: // |
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344 | | #H1 #H2 <plus_minus_m_m // |
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345 | | #H1 #H2 #H3 <plus_minus_m_m //] |
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346 | qed. |
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347 | |
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348 | (* This is taken by sigma_pi.ma that does not compile now *) |
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349 | definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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350 | (λa,b,c.sym_eq ??? (associative_plus a b c)). |
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351 | |
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352 | definition natACop ≝ mk_ACop nat 0 natAop commutative_plus. |
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353 | |
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354 | definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n))) |
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355 | distributive_times_plus. |
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356 | |
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357 | unification hint 0 ≔ ; |
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358 | S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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359 | (λa,b,c.sym_eq ??? (associative_plus a b c)) |
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360 | (* ---------------------------------------- *) ⊢ |
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361 | plus ≡ op ? ? S. |
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362 | |
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363 | unification hint 0 ≔ ; |
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364 | S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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365 | (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus |
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366 | (* ---------------------------------------- *) ⊢ |
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367 | plus ≡ op ? ? S. |
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368 | |
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369 | unification hint 0 ≔ ; |
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370 | S ≟ natDop |
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371 | (* ---------------------------------------- *) ⊢ |
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372 | plus ≡ sum ? ? S. |
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373 | |
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374 | unification hint 0 ≔ ; |
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375 | S ≟ natDop |
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376 | (* ---------------------------------------- *) ⊢ |
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377 | times ≡ prod ? ? S. |
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378 | |
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379 | notation > "Σ_{ ident i < n } f" |
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380 | with precedence 20 |
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381 | for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}. |
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382 | |
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383 | notation < "Σ_{ ident i < n } f" |
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384 | with precedence 20 |
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385 | for @{'bigop $n plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f)}. |
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386 | |
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387 | definition tech_cost_of_label0: |
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388 | ∀cost_labels: costlabel_map. |
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389 | ∀cost_map: identifier_map CostTag nat. |
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390 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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391 | ∀ctrace:list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k). |
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392 | ∀i,p. present … cost_map (nth_safe ? i ctrace p). |
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393 | #cost_labels #cost_map #codom_dom #ctrace #i #p |
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394 | cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K |
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395 | lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres) |
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396 | % #abs destruct (abs) |
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397 | qed. |
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398 | |
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399 | include alias "arithmetics/nat.ma". |
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400 | include alias "basics/logic.ma". |
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401 | |
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402 | lemma ltb_rect: |
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403 | ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P. |
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404 | #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2 |
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405 | [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ] |
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406 | qed. |
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407 | |
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408 | lemma same_ltb_rect: |
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409 | ∀P,n,m,H1,H2,n',m',H1',H2'. |
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410 | ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) → |
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411 | ltb_rect P n m H1 H2 = |
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412 | ltb_rect P n' m' H1' H2'. |
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413 | #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?); |
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414 | cut (∀xxx,yyy,xxx',yyy'. |
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415 | match ltb n m |
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416 | return λx:bool. |
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417 | eq bool (ltb n m) x |
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418 | → (lt n m → P) → (Not (lt n m) → P) → P |
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419 | with |
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420 | [ true ⇒ |
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421 | λE0:eq bool (ltb n m) true. |
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422 | λH10:lt n m → P. |
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423 | λH20:Not (lt n m) → P. H10 (xxx E0) |
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424 | | false ⇒ |
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425 | λE0:eq bool (ltb n m) false. |
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426 | λH10:lt n m → P. |
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427 | λH20:Not (lt n m) → P. H20 (yyy E0)] |
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428 | (refl … (ltb n m)) H1 H2 = |
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429 | match ltb n' m' |
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430 | return λx:bool. |
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431 | eq bool (ltb n' m') x |
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432 | → (lt n' m' → P) → (Not (lt n' m') → P) → P |
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433 | with |
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434 | [ true ⇒ |
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435 | λE0:eq bool (ltb n' m') true. |
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436 | λH10:lt n' m' → P. |
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437 | λH20:Not (lt n' m') → P. H10 (xxx' E0) |
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438 | | false ⇒ |
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439 | λE0:eq bool (ltb n' m') false. |
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440 | λH10:lt n' m' → P. |
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441 | λH20:Not (lt n' m') → P. H20 (yyy' E0)] |
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442 | (refl … (ltb n' m')) H1' H2' |
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443 | ) [2: #X @X] |
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444 | >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize |
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445 | [ @K1 | @K2 ] |
---|
446 | qed. |
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447 | |
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448 | definition tech_cost_of_label: |
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449 | ∀cost_labels: costlabel_map. |
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450 | ∀cost_map: identifier_map CostTag nat. |
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451 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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452 | list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k) → |
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453 | nat → nat |
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454 | ≝ λcost_labels,cost_map,codom_dom,ctrace,i. |
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455 | ltb_rect ? i (|ctrace|) |
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456 | (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?) |
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457 | (λ_.0). |
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458 | @tech_cost_of_label0 @codom_dom |
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459 | qed. |
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460 | |
---|
461 | lemma tech_cost_of_label_shift: |
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462 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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463 | i < |l2| → |
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464 | tech_cost_of_label cost_labels cost_map codom_dom l2 i = |
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465 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|). |
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466 | #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H |
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467 | whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect |
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468 | [ @(ltb_rect ? i (|l2|)) @(ltb_rect ? (i+|l1|) (|l1@l2|)) #K1 #K2 |
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469 | whd in match ltb; normalize nodelta |
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470 | [1: >le_to_leb_true try assumption applyS le_to_leb_true / by / |
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471 | |4: >not_le_to_leb_false try assumption applyS not_le_to_leb_false |
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472 | change with (¬ ? ≤ ?) in K1; applyS K1 |
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473 | |2: @⊥ @(absurd (i+|l1| < |l1@l2|)) / by / >length_append |
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474 | applyS (monotonic_lt_plus_r … (|l1|)) / by / |
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475 | |3: @⊥ @(absurd ?? K2) >length_append in K1; #K1 /2 by lt_plus_to_lt_l/ ] |
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476 | | #H1 #H2 |
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477 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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478 | <(shift_nth_safe … H1) #p % |
---|
479 | | / by / ] |
---|
480 | qed. |
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481 | |
---|
482 | lemma tech_cost_of_label_prefix: |
---|
483 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
---|
484 | i < |l1| → |
---|
485 | tech_cost_of_label cost_labels cost_map codom_dom l1 i = |
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486 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) i. |
---|
487 | #cost_labels #cost_map #codom_dom #l1 #l2 #i #H |
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488 | whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect |
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489 | [1: |
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490 | whd in match ltb; normalize nodelta |
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491 | >(le_to_leb_true … H) applyS le_to_leb_true |
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492 | >length_append whd in H; >commutative_plus @le_plus_a assumption |
---|
493 | |2: |
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494 | #K1 #K2 |
---|
495 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
---|
496 | <(shift_nth_prefix … l1 i l2 K1 K2) // |
---|
497 | |3: |
---|
498 | #_ #_ % |
---|
499 | ] |
---|
500 | qed. |
---|
501 | |
---|
502 | let rec compute_trace_label_return_using_paid_ok_with_trace |
---|
503 | (cm: ?) (cost_labels: costlabel_map) |
---|
504 | (cost_map: identifier_map CostTag nat) |
---|
505 | (initial: Status cm) (final: Status cm) |
---|
506 | (trace: trace_label_return (ASM_abstract_status cm cost_labels) initial final) |
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507 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
---|
508 | on trace: |
---|
509 | ∀unrepeating_witness: tlr_unrepeating … trace. |
---|
510 | ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
---|
511 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres). |
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512 | let ctrace ≝ flatten_trace_label_return (ASM_abstract_status cm cost_labels) … trace in |
---|
513 | compute_trace_label_return_cost_using_paid cm … trace = |
---|
514 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
---|
515 | ≝ ? |
---|
516 | and compute_trace_any_label_using_paid_ok_with_trace |
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517 | (cm: ?) (cost_labels: costlabel_map) |
---|
518 | (trace_ends_flag: trace_ends_with_ret) |
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519 | (cost_map: identifier_map CostTag nat) |
---|
520 | (initial: Status cm) (final: Status cm) |
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521 | (trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final) |
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522 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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523 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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524 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
---|
525 | on trace: |
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526 | ∀unrepeating_witness: tal_unrepeating … trace. |
---|
527 | let ctrace ≝ flatten_trace_any_label (ASM_abstract_status cm cost_labels) … trace_ends_flag … trace in |
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528 | compute_trace_any_label_cost_using_paid … trace = |
---|
529 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
---|
530 | ≝ ? |
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531 | and compute_trace_label_label_using_paid_ok_with_trace |
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532 | (cm: ?) (cost_labels: costlabel_map) |
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533 | (trace_ends_flag: trace_ends_with_ret) |
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534 | (cost_map: identifier_map CostTag nat) |
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535 | (initial: Status cm) (final: Status cm) |
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536 | (trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final) |
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537 | (unrepeating_witness: tll_unrepeating … trace) |
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538 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
---|
539 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
---|
540 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
---|
541 | on trace: |
---|
542 | ∀unrepeating_witness: tll_unrepeating … trace. |
---|
543 | let ctrace ≝ flatten_trace_label_label (ASM_abstract_status cm cost_labels) … trace in |
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544 | compute_trace_label_label_cost_using_paid … trace = |
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545 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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546 | ≝ ?. |
---|
547 | cases trace normalize nodelta |
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548 | [ #sb #sa #tr #unrepeating_witness #dom_codom whd in ⊢ (??%?); |
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549 | whd in match (flatten_trace_label_return ????); |
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550 | @compute_trace_label_label_using_paid_ok_with_trace |
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551 | assumption |
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552 | | #si #sl #sf #tr1 #tr2 #unrepeating_witness #dom_codom |
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553 | whd in ⊢ (??%?); |
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554 | whd in match (flatten_trace_label_return ????); |
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555 | >append_length >bigop_sum_rev >commutative_plus @eq_f2 |
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556 | [ >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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557 | -compute_trace_label_return_using_paid_ok_with_trace |
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558 | [1: |
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559 | @same_bigop [//] #i #H #_ -dom_codom @tech_cost_of_label_shift // |
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560 | |2: |
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561 | inversion unrepeating_witness |
---|
562 | #tll_unrepeating #tlr_unrepeating #_ assumption |
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563 | ] |
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564 | | >(compute_trace_label_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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565 | -compute_trace_label_label_using_paid_ok_with_trace |
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566 | [1: |
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567 | @same_bigop [//] #i #H #_ @(tech_cost_of_label_prefix … H) |
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568 | |2,3: |
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569 | inversion unrepeating_witness |
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570 | #tll_unrepeating #tlr_unrepeating #_ assumption |
---|
571 | ] |
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572 | ] |
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573 | |8: |
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574 | #end_flag #start_status #end_status #trace_any_label #costed_assm |
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575 | #unrepeating_witness' |
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576 | whd in ⊢ (??%?); whd in ⊢ (??(?%?)?); |
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577 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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578 | [1: |
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579 | whd in match (flatten_trace_label_label ?????); |
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580 | >bigop_0 in ⊢ (???%); >commutative_plus @eq_f2 |
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581 | [1: |
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582 | @same_bigop [//] #i #H #_ -dom_codom >(plus_n_O i) >plus_n_Sm |
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583 | <(tech_cost_of_label_shift ??? [?] ? i) try assumption <(plus_n_O i) % |
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584 | |2: |
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585 | change with (? = lookup_present ? ? ? ? ?) |
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586 | generalize in match (tech_cost_of_label0 ? ? ? ? ? ?); |
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587 | normalize in match (nth_safe ? ? ? ?); |
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588 | whd in costed_assm; lapply costed_assm whd in match (as_label ??); |
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589 | inversion (lookup_opt ? ? (program_counter … cm start_status) cost_labels) |
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590 | [1: |
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591 | #_ #absurd @⊥ cases absurd #absurd @absurd % |
---|
592 | |2: |
---|
593 | normalize nodelta #cost_label #Some_assm #_ #p |
---|
594 | cases (dom_codom ? p ? Some_assm) |
---|
595 | cases (block_cost ???) |
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596 | #cost #block_cost_assm |
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597 | cases (block_cost_assm ??? trace_any_label ??) |
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598 | try @refl assumption |
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599 | ] |
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600 | ] |
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601 | |2: |
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602 | assumption |
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603 | ] |
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604 | |3: |
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605 | #start_status #final_status #execute_assm #classifier_assm #costed_assm |
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606 | #unrepeating_witness |
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607 | % |
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608 | |4: |
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609 | #start_status #final_status #execute_assm #classifier_assm #unrepeating_witness |
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610 | % |
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611 | |5: |
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612 | #status_pre_fun_call #status_start_fun_call #status_final #execute_assm |
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613 | #classifier_assm #after_return_assm #trace_label_return #costed_assm #unrepeating_witness |
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614 | whd in ⊢ (??%?); |
---|
615 | @(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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616 | assumption |
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617 | |6: |
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618 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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619 | #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return |
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620 | #costed_assm #trace_any_label #unrepeating_witness |
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621 | whd in ⊢ (??%?); |
---|
622 | >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
---|
623 | [1: |
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624 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
---|
625 | [1: |
---|
626 | >length_append >bigop_sum_rev >commutative_plus @eq_f2 |
---|
627 | [ @same_bigop [2: #i #H #_ -dom_codom @tech_cost_of_label_shift assumption |
---|
628 | |1: #i #H % ] |
---|
629 | | @same_bigop [#i #H %] #i #H #_ @(tech_cost_of_label_prefix … H) |
---|
630 | ] |
---|
631 | |2: |
---|
632 | inversion unrepeating_witness |
---|
633 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
---|
634 | ] |
---|
635 | |2: |
---|
636 | inversion unrepeating_witness |
---|
637 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
---|
638 | ] |
---|
639 | |7: |
---|
640 | #end_flag #status_pre_fun_call #status_start_fun_call #status_final |
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641 | #execute_assm #trace_any_label #classifier_assm #costed_assm #unrepeating_witness |
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642 | whd in ⊢ (??%?); |
---|
643 | @(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
---|
644 | inversion unrepeating_witness |
---|
645 | #memb_1 #tal_unrepeating #_ assumption |
---|
646 | ] |
---|
647 | qed. |
---|
648 | |
---|
649 | lemma compute_max_trace_label_return_cost_ok_with_trace_aux: |
---|
650 | ∀code_memory: BitVectorTrie Byte 16. |
---|
651 | ∀cost_labels: costlabel_map. |
---|
652 | ∀cost_map: identifier_map CostTag nat. |
---|
653 | ∀initial: Status code_memory. |
---|
654 | ∀final: Status code_memory. |
---|
655 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
---|
656 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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657 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
---|
658 | ∀dom_codom: (∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
---|
659 | pi1 … (block_cost code_memory pc cost_labels) = lookup_present … k_pres). |
---|
660 | let ctrace ≝ flatten_trace_label_return … trace in |
---|
661 | clock … code_memory … final = |
---|
662 | clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)). |
---|
663 | #code_memory #cost_labels #cost_map |
---|
664 | #initial #final #trace #unrepeating_witness #codom_dom #dom_codom normalize nodelta |
---|
665 | <compute_trace_label_return_using_paid_ok_with_trace try assumption |
---|
666 | >commutative_plus >compute_trace_label_return_cost_using_paid_ok // |
---|
667 | qed. |
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668 | |
---|
669 | theorem compute_max_trace_label_return_cost_ok_with_trace: |
---|
670 | ∀code_memory: BitVectorTrie Byte 16. |
---|
671 | ∀cost_labels: costlabel_map. |
---|
672 | ∀cost_labels_injective: |
---|
673 | (∀pc,pc',l. |
---|
674 | lookup_opt costlabel 16 pc cost_labels=Some costlabel l |
---|
675 | →lookup_opt costlabel 16 pc' cost_labels=Some costlabel l→pc=pc'). |
---|
676 | ∀initial: Status code_memory. |
---|
677 | ∀final: Status code_memory. |
---|
678 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
---|
679 | ∀unrepeating_witness: tlr_unrepeating … trace. |
---|
680 | let cost_map ≝ traverse_code code_memory cost_labels cost_labels_injective in |
---|
681 | let ctrace ≝ flatten_trace_label_return … trace in |
---|
682 | clock … code_memory … final = clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map ? ctrace i)). |
---|
683 | [1: |
---|
684 | #code_memory #cost_labels #cost_labels_injective #initial #final #trace #unrepeating_witness |
---|
685 | @compute_max_trace_label_return_cost_ok_with_trace_aux try assumption |
---|
686 | |2: |
---|
687 | skip |
---|
688 | ] |
---|
689 | normalize nodelta |
---|
690 | cases (traverse_code ???) |
---|
691 | #cost_map * #dom_codom #codom_dom try assumption |
---|
692 | #pc #k #lookup_opt_assm @(dom_codom … lookup_opt_assm) |
---|
693 | lapply (sym_eq ? ? ? lookup_opt_assm) |
---|
694 | -lookup_opt_assm #lookup_opt_assm |
---|
695 | @nat_of_bitvector_lt_bound |
---|
696 | qed. |
---|
697 | |
---|
698 | include "utilities/permutations.ma". |
---|
699 | (* |
---|
700 | lemma tech_cost_sum_eq_as_cost : |
---|
701 | ∀code_memory: object_code. |
---|
702 | ∀cost_labels: costlabel_map. |
---|
703 | ∀cost_map: identifier_map CostTag nat. |
---|
704 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
---|
705 | ∀trace. |
---|
706 | (Σ_{i < |trace|}(tech_cost_of_label cost_labels cost_map codom_dom trace i)) = |
---|
707 | (Σ_{l ∈ trace}(ASM_cost_map 〈code_memory, cost_labels〉 … codom_dom l)). |
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708 | #cmem #clab #cmap #codom_dom #trace |
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709 | @(list_elim_left … trace) |
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710 | [ % |
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711 | | #tl #hd #IH >append_length >commutative_plus |
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712 | >(fold_permute … (hd@[tl]) (tl::hd)) [2: @perm_swap_append ] |
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713 | whd in ⊢ (??%%); <IH |
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714 | <tech_cost_of_label_shift [2: %] |
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715 | cut (∀x,y,z,w.x = y → z = w → x + z = y + w) [ // ] #APP @APP -APP |
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716 | [ % |
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717 | | @sym_eq @same_bigop /2 by tech_cost_of_label_prefix/ |
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718 | ] |
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719 | ] |
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720 | qed. |
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721 | *) |
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722 | lemma tech_cost_sum_eq_as_cost : |
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723 | ∀code_memory: BitVectorTrie Byte 16. |
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724 | ∀cost_labels: costlabel_map. |
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725 | ∀cost_map: identifier_map CostTag nat. |
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726 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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727 | ∀trace. |
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728 | (Σ_{i < |trace|}(tech_cost_of_label cost_labels cost_map codom_dom trace i)) = |
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729 | (Σ_{l ∈ trace}(ASM_cost_map code_memory … codom_dom l)). |
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730 | #cmem #clab #cmap #codom_dom #trace |
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731 | @(list_elim_left … trace) |
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732 | [ % |
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733 | | #tl #hd #IH >append_length >commutative_plus |
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734 | >(fold_permute … (hd@[tl]) (tl::hd)) [2: @perm_swap_append ] |
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735 | whd in ⊢ (??%%); <IH |
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736 | <tech_cost_of_label_shift [2: %] |
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737 | cut (∀x,y,z,w.x = y → z = w → x + z = y + w) [ // ] #APP @APP -APP |
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738 | [ % |
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739 | | @sym_eq @same_bigop /2 by tech_cost_of_label_prefix/ |
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740 | ] |
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741 | ] |
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742 | qed. |
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