1 | include "ASM/Assembly.ma". |
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2 | include "ASM/Interpret.ma". |
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3 | |
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4 | (* RUSSEL **) |
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5 | |
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6 | include "basics/jmeq.ma". |
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7 | |
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8 | definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p. |
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9 | definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w]. |
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10 | |
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11 | coercion inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?. |
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12 | coercion eject : ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?. |
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13 | |
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14 | axiom VOID: Type[0]. |
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15 | axiom assert_false: VOID. |
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16 | definition bigbang: ∀A:Type[0].∀P:A → Prop. False → VOID → Σx:A.P x. |
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17 | #A #P #abs cases abs |
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18 | qed. |
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19 | |
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20 | (*coercion bigbang : ∀A.∀P:A → Prop.False → ∀v:VOID.Σx:A.P x ≝ bigbang on _v:VOID to Σx:?.?.*) |
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21 | |
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22 | lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p). |
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23 | #A #P #p cases p #w #q @q |
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24 | qed. |
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25 | |
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26 | (* END RUSSELL **) |
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27 | |
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28 | (* This establishes the correspondence between pseudo program counters and |
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29 | program counters. It is at the heart of the proof. *) |
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30 | (*CSC: code taken from build_maps *) |
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31 | definition sigma0: pseudo_assembly_program → option (nat × (nat × (BitVectorTrie Word 16))) ≝ |
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32 | λinstr_list. |
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33 | foldl ?? |
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34 | (λt. λi. |
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35 | match t with |
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36 | [ None ⇒ None ? |
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37 | | Some ppc_pc_map ⇒ |
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38 | let 〈ppc,pc_map〉 ≝ ppc_pc_map in |
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39 | let 〈program_counter, sigma_map〉 ≝ pc_map in |
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40 | let 〈label, i〉 ≝ i in |
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41 | match construct_costs instr_list program_counter (λx. zero ?) (λx. zero ?) (Stub …) i with |
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42 | [ None ⇒ None ? |
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43 | | Some pc_ignore ⇒ |
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44 | let 〈pc,ignore〉 ≝ pc_ignore in |
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45 | Some … 〈S ppc,〈pc, insert ? ? (bitvector_of_nat ? ppc) (bitvector_of_nat ? pc) sigma_map〉〉 ] |
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46 | ]) (Some ? 〈0, 〈0, (Stub ? ?)〉〉) (\snd instr_list). |
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47 | |
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48 | definition tech_pc_sigma0: pseudo_assembly_program → option nat ≝ |
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49 | λinstr_list. |
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50 | match sigma0 instr_list with |
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51 | [ None ⇒ None … |
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52 | | Some result ⇒ |
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53 | let 〈ppc,pc_sigma_map〉 ≝ result in |
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54 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
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55 | Some … pc ]. |
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56 | |
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57 | definition sigma: pseudo_assembly_program → option (Word → Word) ≝ |
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58 | λinstr_list. |
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59 | match sigma0 instr_list with |
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60 | [ None ⇒ None ? |
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61 | | Some result ⇒ |
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62 | let 〈ppc,pc_sigma_map〉 ≝ result in |
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63 | let 〈pc, sigma_map〉 ≝ pc_sigma_map in |
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64 | if gtb pc (2^16) then |
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65 | None ? |
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66 | else |
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67 | Some ? (λx.lookup ?? x sigma_map (zero …)) ]. |
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68 | |
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69 | axiom policy_ok: ∀p. sigma p ≠ None …. |
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70 | |
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71 | axiom is_prefix: ∀A:Type[0]. list A → list A → Prop. |
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72 | |
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73 | definition build_maps' ≝ |
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74 | λpseudo_program. |
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75 | let 〈preamble,instr_list〉 ≝ pseudo_program in |
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76 | let result ≝ |
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77 | foldl |
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78 | (Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
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79 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
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80 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t))) |
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81 | (Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
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82 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
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83 | is_prefix ? instr_list_prefix' instr_list ∧ |
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84 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ?) |
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85 | (λt: Σt:((BitVectorTrie Word 16) × (nat × (BitVectorTrie Word 16))). |
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86 | ∃instr_list_prefix. is_prefix ? instr_list_prefix instr_list ∧ |
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87 | tech_pc_sigma0 〈preamble,instr_list_prefix〉 = Some ? (\fst (\snd t)). |
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88 | λi: Σi:option Identifier × pseudo_instruction. ∀instr_list_prefix. |
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89 | let instr_list_prefix' ≝ instr_list_prefix @ [i] in |
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90 | is_prefix ? instr_list_prefix' instr_list ∧ |
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91 | tech_pc_sigma0 〈preamble,instr_list_prefix'〉 ≠ None ? . |
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92 | let 〈labels, pc_costs〉 ≝ t in |
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93 | let 〈program_counter, costs〉 ≝ pc_costs in |
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94 | let 〈label, i'〉 ≝ i in |
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95 | let labels ≝ |
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96 | match label with |
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97 | [ None ⇒ labels |
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98 | | Some label ⇒ |
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99 | let program_counter_bv ≝ bitvector_of_nat ? program_counter in |
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100 | insert ? ? label program_counter_bv labels |
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101 | ] |
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102 | in |
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103 | match construct_costs pseudo_program program_counter (λx. zero ?) (λx. zero ?) costs i' with |
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104 | [ None ⇒ 〈labels,pc_costs〉 (*assert_false*) |
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105 | | Some construct ⇒ 〈labels, construct〉 |
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106 | ] |
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107 | ) 〈(Stub ? ?), 〈0, (Stub ? ?)〉〉 (\snd ?(*instr_list*)) |
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108 | in |
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109 | let 〈labels, pc_costs〉 ≝ result in |
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110 | let 〈pc, costs〉 ≝ pc_costs in |
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111 | 〈labels, costs〉. |
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112 | [ cases t in c2 ⊢ % #t' * #LIST_PREFIX * #H1t' #H2t' #HJMt' |
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113 | % [@ (LIST_PREFIX @ [i])] % |
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114 | [ cases (sig2 … i LIST_PREFIX) #K1 #K2 @K1 |
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115 | | (* DOABLE IN PRINCIPLE *) |
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116 | ] |
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117 | | (* assert false case *) |
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118 | |3: % [@ ([ ])] % [2: % | (* DOABLE *)] |
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119 | | |
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120 | |
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121 | |
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122 | |
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123 | |
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124 | let rec encoding_check (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
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125 | (encoding: list Byte) on encoding: Prop ≝ |
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126 | match encoding with |
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127 | [ nil ⇒ final_pc = pc |
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128 | | cons hd tl ⇒ |
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129 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
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130 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
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131 | ]. |
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132 | |
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133 | axiom sigma': pseudo_assembly_program → Word → Word. |
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134 | |
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135 | definition assembly_specification: |
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136 | ∀assembly_program: pseudo_assembly_program. |
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137 | ∀code_mem: BitVectorTrie Byte 16. Prop ≝ |
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138 | λpseudo_assembly_program. |
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139 | λcode_mem. |
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140 | ∀pc: Word. |
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141 | let 〈preamble, instr_list〉 ≝ pseudo_assembly_program in |
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142 | let 〈pre_instr, pre_new_pc〉 ≝ fetch_pseudo_instruction instr_list pc in |
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143 | let labels ≝ λx. sigma' pseudo_assembly_program (address_of_word_labels_code_mem instr_list x) in |
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144 | let datalabels ≝ λx. sigma' pseudo_assembly_program (lookup ? ? x (construct_datalabels preamble) (zero ?)) in |
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145 | let pre_assembled ≝ assembly_1_pseudoinstruction pseudo_assembly_program |
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146 | (sigma' pseudo_assembly_program pc) labels datalabels pre_instr in |
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147 | match pre_assembled with |
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148 | [ None ⇒ True |
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149 | | Some pc_code ⇒ |
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150 | let 〈new_pc,code〉 ≝ pc_code in |
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151 | encoding_check code_mem pc (sigma' pseudo_assembly_program pre_new_pc) code ]. |
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152 | |
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153 | axiom assembly_meets_specification: |
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154 | ∀pseudo_assembly_program. |
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155 | match assembly pseudo_assembly_program with |
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156 | [ None ⇒ True |
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157 | | Some code_mem_cost ⇒ |
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158 | let 〈code_mem, cost〉 ≝ code_mem_cost in |
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159 | assembly_specification pseudo_assembly_program (load_code_memory code_mem) |
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160 | ]. |
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161 | (* |
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162 | # PROGRAM |
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163 | [ cases PROGRAM |
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164 | # PREAMBLE |
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165 | # INSTR_LIST |
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166 | elim INSTR_LIST |
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167 | [ whd |
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168 | whd in ⊢ (∀_. %) |
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169 | # PC |
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170 | whd |
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171 | | # INSTR |
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172 | # INSTR_LIST_TL |
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173 | # H |
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174 | whd |
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175 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ?]) |
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176 | ] |
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177 | | cases not_implemented |
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178 | ] *) |
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179 | |
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180 | definition status_of_pseudo_status: PseudoStatus → option Status ≝ |
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181 | λps. |
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182 | let pap ≝ code_memory … ps in |
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183 | match assembly pap with |
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184 | [ None ⇒ None … |
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185 | | Some p ⇒ |
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186 | let cm ≝ load_code_memory (\fst p) in |
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187 | let pc ≝ sigma' pap (program_counter ? ps) in |
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188 | Some … |
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189 | (mk_PreStatus (BitVectorTrie Byte 16) |
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190 | cm |
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191 | (low_internal_ram … ps) |
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192 | (high_internal_ram … ps) |
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193 | (external_ram … ps) |
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194 | pc |
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195 | (special_function_registers_8051 … ps) |
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196 | (special_function_registers_8052 … ps) |
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197 | (p1_latch … ps) |
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198 | (p3_latch … ps) |
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199 | (clock … ps)) ]. |
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200 | (* |
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201 | definition write_at_stack_pointer': |
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202 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
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203 | λM: Type[0]. |
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204 | λs: PreStatus M. |
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205 | λv: Byte. |
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206 | let 〈 nu, nl 〉 ≝ split … 4 4 (get_8051_sfr ? s SFR_SP) in |
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207 | let bit_zero ≝ get_index_v… nu O ? in |
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208 | let bit_1 ≝ get_index_v… nu 1 ? in |
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209 | let bit_2 ≝ get_index_v… nu 2 ? in |
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210 | let bit_3 ≝ get_index_v… nu 3 ? in |
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211 | if bit_zero then |
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212 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
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213 | v (low_internal_ram ? s) in |
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214 | set_low_internal_ram ? s memory |
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215 | else |
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216 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
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217 | v (high_internal_ram ? s) in |
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218 | set_high_internal_ram ? s memory. |
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219 | [ cases l0 % |
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220 | |2,3,4,5: cases not_implemented (* CSC: ???? normalize repeat (@ le_S_S) @ le_O_n *) |
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221 | ] |
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222 | qed. |
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223 | |
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224 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
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225 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
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226 | ≝ |
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227 | λticks_of. |
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228 | λs. |
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229 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
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230 | let ticks ≝ ticks_of (program_counter ? s) in |
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231 | let s ≝ set_clock ? s (clock ? s + ticks) in |
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232 | let s ≝ set_program_counter ? s pc in |
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233 | match instr with |
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234 | [ Instruction instr ⇒ |
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235 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
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236 | | Comment cmt ⇒ s |
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237 | | Cost cst ⇒ s |
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238 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
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239 | | Call call ⇒ |
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240 | let a ≝ address_of_word_labels s call in |
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241 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
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242 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
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243 | let 〈pc_bu, pc_bl〉 ≝ split ? 8 8 (program_counter ? s) in |
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244 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
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245 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
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246 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
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247 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
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248 | set_program_counter ? s a |
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249 | | Mov dptr ident ⇒ |
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250 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
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251 | ]. |
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252 | [ |
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253 | |2,3,4: % |
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254 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
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255 | | |
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256 | | (*CSC: ??? *) |
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257 | | (*CSC: ??? *) |
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258 | | % |
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259 | ] |
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260 | cases not_implemented |
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261 | qed. |
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262 | |
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263 | (* |
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264 | lemma execute_code_memory_unchanged: |
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265 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
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266 | #ticks #ps whd in ⊢ (??? (??%)) |
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267 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
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268 | (program_counter pseudo_assembly_program ps)) #instr #pc |
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269 | whd in ⊢ (??? (??%)) cases instr |
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270 | [ #pre cases pre |
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271 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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272 | cases (split ????) #z1 #z2 % |
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273 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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274 | cases (split ????) #z1 #z2 % |
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275 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
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276 | cases (split ????) #z1 #z2 % |
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277 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
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278 | [ #x1 whd in ⊢ (??? (??%)) |
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279 | | *: cases not_implemented |
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280 | ] |
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281 | | #comment % |
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282 | | #cost % |
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283 | | #label % |
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284 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
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285 | cases (split ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
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286 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (split ????) #w1 #w2 |
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287 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
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288 | (* CSC: ??? *) |
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289 | | #dptr #label (* CSC: ??? *) |
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290 | ] |
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291 | cases not_implemented |
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292 | qed. |
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293 | *) |
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294 | |
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295 | lemma status_of_pseudo_status_failure_depends_only_on_code_memory: |
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296 | ∀ps,ps': PseudoStatus. |
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297 | code_memory … ps = code_memory … ps' → |
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298 | match status_of_pseudo_status ps with |
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299 | [ None ⇒ status_of_pseudo_status ps' = None … |
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300 | | Some _ ⇒ ∃w. status_of_pseudo_status ps' = Some … w |
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301 | ]. |
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302 | #ps #ps' #H whd in ⊢ (mat |
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303 | ch % with [ _ ⇒ ? | _ ⇒ ? ]) |
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304 | generalize in match (refl … (assembly (code_memory … ps))) |
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305 | cases (assembly ?) in ⊢ (???% → %) |
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306 | [ #K whd whd in ⊢ (??%?) <H >K % |
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307 | | #x #K whd whd in ⊢ (?? (λ_.??%?)) <H >K % [2: % ] ] |
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308 | qed.*) |
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309 | |
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310 | let rec encoding_check' (code_memory: BitVectorTrie Byte 16) (pc: Word) (encoding: list Byte) on encoding: Prop ≝ |
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311 | match encoding with |
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312 | [ nil ⇒ True |
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313 | | cons hd tl ⇒ |
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314 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
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315 | hd = byte ∧ encoding_check' code_memory new_pc tl |
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316 | ]. |
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317 | |
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318 | lemma test: |
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319 | ∀i: instruction. |
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320 | ∃pc. |
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321 | let assembled ≝ assembly1 i in |
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322 | let code_memory ≝ load_code_memory assembled in |
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323 | let fetched ≝ fetch code_memory pc in |
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324 | let 〈instr_pc, ticks〉 ≝ fetched in |
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325 | \fst instr_pc = i. |
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326 | # INSTR |
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327 | % |
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328 | [ @ (zero 16) |
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329 | | cases INSTR |
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330 | ]. |
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331 | |
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332 | lemma test: |
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333 | ∀pc: Word. |
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334 | ∀code_memory: BitVectorTrie Byte 16. |
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335 | ∀i: instruction. |
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336 | let assembled ≝ assembly1 i in |
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337 | encoding_check' code_memory pc assembled → |
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338 | let 〈instr_pc, ignore〉 ≝ fetch code_memory pc in |
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339 | let 〈instr, pc〉 ≝ instr_pc in |
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340 | instr = i. |
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341 | # PC # CODE_MEMORY # INSTRUCTION |
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342 | whd |
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343 | whd in ⊢ (? → match % with [ _ ⇒ ? ]) |
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344 | cases (next CODE_MEMORY PC) |
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345 | whd |
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346 | # PC1 # V1 |
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347 | |
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348 | |
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349 | cases (fetch CODE_MEMORY PC) |
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350 | # INSTR_PC |
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351 | # TICKS |
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352 | cases (INSTRUCTION) |
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353 | [ # ADDR11 |
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354 | # ENCODING_CHECK |
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355 | whd in ENCODING_CHECK; |
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356 | normalize |
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357 | cases (INSTR_PC) |
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358 | # INSTR |
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359 | # PC |
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360 | normalize |
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361 | |
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362 | |
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363 | # ENCODING |
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364 | normalize |
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365 | [ cases INSTR_PC |
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366 | # NEW_INSTRUCTION |
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367 | # PC |
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368 | normalize |
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369 | normalize in ENCODING; |
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370 | | # ASSEMBLED |
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371 | whd |
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372 | |
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373 | lemma main_thm: |
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374 | ∀ticks_of. |
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375 | ∀ps: PseudoStatus. |
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376 | match status_of_pseudo_status ps with [ None ⇒ True | Some s ⇒ |
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377 | let ps' ≝ execute_1_pseudo_instruction ticks_of ps in |
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378 | match status_of_pseudo_status ps' with [ None ⇒ True | Some s'' ⇒ |
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379 | let s' ≝ execute_1 s in |
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380 | s = s'']]. |
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381 | #ticks_of #ps |
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382 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
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383 | cases (assembly (code_memory pseudo_assembly_program ps)) [%] * #cm #costs whd |
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384 | whd in ⊢ (match % with [ _ ⇒ ? | _ ⇒ ? ]) |
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385 | generalize in match (sig2 … (execute_1_pseudo_instruction' ticks_of ps)) |
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386 | |
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387 | cases (status_of_pseudo_status (execute_1_pseudo_instruction ticks_of ps)) [%] #s'' whd |
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