[1898] | 1 | include "ASM/ASMCostsSplit.ma". |
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[1499] | 2 | include "ASM/WellLabeled.ma". |
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| 3 | include "ASM/Status.ma". |
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[1544] | 4 | include "common/StructuredTraces.ma". |
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[1619] | 5 | include "arithmetics/bigops.ma". |
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[1898] | 6 | |
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[1619] | 7 | include alias "arithmetics/nat.ma". |
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| 8 | include alias "basics/logic.ma". |
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[1498] | 9 | |
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[1506] | 10 | let rec compute_max_trace_label_label_cost |
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[1692] | 11 | (cm: ?) |
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[2498] | 12 | (cost_labels: costlabel_map) |
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[1544] | 13 | (trace_ends_flag: trace_ends_with_ret) |
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[1692] | 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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[1544] | 16 | start_status final_status) on the_trace: nat ≝ |
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[1506] | 17 | match the_trace with |
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| 18 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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[1544] | 19 | compute_max_trace_any_label_cost … given_trace |
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[1506] | 20 | ] |
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[1544] | 21 | and compute_max_trace_any_label_cost |
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[1692] | 22 | (cm: ?) |
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[2498] | 23 | (cost_labels: costlabel_map) |
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[1544] | 24 | (trace_ends_flag: trace_ends_with_ret) |
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[1692] | 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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[1544] | 27 | on the_trace: nat ≝ |
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[1506] | 28 | match the_trace with |
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[1693] | 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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[1692] | 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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[1544] | 35 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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[1692] | 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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[1544] | 38 | let call_trace_cost ≝ compute_max_trace_label_return_cost … call_trace in |
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[1693] | 39 | let final_trace_cost ≝ compute_max_trace_any_label_cost cm cost_labels end_flag … final_trace in |
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[1506] | 40 | call_trace_cost + current_instruction_cost + final_trace_cost |
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[1544] | 41 | | tal_step_default end_flag status_pre status_init status_end _ tail_trace _ _ ⇒ |
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[1692] | 42 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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[1544] | 43 | let tail_trace_cost ≝ |
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[1693] | 44 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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[1544] | 45 | status_init status_end tail_trace |
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| 46 | in |
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[1506] | 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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[1692] | 50 | (cm: ?) |
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[2498] | 51 | (cost_labels: costlabel_map) |
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[1692] | 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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[1506] | 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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[1500] | 62 | |
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[1544] | 63 | include alias "arithmetics/nat.ma". |
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| 64 | |
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[1579] | 65 | let rec compute_max_trace_label_label_cost_is_ok |
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[1693] | 66 | (cm: ?) |
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[2498] | 67 | (cost_labels: costlabel_map) |
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[1579] | 68 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1579] | 71 | start_status final_status) on the_trace: |
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[1693] | 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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[1579] | 73 | and compute_max_trace_any_label_cost_is_ok |
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[1693] | 74 | (cm: ?) |
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[2498] | 75 | (cost_labels: costlabel_map) |
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[1579] | 76 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1579] | 79 | on the_trace: |
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[1693] | 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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[1579] | 81 | and compute_max_trace_label_return_cost_is_ok |
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[1693] | 82 | (cm: ?) |
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[2498] | 83 | (cost_labels: costlabel_map) |
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[1693] | 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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[1579] | 86 | on the_trace: |
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[1693] | 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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[1579] | 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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[1693] | 96 | change with (current_instruction_cost cm start_status) in ⊢ (???(?%?)); |
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[1898] | 97 | cases(is_next) @execute_1_ok_clock |
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[1579] | 98 | |3: |
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[1693] | 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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[1898] | 104 | @execute_1_ok_clock |
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[1693] | 105 | |4: |
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[1579] | 106 | #end_flag #status_pre_fun_call #status_start_fun_call #status_after_fun_call |
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[1693] | 107 | #status_final #is_next #is_call #is_after_return #call_trace #not_costed #final_trace |
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[1579] | 108 | change with ( |
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[1693] | 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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[1579] | 112 | call_trace_cost + current_instruction_cost + final_trace_cost) in ⊢ (???(?%?)); |
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| 113 | normalize nodelta; |
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[1693] | 114 | >(compute_max_trace_any_label_cost_is_ok … cost_labels end_flag status_after_fun_call |
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[1579] | 115 | status_final final_trace) |
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[1693] | 116 | >(compute_max_trace_label_return_cost_is_ok … cost_labels status_start_fun_call |
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[1579] | 117 | status_after_fun_call call_trace) |
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[1693] | 118 | cases(is_next) in match (clock … cm status_start_fun_call); |
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[1898] | 119 | >(execute_1_ok_clock cm status_pre_fun_call) |
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[1579] | 120 | <associative_plus in ⊢ (??%?); |
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| 121 | <commutative_plus in match ( |
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[1693] | 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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[1579] | 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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[1693] | 131 | let current_instruction_cost ≝ current_instruction_cost cm status_pre in |
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[1579] | 132 | let tail_trace_cost ≝ |
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[1693] | 133 | compute_max_trace_any_label_cost cm cost_labels end_flag |
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[1579] | 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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[1693] | 138 | >(compute_max_trace_any_label_cost_is_ok cm cost_labels end_flag |
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[1579] | 139 | status_init status_end trace_any_label) |
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[1693] | 140 | cases(is_next) in match (clock … cm status_init); |
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[1898] | 141 | >(execute_1_ok_clock … status_pre) |
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[1619] | 142 | >commutative_plus >associative_plus >associative_plus @eq_f |
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| 143 | @commutative_plus |
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[1579] | 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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[1693] | 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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[1579] | 155 | <associative_plus in ⊢ (??%?); |
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| 156 | >commutative_plus in match ( |
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[1693] | 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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[1579] | 159 | % |
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| 160 | ] |
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[1693] | 161 | ] |
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[1579] | 162 | qed. |
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[1549] | 163 | |
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[1554] | 164 | let rec compute_trace_label_label_cost_using_paid |
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[1693] | 165 | (cm: ?) |
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[2498] | 166 | (cost_labels: costlabel_map) |
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[1554] | 167 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1554] | 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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[1693] | 173 | compute_paid_trace_label_label cm cost_labels … the_trace + |
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[1554] | 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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[1693] | 177 | (cm: ?) |
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[2498] | 178 | (cost_labels: costlabel_map) |
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[1554] | 179 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1554] | 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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[1693] | 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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[1554] | 188 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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[1693] | 189 | _ _ _ call_trace _ final_trace ⇒ |
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[1554] | 190 | let call_trace_cost ≝ compute_trace_label_return_cost_using_paid … call_trace in |
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[1693] | 191 | let final_trace_cost ≝ compute_trace_any_label_cost_using_paid cm cost_labels end_flag … final_trace in |
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[1554] | 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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[1693] | 194 | compute_trace_any_label_cost_using_paid cm cost_labels end_flag |
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[1554] | 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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[1693] | 198 | (cm: ?) |
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[2498] | 199 | (cost_labels: costlabel_map) |
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[1693] | 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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[1554] | 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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[1693] | 212 | (cm: ?) |
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[2498] | 213 | (cost_labels: costlabel_map) |
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[1554] | 214 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1554] | 217 | start_status final_status) on the_trace: |
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[1693] | 218 | compute_trace_label_label_cost_using_paid cm cost_labels … the_trace = |
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[1554] | 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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[1693] | 221 | (cm: ?) |
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[2498] | 222 | (cost_labels: costlabel_map) |
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[1554] | 223 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1554] | 226 | trace_ends_flag start_status final_status) on the_trace: |
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[1693] | 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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[1554] | 230 | and compute_trace_label_return_cost_using_paid_ok |
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[1693] | 231 | (cm: ?) |
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[2498] | 232 | (cost_labels: costlabel_map) |
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[1693] | 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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[1554] | 235 | start_status final_status) on the_trace: |
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[1693] | 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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[1554] | 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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[1693] | 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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[1554] | 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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[1506] | 260 | ] |
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[1554] | 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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[1506] | 267 | |
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[1693] | 268 | include alias "ASM/BitVectorTrie.ma". |
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| 269 | |
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[1935] | 270 | (* |
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[1554] | 271 | let rec compute_cost_trace_label_label |
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[1693] | 272 | (cm: BitVectorTrie Byte 16) |
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[2498] | 273 | (cost_labels: costlabel_map) |
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[1554] | 274 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1561] | 277 | start_status final_status) on the_trace: |
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[1693] | 278 | list (Σk:costlabel. ∃pc: Word. lookup_opt … pc cost_labels = Some … k) ≝ |
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[1554] | 279 | match the_trace with |
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| 280 | [ tll_base ends_flag initial final given_trace labelled_proof ⇒ |
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[1693] | 281 | let pc ≝ program_counter ? cm initial in |
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[1554] | 282 | let label ≝ |
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[1927] | 283 | match lookup_opt … pc cost_labels return λx: option ?. x ≠ None … → costlabel with |
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[1554] | 284 | [ None ⇒ λabs. ⊥ |
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| 285 | | Some l ⇒ λ_. l ] labelled_proof in |
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[1693] | 286 | (mk_Sig ?? label ?)::compute_cost_trace_any_label cm cost_labels ends_flag initial final … given_trace |
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[1554] | 287 | ] |
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| 288 | and compute_cost_trace_any_label |
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[1693] | 289 | (cm: BitVectorTrie Byte 16) |
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[2498] | 290 | (cost_labels: costlabel_map) |
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[1554] | 291 | (trace_ends_flag: trace_ends_with_ret) |
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[1693] | 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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[1561] | 294 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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[1554] | 295 | match the_trace with |
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[1693] | 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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[1554] | 300 | | tal_step_call end_flag pre_fun_call start_fun_call after_fun_call final |
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[1693] | 301 | _ _ _ call_trace _ final_trace ⇒ |
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[1554] | 302 | let call_cost_trace ≝ compute_cost_trace_label_return … call_trace in |
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[1693] | 303 | let final_cost_trace ≝ compute_cost_trace_any_label cm cost_labels end_flag … final_trace in |
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[1554] | 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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[1693] | 306 | compute_cost_trace_any_label cm cost_labels end_flag |
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[1554] | 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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[1693] | 310 | (cm: BitVectorTrie Byte 16) |
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[2498] | 311 | (cost_labels: costlabel_map) |
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[1693] | 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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[1561] | 314 | on the_trace: list (Σk:costlabel. ∃pc. lookup_opt … pc cost_labels = Some … k) ≝ |
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[1554] | 315 | match the_trace with |
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[1693] | 316 | [ tlr_base before after trace_to_lift ⇒ compute_cost_trace_label_label cm … trace_to_lift |
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[1554] | 317 | | tlr_step initial labelled final labelled_trace ret_trace ⇒ |
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[1693] | 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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[1554] | 320 | labelled_cost @ return_cost |
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| 321 | ]. |
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[1927] | 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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[1554] | 329 | qed. |
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[1935] | 330 | *) |
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[1500] | 331 | |
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[1650] | 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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[1693] | 335 | include alias "arithmetics/bigops.ma". |
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| 336 | |
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[1570] | 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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[1693] | 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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[1570] | 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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[1561] | 346 | qed. |
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| 347 | |
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[1570] | 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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[1619] | 379 | notation > "Σ_{ ident i < n } f" |
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[1570] | 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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[1619] | 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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[1570] | 387 | definition tech_cost_of_label0: |
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[2498] | 388 | ∀cost_labels: costlabel_map. |
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[1570] | 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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[1964] | 391 | ∀ctrace:list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k). |
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[1570] | 392 | ∀i,p. present … cost_map (nth_safe ? i ctrace p). |
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[1964] | 393 | #cost_labels #cost_map #codom_dom #ctrace #i #p |
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[1570] | 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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[1564] | 397 | qed. |
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| 398 | |
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[1619] | 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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[1570] | 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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[1564] | 406 | qed. |
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| 407 | |
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[1570] | 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 ] |
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[1564] | 446 | qed. |
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| 447 | |
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[1570] | 448 | definition tech_cost_of_label: |
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[2498] | 449 | ∀cost_labels: costlabel_map. |
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[1570] | 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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[1964] | 452 | list (Σk:costlabel.∃b.lookup_opt … b cost_labels = Some ? k) → |
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[1570] | 453 | nat → nat |
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[1964] | 454 | ≝ λcost_labels,cost_map,codom_dom,ctrace,i. |
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[1570] | 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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[1564] | 459 | qed. |
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| 460 | |
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[1570] | 461 | lemma tech_cost_of_label_shift: |
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[1964] | 462 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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[1620] | 463 | i < |l2| → |
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[1964] | 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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[1619] | 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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[1964] | 470 | [1: >le_to_leb_true try assumption applyS le_to_leb_true / by / |
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[1619] | 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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[1964] | 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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[1619] | 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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[1964] | 477 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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[1619] | 478 | <(shift_nth_safe … H1) #p % |
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[1964] | 479 | | / by / ] |
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[1619] | 480 | qed. |
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[1570] | 481 | |
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[1695] | 482 | lemma tech_cost_of_label_prefix: |
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[1964] | 483 | ∀cost_labels,cost_map,codom_dom,l1,l2,i. |
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[1695] | 484 | i < |l1| → |
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[1964] | 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. |
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| 487 | #cost_labels #cost_map #codom_dom #l1 #l2 #i #H |
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[1695] | 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 |
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| 493 | |2: |
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| 494 | #K1 #K2 |
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[1964] | 495 | generalize in match (tech_cost_of_label0 ??? (l1@l2) ??); |
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[1695] | 496 | <(shift_nth_prefix … l1 i l2 K1 K2) // |
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| 497 | |3: |
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| 498 | #_ #_ % |
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| 499 | ] |
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| 500 | qed. |
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| 501 | |
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| 502 | let rec compute_trace_label_return_using_paid_ok_with_trace |
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[2498] | 503 | (cm: ?) (cost_labels: costlabel_map) |
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[1570] | 504 | (cost_map: identifier_map CostTag nat) |
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[1693] | 505 | (initial: Status cm) (final: Status cm) |
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| 506 | (trace: trace_label_return (ASM_abstract_status cm cost_labels) initial final) |
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[1570] | 507 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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| 508 | on trace: |
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[1921] | 509 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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[1900] | 510 | ∀dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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[1929] | 511 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres). |
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[1964] | 512 | let ctrace ≝ flatten_trace_label_return (ASM_abstract_status cm cost_labels) … trace in |
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[1695] | 513 | compute_trace_label_return_cost_using_paid cm … trace = |
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[1964] | 514 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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[1620] | 515 | ≝ ? |
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[1695] | 516 | and compute_trace_any_label_using_paid_ok_with_trace |
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[2498] | 517 | (cm: ?) (cost_labels: costlabel_map) |
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[1620] | 518 | (trace_ends_flag: trace_ends_with_ret) |
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| 519 | (cost_map: identifier_map CostTag nat) |
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[1693] | 520 | (initial: Status cm) (final: Status cm) |
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[1695] | 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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[1900] | 523 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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[1929] | 524 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
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[1695] | 525 | on trace: |
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[1921] | 526 | ∀unrepeating_witness: tal_unrepeating … trace. |
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[1964] | 527 | let ctrace ≝ flatten_trace_any_label (ASM_abstract_status cm cost_labels) … trace_ends_flag … trace in |
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[1695] | 528 | compute_trace_any_label_cost_using_paid … trace = |
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[1964] | 529 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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[1695] | 530 | ≝ ? |
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| 531 | and compute_trace_label_label_using_paid_ok_with_trace |
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[2498] | 532 | (cm: ?) (cost_labels: costlabel_map) |
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[1695] | 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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[1693] | 536 | (trace: trace_label_label (ASM_abstract_status cm cost_labels) trace_ends_flag initial final) |
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[1921] | 537 | (unrepeating_witness: tll_unrepeating … trace) |
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[1620] | 538 | (codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k)) |
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[1900] | 539 | (dom_codom:(∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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[1929] | 540 | pi1 … (block_cost cm pc cost_labels) = lookup_present … k_pres)) |
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[1620] | 541 | on trace: |
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[1921] | 542 | ∀unrepeating_witness: tll_unrepeating … trace. |
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[1964] | 543 | let ctrace ≝ flatten_trace_label_label (ASM_abstract_status cm cost_labels) … trace in |
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[1695] | 544 | compute_trace_label_label_cost_using_paid … trace = |
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[1964] | 545 | (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)) |
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[1570] | 546 | ≝ ?. |
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[1695] | 547 | cases trace normalize nodelta |
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[1921] | 548 | [ #sb #sa #tr #unrepeating_witness #dom_codom whd in ⊢ (??%?); |
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[1964] | 549 | whd in match (flatten_trace_label_return ????); |
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[1929] | 550 | @compute_trace_label_label_using_paid_ok_with_trace |
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[1921] | 551 | assumption |
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| 552 | | #si #sl #sf #tr1 #tr2 #unrepeating_witness #dom_codom |
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[1695] | 553 | whd in ⊢ (??%?); |
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[1964] | 554 | whd in match (flatten_trace_label_return ????); |
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[1695] | 555 | >append_length >bigop_sum_rev >commutative_plus @eq_f2 |
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[1929] | 556 | [ >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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[1695] | 557 | -compute_trace_label_return_using_paid_ok_with_trace |
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[1921] | 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 |
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| 562 | #tll_unrepeating #tlr_unrepeating #_ assumption |
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| 563 | ] |
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[1929] | 564 | | >(compute_trace_label_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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[1695] | 565 | -compute_trace_label_label_using_paid_ok_with_trace |
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[1921] | 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 |
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| 571 | ] |
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[1695] | 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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[1923] | 575 | #unrepeating_witness' |
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[1695] | 576 | whd in ⊢ (??%?); whd in ⊢ (??(?%?)?); |
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[1929] | 577 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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[1695] | 578 | [1: |
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[1964] | 579 | whd in match (flatten_trace_label_label ?????); |
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[1921] | 580 | >bigop_0 in ⊢ (???%); >commutative_plus @eq_f2 |
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[1900] | 581 | [1: |
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[1921] | 582 | @same_bigop [//] #i #H #_ -dom_codom >(plus_n_O i) >plus_n_Sm |
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[1964] | 583 | <(tech_cost_of_label_shift ??? [?] ? i) try assumption <(plus_n_O i) % |
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[1900] | 584 | |2: |
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[1921] | 585 | change with (? = lookup_present ? ? ? ? ?) |
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[1964] | 586 | generalize in match (tech_cost_of_label0 ? ? ? ? ? ?); |
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[1921] | 587 | normalize in match (nth_safe ? ? ? ?); |
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[1927] | 588 | whd in costed_assm; lapply costed_assm whd in match (as_label ??); |
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[1921] | 589 | inversion (lookup_opt ? ? (program_counter … cm start_status) cost_labels) |
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| 590 | [1: |
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[1927] | 591 | #_ #absurd @⊥ cases absurd #absurd @absurd % |
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[1921] | 592 | |2: |
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| 593 | normalize nodelta #cost_label #Some_assm #_ #p |
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| 594 | cases (dom_codom ? p ? Some_assm) |
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[1929] | 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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[1923] | 598 | try @refl assumption |
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[1921] | 599 | ] |
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[1900] | 600 | ] |
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[1921] | 601 | |2: |
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| 602 | assumption |
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[1695] | 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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[1921] | 606 | #unrepeating_witness |
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[1695] | 607 | % |
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| 608 | |4: |
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[1921] | 609 | #start_status #final_status #execute_assm #classifier_assm #unrepeating_witness |
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| 610 | % |
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[1695] | 611 | |5: |
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| 612 | #status_pre_fun_call #status_start_fun_call #status_final #execute_assm |
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[1921] | 613 | #classifier_assm #after_return_assm #trace_label_return #costed_assm #unrepeating_witness |
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[1695] | 614 | whd in ⊢ (??%?); |
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[1929] | 615 | @(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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[1921] | 616 | assumption |
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[1695] | 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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[1921] | 620 | #costed_assm #trace_any_label #unrepeating_witness |
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[1695] | 621 | whd in ⊢ (??%?); |
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[1929] | 622 | >(compute_trace_label_return_using_paid_ok_with_trace … cost_map … codom_dom ? dom_codom) |
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[1921] | 623 | [1: |
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[1929] | 624 | >(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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[1921] | 625 | [1: |
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| 626 | >length_append >bigop_sum_rev >commutative_plus @eq_f2 |
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| 627 | [ @same_bigop [2: #i #H #_ -dom_codom @tech_cost_of_label_shift assumption |
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| 628 | |1: #i #H % ] |
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| 629 | | @same_bigop [#i #H %] #i #H #_ @(tech_cost_of_label_prefix … H) |
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| 630 | ] |
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| 631 | |2: |
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| 632 | inversion unrepeating_witness |
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| 633 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
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| 634 | ] |
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| 635 | |2: |
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| 636 | inversion unrepeating_witness |
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| 637 | * #memb_1 #tal_unrepeating #tlr_unrepeating #_ assumption |
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[1695] | 638 | ] |
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| 639 | |7: |
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| 640 | #end_flag #status_pre_fun_call #status_start_fun_call #status_final |
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[1921] | 641 | #execute_assm #trace_any_label #classifier_assm #costed_assm #unrepeating_witness |
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[1695] | 642 | whd in ⊢ (??%?); |
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[1929] | 643 | @(compute_trace_any_label_using_paid_ok_with_trace … cost_map … codom_dom … dom_codom) |
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[1921] | 644 | inversion unrepeating_witness |
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| 645 | #memb_1 #tal_unrepeating #_ assumption |
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[1554] | 646 | ] |
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[1695] | 647 | qed. |
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| 648 | |
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[1898] | 649 | lemma compute_max_trace_label_return_cost_ok_with_trace_aux: |
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[1695] | 650 | ∀code_memory: BitVectorTrie Byte 16. |
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[2498] | 651 | ∀cost_labels: costlabel_map. |
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[1695] | 652 | ∀cost_map: identifier_map CostTag nat. |
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| 653 | ∀initial: Status code_memory. |
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| 654 | ∀final: Status code_memory. |
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| 655 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
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[1921] | 656 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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[1695] | 657 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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[1900] | 658 | ∀dom_codom: (∀k. ∀k_pres:present … cost_map k. ∀pc. lookup_opt … pc cost_labels = Some … k → |
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[1929] | 659 | pi1 … (block_cost code_memory pc cost_labels) = lookup_present … k_pres). |
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[1964] | 660 | let ctrace ≝ flatten_trace_label_return … trace in |
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[1695] | 661 | clock … code_memory … final = |
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[1964] | 662 | clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map codom_dom ctrace i)). |
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[1929] | 663 | #code_memory #cost_labels #cost_map |
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[1921] | 664 | #initial #final #trace #unrepeating_witness #codom_dom #dom_codom normalize nodelta |
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[1695] | 665 | <compute_trace_label_return_using_paid_ok_with_trace try assumption |
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| 666 | >commutative_plus >compute_trace_label_return_cost_using_paid_ok // |
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[1898] | 667 | qed. |
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| 668 | |
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| 669 | theorem compute_max_trace_label_return_cost_ok_with_trace: |
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| 670 | ∀code_memory: BitVectorTrie Byte 16. |
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[2498] | 671 | ∀cost_labels: costlabel_map. |
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[1900] | 672 | ∀cost_labels_injective: |
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| 673 | (∀pc,pc',l. |
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| 674 | lookup_opt costlabel 16 pc cost_labels=Some costlabel l |
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| 675 | →lookup_opt costlabel 16 pc' cost_labels=Some costlabel l→pc=pc'). |
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[1898] | 676 | ∀initial: Status code_memory. |
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| 677 | ∀final: Status code_memory. |
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| 678 | ∀trace: trace_label_return (ASM_abstract_status code_memory cost_labels) initial final. |
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[1921] | 679 | ∀unrepeating_witness: tlr_unrepeating … trace. |
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[1929] | 680 | let cost_map ≝ traverse_code code_memory cost_labels cost_labels_injective in |
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[1964] | 681 | let ctrace ≝ flatten_trace_label_return … trace in |
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| 682 | clock … code_memory … final = clock … code_memory … initial + (Σ_{i < |ctrace|} (tech_cost_of_label cost_labels cost_map ? ctrace i)). |
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[1898] | 683 | [1: |
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[1929] | 684 | #code_memory #cost_labels #cost_labels_injective #initial #final #trace #unrepeating_witness |
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[1898] | 685 | @compute_max_trace_label_return_cost_ok_with_trace_aux try assumption |
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| 686 | |2: |
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| 687 | skip |
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| 688 | ] |
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| 689 | normalize nodelta |
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[1929] | 690 | cases (traverse_code ???) |
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[1898] | 691 | #cost_map * #dom_codom #codom_dom try assumption |
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| 692 | #pc #k #lookup_opt_assm @(dom_codom … lookup_opt_assm) |
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| 693 | lapply (sym_eq ? ? ? lookup_opt_assm) |
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| 694 | -lookup_opt_assm #lookup_opt_assm |
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[1929] | 695 | @nat_of_bitvector_lt_bound |
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[1976] | 696 | qed. |
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| 697 | |
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| 698 | include "utilities/permutations.ma". |
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[2498] | 699 | (* |
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[1976] | 700 | lemma tech_cost_sum_eq_as_cost : |
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[2498] | 701 | ∀code_memory: object_code. |
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| 702 | ∀cost_labels: costlabel_map. |
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| 703 | ∀cost_map: identifier_map CostTag nat. |
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| 704 | ∀codom_dom: (∀pc,k. lookup_opt … pc cost_labels = Some … k → present … cost_map k). |
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| 705 | ∀trace. |
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| 706 | (Σ_{i < |trace|}(tech_cost_of_label cost_labels cost_map codom_dom trace i)) = |
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| 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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[1976] | 723 | ∀code_memory: BitVectorTrie Byte 16. |
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[2498] | 724 | ∀cost_labels: costlabel_map. |
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[1976] | 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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[2057] | 742 | qed. |
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