1 | include "ASM/Assembly.ma". |
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2 | include "ASM/Interpret.ma". |
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3 | include "ASM/StatusProofs.ma". |
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4 | include alias "arithmetics/nat.ma". |
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5 | |
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6 | let rec encoding_check |
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7 | (code_memory: BitVectorTrie Byte 16) (pc: Word) (final_pc: Word) |
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8 | (encoding: list Byte) |
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9 | on encoding: Prop ≝ |
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10 | match encoding with |
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11 | [ nil ⇒ final_pc = pc |
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12 | | cons hd tl ⇒ |
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13 | let 〈new_pc, byte〉 ≝ next code_memory pc in |
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14 | hd = byte ∧ encoding_check code_memory new_pc final_pc tl |
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15 | ]. |
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16 | |
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17 | lemma encoding_check_append: |
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18 | ∀code_memory: BitVectorTrie Byte 16. |
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19 | ∀final_pc: Word. |
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20 | ∀l1: list Byte. |
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21 | ∀pc: Word. |
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22 | ∀l2: list Byte. |
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23 | encoding_check code_memory pc final_pc (l1@l2) → |
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24 | let pc_plus_len ≝ add … pc (bitvector_of_nat … (length … l1)) in |
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25 | encoding_check code_memory pc pc_plus_len l1 ∧ |
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26 | encoding_check code_memory pc_plus_len final_pc l2. |
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27 | #code_memory #final_pc #l1 elim l1 |
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28 | [1: |
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29 | #pc #l2 |
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30 | whd in ⊢ (????% → ?); #H |
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31 | <add_zero |
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32 | whd whd in ⊢ (?%?); /2/ |
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33 | |2: |
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34 | #hd #tl #IH #pc #l2 * #H1 #H2 |
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35 | (* >add_SO in H2; #H2 *) |
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36 | cases (IH … H2) #E1 #E2 % |
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37 | [1: |
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38 | % try @H1 |
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39 | <(add_bitvector_of_nat_Sm 16 (|tl|)) in E1; |
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40 | <add_associative #assm assumption |
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41 | |2: |
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42 | <add_associative in E2; |
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43 | <(add_bitvector_of_nat_Sm 16 (|tl|)) #assm |
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44 | assumption |
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45 | ] |
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46 | ] |
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47 | qed. |
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48 | |
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49 | definition ticks_of_instruction ≝ |
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50 | λi. |
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51 | let trivial_code_memory ≝ assembly1 i in |
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52 | let trivial_status ≝ load_code_memory trivial_code_memory in |
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53 | \snd (fetch trivial_status (zero ?)). |
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54 | |
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55 | lemma fetch_assembly: |
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56 | ∀pc: Word. |
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57 | ∀i: instruction. |
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58 | ∀code_memory: BitVectorTrie Byte 16. |
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59 | ∀assembled: list Byte. |
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60 | assembled = assembly1 i → |
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61 | let len ≝ length … assembled in |
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62 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
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63 | encoding_check code_memory pc pc_plus_len assembled → |
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64 | let 〈instr, pc', ticks〉 ≝ fetch code_memory pc in |
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65 | (eq_instruction instr i ∧ eqb ticks (ticks_of_instruction instr) ∧ eq_bv … pc' pc_plus_len) = true. |
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66 | cases daemon |
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67 | (* XXX: commented out as takes ages to typecheck |
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68 | #pc #i #code_memory #assembled cases i [8: *] |
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69 | [16,20,29: * * |18,19: * * [1,2,4,5: *] |28: * * [1,2: * [1,2: * [1,2: * [1,2: *]]]]] |
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70 | [47,48,49: |
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71 | |*: #arg @(subaddressing_mode_elim … arg) |
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72 | [2,3,5,7,10,12,16,17,18,21,26,27,28,31,32,33,37,38,39,40,41,42,43,44,45,48,51,58, |
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73 | 59,60,63,64,65,66,67: #ARG]] |
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74 | [4,5,6,7,8,9,10,11,12,13,22,23,24,27,28,39,40,41,42,43,44,45,46,47,48,49,50,51,52, |
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75 | 56,57,69,70,72: #arg2 @(subaddressing_mode_elim … arg2) |
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76 | [1,2,4,7,9,11,12,14,15,17,18,19,20,22,23,24,25,26,27,28,29,30,31,32,33,36,37,38, |
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77 | 39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65, |
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78 | 68,69,70,71: #ARG2]] |
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79 | [1,2,19,20: #arg3 @(subaddressing_mode_elim … arg3) #ARG3] |
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80 | normalize in ⊢ (???% → ?); |
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81 | [92,94,42,93,95: @vsplit_elim #vl #vm #E >E -E; [2,4: @(bitvector_3_elim_prop … vl)] |
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82 | normalize in ⊢ (???% → ?);] |
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83 | #H >H * #H1 try (whd in ⊢ (% → ?); * #H2) |
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84 | try (whd in ⊢ (% → ?); * #H3) whd in ⊢ (% → ?); #H4 |
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85 | whd in match fetch; normalize nodelta <H1 whd in ⊢ (match % with [ _ ⇒ ? ]); |
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86 | [17,18,19,20,21,22,23,24,25,26,32,34,35,36,37,39: <H3] |
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87 | try <H2 whd >eq_instruction_refl >H4 @eq_bv_refl *) |
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88 | qed. |
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89 | |
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90 | let rec fetch_many |
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91 | (code_memory: BitVectorTrie Byte 16) (final_pc: Word) (pc: Word) |
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92 | (expected: list instruction) |
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93 | on expected: Prop ≝ |
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94 | match expected with |
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95 | [ nil ⇒ eq_bv … pc final_pc = true |
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96 | | cons i tl ⇒ |
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97 | let pc' ≝ add 16 pc (bitvector_of_nat 16 (|assembly1 … i|)) in |
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98 | (〈i, pc', ticks_of_instruction i〉 = fetch code_memory pc ∧ |
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99 | fetch_many code_memory final_pc pc' tl) |
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100 | ]. |
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101 | |
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102 | lemma fetch_assembly_pseudo': |
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103 | ∀lookup_labels. |
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104 | ∀sigma: Word → Word. |
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105 | ∀policy: Word → bool. |
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106 | ∀ppc. |
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107 | ∀lookup_datalabels. |
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108 | ∀pi. |
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109 | ∀code_memory. |
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110 | ∀len. |
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111 | ∀assembled. |
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112 | ∀instructions. |
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113 | let pc ≝ sigma ppc in |
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114 | instructions = expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi → |
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115 | 〈len,assembled〉 = assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi → |
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116 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
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117 | encoding_check code_memory pc pc_plus_len assembled → |
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118 | fetch_many code_memory pc_plus_len pc instructions. |
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119 | #lookup_labels #sigma #policy #ppc #lookup_datalabels #pi #code_memory #len #assembled #instructions |
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120 | normalize nodelta #instructions_refl whd in ⊢ (???% → ?); <instructions_refl whd in ⊢ (???% → ?); #assembled_refl |
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121 | cases (pair_destruct ?????? assembled_refl) -assembled_refl #len_refl #assembled_refl |
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122 | >len_refl >assembled_refl -len_refl |
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123 | generalize in match (add 16 (sigma ppc) |
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124 | (bitvector_of_nat 16 |
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125 | (|flatten (Vector bool 8) |
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126 | (map instruction (list (Vector bool 8)) assembly1 instructions)|))); |
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127 | #final_pc |
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128 | generalize in match (sigma ppc); elim instructions |
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129 | [1: |
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130 | #pc whd in ⊢ (% → %); #H >H @eq_bv_refl |
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131 | |2: |
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132 | #i #tl #IH #pc #H whd |
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133 | cases (encoding_check_append ????? H) -H #H1 #H2 |
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134 | lapply (fetch_assembly pc i code_memory (assembly1 i) (refl …)) whd in ⊢ (% → ?); |
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135 | cases (fetch ??) * #instr #pc' #ticks |
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136 | #H3 lapply (H3 H1) -H3 normalize nodelta #H3 |
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137 | lapply (conjunction_true ?? H3) * #H4 #H5 |
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138 | lapply (conjunction_true … H4) * #B1 #B2 |
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139 | >(eq_instruction_to_eq … B1) >(eq_bv_eq … H5) % |
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140 | >(eqb_true_to_refl … B2) >(eq_instruction_to_eq … B1) try % @IH @H2 |
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141 | ] |
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142 | qed. |
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143 | |
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144 | lemma fetch_assembly_pseudo: |
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145 | ∀program: pseudo_assembly_program. |
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146 | ∀sigma: Word → Word. |
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147 | ∀policy: Word → bool. |
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148 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
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149 | let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in |
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150 | ∀ppc.∀ppc_ok. |
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151 | ∀code_memory. |
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152 | let lookup_datalabels ≝ λx:Identifier. lookup_def … (construct_datalabels (\fst program)) x (zero 16) in |
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153 | let pi ≝ \fst (fetch_pseudo_instruction (\snd program) ppc ppc_ok) in |
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154 | let pc ≝ sigma ppc in |
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155 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
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156 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
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157 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
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158 | encoding_check code_memory pc pc_plus_len assembled → |
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159 | fetch_many code_memory pc_plus_len pc instructions. |
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160 | #program #sigma #policy |
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161 | @pair_elim #labels #costs #create_label_cost_map_refl |
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162 | letin lookup_labels ≝ (λx.?) #ppc #ppc_ok #code_memory |
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163 | letin lookup_datalabels ≝ (λx.?) |
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164 | letin pi ≝ (fst ???) |
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165 | letin pc ≝ (sigma ?) |
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166 | letin instructions ≝ (expand_pseudo_instruction ??????) |
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167 | @pair_elim #len #assembled #assembled_refl normalize nodelta |
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168 | #H |
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169 | generalize in match |
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170 | (fetch_assembly_pseudo' lookup_labels sigma policy ppc lookup_datalabels pi code_memory len assembled instructions) in ⊢ ?; |
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171 | #X destruct normalize nodelta @X try % <assembled_refl try % assumption |
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172 | qed. |
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173 | |
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174 | definition is_present_in_machine_code_image_p: ∀pseudo_instruction. Prop ≝ |
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175 | λpseudo_instruction. |
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176 | match pseudo_instruction with |
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177 | [ Comment c ⇒ False |
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178 | | Cost c ⇒ False |
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179 | | _ ⇒ True |
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180 | ]. |
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181 | |
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182 | lemma monotone_sigma: |
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183 | ∀program. |\snd program| ≤ 2^16 → |
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184 | ∀sigma: Word → Word. ∀policy: Word → bool. |
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185 | sigma_policy_specification program sigma policy → |
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186 | ∀ppc1,ppc2. |
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187 | nat_of_bitvector … ppc2 ≤ |\snd program| → |
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188 | nat_of_bitvector … ppc1 ≤ nat_of_bitvector … ppc2 → |
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189 | (nat_of_bitvector … (sigma ppc1) ≤ nat_of_bitvector … (sigma ppc2) ∨ |
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190 | sigma ppc2 = zero … ∧ nat_of_bitvector … ppc2 = |\snd program|). |
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191 | #program #program_bounded #sigma #policy * #SIGMAOK1 #SIGMAOK2 #ppc1 #ppc2 |
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192 | #ppc2_ok #LE |
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193 | <(bitvector_of_nat_inverse_nat_of_bitvector … ppc1) |
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194 | <(bitvector_of_nat_inverse_nat_of_bitvector … ppc2) in ⊢ (?%(?%?)); |
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195 | lapply (le_n … (nat_of_bitvector … ppc2)) |
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196 | elim LE in ⊢ (?%? → %); |
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197 | [ #_ % % |
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198 | | #m #LE_ppc1_m #IH #BOUNDED |
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199 | cases (IH ?) |
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200 | [3: @(transitive_le … BOUNDED) %2 % |
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201 | |2: * #_ #abs @⊥ <abs in ppc2_ok; #abs' |
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202 | @(absurd ?? (not_le_Sn_n m)) |
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203 | @(transitive_le … BOUNDED abs') ] |
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204 | -IH #IH |
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205 | cut (m < |\snd program|) [ @(lt_to_le_to_lt … ppc2_ok) assumption ] #LT1 |
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206 | cut (m < 2^16) [ @(lt_to_le_to_lt … program_bounded) assumption ] #LT2 |
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207 | cases (SIGMAOK2 (bitvector_of_nat … m) ?) -SIGMAOK2 |
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208 | [2: >nat_of_bitvector_bitvector_of_nat_inverse assumption ] |
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209 | #H * |
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210 | [ #LTsigma_m %1 @(transitive_le … IH) -IH |
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211 | <add_bitvector_of_nat_Sm >add_commutative >H -H |
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212 | >nat_of_bitvector_add |
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213 | [ // |
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214 | | >nat_of_bitvector_bitvector_of_nat_inverse try assumption lapply LTsigma_m |
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215 | whd in match instruction_size; normalize nodelta |
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216 | inversion (assembly_1_pseudoinstruction ??????) #len #assembled #EQ #X |
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217 | @(transitive_le … X) @le_S_S // |
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218 | ] |
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219 | | * #EQ1 #EQ2 %2 % |
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220 | [ <add_bitvector_of_nat_Sm >add_commutative >H |
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221 | lapply (eq_f … (bitvector_of_nat 16) … EQ2) |
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222 | <add_bitvector_of_nat_plus |
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223 | >bitvector_of_nat_inverse_nat_of_bitvector #X >X @bitvector_of_nat_exp_zero |
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224 | | <(nat_of_bitvector_bitvector_of_nat_inverse 16 m) try assumption |
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225 | ]]] |
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226 | qed. |
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227 | |
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228 | (* This is a non trivial consequence of fetch_assembly_pseudo + |
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229 | load_code_memory_ok because of the finite amount of memory. In |
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230 | particular the case when the compiled program exhausts the |
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231 | code memory is very tricky. It also requires monotonicity of |
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232 | sigma in the out-of-bounds region too. *) |
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233 | lemma assembly_ok: |
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234 | ∀program. |
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235 | ∀length_proof: |\snd program| ≤ 2^16. |
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236 | ∀sigma: Word → Word. |
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237 | ∀policy: Word → bool. |
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238 | ∀sigma_policy_witness: sigma_policy_specification program sigma policy. |
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239 | ∀assembled. |
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240 | ∀costs': BitVectorTrie costlabel 16. |
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241 | let 〈preamble, instr_list〉 ≝ program in |
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242 | let 〈labels, costs〉 ≝ create_label_cost_map instr_list in |
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243 | let datalabels ≝ construct_datalabels preamble in |
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244 | let lookup_datalabels ≝ λx. lookup_def … datalabels x (zero …) in |
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245 | 〈assembled,costs'〉 = assembly program sigma policy → |
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246 | (* costs = costs' ∧ CSC: costs != costs', costs' = sigma costs! *) |
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247 | let code_memory ≝ load_code_memory assembled in |
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248 | let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in |
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249 | ∀ppc.∀ppc_ok. |
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250 | let 〈pi, newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
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251 | let 〈len,assembled〉 ≝ assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi in |
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252 | let pc ≝ sigma ppc in |
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253 | let pc_plus_len ≝ add … pc (bitvector_of_nat … len) in |
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254 | encoding_check code_memory pc pc_plus_len assembled ∧ |
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255 | sigma newppc = add … pc (bitvector_of_nat … len). |
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256 | #program #length_proof #sigma #policy #sigma_policy_witness #assembled #costs' |
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257 | cases (assembly program sigma policy) * #assembled' #costs'' |
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258 | @pair_elim #preamble #instr_list #EQprogram whd in ⊢ (% → %); |
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259 | cut (|instr_list| ≤ 2^16) [ >EQprogram in length_proof; // ] #instr_list_ok |
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260 | #H lapply (H sigma_policy_witness instr_list_ok) -H whd in ⊢ (% → ?); |
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261 | @pair_elim #labels #costs #create_label_cost_refl whd in ⊢ (% → %); |
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262 | * * #assembly_ok1 #assembly_ok3 #assembly_ok2 #Pair_eq destruct(Pair_eq) whd |
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263 | #ppc #ppc_ok @pair_elim #pi #newppc #eq_fetch_pseudo_instruction |
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264 | @pair_elim #len #assembled #eq_assembly_1_pseudoinstruction whd |
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265 | lapply (assembly_ok2 ppc ?) try // -assembly_ok2 |
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266 | >eq_fetch_pseudo_instruction |
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267 | change with ((let 〈len0,assembledi〉 ≝ assembly_1_pseudoinstruction ????? pi in ? ∧ ∀j.∀H:j<|assembledi|.?) → ?) |
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268 | >eq_assembly_1_pseudoinstruction |
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269 | whd in ⊢ (% → ?); * #assembly_ok4 #assembly_ok |
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270 | % |
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271 | [2: >(pair_destruct_2 ????? (sym_eq … eq_fetch_pseudo_instruction)) |
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272 | >snd_fetch_pseudo_instruction |
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273 | cases sigma_policy_witness #_ >EQprogram #H cases (H ? ppc_ok) -H |
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274 | #spw1 #_ >spw1 -spw1 @eq_f @eq_f |
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275 | >eq_fetch_pseudo_instruction whd in match instruction_size; |
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276 | normalize nodelta >create_label_cost_refl |
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277 | >fst_assembly_1_pseudoinstruction_insensible_to_lookup_datalabels |
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278 | [>eq_assembly_1_pseudoinstruction % | skip] |
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279 | | lapply (load_code_memory_ok assembled' ?) [ assumption ] |
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280 | #load_code_memory_ok |
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281 | lapply (fst_snd_assembly_1_pseudoinstruction … eq_assembly_1_pseudoinstruction) #EQlen |
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282 | (* Nice statement here *) |
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283 | cut (∀j. ∀H: j < |assembled|. |
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284 | nth_safe Byte j assembled H = |
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285 | \snd (next (load_code_memory assembled') |
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286 | (bitvector_of_nat 16 |
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287 | (nat_of_bitvector … |
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288 | (add … (sigma ppc) (bitvector_of_nat … j)))))) |
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289 | [1: #j #H <load_code_memory_ok |
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290 | [ @assembly_ok |
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291 | | cut (0=|assembled'| → False) |
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292 | [ #abs <abs in assembly_ok4; >EQlen #abs' |
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293 | @(absurd ?? (not_le_Sn_O j)) @(transitive_le … H) assumption ] #NOT_EMPTY |
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294 | cases assembly_ok3 -assembly_ok3 |
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295 | [2: * #_ #EQ2 >EQ2 @lt_nat_of_bitvector ] |
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296 | #EQlen_assembled' <EQlen_assembled' |
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297 | cases sigma_policy_witness #sigma_zero >EQprogram #sigma_ok |
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298 | cases (sigma_ok … ppc_ok) -sigma_ok >eq_fetch_pseudo_instruction |
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299 | whd in match instruction_size; normalize nodelta |
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300 | >create_label_cost_refl |
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301 | >fst_assembly_1_pseudoinstruction_insensible_to_lookup_datalabels |
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302 | [>eq_assembly_1_pseudoinstruction |2: skip] #OK >OK <EQlen * |
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303 | [2: * #EQ1 #EQ2 @⊥ <EQ1 in EQlen_assembled'; |
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304 | <add_bitvector_of_nat_Sm >add_commutative |
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305 | >bitvector_of_nat_inverse_nat_of_bitvector >OK >EQ2 |
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306 | lapply (eq_f … (bitvector_of_nat 16) … EQ2) -EQ2 |
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307 | <add_bitvector_of_nat_plus >bitvector_of_nat_inverse_nat_of_bitvector |
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308 | #X >X #Y @NOT_EMPTY <Y >bitvector_of_nat_exp_zero % ] |
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309 | cases (le_to_or_lt_eq … instr_list_ok) |
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310 | [2: #abs @⊥ >abs in EQlen_assembled'; >bitvector_of_nat_exp_zero >sigma_zero |
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311 | assumption ] |
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312 | #instr_list_ok' #NO_OVERFLOW |
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313 | @(lt_to_le_to_lt … (nat_of_bitvector … (add … (sigma ppc) (bitvector_of_nat … (|assembled|))))) |
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314 | [ @lt_to_lt_nat_of_bitvector_add |
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315 | [ @(eq_ind ?? (λp.λ_. |\snd p| < 2^16) ?? eq_assembly_1_pseudoinstruction) |
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316 | / by / |
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317 | | <EQlen assumption |
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318 | | assumption |
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319 | ] |
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320 | | <EQlen <OK |
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321 | cases (monotone_sigma ???? sigma_policy_witness |
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322 | (add … ppc (bitvector_of_nat … 1)) |
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323 | (bitvector_of_nat … (|instr_list|)) ??) try assumption |
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324 | [ #H @H |
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325 | |3: >EQprogram >nat_of_bitvector_bitvector_of_nat_inverse |
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326 | try % assumption |
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327 | |4: >nat_of_bitvector_bitvector_of_nat_inverse try assumption |
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328 | >nat_of_bitvector_add <plus_n_Sm <plus_n_O try assumption |
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329 | @(transitive_le … instr_list_ok') @le_S_S assumption |
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330 | | * #EQ1 @⊥ >EQ1 in EQlen_assembled'; assumption ]]] |
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331 | |2: |
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332 | -assembly_ok -load_code_memory_ok generalize in match (sigma ppc); >EQlen -len |
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333 | elim assembled |
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334 | [1: |
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335 | #pc #_ whd <add_zero % |
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336 | | #hd #tl #IH #pc #H % |
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337 | [ lapply (H 0 ?) [ normalize @le_S_S @le_O_n ] whd in ⊢ (??%? → ?); -H #H |
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338 | >H -H whd in ⊢ (??%?); <add_zero // |
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339 | | >(?: add … pc (bitvector_of_nat … (S (|tl|))) = add … (add … pc (bitvector_of_nat … 1)) (bitvector_of_nat … (|tl|))) |
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340 | [2: <add_bitvector_of_nat_Sm @add_associative ] |
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341 | @IH -IH #j #LTj lapply (H (S j) ?) [ @le_S_S @LTj ] |
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342 | <(nth_safe_prepend … [hd] … LTj) #IH >IH <add_bitvector_of_nat_Sm |
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343 | >add_associative % ]]]] |
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344 | qed. |
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345 | |
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346 | (* XXX: should we add that costs = costs'? *) |
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347 | lemma fetch_assembly_pseudo2: |
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348 | ∀program. |
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349 | ∀length_proof: |\snd program| ≤ 2^16. |
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350 | ∀sigma. |
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351 | ∀policy. |
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352 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
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353 | ∀ppc.∀ppc_ok. |
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354 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
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355 | let 〈assembled, costs'〉 ≝ pi1 … (assembly program sigma policy) in |
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356 | let code_memory ≝ load_code_memory assembled in |
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357 | let data_labels ≝ construct_datalabels (\fst program) in |
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358 | let lookup_labels ≝ λx. bitvector_of_nat 16 (lookup_def … labels x 0) in |
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359 | let lookup_datalabels ≝ λx. lookup_def ? ? data_labels x (zero ?) in |
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360 | let 〈pi,newppc〉 ≝ fetch_pseudo_instruction (\snd program) ppc ppc_ok in |
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361 | let instructions ≝ expand_pseudo_instruction lookup_labels sigma policy ppc lookup_datalabels pi in |
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362 | fetch_many code_memory (sigma newppc) (sigma ppc) instructions. |
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363 | * #preamble #instr_list #length_proof #sigma #policy #sigma_policy_specification_witness #ppc #ppc_ok |
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364 | @pair_elim #labels #costs #create_label_map_refl |
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365 | @pair_elim #assembled #costs' #assembled_refl |
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366 | letin code_memory ≝ (load_code_memory ?) |
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367 | letin data_labels ≝ (construct_datalabels ?) |
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368 | letin lookup_labels ≝ (λx. ?) |
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369 | letin lookup_datalabels ≝ (λx. ?) |
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370 | @pair_elim #pi #newppc #fetch_pseudo_refl |
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371 | lapply (assembly_ok 〈preamble, instr_list〉 ? sigma policy sigma_policy_specification_witness assembled costs') |
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372 | normalize nodelta try assumption |
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373 | >create_label_map_refl in ⊢ (match % with [_ ⇒ ?] → ?); #H |
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374 | lapply (H (sym_eq … assembled_refl)) -H |
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375 | lapply (refl … (assembly_1_pseudoinstruction lookup_labels sigma policy ppc lookup_datalabels pi)) |
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376 | cases (assembly_1_pseudoinstruction ??????) in ⊢ (???% → ?); |
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377 | #len #assembledi #EQ4 #H |
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378 | lapply (H ppc) >fetch_pseudo_refl #H |
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379 | lapply (fetch_assembly_pseudo 〈preamble,instr_list〉 sigma policy) |
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380 | >create_label_map_refl #X lapply (X ppc ppc_ok (load_code_memory assembled)) -X |
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381 | >EQ4 #H1 cases (H ppc_ok) |
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382 | #H2 #H3 >H3 normalize nodelta in H1; normalize nodelta |
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383 | >fetch_pseudo_refl in H1; #assm @assm assumption |
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384 | qed. |
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385 | |
---|
386 | inductive upper_lower: Type[0] ≝ |
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387 | | upper: upper_lower |
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388 | | lower: upper_lower. |
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389 | |
---|
390 | definition eq_upper_lower: |
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391 | upper_lower → upper_lower → bool ≝ |
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392 | λleft: upper_lower. |
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393 | λright: upper_lower. |
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394 | match left with |
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395 | [ upper ⇒ |
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396 | match right with |
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397 | [ upper ⇒ true |
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398 | | _ ⇒ false |
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399 | ] |
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400 | | lower ⇒ |
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401 | match right with |
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402 | [ lower ⇒ true |
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403 | | _ ⇒ false |
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404 | ] |
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405 | ]. |
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406 | |
---|
407 | lemma eq_upper_lower_true_to_eq: |
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408 | ∀left: upper_lower. |
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409 | ∀right: upper_lower. |
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410 | eq_upper_lower left right = true → left = right. |
---|
411 | * * normalize try (#_ %) |
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412 | #absurd destruct(absurd) |
---|
413 | qed. |
---|
414 | |
---|
415 | lemma eq_upper_lower_false_to_neq: |
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416 | ∀left: upper_lower. |
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417 | ∀right: upper_lower. |
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418 | eq_upper_lower left right = false → left ≠ right. |
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419 | * * normalize try (#absurd destruct(absurd)) |
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420 | @nmk #absurd destruct(absurd) |
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421 | qed. |
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422 | |
---|
423 | inductive accumulator_address_map_entry: Type[0] ≝ |
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424 | | data: accumulator_address_map_entry |
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425 | | address: upper_lower → Word → accumulator_address_map_entry. |
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426 | |
---|
427 | definition eq_accumulator_address_map_entry: |
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428 | accumulator_address_map_entry → accumulator_address_map_entry → bool ≝ |
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429 | λleft: accumulator_address_map_entry. |
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430 | λright: accumulator_address_map_entry. |
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431 | match left with |
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432 | [ data ⇒ |
---|
433 | match right with |
---|
434 | [ data ⇒ true |
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435 | | _ ⇒ false |
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436 | ] |
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437 | | address upper_lower word ⇒ |
---|
438 | match right with |
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439 | [ address upper_lower' word' ⇒ |
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440 | eq_upper_lower upper_lower upper_lower' ∧ eq_bv … word word' |
---|
441 | | _ ⇒ false |
---|
442 | ] |
---|
443 | ]. |
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444 | |
---|
445 | lemma eq_accumulator_address_map_entry_true_to_eq: |
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446 | ∀left: accumulator_address_map_entry. |
---|
447 | ∀right: accumulator_address_map_entry. |
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448 | eq_accumulator_address_map_entry left right = true → left = right. |
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449 | #left #right cases left cases right |
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450 | [1: |
---|
451 | #_ % |
---|
452 | |2,3: |
---|
453 | #upper_lower #word normalize #absurd destruct(absurd) |
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454 | |4: |
---|
455 | #upper_lower #word #upper_lower' #word' normalize |
---|
456 | cases upper_lower' normalize nodelta |
---|
457 | cases upper_lower normalize |
---|
458 | [2,3: |
---|
459 | #absurd destruct(absurd) |
---|
460 | ] |
---|
461 | change with (eq_bv 16 ?? = true → ?) |
---|
462 | #relevant lapply (eq_bv_eq … relevant) |
---|
463 | #word_refl >word_refl % |
---|
464 | ] |
---|
465 | qed. |
---|
466 | |
---|
467 | lemma eq_bv_false_to_neq: |
---|
468 | ∀n: nat. |
---|
469 | ∀left, right: BitVector n. |
---|
470 | eq_bv n left right = false → left ≠ right. |
---|
471 | #n #left elim left |
---|
472 | [1: |
---|
473 | #right >(BitVector_O … right) normalize #absurd destruct(absurd) |
---|
474 | |2: |
---|
475 | #n' #hd #tl #inductive_hypothesis #right |
---|
476 | cases (BitVector_Sn … right) #hd' * #tl' #right_refl |
---|
477 | >right_refl normalize |
---|
478 | cases hd normalize nodelta |
---|
479 | cases hd' normalize nodelta |
---|
480 | [2,3: |
---|
481 | #_ @nmk #absurd destruct(absurd) |
---|
482 | ] |
---|
483 | change with (eq_bv ??? = false → ?) |
---|
484 | #relevant lapply (inductive_hypothesis … relevant) |
---|
485 | #tl_neq_assm @nmk #tl_eq_assm cases tl_neq_assm |
---|
486 | #tl_eq_assm' @tl_eq_assm' destruct(tl_eq_assm) % |
---|
487 | ] |
---|
488 | qed. |
---|
489 | |
---|
490 | lemma eq_accumulator_address_map_entry_false_to_neq: |
---|
491 | ∀left: accumulator_address_map_entry. |
---|
492 | ∀right: accumulator_address_map_entry. |
---|
493 | eq_accumulator_address_map_entry left right = false → left ≠ right. |
---|
494 | #left #right cases left cases right |
---|
495 | [1: |
---|
496 | normalize #absurd destruct(absurd) |
---|
497 | |2,3: |
---|
498 | #upper_lower #word normalize #_ |
---|
499 | @nmk #absurd destruct(absurd) |
---|
500 | |4: |
---|
501 | #upper_lower #word #upper_lower' #word' normalize |
---|
502 | cases upper_lower' normalize nodelta |
---|
503 | cases upper_lower normalize nodelta |
---|
504 | [2,3: |
---|
505 | #_ @nmk #absurd destruct(absurd) |
---|
506 | ] |
---|
507 | change with (eq_bv ??? = false → ?) |
---|
508 | #eq_bv_false_assm lapply(eq_bv_false_to_neq … eq_bv_false_assm) |
---|
509 | #word_neq_assm @nmk #address_eq_assm cases word_neq_assm |
---|
510 | #word_eq_assm @word_eq_assm destruct(address_eq_assm) % |
---|
511 | ] |
---|
512 | qed. |
---|
513 | |
---|
514 | (* XXX: changed this type. Bool specifies whether byte is first or second |
---|
515 | component of an address, and the Word is the pseudoaddress that it |
---|
516 | corresponds to. Second component is the same principle for the accumulator |
---|
517 | A. |
---|
518 | *) |
---|
519 | definition internal_pseudo_address_map ≝ list (Byte × (upper_lower × Word)) × accumulator_address_map_entry. |
---|
520 | |
---|
521 | include alias "ASM/BitVectorTrie.ma". |
---|
522 | |
---|
523 | include "common/AssocList.ma". |
---|
524 | |
---|
525 | axiom low_internal_ram_of_pseudo_low_internal_ram: |
---|
526 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
527 | |
---|
528 | axiom high_internal_ram_of_pseudo_high_internal_ram: |
---|
529 | ∀M:internal_pseudo_address_map.∀ram:BitVectorTrie Byte 7.BitVectorTrie Byte 7. |
---|
530 | |
---|
531 | axiom low_internal_ram_of_pseudo_internal_ram_hit: |
---|
532 | ∀M:internal_pseudo_address_map.∀cm.∀s:PseudoStatus cm.∀sigma:Word → Word × bool.∀upper_lower: upper_lower. ∀points_to: Word. ∀addr:BitVector 7. |
---|
533 | assoc_list_lookup ?? (false:::addr) (eq_bv 8) (\fst M) = Some … 〈upper_lower, points_to〉 → |
---|
534 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
535 | let pbl ≝ lookup ? 7 addr (low_internal_ram … s) (zero 8) in |
---|
536 | let bl ≝ lookup ? 7 addr ram (zero 8) in |
---|
537 | let 〈lower, higher〉 ≝ vsplit ? 8 8 points_to in |
---|
538 | let 〈plower, phigher〉 ≝ vsplit ? 8 8 (\fst (sigma points_to)) in |
---|
539 | if eq_upper_lower upper_lower upper then |
---|
540 | (pbl = higher) ∧ (bl = phigher) |
---|
541 | else |
---|
542 | (pbl = lower) ∧ (bl = plower). |
---|
543 | |
---|
544 | (* changed from add to sub *) |
---|
545 | axiom low_internal_ram_of_pseudo_internal_ram_miss: |
---|
546 | ∀T.∀M:internal_pseudo_address_map.∀cm.∀s:PreStatus T cm.∀addr:BitVector 7. |
---|
547 | let ram ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s) in |
---|
548 | assoc_list_exists ?? (false:::addr) (eq_bv 8) (\fst M) = false → |
---|
549 | lookup ? 7 addr ram (zero ?) = lookup ? 7 addr (low_internal_ram … s) (zero ?). |
---|
550 | |
---|
551 | definition addressing_mode_ok ≝ |
---|
552 | λT.λM:internal_pseudo_address_map.λcm.λs:PreStatus T cm. |
---|
553 | λaddr:addressing_mode. |
---|
554 | match addr with |
---|
555 | [ DIRECT d ⇒ |
---|
556 | if eq_bv … d (bitvector_of_nat … 224) then |
---|
557 | ¬assoc_list_exists ?? d (eq_bv 8) (\fst M) ∧ eq_accumulator_address_map_entry (\snd M) data |
---|
558 | else |
---|
559 | ¬assoc_list_exists ?? d (eq_bv 8) (\fst M) |
---|
560 | | INDIRECT i ⇒ |
---|
561 | let d ≝ get_register … s [[false;false;i]] in |
---|
562 | ¬(assoc_list_exists ?? d (eq_bv 8) (\fst M)) |
---|
563 | | EXT_INDIRECT _ ⇒ true |
---|
564 | | REGISTER r ⇒ |
---|
565 | let address ≝ bit_address_of_register … s r in |
---|
566 | ¬assoc_list_exists … (false:::address) (eq_bv 8) (\fst M) |
---|
567 | | ACC_A ⇒ match \snd M with [ data ⇒ true | _ ⇒ false ] |
---|
568 | | ACC_B ⇒ true |
---|
569 | | DPTR ⇒ true |
---|
570 | | DATA _ ⇒ true |
---|
571 | | DATA16 _ ⇒ true |
---|
572 | | ACC_DPTR ⇒ true |
---|
573 | | ACC_PC ⇒ true |
---|
574 | | EXT_INDIRECT_DPTR ⇒ true |
---|
575 | | INDIRECT_DPTR ⇒ true |
---|
576 | | CARRY ⇒ true |
---|
577 | | BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
578 | | N_BIT_ADDR _ ⇒ ¬true (* TO BE COMPLETED *) |
---|
579 | | RELATIVE _ ⇒ true |
---|
580 | | ADDR11 _ ⇒ true |
---|
581 | | ADDR16 _ ⇒ true |
---|
582 | ]. |
---|
583 | |
---|
584 | definition next_internal_pseudo_address_map0 ≝ |
---|
585 | λT. |
---|
586 | λcm:T. |
---|
587 | λaddr_of: Identifier → PreStatus T cm → Word. |
---|
588 | λfetched. |
---|
589 | λM: internal_pseudo_address_map. |
---|
590 | λs: PreStatus T cm. |
---|
591 | match fetched with |
---|
592 | [ Comment _ ⇒ Some ? M |
---|
593 | | Cost _ ⇒ Some … M |
---|
594 | | Jmp _ ⇒ Some … M |
---|
595 | | Call a ⇒ |
---|
596 | let a' ≝ addr_of a s in |
---|
597 | let 〈callM, accM〉 ≝ M in |
---|
598 | Some … 〈〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)), 〈lower, a'〉〉:: |
---|
599 | 〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 2)), 〈upper, a'〉〉::callM, accM〉 |
---|
600 | | Mov _ _ ⇒ Some … M |
---|
601 | | Instruction instr ⇒ |
---|
602 | match instr return λx. option internal_pseudo_address_map with |
---|
603 | [ ADD addr1 addr2 ⇒ |
---|
604 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
605 | Some ? M |
---|
606 | else |
---|
607 | None ? |
---|
608 | | ADDC addr1 addr2 ⇒ |
---|
609 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
610 | Some ? M |
---|
611 | else |
---|
612 | None ? |
---|
613 | | SUBB addr1 addr2 ⇒ |
---|
614 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
615 | Some ? M |
---|
616 | else |
---|
617 | None ? |
---|
618 | | INC addr1 ⇒ |
---|
619 | if addressing_mode_ok T M ? s addr1 then |
---|
620 | Some ? M |
---|
621 | else |
---|
622 | None ? |
---|
623 | | DEC addr1 ⇒ |
---|
624 | if addressing_mode_ok T M … s addr1 then |
---|
625 | Some ? M |
---|
626 | else |
---|
627 | None ? |
---|
628 | | MUL addr1 addr2 ⇒ |
---|
629 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
630 | Some ? M |
---|
631 | else |
---|
632 | None ? |
---|
633 | | DIV addr1 addr2 ⇒ |
---|
634 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
635 | Some ? M |
---|
636 | else |
---|
637 | None ? |
---|
638 | | DA addr1 ⇒ |
---|
639 | if addressing_mode_ok T M … s addr1 then |
---|
640 | Some ? M |
---|
641 | else |
---|
642 | None ? |
---|
643 | | JC addr1 ⇒ Some ? M |
---|
644 | | JNC addr1 ⇒ Some ? M |
---|
645 | | JB addr1 addr2 ⇒ Some ? M |
---|
646 | | JNB addr1 addr2 ⇒ Some ? M |
---|
647 | | JBC addr1 addr2 ⇒ Some ? M |
---|
648 | | JZ addr1 ⇒ Some ? M |
---|
649 | | JNZ addr1 ⇒ Some ? M |
---|
650 | | CJNE addr1 addr2 ⇒ |
---|
651 | match addr1 with |
---|
652 | [ inl l ⇒ |
---|
653 | let 〈left, right〉 ≝ l in |
---|
654 | if addressing_mode_ok T M … s left ∧ addressing_mode_ok T M … s right then |
---|
655 | Some ? M |
---|
656 | else if ¬(addressing_mode_ok T M … s left) ∧ ¬(addressing_mode_ok T M … s right) then |
---|
657 | Some ? M |
---|
658 | else |
---|
659 | None ? |
---|
660 | | inr r ⇒ |
---|
661 | let 〈left, right〉 ≝ r in |
---|
662 | if addressing_mode_ok T M … s left ∧ addressing_mode_ok T M … s right then |
---|
663 | Some ? M |
---|
664 | else if ¬(addressing_mode_ok T M … s left) ∧ ¬(addressing_mode_ok T M … s right) then |
---|
665 | Some ? M |
---|
666 | else |
---|
667 | None ? |
---|
668 | ] |
---|
669 | | DJNZ addr1 addr2 ⇒ |
---|
670 | if addressing_mode_ok T M … s addr1 then |
---|
671 | Some ? M |
---|
672 | else |
---|
673 | None ? |
---|
674 | | CLR addr1 ⇒ |
---|
675 | if addressing_mode_ok T M … s addr1 then |
---|
676 | Some ? M |
---|
677 | else |
---|
678 | None ? |
---|
679 | | CPL addr1 ⇒ |
---|
680 | if addressing_mode_ok T M … s addr1 then |
---|
681 | Some ? M |
---|
682 | else |
---|
683 | None ? |
---|
684 | | RL addr1 ⇒ |
---|
685 | if addressing_mode_ok T M … s addr1 then |
---|
686 | Some ? M |
---|
687 | else |
---|
688 | None ? |
---|
689 | | RLC addr1 ⇒ |
---|
690 | if addressing_mode_ok T M … s addr1 then |
---|
691 | Some ? M |
---|
692 | else |
---|
693 | None ? |
---|
694 | | RR addr1 ⇒ |
---|
695 | if addressing_mode_ok T M … s addr1 then |
---|
696 | Some ? M |
---|
697 | else |
---|
698 | None ? |
---|
699 | | RRC addr1 ⇒ |
---|
700 | if addressing_mode_ok T M … s addr1 then |
---|
701 | Some ? M |
---|
702 | else |
---|
703 | None ? |
---|
704 | | SWAP addr1 ⇒ |
---|
705 | if addressing_mode_ok T M … s addr1 then |
---|
706 | Some ? M |
---|
707 | else |
---|
708 | None ? |
---|
709 | | SETB addr1 ⇒ |
---|
710 | if addressing_mode_ok T M … s addr1 then |
---|
711 | Some ? M |
---|
712 | else |
---|
713 | None ? |
---|
714 | (* XXX: need to track addresses pushed and popped off the stack? *) |
---|
715 | | PUSH addr1 ⇒ |
---|
716 | let 〈callM, accM〉 ≝ M in |
---|
717 | match addr1 return λx. bool_to_Prop (is_in … [[direct]] x) → ? with |
---|
718 | [ DIRECT d ⇒ λproof. |
---|
719 | let extended ≝ pad 8 8 d in |
---|
720 | Some … 〈〈(add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)), 〈lower, extended〉〉::callM, accM〉 |
---|
721 | | _ ⇒ λother: False. ⊥ |
---|
722 | ] (subaddressing_modein … addr1) |
---|
723 | | POP addr1 ⇒ Some … M |
---|
724 | | XCH addr1 addr2 ⇒ |
---|
725 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
726 | Some ? M |
---|
727 | else |
---|
728 | None ? |
---|
729 | | XCHD addr1 addr2 ⇒ |
---|
730 | if addressing_mode_ok T M … s addr1 ∧ addressing_mode_ok T M … s addr2 then |
---|
731 | Some ? M |
---|
732 | else |
---|
733 | None ? |
---|
734 | | RET ⇒ |
---|
735 | let 〈callM, accM〉 ≝ M in |
---|
736 | let sp_plus_1 ≝ assoc_list_exists ?? (add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)) (eq_bv 8) callM in |
---|
737 | let sp_plus_2 ≝ assoc_list_exists ?? (add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 2)) (eq_bv 8) callM in |
---|
738 | if sp_plus_1 ∧ sp_plus_2 then |
---|
739 | Some … M |
---|
740 | else |
---|
741 | None ? |
---|
742 | | RETI ⇒ |
---|
743 | let 〈callM, accM〉 ≝ M in |
---|
744 | let sp_plus_1 ≝ assoc_list_exists ?? (add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 1)) (eq_bv 8) callM in |
---|
745 | let sp_plus_2 ≝ assoc_list_exists ?? (add 8 (get_8051_sfr ?? s SFR_SP) (bitvector_of_nat 8 2)) (eq_bv 8) callM in |
---|
746 | if sp_plus_1 ∧ sp_plus_2 then |
---|
747 | Some … M |
---|
748 | else |
---|
749 | None ? |
---|
750 | | NOP ⇒ Some … M |
---|
751 | | MOVX addr1 ⇒ Some … M |
---|
752 | | XRL addr1 ⇒ |
---|
753 | match addr1 with |
---|
754 | [ inl l ⇒ |
---|
755 | let 〈acc_a, others〉 ≝ l in |
---|
756 | let acc_a_ok ≝ addressing_mode_ok T M … s acc_a in |
---|
757 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
758 | if acc_a_ok ∧ others_ok then |
---|
759 | Some ? M |
---|
760 | else |
---|
761 | None ? |
---|
762 | | inr r ⇒ |
---|
763 | let 〈direct, others〉 ≝ r in |
---|
764 | let direct_ok ≝ addressing_mode_ok T M … s direct in |
---|
765 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
766 | if direct_ok ∧ others_ok then |
---|
767 | Some ? M |
---|
768 | else |
---|
769 | None ? |
---|
770 | ] |
---|
771 | | ORL addr1 ⇒ |
---|
772 | match addr1 with |
---|
773 | [ inl l ⇒ |
---|
774 | match l with |
---|
775 | [ inl l ⇒ |
---|
776 | let 〈acc_a, others〉 ≝ l in |
---|
777 | let acc_a_ok ≝ addressing_mode_ok T M … s acc_a in |
---|
778 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
779 | if acc_a_ok ∧ others_ok then |
---|
780 | Some ? M |
---|
781 | else |
---|
782 | None ? |
---|
783 | | inr r ⇒ |
---|
784 | let 〈direct, others〉 ≝ r in |
---|
785 | let direct_ok ≝ addressing_mode_ok T M … s direct in |
---|
786 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
787 | if direct_ok ∧ others_ok then |
---|
788 | Some ? M |
---|
789 | else |
---|
790 | None ? |
---|
791 | ] |
---|
792 | | inr r ⇒ |
---|
793 | let 〈carry, others〉 ≝ r in |
---|
794 | let carry_ok ≝ addressing_mode_ok T M … s carry in |
---|
795 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
796 | if carry_ok ∧ others_ok then |
---|
797 | Some ? M |
---|
798 | else |
---|
799 | None ? |
---|
800 | ] |
---|
801 | | ANL addr1 ⇒ |
---|
802 | match addr1 with |
---|
803 | [ inl l ⇒ |
---|
804 | match l with |
---|
805 | [ inl l ⇒ |
---|
806 | let 〈acc_a, others〉 ≝ l in |
---|
807 | let acc_a_ok ≝ addressing_mode_ok T M … s acc_a in |
---|
808 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
809 | if acc_a_ok ∧ others_ok then |
---|
810 | Some ? M |
---|
811 | else |
---|
812 | None ? |
---|
813 | | inr r ⇒ |
---|
814 | let 〈direct, others〉 ≝ r in |
---|
815 | let direct_ok ≝ addressing_mode_ok T M … s direct in |
---|
816 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
817 | if direct_ok ∧ others_ok then |
---|
818 | Some ? M |
---|
819 | else |
---|
820 | None ? |
---|
821 | ] |
---|
822 | | inr r ⇒ |
---|
823 | let 〈carry, others〉 ≝ r in |
---|
824 | let carry_ok ≝ addressing_mode_ok T M … s carry in |
---|
825 | let others_ok ≝ addressing_mode_ok T M … s others in |
---|
826 | if carry_ok ∧ others_ok then |
---|
827 | Some ? M |
---|
828 | else |
---|
829 | None ? |
---|
830 | ] |
---|
831 | | MOV addr1 ⇒ Some … M |
---|
832 | ] |
---|
833 | ]. |
---|
834 | cases other |
---|
835 | qed. |
---|
836 | |
---|
837 | definition next_internal_pseudo_address_map ≝ |
---|
838 | λM:internal_pseudo_address_map. |
---|
839 | λcm. |
---|
840 | λaddr_of. |
---|
841 | λs:PseudoStatus cm. |
---|
842 | λppc_ok. |
---|
843 | next_internal_pseudo_address_map0 ? cm addr_of |
---|
844 | (\fst (fetch_pseudo_instruction (\snd cm) (program_counter … s) ppc_ok)) M s. |
---|
845 | |
---|
846 | definition code_memory_of_pseudo_assembly_program: |
---|
847 | ∀pap:pseudo_assembly_program. |
---|
848 | (Word → Word) → (Word → bool) → BitVectorTrie Byte 16 ≝ |
---|
849 | λpap. |
---|
850 | λsigma. |
---|
851 | λpolicy. |
---|
852 | let p ≝ pi1 … (assembly pap sigma policy) in |
---|
853 | load_code_memory (\fst p). |
---|
854 | |
---|
855 | definition sfr_8051_of_pseudo_sfr_8051 ≝ |
---|
856 | λM: internal_pseudo_address_map. |
---|
857 | λsfrs: Vector Byte 19. |
---|
858 | λsigma: Word → Word. |
---|
859 | match \snd M with |
---|
860 | [ data ⇒ sfrs |
---|
861 | | address upper_lower address ⇒ |
---|
862 | let index ≝ sfr_8051_index SFR_ACC_A in |
---|
863 | let 〈high, low〉 ≝ vsplit ? 8 8 (sigma address) in |
---|
864 | if eq_upper_lower upper_lower upper then |
---|
865 | set_index Byte 19 sfrs index high ? |
---|
866 | else |
---|
867 | set_index Byte 19 sfrs index low ? |
---|
868 | ]. |
---|
869 | // |
---|
870 | qed. |
---|
871 | |
---|
872 | definition status_of_pseudo_status: |
---|
873 | internal_pseudo_address_map → ∀pap. ∀ps: PseudoStatus pap. |
---|
874 | ∀sigma: Word → Word. ∀policy: Word → bool. |
---|
875 | Status (code_memory_of_pseudo_assembly_program pap sigma policy) ≝ |
---|
876 | λM. |
---|
877 | λpap. |
---|
878 | λps. |
---|
879 | λsigma. |
---|
880 | λpolicy. |
---|
881 | let cm ≝ code_memory_of_pseudo_assembly_program … sigma policy in |
---|
882 | let pc ≝ sigma (program_counter … ps) in |
---|
883 | let lir ≝ low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … ps) in |
---|
884 | let hir ≝ high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … ps) in |
---|
885 | mk_PreStatus (BitVectorTrie Byte 16) |
---|
886 | cm |
---|
887 | lir |
---|
888 | hir |
---|
889 | (external_ram … ps) |
---|
890 | pc |
---|
891 | (sfr_8051_of_pseudo_sfr_8051 M (special_function_registers_8051 … ps) sigma) |
---|
892 | (special_function_registers_8052 … ps) |
---|
893 | (p1_latch … ps) |
---|
894 | (p3_latch … ps) |
---|
895 | (clock … ps). |
---|
896 | |
---|
897 | (* |
---|
898 | definition write_at_stack_pointer': |
---|
899 | ∀M. ∀ps: PreStatus M. Byte → Σps':PreStatus M.(code_memory … ps = code_memory … ps') ≝ |
---|
900 | λM: Type[0]. |
---|
901 | λs: PreStatus M. |
---|
902 | λv: Byte. |
---|
903 | let 〈 nu, nl 〉 ≝ vsplit … 4 4 (get_8051_sfr ? s SFR_SP) in |
---|
904 | let bit_zero ≝ get_index_v… nu O ? in |
---|
905 | let bit_1 ≝ get_index_v… nu 1 ? in |
---|
906 | let bit_2 ≝ get_index_v… nu 2 ? in |
---|
907 | let bit_3 ≝ get_index_v… nu 3 ? in |
---|
908 | if bit_zero then |
---|
909 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
910 | v (low_internal_ram ? s) in |
---|
911 | set_low_internal_ram ? s memory |
---|
912 | else |
---|
913 | let memory ≝ insert … ([[ bit_1 ; bit_2 ; bit_3 ]] @@ nl) |
---|
914 | v (high_internal_ram ? s) in |
---|
915 | set_high_internal_ram ? s memory. |
---|
916 | [ cases l0 % |
---|
917 | |2,3,4,5: normalize repeat (@ le_S_S) @ le_O_n ] |
---|
918 | qed. |
---|
919 | |
---|
920 | definition execute_1_pseudo_instruction': (Word → nat) → ∀ps:PseudoStatus. |
---|
921 | Σps':PseudoStatus.(code_memory … ps = code_memory … ps') |
---|
922 | ≝ |
---|
923 | λticks_of. |
---|
924 | λs. |
---|
925 | let 〈instr, pc〉 ≝ fetch_pseudo_instruction (\snd (code_memory ? s)) (program_counter ? s) in |
---|
926 | let ticks ≝ ticks_of (program_counter ? s) in |
---|
927 | let s ≝ set_clock ? s (clock ? s + ticks) in |
---|
928 | let s ≝ set_program_counter ? s pc in |
---|
929 | match instr with |
---|
930 | [ Instruction instr ⇒ |
---|
931 | execute_1_preinstruction … (λx, y. address_of_word_labels y x) instr s |
---|
932 | | Comment cmt ⇒ s |
---|
933 | | Cost cst ⇒ s |
---|
934 | | Jmp jmp ⇒ set_program_counter ? s (address_of_word_labels s jmp) |
---|
935 | | Call call ⇒ |
---|
936 | let a ≝ address_of_word_labels s call in |
---|
937 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
938 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
939 | let 〈pc_bu, pc_bl〉 ≝ vsplit ? 8 8 (program_counter ? s) in |
---|
940 | let s ≝ write_at_stack_pointer' ? s pc_bl in |
---|
941 | let 〈carry, new_sp〉 ≝ half_add ? (get_8051_sfr ? s SFR_SP) (bitvector_of_nat 8 1) in |
---|
942 | let s ≝ set_8051_sfr ? s SFR_SP new_sp in |
---|
943 | let s ≝ write_at_stack_pointer' ? s pc_bu in |
---|
944 | set_program_counter ? s a |
---|
945 | | Mov dptr ident ⇒ |
---|
946 | set_arg_16 ? s (get_arg_16 ? s (DATA16 (address_of_word_labels s ident))) dptr |
---|
947 | ]. |
---|
948 | [ |
---|
949 | |2,3,4: % |
---|
950 | | <(sig2 … l7) whd in ⊢ (??? (??%)) <(sig2 … l5) % |
---|
951 | | |
---|
952 | | % |
---|
953 | ] |
---|
954 | cases not_implemented |
---|
955 | qed. |
---|
956 | *) |
---|
957 | |
---|
958 | (* |
---|
959 | lemma execute_code_memory_unchanged: |
---|
960 | ∀ticks_of,ps. code_memory ? ps = code_memory ? (execute_1_pseudo_instruction ticks_of ps). |
---|
961 | #ticks #ps whd in ⊢ (??? (??%)) |
---|
962 | cases (fetch_pseudo_instruction (\snd (code_memory pseudo_assembly_program ps)) |
---|
963 | (program_counter pseudo_assembly_program ps)) #instr #pc |
---|
964 | whd in ⊢ (??? (??%)) cases instr |
---|
965 | [ #pre cases pre |
---|
966 | [ #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
967 | cases (vsplit ????) #z1 #z2 % |
---|
968 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (add_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
969 | cases (vsplit ????) #z1 #z2 % |
---|
970 | | #a1 #a2 whd in ⊢ (??? (??%)) cases (sub_8_with_carry ???) #y1 #y2 whd in ⊢ (??? (??%)) |
---|
971 | cases (vsplit ????) #z1 #z2 % |
---|
972 | | #a1 whd in ⊢ (??? (??%)) cases a1 #x #H whd in ⊢ (??? (??%)) cases x |
---|
973 | [ #x1 whd in ⊢ (??? (??%)) |
---|
974 | | *: cases not_implemented |
---|
975 | ] |
---|
976 | | #comment % |
---|
977 | | #cost % |
---|
978 | | #label % |
---|
979 | | #label whd in ⊢ (??? (??%)) cases (half_add ???) #x1 #x2 whd in ⊢ (??? (??%)) |
---|
980 | cases (vsplit ????) #y1 #y2 whd in ⊢ (??? (??%)) cases (half_add ???) #z1 #z2 |
---|
981 | whd in ⊢ (??? (??%)) whd in ⊢ (??? (??%)) cases (vsplit ????) #w1 #w2 |
---|
982 | whd in ⊢ (??? (??%)) cases (get_index_v bool ????) whd in ⊢ (??? (??%)) |
---|
983 | (* CSC: ??? *) |
---|
984 | | #dptr #label (* CSC: ??? *) |
---|
985 | ] |
---|
986 | cases not_implemented |
---|
987 | qed. |
---|
988 | *) |
---|
989 | |
---|
990 | (* XXX: check values returned for conditional jumps below! They are wrong, |
---|
991 | find correct values *) |
---|
992 | definition ticks_of0: |
---|
993 | ∀p:pseudo_assembly_program. |
---|
994 | (Identifier → Word) → (Word → Word) → (Word → bool) → Word → pseudo_instruction → nat × nat ≝ |
---|
995 | λprogram: pseudo_assembly_program. |
---|
996 | λlookup_labels: Identifier → Word. |
---|
997 | λsigma. |
---|
998 | λpolicy. |
---|
999 | λppc: Word. |
---|
1000 | λfetched. |
---|
1001 | match fetched with |
---|
1002 | [ Instruction instr ⇒ |
---|
1003 | match instr with |
---|
1004 | [ JC lbl ⇒ |
---|
1005 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1006 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1007 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1008 | if sj_possible then |
---|
1009 | 〈2, 2〉 |
---|
1010 | else |
---|
1011 | 〈4, 4〉 |
---|
1012 | | JNC lbl ⇒ |
---|
1013 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1014 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1015 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1016 | if sj_possible then |
---|
1017 | 〈2, 2〉 |
---|
1018 | else |
---|
1019 | 〈4, 4〉 |
---|
1020 | | JB bit lbl ⇒ |
---|
1021 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1022 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1023 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1024 | if sj_possible then |
---|
1025 | 〈2, 2〉 |
---|
1026 | else |
---|
1027 | 〈4, 4〉 |
---|
1028 | | JNB bit lbl ⇒ |
---|
1029 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1030 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1031 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1032 | if sj_possible then |
---|
1033 | 〈2, 2〉 |
---|
1034 | else |
---|
1035 | 〈4, 4〉 |
---|
1036 | | JBC bit lbl ⇒ |
---|
1037 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1038 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1039 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1040 | if sj_possible then |
---|
1041 | 〈2, 2〉 |
---|
1042 | else |
---|
1043 | 〈4, 4〉 |
---|
1044 | | JZ lbl ⇒ |
---|
1045 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1046 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1047 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1048 | if sj_possible then |
---|
1049 | 〈2, 2〉 |
---|
1050 | else |
---|
1051 | 〈4, 4〉 |
---|
1052 | | JNZ lbl ⇒ |
---|
1053 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1054 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1055 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1056 | if sj_possible then |
---|
1057 | 〈2, 2〉 |
---|
1058 | else |
---|
1059 | 〈4, 4〉 |
---|
1060 | | CJNE arg lbl ⇒ |
---|
1061 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1062 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1063 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1064 | if sj_possible then |
---|
1065 | 〈2, 2〉 |
---|
1066 | else |
---|
1067 | 〈4, 4〉 |
---|
1068 | | DJNZ arg lbl ⇒ |
---|
1069 | let lookup_address ≝ sigma (lookup_labels lbl) in |
---|
1070 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1071 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1072 | if sj_possible then |
---|
1073 | 〈2, 2〉 |
---|
1074 | else |
---|
1075 | 〈4, 4〉 |
---|
1076 | | ADD arg1 arg2 ⇒ |
---|
1077 | let ticks ≝ ticks_of_instruction (ADD ? arg1 arg2) in |
---|
1078 | 〈ticks, ticks〉 |
---|
1079 | | ADDC arg1 arg2 ⇒ |
---|
1080 | let ticks ≝ ticks_of_instruction (ADDC ? arg1 arg2) in |
---|
1081 | 〈ticks, ticks〉 |
---|
1082 | | SUBB arg1 arg2 ⇒ |
---|
1083 | let ticks ≝ ticks_of_instruction (SUBB ? arg1 arg2) in |
---|
1084 | 〈ticks, ticks〉 |
---|
1085 | | INC arg ⇒ |
---|
1086 | let ticks ≝ ticks_of_instruction (INC ? arg) in |
---|
1087 | 〈ticks, ticks〉 |
---|
1088 | | DEC arg ⇒ |
---|
1089 | let ticks ≝ ticks_of_instruction (DEC ? arg) in |
---|
1090 | 〈ticks, ticks〉 |
---|
1091 | | MUL arg1 arg2 ⇒ |
---|
1092 | let ticks ≝ ticks_of_instruction (MUL ? arg1 arg2) in |
---|
1093 | 〈ticks, ticks〉 |
---|
1094 | | DIV arg1 arg2 ⇒ |
---|
1095 | let ticks ≝ ticks_of_instruction (DIV ? arg1 arg2) in |
---|
1096 | 〈ticks, ticks〉 |
---|
1097 | | DA arg ⇒ |
---|
1098 | let ticks ≝ ticks_of_instruction (DA ? arg) in |
---|
1099 | 〈ticks, ticks〉 |
---|
1100 | | ANL arg ⇒ |
---|
1101 | let ticks ≝ ticks_of_instruction (ANL ? arg) in |
---|
1102 | 〈ticks, ticks〉 |
---|
1103 | | ORL arg ⇒ |
---|
1104 | let ticks ≝ ticks_of_instruction (ORL ? arg) in |
---|
1105 | 〈ticks, ticks〉 |
---|
1106 | | XRL arg ⇒ |
---|
1107 | let ticks ≝ ticks_of_instruction (XRL ? arg) in |
---|
1108 | 〈ticks, ticks〉 |
---|
1109 | | CLR arg ⇒ |
---|
1110 | let ticks ≝ ticks_of_instruction (CLR ? arg) in |
---|
1111 | 〈ticks, ticks〉 |
---|
1112 | | CPL arg ⇒ |
---|
1113 | let ticks ≝ ticks_of_instruction (CPL ? arg) in |
---|
1114 | 〈ticks, ticks〉 |
---|
1115 | | RL arg ⇒ |
---|
1116 | let ticks ≝ ticks_of_instruction (RL ? arg) in |
---|
1117 | 〈ticks, ticks〉 |
---|
1118 | | RLC arg ⇒ |
---|
1119 | let ticks ≝ ticks_of_instruction (RLC ? arg) in |
---|
1120 | 〈ticks, ticks〉 |
---|
1121 | | RR arg ⇒ |
---|
1122 | let ticks ≝ ticks_of_instruction (RR ? arg) in |
---|
1123 | 〈ticks, ticks〉 |
---|
1124 | | RRC arg ⇒ |
---|
1125 | let ticks ≝ ticks_of_instruction (RRC ? arg) in |
---|
1126 | 〈ticks, ticks〉 |
---|
1127 | | SWAP arg ⇒ |
---|
1128 | let ticks ≝ ticks_of_instruction (SWAP ? arg) in |
---|
1129 | 〈ticks, ticks〉 |
---|
1130 | | MOV arg ⇒ |
---|
1131 | let ticks ≝ ticks_of_instruction (MOV ? arg) in |
---|
1132 | 〈ticks, ticks〉 |
---|
1133 | | MOVX arg ⇒ |
---|
1134 | let ticks ≝ ticks_of_instruction (MOVX ? arg) in |
---|
1135 | 〈ticks, ticks〉 |
---|
1136 | | SETB arg ⇒ |
---|
1137 | let ticks ≝ ticks_of_instruction (SETB ? arg) in |
---|
1138 | 〈ticks, ticks〉 |
---|
1139 | | PUSH arg ⇒ |
---|
1140 | let ticks ≝ ticks_of_instruction (PUSH ? arg) in |
---|
1141 | 〈ticks, ticks〉 |
---|
1142 | | POP arg ⇒ |
---|
1143 | let ticks ≝ ticks_of_instruction (POP ? arg) in |
---|
1144 | 〈ticks, ticks〉 |
---|
1145 | | XCH arg1 arg2 ⇒ |
---|
1146 | let ticks ≝ ticks_of_instruction (XCH ? arg1 arg2) in |
---|
1147 | 〈ticks, ticks〉 |
---|
1148 | | XCHD arg1 arg2 ⇒ |
---|
1149 | let ticks ≝ ticks_of_instruction (XCHD ? arg1 arg2) in |
---|
1150 | 〈ticks, ticks〉 |
---|
1151 | | RET ⇒ |
---|
1152 | let ticks ≝ ticks_of_instruction (RET ?) in |
---|
1153 | 〈ticks, ticks〉 |
---|
1154 | | RETI ⇒ |
---|
1155 | let ticks ≝ ticks_of_instruction (RETI ?) in |
---|
1156 | 〈ticks, ticks〉 |
---|
1157 | | NOP ⇒ |
---|
1158 | let ticks ≝ ticks_of_instruction (NOP ?) in |
---|
1159 | 〈ticks, ticks〉 |
---|
1160 | ] |
---|
1161 | | Comment comment ⇒ 〈0, 0〉 |
---|
1162 | | Cost cost ⇒ 〈0, 0〉 |
---|
1163 | | Jmp jmp ⇒ |
---|
1164 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1165 | let do_a_long ≝ policy ppc in |
---|
1166 | let lookup_address ≝ sigma (lookup_labels jmp) in |
---|
1167 | let 〈sj_possible, disp〉 ≝ short_jump_cond pc_plus_jmp_length lookup_address in |
---|
1168 | if sj_possible ∧ ¬ do_a_long then |
---|
1169 | 〈2, 2〉 (* XXX: SJMP *) |
---|
1170 | else |
---|
1171 | let 〈mj_possible, disp2〉 ≝ absolute_jump_cond pc_plus_jmp_length lookup_address in |
---|
1172 | if mj_possible ∧ ¬ do_a_long then |
---|
1173 | 〈2, 2〉 (* XXX: AJMP *) |
---|
1174 | else |
---|
1175 | 〈2, 2〉 (* XXX: LJMP *) |
---|
1176 | | Call call ⇒ |
---|
1177 | (* XXX: collapse the branches as they are identical? *) |
---|
1178 | let pc_plus_jmp_length ≝ sigma (add … ppc (bitvector_of_nat … 1)) in |
---|
1179 | let lookup_address ≝ sigma (lookup_labels call) in |
---|
1180 | let 〈mj_possible, disp〉 ≝ absolute_jump_cond pc_plus_jmp_length lookup_address in |
---|
1181 | let do_a_long ≝ policy ppc in |
---|
1182 | if mj_possible ∧ ¬ do_a_long then |
---|
1183 | 〈2, 2〉 (* ACALL *) |
---|
1184 | else |
---|
1185 | 〈2, 2〉 (* LCALL *) |
---|
1186 | | Mov dptr tgt ⇒ 〈2, 2〉 |
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1187 | ]. |
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1188 | |
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1189 | definition ticks_of: |
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1190 | ∀p:pseudo_assembly_program. |
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1191 | (Identifier → Word) → (Word → Word) → (Word → bool) → ∀ppc:Word. |
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1192 | nat_of_bitvector … ppc < |\snd p| → nat × nat ≝ |
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1193 | λprogram: pseudo_assembly_program. |
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1194 | λlookup_labels. |
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1195 | λsigma. |
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1196 | λpolicy. |
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1197 | λppc: Word. λppc_ok. |
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1198 | let pseudo ≝ \snd program in |
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1199 | let 〈fetched, new_ppc〉 ≝ fetch_pseudo_instruction pseudo ppc ppc_ok in |
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1200 | ticks_of0 program lookup_labels sigma policy ppc fetched. |
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1201 | |
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1202 | (* |
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1203 | lemma blah: |
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1204 | ∀m: internal_pseudo_address_map. |
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1205 | ∀s: PseudoStatus. |
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1206 | ∀arg: Byte. |
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1207 | ∀b: bool. |
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1208 | addressing_mode_ok m s (DIRECT arg) = true → |
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1209 | get_arg_8 ? (set_low_internal_ram ? s (low_internal_ram_of_pseudo_low_internal_ram m (low_internal_ram ? s))) b (DIRECT arg) = |
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1210 | get_arg_8 ? s b (DIRECT arg). |
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1211 | [2, 3: normalize % ] |
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1212 | #m #s #arg #b #hyp |
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1213 | whd in ⊢ (??%%) |
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1214 | @vsplit_elim'' |
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1215 | #nu' #nl' #arg_nu_nl_eq |
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1216 | normalize nodelta |
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1217 | generalize in match (refl ? (get_index_v bool 4 nu' ? ?)) |
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1218 | cases (get_index_v bool 4 nu' ? ?) in ⊢ (??%? → %) |
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1219 | #get_index_v_eq |
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1220 | normalize nodelta |
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1221 | [2: |
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1222 | normalize nodelta |
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1223 | @vsplit_elim'' |
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1224 | #bit_one' #three_bits' #bit_one_three_bit_eq |
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1225 | generalize in match (low_internal_ram_of_pseudo_internal_ram_miss m s (three_bits'@@nl')) |
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1226 | normalize nodelta |
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1227 | generalize in match (refl ? (sub_7_with_carry ? ? ?)) |
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1228 | cases (sub_7_with_carry ? ? ?) in ⊢ (??%? → %) |
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1229 | #Saddr #carr' #Saddr_carr_eq |
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1230 | normalize nodelta |
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1231 | #carr_hyp' |
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1232 | @carr_hyp' |
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1233 | [1: |
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1234 | |2: whd in hyp:(??%?); generalize in match hyp; -hyp; |
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1235 | generalize in match (refl ? (¬(member (BitVector 8) ? arg m))) |
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1236 | cases (¬(member (BitVector 8) ? arg m)) in ⊢ (??%? → %) |
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1237 | #member_eq |
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1238 | normalize nodelta |
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1239 | [2: #destr destruct(destr) |
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1240 | |1: -carr_hyp'; |
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1241 | >arg_nu_nl_eq |
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1242 | <(vsplit_vector_singleton ? ? nu' ? ? ? bit_one_three_bit_eq) |
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1243 | [1: >get_index_v_eq in ⊢ (??%? → ?) |
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1244 | |2: @le_S @le_S @le_S @le_n |
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1245 | ] |
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1246 | cases (member (BitVector 8) ? (\fst ?) ?) |
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1247 | [1: #destr normalize in destr; destruct(destr) |
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1248 | |2: |
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1249 | ] |
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1250 | ] |
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1251 | |3: >get_index_v_eq in ⊢ (??%?) |
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1252 | change in ⊢ (??(???%?)?) with ((? ::: three_bits') @@ nl') |
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1253 | >(vsplit_vector_singleton … bit_one_three_bit_eq) |
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1254 | <arg_nu_nl_eq |
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1255 | whd in hyp:(??%?) |
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1256 | cases (member (BitVector 8) (eq_bv 8) arg m) in hyp |
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1257 | normalize nodelta [*: #ignore @sym_eq ] |
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1258 | ] |
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1259 | | |
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1260 | ]. |
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1261 | *) |
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1262 | (* |
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1263 | map_address0 ... (DIRECT arg) = Some .. → |
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1264 | get_arg_8 (map_address0 ... (internal_ram ...) (DIRECT arg) = |
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1265 | get_arg_8 (internal_ram ...) (DIRECT arg) |
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1266 | |
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1267 | ((if addressing_mode_ok M ps ACC_A∧addressing_mode_ok M ps (DIRECT ARG2) |
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1268 | then Some internal_pseudo_address_map M |
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1269 | else None internal_pseudo_address_map ) |
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1270 | =Some internal_pseudo_address_map M') |
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1271 | |
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1272 | axiom low_internal_ram_write_at_stack_pointer: |
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1273 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma: Word → Word.∀policy: Word → bool. |
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1274 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
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1275 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
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1276 | low_internal_ram ? cm2 s2 = low_internal_ram T2 cm3 s3 → |
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1277 | sp1 = add ? (get_8051_sfr … cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
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1278 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
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1279 | bu@@bl = sigma (pbu@@pbl) → |
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1280 | low_internal_ram T1 cm1 |
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1281 | (write_at_stack_pointer … |
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1282 | (set_8051_sfr … |
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1283 | (write_at_stack_pointer … |
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1284 | (set_8051_sfr … |
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1285 | (set_low_internal_ram … s1 |
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1286 | (low_internal_ram_of_pseudo_low_internal_ram M (low_internal_ram … s2))) |
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1287 | SFR_SP sp1) |
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1288 | bl) |
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1289 | SFR_SP sp2) |
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1290 | bu) |
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1291 | = low_internal_ram_of_pseudo_low_internal_ram (sp1::M) |
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1292 | (low_internal_ram … |
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1293 | (write_at_stack_pointer … |
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1294 | (set_8051_sfr … |
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1295 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
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1296 | SFR_SP sp2) |
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1297 | pbu)). |
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1298 | |
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1299 | lemma high_internal_ram_write_at_stack_pointer: |
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1300 | ∀T1,T2,M,cm1,s1,cm2,s2,cm3,s3.∀sigma:Word → Word.∀policy: Word → bool. |
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1301 | ∀pbu,pbl,bu,bl,sp1,sp2:BitVector 8. |
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1302 | get_8051_sfr T2 cm2 s2 SFR_SP = get_8051_sfr ? cm3 s3 SFR_SP → |
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1303 | high_internal_ram ?? s2 = high_internal_ram T2 cm3 s3 → |
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1304 | sp1 = add ? (get_8051_sfr ? cm1 s1 SFR_SP) (bitvector_of_nat 8 1) → |
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1305 | sp2 = add ? sp1 (bitvector_of_nat 8 1) → |
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1306 | bu@@bl = sigma (pbu@@pbl) → |
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1307 | high_internal_ram T1 cm1 |
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1308 | (write_at_stack_pointer … |
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1309 | (set_8051_sfr … |
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1310 | (write_at_stack_pointer … |
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1311 | (set_8051_sfr … |
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1312 | (set_high_internal_ram … s1 |
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1313 | (high_internal_ram_of_pseudo_high_internal_ram M (high_internal_ram … s2))) |
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1314 | SFR_SP sp1) |
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1315 | bl) |
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1316 | SFR_SP sp2) |
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1317 | bu) |
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1318 | = high_internal_ram_of_pseudo_high_internal_ram (sp1::M) |
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1319 | (high_internal_ram … |
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1320 | (write_at_stack_pointer … |
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1321 | (set_8051_sfr … |
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1322 | (write_at_stack_pointer … (set_8051_sfr … s3 SFR_SP sp1) pbl) |
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1323 | SFR_SP sp2) |
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1324 | pbu)). |
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1325 | #T1 #T2 #M #cm1 #s1 #cm2 #s2 #cm3 #s3 #sigma #policy #pbu #pbl #bu #bl #sp1 #sp2 |
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1326 | #get_8051_sfr_refl #high_internal_ram_refl #sp1_refl #sp2_refl #sigma_refl |
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1327 | cases daemon (* XXX: !!! *) |
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1328 | qed. |
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1329 | *) |
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1330 | |
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1331 | lemma snd_assembly_1_pseudoinstruction_ok: |
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1332 | ∀program: pseudo_assembly_program. |
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1333 | ∀program_length_proof: |\snd program| ≤ 2^16. |
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1334 | ∀sigma: Word → Word. |
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1335 | ∀policy: Word → bool. |
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1336 | ∀sigma_policy_specification_witness: sigma_policy_specification program sigma policy. |
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1337 | ∀ppc: Word. |
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1338 | ∀ppc_in_bounds: nat_of_bitvector 16 ppc < |\snd program|. |
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1339 | ∀pi. |
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1340 | ∀lookup_labels. |
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1341 | ∀lookup_datalabels. |
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1342 | let 〈labels, costs〉 ≝ create_label_cost_map (\snd program) in |
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1343 | lookup_labels = (λx. bitvector_of_nat 16 (lookup_def … labels x 0)) → |
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1344 | lookup_datalabels = (λx. lookup_def … (construct_datalabels (\fst program)) x (zero ?)) → |
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1345 | \fst (fetch_pseudo_instruction (\snd program) ppc ppc_in_bounds) = pi → |
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1346 | let len ≝ \fst (assembly_1_pseudoinstruction lookup_labels sigma policy (*(sigma ppc)*) ppc lookup_datalabels pi) in |
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1347 | sigma (add … ppc (bitvector_of_nat ? 1)) = add … (sigma ppc) (bitvector_of_nat ? len). |
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1348 | #program #program_length_proof #sigma #policy #sigma_policy_specification_witness #ppc #ppc_in_bounds #pi |
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1349 | #lookup_labels #lookup_datalabels |
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1350 | @pair_elim #labels #costs #create_label_cost_map_refl |
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1351 | #lookup_labels_refl #lookup_datalabels_refl #fetch_pseudo_refl |
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1352 | normalize nodelta |
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1353 | generalize in match fetch_pseudo_refl; -fetch_pseudo_refl |
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1354 | #fetch_pseudo_refl |
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1355 | letin assembled ≝ (\fst (pi1 … (assembly program sigma policy))) |
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1356 | letin costs ≝ (\snd (pi1 … (assembly program sigma policy))) |
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1357 | lapply (assembly_ok program program_length_proof sigma policy sigma_policy_specification_witness assembled costs) |
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1358 | @pair_elim #preamble #instr_list #program_refl |
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1359 | lapply create_label_cost_map_refl; -create_label_cost_map_refl |
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1360 | >program_refl in ⊢ (% → ?); #create_label_cost_map_refl |
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1361 | >create_label_cost_map_refl |
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1362 | <eq_pair_fst_snd #H lapply (H (refl …)) -H #H |
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1363 | lapply (H ppc ppc_in_bounds) -H |
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1364 | @pair_elim #pi' #newppc #fetch_pseudo_refl' |
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1365 | @pair_elim #len #assembled #assembly1_refl #H |
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1366 | cases H |
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1367 | #encoding_check_assm #sigma_newppc_refl |
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1368 | >fetch_pseudo_refl' in fetch_pseudo_refl; #pi_refl' |
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1369 | >pi_refl' in assembly1_refl; #assembly1_refl |
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1370 | >lookup_labels_refl >lookup_datalabels_refl |
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1371 | >program_refl normalize nodelta |
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1372 | >assembly1_refl |
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1373 | <sigma_newppc_refl |
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1374 | generalize in match fetch_pseudo_refl'; |
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1375 | whd in match (fetch_pseudo_instruction ???); |
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1376 | @pair_elim #lbl #instr #nth_refl normalize nodelta |
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1377 | #G cases (pair_destruct_right ?????? G) % |
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1378 | qed. |
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1379 | |
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1380 | (* To be moved in ProofStatus *) |
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1381 | lemma program_counter_set_program_counter: |
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1382 | ∀T. |
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1383 | ∀cm. |
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1384 | ∀s. |
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1385 | ∀x. |
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1386 | program_counter T cm (set_program_counter T cm s x) = x. |
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1387 | // |
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1388 | qed. |
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