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: BitVectorTrie costlabel 16) |
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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: BitVectorTrie costlabel 16) |
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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_base_tailcall pre_fun_call start_fun_call final _ _ call_trace ⇒ |
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36 | let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in |
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37 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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38 | call_trace_cost + current_instruction_cost |
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39 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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40 | _ _ _ call_trace _ final_trace ⇒ |
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41 | let current_instruction_cost ≝ current_instruction_cost cm pre_fun_call in |
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42 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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43 | let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in |
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44 | call_trace_cost + current_instruction_cost + final_trace_cost |
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45 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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46 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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47 | let tail_trace_cost ≝ |
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48 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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49 | status_init status_end tail_trace |
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50 | in |
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51 | current_instruction_cost + tail_trace_cost |
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52 | ] |
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53 | and compute_max_trace_label_return_cost |
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54 | (cm: ?) |
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55 | (cost_labels: BitVectorTrie costlabel 16) |
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56 | (start_status: Status cm) (final_status: Status cm) |
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57 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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58 | on the_trace: nat ≝ |
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59 | match the_trace with |
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60 | [ tlr_base before after trace_to_lift ⇒ compute_max_trace_label_label_cost … trace_to_lift |
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61 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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62 | let labelled_cost ≝ compute_max_trace_label_label_cost … labelled_trace in |
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63 | let return_cost ≝ compute_max_trace_label_return_cost … ret_trace in |
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64 | labelled_cost + return_cost |
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65 | ]. |
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66 | |
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67 | include alias "arithmetics/nat.ma". |
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68 | |
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69 | let rec compute_max_trace_label_label_cost_is_ok |
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70 | (cm: ?) |
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71 | (cost_labels: BitVectorTrie costlabel 16) |
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72 | (trace_ends_flag: trace_ends_with_ret) |
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73 | (start_status: Status cm) (final_status: Status cm) |
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74 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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75 | start_status final_status) on the_trace: |
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76 | 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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77 | and compute_max_trace_any_label_cost_is_ok |
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78 | (cm: ?) |
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79 | (cost_labels: BitVectorTrie costlabel 16) |
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80 | (trace_ends_flag: trace_ends_with_ret) |
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81 | (start_status: Status cm) (final_status: Status cm) |
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82 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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83 | on the_trace: |
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84 | 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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85 | and compute_max_trace_label_return_cost_is_ok |
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86 | (cm: ?) |
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87 | (cost_labels: BitVectorTrie costlabel 16) |
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88 | (start_status: Status cm) (final_status: Status cm) |
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89 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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90 | on the_trace: |
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91 | 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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92 | [1: |
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93 | cases the_trace |
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94 | #ends_flag #start_status #end_status #any_label_trace #is_costed |
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95 | normalize @compute_max_trace_any_label_cost_is_ok |
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96 | |2: |
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97 | cases the_trace |
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98 | [1,2: |
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99 | #start_status #final_status #is_next #is_not_return try (#is_costed) |
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100 | change with (current_instruction_cost cm start_status) in ⊢ (???(?%?)); |
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101 | cases(is_next) @execute_1_ok_clock |
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102 | |3: |
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103 | #status_pre_fun_call #status_start_fun_call #status_final #is_next |
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104 | #classifier_assm #after_return_assm #call_trace #costed_assm |
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105 | whd in match (compute_max_trace_any_label_cost … (tal_base_call …)); |
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106 | >(compute_max_trace_label_return_cost_is_ok … call_trace) |
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107 | >associative_plus @eq_f cases(is_next) |
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108 | @execute_1_ok_clock |
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109 | |4: |
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110 | #status_pre_fun_call #status_start_fun_call #status_final #is_next |
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111 | #classifier_assm #call_trace |
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112 | whd in match (compute_max_trace_any_label_cost … (tal_base_tailcall …)); |
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113 | >(compute_max_trace_label_return_cost_is_ok … call_trace) |
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114 | >associative_plus @eq_f cases(is_next) |
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115 | @execute_1_ok_clock |
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116 | |5: |
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117 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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118 | #status_final #is_next #is_call #is_after_return #call_trace #not_costed #final_trace |
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119 | change with ( |
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120 | let current_instruction_cost ≝ current_instruction_cost cm status_pre_fun_call in |
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121 | let call_trace_cost ≝ compute_max_trace_label_return_cost cm … call_trace in |
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122 | let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in |
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123 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?)); |
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124 | normalize nodelta; |
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125 | >(compute_max_trace_any_label_cost_is_ok … cost_labels end_flag status_after_fun_call |
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126 | status_final final_trace) |
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127 | >(compute_max_trace_label_return_cost_is_ok … cost_labels status_start_fun_call |
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128 | status_after_fun_call call_trace) |
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129 | cases(is_next) in match (clock … cm status_start_fun_call); |
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130 | >(execute_1_ok_clock cm status_pre_fun_call) |
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131 | <associative_plus in ⊢ (??%?); |
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132 | <commutative_plus in match ( |
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133 | compute_max_trace_any_label_cost cm cost_labels end_flag status_after_fun_call status_final final_trace |
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134 | + compute_max_trace_label_return_cost cm cost_labels status_start_fun_call status_after_fun_call call_trace); |
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135 | >associative_plus in ⊢ (??%?); >associative_plus in ⊢ (???%); >associative_plus in ⊢ (???%); |
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136 | @eq_f >commutative_plus in ⊢ (??%?); >associative_plus in ⊢ (??%?); |
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137 | @eq_f @commutative_plus |
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138 | |6: |
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139 | #end_flag #status_pre #status_init #status_end #is_next |
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140 | #trace_any_label #is_other #is_not_costed |
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141 | change with ( |
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142 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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143 | let tail_trace_cost ≝ |
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144 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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145 | status_init status_end trace_any_label |
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146 | in |
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147 | current_instruction_cost + tail_trace_cost) in ⊢ (???(?%?)); |
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148 | normalize nodelta; |
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149 | >(compute_max_trace_any_label_cost_is_ok cm cost_labels end_flag |
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150 | status_init status_end trace_any_label) |
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151 | cases(is_next) in match (clock … cm status_init); |
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152 | >(execute_1_ok_clock … status_pre) |
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153 | >commutative_plus >associative_plus >associative_plus @eq_f |
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154 | @commutative_plus |
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155 | ] |
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156 | |3: |
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157 | cases the_trace |
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158 | [1: |
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159 | #status_before #status_after #trace_to_lift |
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160 | normalize @compute_max_trace_label_label_cost_is_ok |
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161 | |2: |
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162 | #status_initial #status_labelled #status_final #labelled_trace #ret_trace |
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163 | normalize |
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164 | >(compute_max_trace_label_return_cost_is_ok cm cost_labels status_labelled status_final ret_trace); |
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165 | >(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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166 | <associative_plus in ⊢ (??%?); |
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167 | >commutative_plus in match ( |
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168 | compute_max_trace_label_return_cost cm cost_labels status_labelled status_final ret_trace |
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169 | + compute_max_trace_label_label_cost cm cost_labels doesnt_end_with_ret status_initial status_labelled labelled_trace); |
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170 | % |
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171 | ] |
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172 | ] |
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173 | qed. |
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174 | |
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175 | let rec compute_trace_label_label_cost_using_paid |
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176 | (cm: ?) |
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177 | (cost_labels: BitVectorTrie costlabel 16) |
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178 | (trace_ends_flag: trace_ends_with_ret) |
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179 | (start_status: Status cm) (final_status: Status cm) |
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180 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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181 | start_status final_status) on the_trace: nat ≝ |
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182 | match the_trace with |
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183 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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184 | compute_paid_trace_label_label cm cost_labels … the_trace + |
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185 | compute_trace_any_label_cost_using_paid … given_trace |
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186 | ] |
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187 | and compute_trace_any_label_cost_using_paid |
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188 | (cm: ?) |
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189 | (cost_labels: BitVectorTrie costlabel 16) |
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190 | (trace_ends_flag: trace_ends_with_ret) |
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191 | (start_status: Status cm) (final_status: Status cm) |
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192 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag start_status final_status) |
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193 | on the_trace: nat ≝ |
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194 | match the_trace with |
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195 | [ tal_base_not_return the_status _ _ _ _ ⇒ 0 |
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196 | | tal_base_return the_status _ _ _ ⇒ 0 |
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197 | | tal_base_call pre_fun_call start_fun_call final _ _ _ call_trace _ ⇒ |
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198 | compute_trace_label_return_cost_using_paid … call_trace |
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199 | | tal_base_tailcall pre_fun_call start_fun_call final _ _ call_trace ⇒ |
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200 | compute_trace_label_return_cost_using_paid … call_trace |
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201 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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202 | _ _ _ call_trace _ final_trace ⇒ |
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203 | let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in |
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204 | let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cm cost_labels end_flag … final_trace in |
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205 | call_trace_cost + final_trace_cost |
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206 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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207 | compute_trace_any_label_cost_using_paid cm cost_labels end_flag |
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208 | status_init status_end tail_trace |
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209 | ] |
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210 | and compute_trace_label_return_cost_using_paid |
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211 | (cm: ?) |
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212 | (cost_labels: BitVectorTrie costlabel 16) |
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213 | (start_status: Status cm) (final_status: Status cm) |
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214 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) start_status final_status) |
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215 | on the_trace: nat ≝ |
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216 | match the_trace with |
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217 | [ tlr_base before after trace_to_lift ⇒ compute_trace_label_label_cost_using_paid … trace_to_lift |
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218 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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219 | let labelled_cost ≝ compute_trace_label_label_cost_using_paid … labelled_trace in |
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220 | let return_cost ≝ compute_trace_label_return_cost_using_paid … ret_trace in |
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221 | labelled_cost + return_cost |
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222 | ]. |
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223 | |
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224 | let rec compute_trace_label_label_cost_using_paid_ok |
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225 | (cm: ?) |
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226 | (cost_labels: BitVectorTrie costlabel 16) |
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227 | (trace_ends_flag: trace_ends_with_ret) |
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228 | (start_status: Status cm) (final_status: Status cm) |
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229 | (the_trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag |
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230 | start_status final_status) on the_trace: |
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231 | compute_trace_label_label_cost_using_paid cm cost_labels … the_trace = |
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232 | compute_max_trace_label_label_cost … the_trace ≝ ? |
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233 | and compute_trace_any_label_cost_using_paid_ok |
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234 | (cm: ?) |
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235 | (cost_labels: BitVectorTrie costlabel 16) |
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236 | (trace_ends_flag: trace_ends_with_ret) |
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237 | (start_status: Status cm) (final_status: Status cm) |
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238 | (the_trace: trace_any_label (ASM_abstract_status cm cost_labels) |
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239 | trace_ends_flag start_status final_status) on the_trace: |
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240 | compute_paid_trace_any_label cm cost_labels trace_ends_flag … the_trace |
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241 | +compute_trace_any_label_cost_using_paid cm cost_labels trace_ends_flag … the_trace |
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242 | =compute_max_trace_any_label_cost cm cost_labels trace_ends_flag … the_trace ≝ ? |
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243 | and compute_trace_label_return_cost_using_paid_ok |
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244 | (cm: ?) |
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245 | (cost_labels: BitVectorTrie costlabel 16) |
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246 | (start_status: Status cm) (final_status: Status cm) |
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247 | (the_trace: trace_label_return (ASM_abstract_status cm cost_labels) |
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248 | start_status final_status) on the_trace: |
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249 | compute_trace_label_return_cost_using_paid cm cost_labels … the_trace = |
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250 | compute_max_trace_label_return_cost cm cost_labels … the_trace ≝ ?. |
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251 | [ cases the_trace #endsf #ss #es #tr #H normalize |
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252 | @compute_trace_any_label_cost_using_paid_ok |
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253 | | cases the_trace |
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254 | [ #ss #fs #H1 #H2 #H3 whd in ⊢ (??(?%%)%); <plus_n_O % |
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255 | | #ss #fs #H1 #H2 whd in ⊢ (??(?%%)%); <plus_n_O % |
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256 | | #sp #ss #sf #H1 #H2 #tr1 #tr2 #H3 |
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257 | whd in ⊢ (???%); whd in ⊢ (??(??%)?); whd in ⊢ (??(?%?)?); |
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258 | >compute_trace_label_return_cost_using_paid_ok in ⊢ (??%?); |
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259 | >commutative_plus in ⊢ (??%?); @eq_f % |
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260 | | #sp #ss #sf #H1 #H2 #tr1 |
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261 | whd in ⊢ (???%); whd in ⊢ (??(??%)?); whd in ⊢ (??(?%?)?); |
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262 | >compute_trace_label_return_cost_using_paid_ok in ⊢ (??%?); |
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263 | >commutative_plus in ⊢ (??%?); @eq_f % |
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264 | | #ef #spfc #ssfc #safc #sf #H1 #H2 #H3 #tr1 #H4 #tr2 whd in ⊢ (??(?%%)%); |
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265 | <compute_trace_any_label_cost_using_paid_ok |
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266 | <compute_trace_label_return_cost_using_paid_ok |
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267 | -compute_trace_label_label_cost_using_paid_ok |
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268 | -compute_trace_label_return_cost_using_paid_ok |
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269 | -compute_trace_any_label_cost_using_paid_ok |
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270 | >commutative_plus in ⊢ (???(?%?)); |
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271 | >commutative_plus in ⊢ (??(??%)?); |
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272 | >associative_plus >associative_plus in ⊢ (???%); @eq_f2 try % |
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273 | <associative_plus <commutative_plus % |
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274 | | #ef #sp #si #se #H1 #tr #H2 #H3 whd in ⊢ (??(?%%)%); >associative_plus @eq_f2 |
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275 | [ % | @compute_trace_any_label_cost_using_paid_ok ] |
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276 | ] |
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277 | | cases the_trace |
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278 | [ #sb #sa #tr normalize @compute_trace_label_label_cost_using_paid_ok |
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279 | | #si #sl #sf #tr1 #tr2 normalize @eq_f2 |
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280 | [ @compute_trace_label_label_cost_using_paid_ok |
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281 | | @compute_trace_label_return_cost_using_paid_ok ]]] |
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282 | qed. |
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283 | |
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284 | include alias "ASM/BitVectorTrie.ma". |
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285 | include alias "arithmetics/nat.ma". |
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286 | include alias "basics/logic.ma". |
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287 | include alias "arithmetics/bigops.ma". |
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288 | |
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289 | (* This shoudl go in bigops! *) |
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290 | theorem bigop_sum_rev: ∀k1,k2,p,B,nil.∀op:Aop B nil.∀f:nat→B. |
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291 | \big[op,nil]_{i<k1+k2|p i} (f i) = |
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292 | 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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293 | #k1 #k2 #p #B #nil #op #f >bigop_sum |
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294 | >commutative_plus @same_bigop #i @leb_elim normalize |
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295 | [2,4: // |
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296 | | #H1 #H2 <plus_minus_m_m // |
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297 | | #H1 #H2 #H3 <plus_minus_m_m //] |
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298 | qed. |
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299 | |
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300 | (* This is taken by sigma_pi.ma that does not compile now *) |
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301 | definition natAop ≝ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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302 | (λa,b,c.sym_eq ??? (associative_plus a b c)). |
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303 | |
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304 | definition natACop ≝ mk_ACop nat 0 natAop commutative_plus. |
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305 | |
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306 | definition natDop ≝ mk_Dop nat 0 natACop times (λn.(sym_eq ??? (times_n_O n))) |
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307 | distributive_times_plus. |
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308 | |
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309 | unification hint 0 ≔ ; |
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310 | S ≟ mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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311 | (λa,b,c.sym_eq ??? (associative_plus a b c)) |
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312 | (* ---------------------------------------- *) ⊢ |
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313 | plus ≡ op ? ? S. |
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314 | |
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315 | unification hint 0 ≔ ; |
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316 | S ≟ mk_ACop nat 0 (mk_Aop nat 0 plus (λa.refl ? a) (λn.sym_eq ??? (plus_n_O n)) |
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317 | (λa,b,c.sym_eq ??? (associative_plus a b c))) commutative_plus |
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318 | (* ---------------------------------------- *) ⊢ |
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319 | plus ≡ op ? ? S. |
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320 | |
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321 | unification hint 0 ≔ ; |
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322 | S ≟ natDop |
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323 | (* ---------------------------------------- *) ⊢ |
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324 | plus ≡ sum ? ? S. |
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325 | |
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326 | unification hint 0 ≔ ; |
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327 | S ≟ natDop |
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328 | (* ---------------------------------------- *) ⊢ |
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329 | times ≡ prod ? ? S. |
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330 | |
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331 | notation > "Σ_{ ident i < n } f" |
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332 | with precedence 20 |
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333 | for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}. |
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334 | |
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335 | notation < "Σ_{ ident i < n } f" |
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336 | with precedence 20 |
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337 | for @{'bigop $n plus 0 (λ${ident i}:$X.true) (λ${ident i}:$Y. $f)}. |
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338 | |
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339 | definition tech_cost_of_label0: |
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340 | ∀cost_labels: BitVectorTrie costlabel 16. |
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341 | ∀cost_map: identifier_map CostTag nat. |
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342 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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343 | ∀ctrace:list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k). |
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344 | ∀i,p. present … cost_map (nth_safe ? i ctrace p). |
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345 | #cost_labels #cost_map #codom_dom #ctrace #i #p |
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346 | cases (nth_safe … i ctrace ?) normalize #id * #id_pc #K |
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347 | lapply (codom_dom … K) #k_pres >(lookup_lookup_present … k_pres) |
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348 | % #abs destruct (abs) |
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349 | qed. |
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350 | |
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351 | include alias "arithmetics/nat.ma". |
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352 | include alias "basics/logic.ma". |
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353 | |
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354 | lemma ltb_rect: |
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355 | ∀P:Type[0].∀n,m. (n < m → P) → (¬ n < m → P) → P. |
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356 | #P #n #m lapply (refl … (ltb n m)) cases (ltb n m) in ⊢ (???% → %); #E #H1 #H2 |
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357 | [ @H1 @leb_true_to_le @E | @H2 @leb_false_to_not_le @E ] |
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358 | qed. |
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359 | |
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360 | lemma same_ltb_rect: |
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361 | ∀P,n,m,H1,H2,n',m',H1',H2'. |
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362 | ltb n m = ltb n' m' → (∀x,y. H1 x = H1' y) → (∀x,y. H2 x = H2' y) → |
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363 | ltb_rect P n m H1 H2 = |
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364 | ltb_rect P n' m' H1' H2'. |
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365 | #P #n #m #H1 #H2 #n' #m' #H1' #H2' #E #K1 #K2 whd in ⊢ (??%?); |
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366 | cut (∀xxx,yyy,xxx',yyy'. |
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367 | match ltb n m |
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368 | return λx:bool. |
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369 | eq bool (ltb n m) x |
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370 | → (lt n m → P) → (Not (lt n m) → P) → P |
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371 | with |
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372 | [ true ⇒ |
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373 | λE0:eq bool (ltb n m) true. |
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374 | λH10:lt n m → P. |
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375 | λH20:Not (lt n m) → P. H10 (xxx E0) |
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376 | | false ⇒ |
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377 | λE0:eq bool (ltb n m) false. |
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378 | λH10:lt n m → P. |
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379 | λH20:Not (lt n m) → P. H20 (yyy E0)] |
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380 | (refl … (ltb n m)) H1 H2 = |
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381 | match ltb n' m' |
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382 | return λx:bool. |
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383 | eq bool (ltb n' m') x |
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384 | → (lt n' m' → P) → (Not (lt n' m') → P) → P |
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385 | with |
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386 | [ true ⇒ |
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387 | λE0:eq bool (ltb n' m') true. |
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388 | λH10:lt n' m' → P. |
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389 | λH20:Not (lt n' m') → P. H10 (xxx' E0) |
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390 | | false ⇒ |
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391 | λE0:eq bool (ltb n' m') false. |
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392 | λH10:lt n' m' → P. |
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393 | λH20:Not (lt n' m') → P. H20 (yyy' E0)] |
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394 | (refl … (ltb n' m')) H1' H2' |
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395 | ) [2: #X @X] |
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396 | >E cases (ltb n' m') #xxx #yyy #xxx' #yyy' normalize |
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397 | [ @K1 | @K2 ] |
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398 | qed. |
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399 | |
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400 | definition tech_cost_of_label: |
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401 | ∀cost_labels: BitVectorTrie costlabel 16. |
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402 | ∀cost_map: identifier_map CostTag nat. |
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403 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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404 | list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k) → |
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405 | nat → nat |
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406 | ≝ λcost_labels,cost_map,codom_dom,ctrace,i. |
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407 | ltb_rect ? i (|ctrace|) |
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408 | (λH. lookup_present ?? cost_map (nth_safe ? i ctrace H) ?) |
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409 | (λ_.0). |
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410 | @tech_cost_of_label0 @codom_dom |
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411 | qed. |
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412 | |
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413 | lemma tech_cost_of_label_shift: |
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414 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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415 | i < |l2| → |
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416 | tech_cost_of_label cost_labels cost_map codom_dom l2 i = |
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417 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) (i+|l1|). |
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418 | #cost_labels #cost_Map #codom_dom #l1 #l2 #i #H |
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419 | whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect |
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420 | [ @(ltb_rect ? i (|l2|)) @(ltb_rect ? (i+|l1|) (|l1@l2|)) #K1 #K2 |
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421 | whd in match ltb; normalize nodelta |
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422 | [1: >le_to_leb_true try assumption applyS le_to_leb_true / by / |
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423 | |4: >not_le_to_leb_false try assumption applyS not_le_to_leb_false |
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424 | change with (¬ ? ≤ ?) in K1; applyS K1 |
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425 | |2: @⊥ @(absurd (i+|l1| < |l1@l2|)) / by / >length_append |
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426 | applyS (monotonic_lt_plus_r … (|l1|)) / by / |
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427 | |3: @⊥ @(absurd ?? K2) >length_append in K1; #K1 /2 by lt_plus_to_lt_l/ ] |
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428 | | #H1 #H2 |
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429 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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430 | <(shift_nth_safe … H1) #p % |
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431 | | / by / ] |
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432 | qed. |
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433 | |
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434 | lemma tech_cost_of_label_prefix: |
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435 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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436 | i < |l1| → |
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437 | tech_cost_of_label cost_labels cost_map codom_dom l1 i = |
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438 | tech_cost_of_label cost_labels cost_map codom_dom (l1@l2) i. |
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439 | #cost_labels #cost_map #codom_dom #l1 #l2 #i #H |
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440 | whd in match tech_cost_of_label; normalize nodelta @same_ltb_rect |
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441 | [1: |
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442 | whd in match ltb; normalize nodelta |
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443 | >(le_to_leb_true … H) applyS le_to_leb_true |
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444 | >length_append whd in H; >commutative_plus @le_plus_a assumption |
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445 | |2: |
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446 | #K1 #K2 |
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447 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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448 | <(shift_nth_prefix … l1 i l2 K1 K2) // |
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449 | |3: |
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450 | #_ #_ % |
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451 | ] |
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452 | qed. |
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453 | |
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454 | let rec compute_trace_label_return_using_paid_ok_with_trace |
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455 | (cm: ?) (cost_labels: BitVectorTrie costlabel 16) |
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456 | (cost_map: identifier_map CostTag nat) |
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457 | (initial: Status cm) (final: Status cm) |
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458 | (trace: trace_label_return (ASM_abstract_status cm cost_labels) initial final) |
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459 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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460 | on trace: |
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461 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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462 | ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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463 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres). |
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464 | let ctrace ≝ flatten_trace_label_return (ASM_abstract_status cm cost_labels) … trace in |
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465 | compute_trace_label_return_cost_using_paid cm … trace = |
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466 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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467 | ≝ ? |
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468 | and compute_trace_any_label_using_paid_ok_with_trace |
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469 | (cm: ?) (cost_labels: BitVectorTrie costlabel 16) |
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470 | (trace_ends_flag: trace_ends_with_ret) |
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471 | (cost_map: identifier_map CostTag nat) |
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472 | (initial: Status cm) (final: Status cm) |
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473 | (trace: trace_any_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final) |
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474 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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475 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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476 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
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477 | on trace: |
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478 | ∀unrepeating_witness: tal_unrepeating … trace. |
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479 | let ctrace ≝ flatten_trace_any_label (ASM_abstract_status cm cost_labels) … trace_ends_flag … trace in |
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480 | compute_trace_any_label_cost_using_paid … trace = |
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481 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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482 | ≝ ? |
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483 | and compute_trace_label_label_using_paid_ok_with_trace |
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484 | (cm: ?) (cost_labels: BitVectorTrie costlabel 16) |
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485 | (trace_ends_flag: trace_ends_with_ret) |
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486 | (cost_map: identifier_map CostTag nat) |
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487 | (initial: Status cm) (final: Status cm) |
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488 | (trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final) |
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489 | (unrepeating_witness: tll_unrepeating … trace) |
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490 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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491 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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492 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
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493 | on trace: |
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494 | ∀unrepeating_witness: tll_unrepeating … trace. |
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495 | let ctrace ≝ flatten_trace_label_label (ASM_abstract_status cm cost_labels) … trace in |
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496 | compute_trace_label_label_cost_using_paid … trace = |
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497 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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498 | ≝ ?. |
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499 | cases trace normalize nodelta |
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500 | [ #sb #sa #tr #unrepeating_witness #dom_codom whd in ⊢ (??%?); |
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501 | whd in match (flatten_trace_label_return ????); |
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502 | @compute_trace_label_label_using_paid_ok_with_trace |
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503 | assumption |
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504 | | #si #sl #sf #tr1 #tr2 #unrepeating_witness #dom_codom |
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505 | whd in ⊢ (??%?); |
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506 | whd in match (flatten_trace_label_return ????); |
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507 | >append_length >bigop_sum_rev >commutative_plus @eq_f2 |
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508 | [ >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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509 | -compute_trace_label_return_using_paid_ok_with_trace |
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510 | [1: |
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511 | @same_bigop [//] #i #H #_ -dom_codom @tech_cost_of_label_shift // |
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512 | |2: |
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513 | inversion unrepeating_witness |
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514 | #tll_unrepeating #tlr_unrepeating #_ assumption |
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515 | ] |
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516 | | >(compute_trace_label_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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517 | -compute_trace_label_label_using_paid_ok_with_trace |
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518 | [1: |
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519 | @same_bigop [//] #i #H #_ @(tech_cost_of_label_prefix … H) |
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520 | |2,3: |
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521 | inversion unrepeating_witness |
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522 | #tll_unrepeating #tlr_unrepeating #_ assumption |
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523 | ] |
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524 | ] |
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525 | |9: |
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526 | #end_flag #start_status #end_status #trace_any_label #costed_assm |
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527 | #unrepeating_witness' |
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528 | whd in ⊢ (??%?); whd in ⊢ (??(?%?)?); |
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529 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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530 | [1: |
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531 | whd in match (flatten_trace_label_label ?????); |
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532 | >bigop_0 in ⊢ (???%); >commutative_plus @eq_f2 |
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533 | [1: |
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534 | @same_bigop [//] #i #H #_ -dom_codom >(plus_n_O i) >plus_n_Sm |
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535 | <(tech_cost_of_label_shift ??? [?] ? i) try assumption <(plus_n_O i) % |
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536 | |2: |
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537 | change with (? = lookup_present ? ? ? ? ?) |
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538 | generalize in match (tech_cost_of_label0 ? ? ? ? ? ?); |
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539 | whd in ⊢ (∀p:????%.???(????%?)); |
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540 | whd in costed_assm; lapply costed_assm whd in match (as_label ??); |
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541 | inversion (lookup_opt ? ? (program_counter … cm start_status) cost_labels) |
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542 | [1: |
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543 | #_ #absurd @⊥ cases absurd #absurd @absurd % |
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544 | |2: |
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545 | normalize nodelta #cost_label #Some_assm #_ #p |
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546 | cases (dom_codom ? p ? Some_assm) |
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547 | cases (block_cost ???) |
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548 | #cost #block_cost_assm |
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549 | cases (block_cost_assm ??? trace_any_label ??) |
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550 | try @refl assumption |
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551 | ] |
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552 | ] |
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553 | |2: |
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554 | assumption |
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555 | ] |
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556 | |3: |
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557 | #start_status #final_status #execute_assm #classifier_assm #costed_assm |
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558 | #unrepeating_witness |
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559 | % |
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560 | |4: |
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561 | #start_status #final_status #execute_assm #classifier_assm #unrepeating_witness |
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562 | % |
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563 | |5: |
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564 | #status_pre_fun_call #status_start_fun_call #status_final #execute_assm |
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565 | #classifier_assm #after_return_assm #trace_label_return #costed_assm #unrepeating_witness |
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566 | whd in ⊢ (??%?); |
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567 | @(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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568 | assumption |
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569 | |6: |
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570 | #status_pre_fun_call #status_start_fun_call #status_final #execute_assm |
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571 | #classifier_assm #trace_label_return #unrepeating_witness |
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572 | whd in ⊢ (??%?); |
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573 | @(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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574 | assumption |
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575 | |7: |
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576 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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577 | #status_final #execute_assm #classifier_assm #after_return_assm #trace_label_return |
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578 | #costed_assm #trace_any_label #unrepeating_witness |
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579 | whd in ⊢ (??%?); |
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580 | >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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581 | [1: |
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582 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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583 | [1: |
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584 | >length_append >bigop_sum_rev >commutative_plus @eq_f2 |
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585 | [ @same_bigop [2: #i #H #_ -dom_codom @tech_cost_of_label_shift assumption |
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586 | |1: #i #H % ] |
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587 | | @same_bigop [#i #H %] #i #H #_ @(tech_cost_of_label_prefix … H) |
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588 | ] |
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589 | |2: |
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590 | inversion unrepeating_witness |
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591 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
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592 | ] |
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593 | |2: |
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594 | inversion unrepeating_witness |
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595 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
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596 | ] |
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597 | |8: |
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598 | #end_flag #status_pre_fun_call #status_start_fun_call #status_final |
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599 | #execute_assm #trace_any_label #classifier_assm #costed_assm #unrepeating_witness |
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600 | whd in ⊢ (??%?); |
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601 | @(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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602 | inversion unrepeating_witness |
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603 | #memb_1 #tal_unrepeating #_ assumption |
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604 | ] |
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605 | qed. |
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606 | |
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607 | lemma compute_max_trace_label_return_cost_ok_with_trace_aux: |
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608 | ∀code_memory: BitVectorTrie Byte 16. |
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609 | ∀cost_labels: BitVectorTrie costlabel 16. |
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610 | ∀cost_map: identifier_map CostTag nat. |
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611 | ∀initial: Status code_memory. |
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612 | ∀final: Status code_memory. |
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613 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
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614 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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615 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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616 | ∀dom_codom: (∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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617 | pi1 … (block_cost code_memory pc cost_labels) = lookup_present … k_pres). |
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618 | let ctrace ≝ flatten_trace_label_return … trace in |
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619 | clock … code_memory … final = |
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620 | clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)). |
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621 | #code_memory #cost_labels #cost_map |
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622 | #initial #final #trace #unrepeating_witness #codom_dom #dom_codom normalize nodelta |
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623 | <compute_trace_label_return_using_paid_ok_with_trace try assumption |
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624 | >commutative_plus >compute_trace_label_return_cost_using_paid_ok // |
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625 | qed. |
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626 | |
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627 | theorem compute_max_trace_label_return_cost_ok_with_trace: |
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628 | ∀code_memory: BitVectorTrie Byte 16. |
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629 | ∀cost_labels: BitVectorTrie costlabel 16. |
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630 | ∀cost_labels_injective: |
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631 | (∀pc,pc',l. |
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632 | lookup_opt costlabel 16 pc cost_labels=Some costlabel l |
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633 | →lookup_opt costlabel 16 pc' cost_labels=Some costlabel l→pc=pc'). |
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634 | ∀initial: Status code_memory. |
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635 | ∀final: Status code_memory. |
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636 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
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637 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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638 | let cost_map ≝ traverse_code code_memory cost_labels cost_labels_injective in |
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639 | let ctrace ≝ flatten_trace_label_return … trace in |
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640 | clock … code_memory … final = clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map ? ctrace i)). |
---|
641 | [1: |
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642 | #code_memory #cost_labels #cost_labels_injective #initial #final #trace #unrepeating_witness |
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643 | @compute_max_trace_label_return_cost_ok_with_trace_aux try assumption |
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644 | |2: |
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645 | skip |
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646 | ] |
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647 | normalize nodelta |
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648 | cases (traverse_code ???) |
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649 | #cost_map * #dom_codom #codom_dom try assumption |
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650 | #pc #k #lookup_opt_assm @(dom_codom … lookup_opt_assm) |
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651 | lapply (sym_eq ? ? ? lookup_opt_assm) |
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652 | -lookup_opt_assm #lookup_opt_assm |
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653 | @nat_of_bitvector_lt_bound |
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654 | qed. |
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655 | |
---|
656 | include "utilities/permutations.ma". |
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657 | |
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658 | definition map_of_sigma_map: |
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659 | ∀P. |
---|
660 | (Σmap:identifier_map CostTag ℕ.P map) → identifier_map CostTag ℕ |
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661 | ≝ λP,x.x. |
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662 | |
---|
663 | lemma costlabel_map_of_as_cost_labels_map_ok: |
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664 | ∀ccode1,ccode2,cost_labels_injective,tl,Htl,PRF,PRF2. |
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665 | lookup_present CostTag ℕ (compute_costs ccode1 ccode2 cost_labels_injective) tl |
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666 | PRF |
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667 | = |
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668 | lookup_present CostTag ℕ (compute_costs ccode1 ccode2 cost_labels_injective) |
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669 | (as_cost_get_label (ASM_abstract_status (load_code_memory ccode1) ccode2) |
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670 | «tl,Htl») PRF2. |
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671 | // |
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672 | qed. |
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673 | |
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674 | lemma tech_cost_sum_eq_as_cost : |
---|
675 | ∀compiled_code: labelled_object_code. |
---|
676 | let cost_labels ≝ \fst (\snd compiled_code) in |
---|
677 | ∀cost_labels_injective: |
---|
678 | (∀pc,pc',l. |
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679 | lookup_opt … pc cost_labels = Some … l |
---|
680 | → lookup_opt … pc' cost_labels = Some … l → pc = pc'). |
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681 | let cost_map ≝ compute_costs (\fst compiled_code) (\fst (\snd compiled_code)) cost_labels_injective in |
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682 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … k). |
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683 | ∀trace. |
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684 | (Σ_{i < |trace|}(tech_cost_of_label cost_labels (map_of_sigma_map … cost_map) codom_dom trace i)) = |
---|
685 | (Σ_{l ∈ trace}(ASM_cost_map compiled_code cost_labels_injective l)). |
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686 | * #ccode1 #ccode2 #cost_labels_injective #codom_dom #trace |
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687 | @(list_elim_left … trace) |
---|
688 | [ % |
---|
689 | | #tl #hd #IH >append_length >commutative_plus |
---|
690 | >(fold_permute … (hd@[tl]) (tl::hd)) [2: @perm_swap_append ] |
---|
691 | whd in ⊢ (??%%); <IH |
---|
692 | <tech_cost_of_label_shift [2: //] |
---|
693 | cut (∀x,y,z,w.x = y → z = w → x + z = y + w) [ // ] #APP @APP -APP |
---|
694 | [ cases tl -tl #tl #Htl |
---|
695 | change with (lookup_present ????? = ?) |
---|
696 | whd in ⊢ (???(%???)); normalize nodelta |
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697 | generalize in ⊢ (???(?????%)); #p |
---|
698 | @costlabel_map_of_as_cost_labels_map_ok |
---|
699 | | @sym_eq @same_bigop /2 by tech_cost_of_label_prefix/ |
---|
700 | ] |
---|
701 | ] |
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702 | qed. |
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