1 | include "RTLabs/syntax.ma". |
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2 | include "RTL/RTL.ma". |
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3 | include "common/AssocList.ma". |
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4 | include "common/FrontEndOps.ma". |
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5 | include "common/Graphs.ma". |
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6 | include "joint/TranslateUtils.ma". |
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7 | |
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8 | let rec register_freshes (runiverse: universe RegisterTag) (n: nat) on n ≝ |
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9 | match n with |
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10 | [ O ⇒ 〈[],runiverse〉 |
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11 | | S n' ⇒ |
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12 | let 〈r,runiverse〉 ≝ fresh … runiverse in |
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13 | let 〈res,runiverse〉 ≝ register_freshes runiverse n' in |
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14 | 〈r::res,runiverse〉 ]. |
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15 | |
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16 | definition complete_regs ≝ |
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17 | λglobals. |
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18 | λdef. |
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19 | λsrcrs1. |
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20 | λsrcrs2. |
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21 | if leb (length … srcrs2) (length … srcrs1) then |
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22 | let 〈def, added_regs〉 ≝ fresh_regs rtl_params0 globals def (minus (length ? srcrs1) (length ? srcrs2)) in |
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23 | 〈〈srcrs1, srcrs2 @ added_regs〉, added_regs〉 |
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24 | else |
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25 | let 〈def, added_regs〉 ≝ fresh_regs rtl_params0 globals def (minus (length ? srcrs2) (length ? srcrs1)) in |
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26 | 〈〈srcrs1 @ added_regs, srcrs2〉, added_regs〉. |
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27 | |
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28 | lemma complete_regs_length: |
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29 | ∀globals,def,left,right. |
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30 | |\fst (\fst (complete_regs globals def left right))| = |\snd (\fst (complete_regs globals def left right))|. |
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31 | #globals #def #left #right |
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32 | whd in match complete_regs normalize nodelta |
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33 | @leb_elim normalize nodelta #H |
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34 | [ generalize in match (fresh_regs_length rtl_params0 globals def (minus (length … left) (length … right))) |
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35 | | generalize in match (fresh_regs_length rtl_params0 globals def (minus (length … right) (length … left)))] |
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36 | cases (fresh_regs ????) #def' #fresh normalize >append_length |
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37 | generalize in match H -H; |
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38 | generalize in match (length … left) generalize in match (length … right) generalize in match (length … fresh) |
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39 | [ /2/ | #x #y #z #H generalize in match (not_le_to_lt … H) -H #H #E >E >commutative_plus |
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40 | <plus_minus_m_m /2/ ] |
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41 | qed. |
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42 | |
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43 | definition size_of_sig_type ≝ |
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44 | λsig. |
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45 | match sig with |
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46 | [ ASTint isize sign ⇒ |
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47 | let isize' ≝ match isize with [ I8 ⇒ 8 | I16 ⇒ 16 | I32 ⇒ 32 ] in |
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48 | isize' ÷ (nat_of_bitvector ? int_size) |
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49 | | ASTfloat _ ⇒ ? (* dpm: not implemented *) |
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50 | | ASTptr rgn ⇒ nat_of_bitvector ? ptr_size |
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51 | ]. |
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52 | cases not_implemented; |
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53 | qed. |
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54 | |
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55 | inductive register_type: Type[0] ≝ |
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56 | | register_int: register → register_type |
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57 | | register_ptr: register → register → register_type. |
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58 | |
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59 | definition local_env ≝ BitVectorTrie (list register) 16. |
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60 | |
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61 | definition mem_local_env : register → local_env → bool ≝ |
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62 | λr. member … (word_of_identifier … r). |
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63 | |
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64 | definition add_local_env : register → list register → local_env → local_env ≝ |
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65 | λr. insert … (word_of_identifier … r). |
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66 | |
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67 | definition find_local_env : register → local_env → list register ≝ |
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68 | λr: register.λbvt. lookup … (word_of_identifier … r) bvt []. |
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69 | |
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70 | definition initialize_local_env_internal ≝ |
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71 | λlenv_runiverse. |
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72 | λr_sig. |
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73 | let 〈lenv,runiverse〉 ≝ lenv_runiverse in |
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74 | let 〈r, sig〉 ≝ r_sig in |
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75 | let size ≝ size_of_sig_type sig in |
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76 | let 〈rs,runiverse〉 ≝ register_freshes runiverse size in |
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77 | 〈add_local_env r rs lenv,runiverse〉. |
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78 | |
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79 | definition initialize_local_env ≝ |
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80 | λruniverse. |
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81 | λregisters. |
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82 | λresult. |
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83 | let registers ≝ registers @ |
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84 | match result with |
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85 | [ None ⇒ [ ] |
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86 | | Some rt ⇒ [ rt ] |
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87 | ] |
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88 | in |
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89 | foldl ? ? initialize_local_env_internal 〈Stub …,runiverse〉 registers. |
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90 | |
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91 | definition map_list_local_env_internal ≝ |
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92 | λlenv,res,r. res @ (find_local_env r lenv). |
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93 | |
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94 | definition map_list_local_env ≝ |
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95 | λlenv,regs. foldl … (map_list_local_env_internal lenv) [ ] regs. |
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96 | |
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97 | definition make_addr ≝ |
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98 | λA. |
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99 | λlst: list A. |
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100 | λprf: 2 ≤ length A lst. |
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101 | match lst return λx. 2 ≤ |x| → A × A with |
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102 | [ nil ⇒ λlst_nil_prf. ? |
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103 | | cons hd tl ⇒ λprf. |
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104 | match tl return λx. 1 ≤ |x| → A × A with |
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105 | [ nil ⇒ λtl_nil_prf. ? |
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106 | | cons hd' tl' ⇒ λtl_cons_prf. 〈hd, hd'〉 |
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107 | ] ? |
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108 | ] prf. |
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109 | [1: normalize in lst_nil_prf; |
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110 | cases(not_le_Sn_O 1); |
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111 | #HYP cases(HYP lst_nil_prf) |
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112 | |2: normalize in prf; |
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113 | @le_S_S_to_le |
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114 | assumption |
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115 | |3: normalize in tl_nil_prf; |
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116 | cases(not_le_Sn_O 0) |
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117 | #HYP cases(HYP tl_nil_prf)] |
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118 | qed. |
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119 | |
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120 | definition find_and_addr ≝ |
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121 | λr,lenv. make_addr ? (find_local_env r lenv). |
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122 | |
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123 | definition rtl_args ≝ |
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124 | λregs_list,lenv. flatten … (map … (λr. find_local_env r lenv) regs_list). |
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125 | |
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126 | definition translate_cst_int_internal ≝ |
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127 | λglobals,dest_lbl,r,i. sequential rtl_params_ globals (INT … r i) dest_lbl. |
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128 | |
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129 | axiom split_into_bytes: |
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130 | ∀size. |
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131 | ∀int: bvint size. |
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132 | Σbytes: list Byte. |bytes| = size_intsize size. |
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133 | |
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134 | lemma eqb_implies_eq: |
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135 | ∀m, n: nat. |
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136 | eqb m n = true → m = n. |
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137 | #M |
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138 | elim M |
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139 | [1: #N normalize |
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140 | cases N |
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141 | [1: normalize // |
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142 | |2: #M' normalize #HYP destruct(HYP) |
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143 | ] |
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144 | |2: #M' #IND_HYP #N |
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145 | normalize |
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146 | cases N |
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147 | [1: normalize #HYP destruct(HYP) |
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148 | |2: #M'' normalize #HYP |
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149 | @eq_f @(IND_HYP M'') |
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150 | assumption |
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151 | ] |
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152 | ] |
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153 | qed. |
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154 | |
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155 | definition translate_op: ∀globals. ? → list register → list register → list register → |
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156 | label → label → rtl_internal_function globals → rtl_internal_function globals ≝ |
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157 | λglobals: list ident. |
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158 | λop. |
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159 | λdestrs: list register. |
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160 | λsrcrs1: list register. |
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161 | λsrcrs2: list register. |
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162 | λstart_lbl: label. |
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163 | λdest_lbl: label. |
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164 | λdef: rtl_internal_function globals. |
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165 | match reduce_strong register register srcrs1 srcrs2 with |
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166 | [ dp reduced first_reduced_proof ⇒ |
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167 | let srcrsl_common ≝ \fst (\fst reduced) in |
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168 | let srcrsr_common ≝ \fst (\snd reduced) in |
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169 | let srcrsl_rest ≝ \snd (\fst reduced) in |
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170 | let srcrsr_rest ≝ \snd (\snd reduced) in |
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171 | let srcrs_rest ≝ srcrsl_rest @ srcrsr_rest in |
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172 | match reduce_strong register register destrs srcrsl_common with |
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173 | [ dp reduced second_reduced_proof ⇒ |
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174 | let destrs_common ≝ \fst (\fst reduced) in |
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175 | let destrs_rest ≝ \snd (\fst reduced) in |
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176 | match reduce_strong register register destrs_rest srcrs_rest with |
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177 | [ dp reduced third_reduced_proof ⇒ |
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178 | let destrs_cted ≝ \fst (\fst reduced) in |
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179 | let destrs_rest ≝ \snd (\fst reduced) in |
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180 | let srcrs_cted ≝ \fst (\snd reduced) in |
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181 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
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182 | let insts_init ≝ [ |
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183 | sequential … (CLEAR_CARRY …); |
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184 | sequential … (INT rtl_params_ globals tmpr (zero …)) |
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185 | ] in |
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186 | let f_add ≝ λdestr. λsrcr1. λsrcr2. sequential … (OP2 rtl_params_ globals op destr srcr1 srcr2) in |
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187 | let insts_add ≝ map3 ? ? ? ? f_add destrs_common srcrsl_common srcrsr_common ? ? in |
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188 | let f_add_cted ≝ λdestr. λsrcr. sequential … (OP2 rtl_params_ globals op destr srcr tmpr) in |
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189 | let insts_add_cted ≝ map2 … f_add_cted destrs_cted srcrs_cted ? in |
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190 | let f_rest ≝ λdestr. sequential … (OP2 rtl_params_ globals op destr tmpr tmpr) in |
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191 | let insts_rest ≝ map … f_rest destrs_rest in |
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192 | adds_graph rtl_params1 globals (insts_init @ insts_add @ insts_add_cted @ insts_rest) start_lbl dest_lbl def |
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193 | ] |
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194 | ] |
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195 | ]. |
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196 | [1: @third_reduced_proof |
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197 | |3: @first_reduced_proof |
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198 | |*: cases daemon (* XXX: some of these look like they may be false *) |
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199 | ] |
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200 | qed. |
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201 | |
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202 | let rec translate_cst |
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203 | (globals: list ident) (cst: constant) (destrs: list register) |
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204 | (start_lbl: label) (dest_lbl: label) (def: rtl_internal_function globals) |
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205 | on cst: rtl_internal_function globals ≝ |
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206 | match cst with |
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207 | [ Ointconst size const ⇒ |
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208 | match destrs with |
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209 | [ nil ⇒ add_graph … start_lbl (GOTO … dest_lbl) def |
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210 | | _ ⇒ |
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211 | let size' ≝ size_intsize size in |
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212 | match eqb size' (|destrs|) return λx. (eqb size' (|destrs|)) = x → rtl_internal_function globals with |
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213 | [ true ⇒ λgood_case. |
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214 | match split_into_bytes size const with |
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215 | [ dp bytes bytes_length_proof ⇒ |
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216 | let mapped ≝ map2 … (λd. λb. (sequential … (INT rtl_params_ globals d b))) destrs bytes ? in |
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217 | adds_graph rtl_params1 globals mapped start_lbl dest_lbl def |
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218 | ] |
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219 | | false ⇒ λbad_case. ? |
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220 | ] (refl … (eqb size' (|destrs|))) |
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221 | ] |
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222 | | Ofloatconst float ⇒ ⊥ |
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223 | | Oaddrsymbol id offset ⇒ |
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224 | let 〈r1, r2〉 ≝ make_addr … destrs ? in |
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225 | let def ≝ add_graph rtl_params1 globals start_lbl (sequential … (ADDRESS rtl_params_ globals id ? r1 r2) dest_lbl) def in |
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226 | let def ≝ translate_op globals Addc [r1] [r1] [r2] start_lbl dest_lbl def in |
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227 | def |
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228 | | Oaddrstack offset ⇒ |
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229 | let 〈r1, r2〉 ≝ make_addr … destrs ? in |
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230 | let def ≝ add_graph rtl_params1 globals start_lbl (sequential … (extension rtl_params_ globals (rtl_st_ext_address r1 r2)) dest_lbl) def in |
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231 | let def ≝ translate_op globals Addc [r1] [r1] [r2] start_lbl dest_lbl def in |
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232 | def |
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233 | ]. |
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234 | [1: >bytes_length_proof |
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235 | cut(size' = |destrs|) |
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236 | [1: @eqb_implies_eq |
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237 | assumption |
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238 | |2: #EQ_HYP |
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239 | <EQ_HYP % |
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240 | ] |
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241 | |2: cases daemon (* XXX: bad case where destrs is of the wrong length *) |
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242 | |3: cases not_implemented (* XXX: float, error_float in o'caml *) |
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243 | |*: cases daemon (* XXX: various proofs to be filled in *) |
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244 | ]. |
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245 | qed. |
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246 | |
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247 | definition translate_move_internal ≝ |
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248 | λglobals. |
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249 | λdestr: register. |
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250 | λsrcr: register. |
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251 | sequential rtl_params_ globals (MOVE … 〈destr,srcr〉). |
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252 | |
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253 | definition translate_move ≝ |
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254 | λglobals. |
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255 | λdestrs: list register. |
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256 | λsrcrs: list register. |
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257 | λstart_lbl: label. |
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258 | match reduce_strong register register destrs srcrs with |
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259 | [ dp crl_crr len_proof ⇒ |
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260 | let commonl ≝ \fst (\fst crl_crr) in |
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261 | let commonr ≝ \fst (\snd crl_crr) in |
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262 | let restl ≝ \snd (\fst crl_crr) in |
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263 | let restr ≝ \snd (\snd crl_crr) in |
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264 | let f_common ≝ translate_move_internal globals in |
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265 | let translate1 ≝ adds_graph rtl_params1 … (map2 … f_common commonl commonr ?) in |
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266 | let translate2 ≝ translate_cst … (Ointconst ? (repr I8 0)) restl in (* should this be 8? *) |
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267 | add_translates … [ translate1 ; translate2 ] start_lbl |
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268 | ]. |
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269 | @len_proof |
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270 | qed. |
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271 | |
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272 | let rec make |
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273 | (A: Type[0]) (elt: A) (n: nat) on n ≝ |
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274 | match n with |
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275 | [ O ⇒ [ ] |
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276 | | S n' ⇒ elt :: make A elt n' |
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277 | ]. |
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278 | |
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279 | lemma make_length: |
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280 | ∀A: Type[0]. |
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281 | ∀elt: A. |
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282 | ∀n: nat. |
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283 | n = length ? (make A elt n). |
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284 | #A #ELT #N |
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285 | elim N |
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286 | [ normalize % |
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287 | | #N #IH |
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288 | normalize <IH % |
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289 | ] |
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290 | qed. |
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291 | |
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292 | definition translate_cast_unsigned ≝ |
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293 | λglobals. |
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294 | λdestrs. |
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295 | λstart_lbl. |
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296 | λdest_lbl. |
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297 | λdef: joint_internal_function … (rtl_params globals). |
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298 | let 〈def, tmp_zero〉 ≝ fresh_reg … def in |
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299 | let zeros ≝ make … tmp_zero (length … destrs) in |
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300 | add_translates … [ |
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301 | adds_graph rtl_params1 … [ |
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302 | sequential rtl_params_ … (INT rtl_params_ ? tmp_zero (bitvector_of_nat ? 0)) |
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303 | ]; |
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304 | translate_move globals destrs zeros |
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305 | ] start_lbl dest_lbl def. |
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306 | |
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307 | definition translate_cast_signed: |
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308 | ∀globals: list ident. list register → ? → label → label → rtl_internal_function globals → rtl_internal_function globals ≝ |
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309 | λglobals: list ident. |
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310 | λdestrs. |
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311 | λsrcr. |
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312 | λstart_lbl. |
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313 | λdest_lbl. |
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314 | λdef. |
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315 | let 〈def, tmp_128〉 ≝ fresh_reg … def in |
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316 | let 〈def, tmp_255〉 ≝ fresh_reg … def in |
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317 | let 〈def, tmpr〉 ≝ fresh_reg … def in |
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318 | let 〈def, dummy〉 ≝ fresh_reg … def in |
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319 | let insts ≝ [ |
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320 | sequential … (INT rtl_params_ globals tmp_128 (bitvector_of_nat ? 128)); |
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321 | sequential … (OP2 rtl_params_ globals And tmpr tmp_128 srcr); |
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322 | sequential … (OPACCS rtl_params_ globals DivuModu tmpr dummy tmpr tmp_128); |
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323 | sequential … (INT rtl_params_ globals tmp_255 (bitvector_of_nat ? 255)); |
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324 | sequential … (OPACCS rtl_params_ globals Mul tmpr dummy tmpr tmp_255) |
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325 | ] |
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326 | in |
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327 | let srcrs ≝ make … tmpr (length … destrs) in |
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328 | add_translates rtl_params1 globals [ |
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329 | adds_graph rtl_params1 globals insts; |
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330 | translate_move globals destrs srcrs |
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331 | ] start_lbl dest_lbl def. |
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332 | |
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333 | definition translate_cast ≝ |
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334 | λglobals: list ident. |
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335 | λsrc_size: nat. |
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336 | λsrc_sign: signedness. |
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337 | λdest_size: nat. |
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338 | λdestrs: list register. |
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339 | λsrcrs: list register. |
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340 | match |srcrs| return λx. |srcrs| = x → ? with |
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341 | [ O ⇒ λzero_prf. adds_graph rtl_params1 globals [ ] |
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342 | | S n' ⇒ λsucc_prf. |
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343 | if ltb dest_size src_size then |
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344 | translate_move globals destrs srcrs |
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345 | else |
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346 | match reduce_strong register register destrs srcrs with |
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347 | [ dp crl len_proof ⇒ |
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348 | let commonl ≝ \fst (\fst crl) in |
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349 | let commonr ≝ \fst (\snd crl) in |
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350 | let restl ≝ \snd (\fst crl) in |
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351 | let restr ≝ \snd (\snd crl) in |
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352 | let insts_common ≝ translate_move globals commonl commonr in |
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353 | let sign_reg ≝ last_safe ? srcrs ? in |
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354 | let insts_sign ≝ |
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355 | match src_sign with |
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356 | [ Unsigned ⇒ translate_cast_unsigned globals restl |
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357 | | Signed ⇒ translate_cast_signed globals restl sign_reg |
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358 | ] |
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359 | in |
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360 | add_translates rtl_params1 globals [ insts_common; insts_sign ] |
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361 | ] |
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362 | ] (refl ? (|srcrs|)). |
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363 | >succ_prf // |
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364 | qed. |
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365 | |
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366 | definition translate_negint ≝ |
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367 | λglobals: list ident. |
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368 | λdestrs: list register. |
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369 | λsrcrs: list register. |
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370 | λstart_lbl: label. |
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371 | λdest_lbl: label. |
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372 | λdef: rtl_internal_function globals. |
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373 | λprf: |destrs| = |srcrs|. (* assert in caml code *) |
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374 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
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375 | let f_cmpl ≝ λdestr. λsrcr. sequential rtl_params_ globals (OP1 rtl_params1 globals Cmpl destr srcr) in |
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376 | let insts_cmpl ≝ map2 … f_cmpl destrs srcrs prf in |
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377 | let insts_init ≝ [ |
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378 | sequential … (SET_CARRY …); |
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379 | sequential … (INT rtl_params_ globals tmpr (zero ?)) |
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380 | ] in |
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381 | let f_add ≝ λdestr. sequential … (OP2 rtl_params_ globals Addc destr destr tmpr) in |
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382 | let insts_add ≝ map … f_add destrs in |
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383 | adds_graph rtl_params1 globals (insts_cmpl @ insts_init @ insts_add) start_lbl dest_lbl def. |
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384 | |
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385 | definition translate_notbool: ∀globals. list register → list register → label → label → rtl_internal_function globals → rtl_internal_function globals ≝ |
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386 | λglobals: list ident. |
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387 | λdestrs: list register. |
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388 | λsrcrs: list register. |
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389 | λstart_lbl: label. |
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390 | λdest_lbl: label. |
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391 | λdef: rtl_internal_function globals. |
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392 | match destrs with |
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393 | [ nil ⇒ add_graph rtl_params1 globals start_lbl (GOTO … start_lbl) def |
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394 | | cons destr destrs ⇒ |
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395 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
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396 | let 〈def, tmp_srcrs〉 ≝ fresh_regs rtl_params0 globals def (length ? srcrs) in |
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397 | let save_srcrs ≝ translate_move globals tmp_srcrs srcrs in |
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398 | let init_destr ≝ sequential … (INT rtl_params_ globals destr (bitvector_of_nat ? 1)) in |
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399 | let f ≝ λtmp_srcr. [ |
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400 | sequential … (CLEAR_CARRY rtl_params_ globals); |
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401 | sequential … (INT rtl_params_ globals tmpr (zero ?)); |
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402 | sequential … (OP2 rtl_params_ globals Sub tmpr tmpr tmp_srcr); |
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403 | sequential … (INT rtl_params_ globals tmpr (zero ?)); |
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404 | sequential … (OP2 rtl_params_ globals Addc tmpr tmpr tmpr); |
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405 | sequential … (OP2 rtl_params_ globals Xor destr destr tmpr) |
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406 | ] in |
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407 | let insts ≝ init_destr :: (flatten … (map … f tmp_srcrs)) in |
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408 | let epilogue ≝ translate_cst globals (Ointconst I8 (zero …)) destrs in |
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409 | add_translates rtl_params1 globals [ |
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410 | save_srcrs; adds_graph rtl_params1 globals insts; epilogue |
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411 | ] start_lbl dest_lbl def |
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412 | ]. |
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413 | |
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414 | (* TODO: examine this properly. This is a change from the O'caml code due |
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415 | to us dropping the explicit use of a cast destination size field. We |
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416 | instead infer the size of the cast's destination from the context. Is |
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417 | this correct? |
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418 | *) |
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419 | definition translate_op1 ≝ |
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420 | λglobals: list ident. |
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421 | λop1: unary_operation. |
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422 | λdestrs: list register. |
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423 | λsrcrs: list register. |
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424 | λprf: |destrs| = |srcrs|. |
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425 | λstart_lbl: label. |
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426 | λdest_lbl: label. |
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427 | λdef: rtl_internal_function globals. |
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428 | match op1 with |
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429 | [ Ocastint src_sign src_size ⇒ |
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430 | let dest_size ≝ |destrs| * 8 in |
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431 | let src_size ≝ bitsize_of_intsize src_size in |
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432 | translate_cast globals src_size src_sign dest_size destrs srcrs start_lbl dest_lbl def |
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433 | | Onegint ⇒ |
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434 | translate_negint globals destrs srcrs start_lbl dest_lbl def prf |
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435 | | Onotbool ⇒ |
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436 | translate_notbool globals destrs srcrs start_lbl dest_lbl def |
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437 | | Onotint ⇒ |
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438 | let f ≝ λdestr. λsrcr. sequential rtl_params_ globals (OP1 … Cmpl destr srcr) in |
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439 | let l ≝ map2 … f destrs srcrs prf in |
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440 | adds_graph rtl_params1 globals l start_lbl dest_lbl def |
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441 | | Optrofint r ⇒ |
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442 | translate_move globals destrs srcrs start_lbl dest_lbl def |
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443 | | Ointofptr r ⇒ |
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444 | translate_move globals destrs srcrs start_lbl dest_lbl def |
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445 | | Oid ⇒ |
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446 | translate_move globals destrs srcrs start_lbl dest_lbl def |
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447 | | _ ⇒ ? (* float operations implemented in runtime *) |
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448 | ]. |
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449 | cases not_implemented |
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450 | qed. |
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451 | |
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452 | let rec translate_mul1 |
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453 | (globals: list ident) (dummy: register) (tmpr: register) |
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454 | (destrs: list register) (srcrs1: list register) (srcr2: register) |
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455 | (start_lbl: label) |
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456 | on srcrs1 ≝ |
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457 | match destrs with |
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458 | [ nil ⇒ adds_graph rtl_params1 globals [ GOTO … ] start_lbl |
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459 | | cons destr tl ⇒ |
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460 | match tl with |
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461 | [ nil ⇒ |
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462 | match srcrs1 with |
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463 | [ nil ⇒ |
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464 | adds_graph rtl_params1 globals [ |
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465 | sequential … (INT rtl_params_ globals tmpr (zero …)); |
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466 | sequential … (OP2 rtl_params_ globals Addc destr destr tmpr) |
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467 | ] start_lbl |
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468 | | cons srcr1 tl' ⇒ |
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469 | adds_graph rtl_params1 globals [ |
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470 | sequential … (OPACCS rtl_params_ globals Mul tmpr dummy srcr2 srcr1); |
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471 | sequential … (OP2 rtl_params_ globals Addc destr destr tmpr) |
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472 | ] start_lbl |
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473 | ] |
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474 | | cons destr2 destrs ⇒ |
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475 | match srcrs1 with |
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476 | [ nil ⇒ |
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477 | add_translates rtl_params1 globals [ |
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478 | adds_graph rtl_params1 globals [ |
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479 | sequential … (INT rtl_params_ globals tmpr (zero …)); |
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480 | sequential … (OP2 rtl_params_ globals Addc destr destr tmpr); |
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481 | sequential … (OP2 rtl_params_ globals Addc destr2 tmpr tmpr) |
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482 | ]; |
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483 | translate_cst globals (Ointconst I8 (zero …)) destrs |
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484 | ] start_lbl |
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485 | | cons srcr1 srcrs1 ⇒ |
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486 | match destrs with |
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487 | [ nil ⇒ |
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488 | add_translates rtl_params1 globals [ |
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489 | adds_graph rtl_params1 globals [ |
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490 | sequential … (INT rtl_params_ globals tmpr (zero …)); |
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491 | sequential … (OP2 rtl_params_ globals Addc destr destr tmpr); |
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492 | sequential … (OP2 rtl_params_ globals Addc destr2 tmpr tmpr) |
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493 | ]; |
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494 | translate_cst globals (Ointconst I8 (zero ?)) destrs |
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495 | ] start_lbl |
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496 | | cons destr2 destrs ⇒ |
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497 | add_translates rtl_params1 globals [ |
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498 | adds_graph rtl_params1 globals [ |
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499 | sequential … (OPACCS rtl_params_ globals Mul tmpr destr2 srcr2 srcr1); |
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500 | sequential … (OP2 rtl_params_ globals Addc destr destr tmpr) |
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501 | ]; |
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502 | translate_mul1 globals dummy tmpr (destr2 :: destrs) srcrs1 srcr2 |
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503 | ] start_lbl |
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504 | ] |
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505 | ] |
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506 | ] |
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507 | ]. |
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508 | |
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509 | definition translate_muli ≝ |
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510 | λglobals: list ident. |
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511 | λdummy: register. |
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512 | λtmpr: register. |
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513 | λdestrs: list register. |
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514 | λtmp_destrs: list register. |
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515 | λsrcrs1: list register. |
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516 | λdummy_lbl: label. |
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517 | λi: nat. |
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518 | λi_prf: i ≤ |tmp_destrs|. |
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519 | λtranslates: list ?. |
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520 | λsrcr2i: register. |
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521 | let 〈tmp_destrs1, tmp_destrs2〉 ≝ split … tmp_destrs i i_prf in |
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522 | let tmp_destrs2' ≝ |
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523 | match tmp_destrs2 with |
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524 | [ nil ⇒ [ ] |
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525 | | cons tmp_destr2 tmp_destrs2 ⇒ [ |
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526 | adds_graph rtl_params1 globals [ |
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527 | sequential rtl_params_ globals (CLEAR_CARRY …); |
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528 | sequential … (INT rtl_params_ globals tmp_destr2 (zero …)) |
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529 | ]; |
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530 | translate_mul1 globals dummy tmpr tmp_destrs2 srcrs1 srcr2i; |
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531 | translate_cst globals (Ointconst I8 (zero …)) tmp_destrs1; |
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532 | adds_graph rtl_params1 globals [ |
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533 | sequential rtl_params_ globals (CLEAR_CARRY …) |
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534 | ]; |
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535 | translate_op globals Addc destrs destrs tmp_destrs |
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536 | ] |
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537 | ] |
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538 | in |
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539 | translates @ tmp_destrs2'. |
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540 | |
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541 | axiom foldi_strong: |
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542 | ∀a: Type[0]. |
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543 | ∀b: Type[0]. |
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544 | ∀the_list: list b. |
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545 | ∀f: ∀n: nat. n ≤ |the_list| → a → b → a. |
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546 | ∀seed: a. |
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547 | a. |
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548 | |
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549 | definition translate_mul ≝ |
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550 | λglobals: list ident. |
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551 | λdestrs: list register. |
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552 | λsrcrs1: list register. |
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553 | λsrcrs2: list register. |
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554 | λregs_proof: |srcrs2| = |destrs|. |
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555 | λstart_lbl: label. |
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556 | λdest_lbl: label. |
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557 | λdef: rtl_internal_function globals. |
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558 | let 〈def, dummy〉 ≝ fresh_reg rtl_params0 globals def in |
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559 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
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560 | match fresh_regs_strong rtl_params0 globals def (|destrs|) with |
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561 | [ dp def_tmp_destrs tmp_destrs_prf ⇒ |
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562 | let def ≝ \fst def_tmp_destrs in |
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563 | let tmp_destrs ≝ \snd def_tmp_destrs in |
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564 | let 〈def, fresh_srcrs1〉 ≝ fresh_regs rtl_params0 globals def (|srcrs1|) in |
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565 | let 〈def, fresh_srcrs2〉 ≝ fresh_regs rtl_params0 globals def (|srcrs2|) in |
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566 | let insts_init ≝ [ |
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567 | translate_move globals fresh_srcrs1 srcrs1; |
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568 | translate_move globals fresh_srcrs2 srcrs2; |
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569 | translate_cst globals (Ointconst I8 (zero …)) destrs |
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570 | ] |
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571 | in |
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572 | let f ≝ translate_muli globals dummy tmpr destrs tmp_destrs fresh_srcrs1 start_lbl in |
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573 | let f' ≝ λi. λproof: i ≤ |srcrs2|. f i ? in |
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574 | let insts_mul ≝ foldi_strong ? ? srcrs2 f' [ ] in |
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575 | add_translates rtl_params1 globals (insts_init @ insts_mul) start_lbl dest_lbl def |
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576 | ]. |
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577 | >tmp_destrs_prf |
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578 | <regs_proof |
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579 | assumption |
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580 | qed. |
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581 | |
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582 | definition translate_divumodu8 ≝ |
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583 | λglobals: list ident. |
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584 | λorder: bool. |
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585 | λdestrs: list register. |
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586 | λsrcr1: register. |
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587 | λsrcr2: register. |
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588 | λstart_lbl: label. |
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589 | λdest_lbl: label. |
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590 | λdef: rtl_internal_function globals. |
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591 | match destrs with |
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592 | [ nil ⇒ add_graph rtl_params1 globals start_lbl (GOTO … dest_lbl) def |
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593 | | cons destr destrs ⇒ |
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594 | let 〈def, dummy〉 ≝ fresh_reg rtl_params0 globals def in |
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595 | let 〈destr1, destr2〉 ≝ match order with [ true ⇒ 〈destr, dummy〉 | _ ⇒ 〈dummy, destr〉 ] in |
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596 | let inst_div ≝ adds_graph rtl_params1 globals [ |
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597 | sequential rtl_params_ globals (OPACCS … DivuModu destr1 destr2 srcr1 srcr2) |
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598 | ] |
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599 | in |
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600 | let insts_rest ≝ translate_cst globals (Ointconst I8 (zero ?)) destrs in |
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601 | add_translates rtl_params1 globals [ inst_div; insts_rest ] start_lbl dest_lbl def |
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602 | ]. |
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603 | |
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604 | definition translate_ne: ∀globals: list ident. ? → ? → ? → ? → ? → ? → rtl_internal_function globals ≝ |
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605 | λglobals: list ident. |
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606 | λdestrs: list register. |
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607 | λsrcrs1: list register. |
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608 | λsrcrs2: list register. |
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609 | λstart_lbl: label. |
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610 | λdest_lbl: label. |
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611 | λdef: rtl_internal_function globals. |
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612 | match destrs with |
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613 | [ nil ⇒ add_graph rtl_params1 globals start_lbl (GOTO … dest_lbl) def |
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614 | | cons destr destrs ⇒ |
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615 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
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616 | let 〈def, tmp_zero〉 ≝ fresh_reg rtl_params0 globals def in |
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617 | let 〈def, tmp_srcrs1〉 ≝ fresh_regs rtl_params0 globals def (|srcrs1|) in |
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618 | let save_srcrs1 ≝ translate_move globals tmp_srcrs1 srcrs1 in |
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619 | let 〈def, tmp_srcrs2〉 ≝ fresh_regs rtl_params0 globals def (|srcrs2|) in |
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620 | let save_srcrs2 ≝ translate_move globals tmp_srcrs2 srcrs2 in |
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621 | match reduce_strong register register tmp_srcrs1 tmp_srcrs2 with |
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622 | [ dp crl their_proof ⇒ |
---|
623 | let commonl ≝ \fst (\fst crl) in |
---|
624 | let commonr ≝ \fst (\snd crl) in |
---|
625 | let restl ≝ \snd (\snd crl) in |
---|
626 | let restr ≝ \snd (\snd crl) in |
---|
627 | let rest ≝ restl @ restr in |
---|
628 | let inits ≝ [ |
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629 | sequential … (INT rtl_params_ globals destr (zero …)); |
---|
630 | sequential … (INT rtl_params_ globals tmp_zero (zero …)) |
---|
631 | ] |
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632 | in |
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633 | let f_common ≝ λtmp_srcr1. λtmp_srcr2. [ |
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634 | sequential … (CLEAR_CARRY …); |
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635 | sequential … (OP2 rtl_params_ globals Sub tmpr tmp_srcr1 tmp_srcr2); |
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636 | sequential … (OP2 rtl_params_ globals Or destr destr tmpr) |
---|
637 | ] |
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638 | in |
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639 | let insts_common ≝ flatten … (map2 … f_common commonl commonr ?) in |
---|
640 | let f_rest ≝ λtmp_srcr. [ |
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641 | sequential … (CLEAR_CARRY …); |
---|
642 | sequential … (OP2 rtl_params_ globals Sub tmpr tmp_zero tmp_srcr); |
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643 | sequential … (OP2 rtl_params_ globals Or destr destr tmpr) |
---|
644 | ] |
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645 | in |
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646 | let insts_rest ≝ flatten … (map … f_rest rest) in |
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647 | let insts ≝ inits @ insts_common @ insts_rest in |
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648 | let epilogue ≝ translate_cst globals (Ointconst I8 (zero …)) destrs in |
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649 | add_translates rtl_params1 globals [ |
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650 | save_srcrs1; save_srcrs2; adds_graph rtl_params1 globals insts; epilogue |
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651 | ] start_lbl dest_lbl def |
---|
652 | ] |
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653 | ]. |
---|
654 | @their_proof |
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655 | qed. |
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656 | |
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657 | definition translate_eq_reg ≝ |
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658 | λglobals: list ident. |
---|
659 | λtmp_zero: register. |
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660 | λtmp_one: register. |
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661 | λtmpr1: register. |
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662 | λtmpr2: register. |
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663 | λdestr: register. |
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664 | λdummy_lbl: label. |
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665 | λsrcr12: register × register. |
---|
666 | let 〈srcr1, srcr2〉 ≝ srcr12 in |
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667 | [ sequential … (CLEAR_CARRY …); |
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668 | sequential rtl_params_ globals (OP2 rtl_params_ globals Sub tmpr1 srcr1 srcr2); |
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669 | sequential … (OP2 rtl_params_ globals Addc tmpr1 tmp_zero tmp_zero); |
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670 | sequential … (OP2 rtl_params_ globals Sub tmpr2 srcr2 srcr1); |
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671 | sequential … (OP2 rtl_params_ globals Addc tmpr2 tmp_zero tmp_zero); |
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672 | sequential … (OP2 rtl_params_ globals Or tmpr1 tmpr1 tmpr2); |
---|
673 | sequential … (OP2 rtl_params_ globals Xor tmpr1 tmpr1 tmp_one); |
---|
674 | sequential … (OP2 rtl_params_ globals And destr destr tmpr1) ]. |
---|
675 | |
---|
676 | definition translate_eq_list ≝ |
---|
677 | λglobals: list ident. |
---|
678 | λtmp_zero: register. |
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679 | λtmp_one: register. |
---|
680 | λtmpr1: register. |
---|
681 | λtmpr2: register. |
---|
682 | λdestr: register. |
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683 | λleq: list (register × register). |
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684 | λdummy_lbl: label. |
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685 | let f ≝ translate_eq_reg globals tmp_zero tmp_one tmpr1 tmpr2 destr dummy_lbl in |
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686 | (sequential … (INT rtl_params_ globals destr (bitvector_of_nat ? 1))) :: |
---|
687 | flatten … (map … f leq). |
---|
688 | |
---|
689 | definition translate_atom ≝ |
---|
690 | λglobals: list ident. |
---|
691 | λtmp_zero: register. |
---|
692 | λtmp_one: register. |
---|
693 | λtmpr1: register. |
---|
694 | λtmpr2: register. |
---|
695 | λtmpr3: register. |
---|
696 | λdestr: register. |
---|
697 | λdummy_lbl: label. |
---|
698 | λleq: list (register × register). |
---|
699 | λsrcr1: register. |
---|
700 | λsrcr2: register. |
---|
701 | translate_eq_list globals tmp_zero tmp_one tmpr1 tmpr2 tmpr3 leq dummy_lbl @ |
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702 | [ sequential … (CLEAR_CARRY …); |
---|
703 | sequential … (OP2 rtl_params_ globals Sub tmpr1 srcr1 srcr2); |
---|
704 | sequential … (OP2 rtl_params_ globals Addc tmpr1 tmp_zero tmp_zero); |
---|
705 | sequential … (OP2 rtl_params_ globals And tmpr3 tmpr3 tmpr1); |
---|
706 | sequential … (OP2 rtl_params_ globals Or destr destr tmpr3) ]. |
---|
707 | |
---|
708 | definition translate_lt_main ≝ |
---|
709 | λglobals: list ident. |
---|
710 | λtmp_zero: register. |
---|
711 | λtmp_one: register. |
---|
712 | λtmpr1: register. |
---|
713 | λtmpr2: register. |
---|
714 | λtmpr3: register. |
---|
715 | λdestr: register. |
---|
716 | λdummy_lbl: label. |
---|
717 | λsrcrs1: list register. |
---|
718 | λsrcrs2: list register. |
---|
719 | λproof: |srcrs1| = |srcrs2|. |
---|
720 | let f ≝ λinsts_leq. λsrcr1. λsrcr2. |
---|
721 | let 〈insts, leq〉 ≝ insts_leq in |
---|
722 | let added_insts ≝ translate_atom globals tmp_zero tmp_one tmpr1 tmpr2 tmpr3 destr dummy_lbl leq srcr1 srcr2 in |
---|
723 | 〈insts @ added_insts, leq @ [〈srcr1, srcr2〉]〉 |
---|
724 | in |
---|
725 | \fst (fold_left2 … f 〈[ ], [ ]〉 srcrs1 srcrs2 proof). |
---|
726 | |
---|
727 | definition fresh_regs_strong: |
---|
728 | ∀globals. rtl_internal_function globals → ∀n: nat. Σfresh: (rtl_internal_function globals) × (list register). |\snd fresh| = n ≝ |
---|
729 | λglobals: list ident. |
---|
730 | λdef. |
---|
731 | λn. |
---|
732 | fresh_regs rtl_params0 globals def n. |
---|
733 | @fresh_regs_length |
---|
734 | qed. |
---|
735 | |
---|
736 | definition complete_regs_strong: |
---|
737 | ∀globals: list ident. rtl_internal_function globals → list register → list register → Σcomplete: (list register) × (list register) × (list register). |\fst (\fst complete)| = |\snd (\fst complete)| ≝ |
---|
738 | λglobals: list ident. |
---|
739 | λdef. |
---|
740 | λleft. |
---|
741 | λright. |
---|
742 | complete_regs globals def left right. |
---|
743 | @complete_regs_length |
---|
744 | qed. |
---|
745 | |
---|
746 | definition translate_lt ≝ |
---|
747 | λglobals: list ident. |
---|
748 | λdestrs: list register. |
---|
749 | λprf_destrs: 0 < |destrs|. |
---|
750 | λsrcrs1: list register. |
---|
751 | λsrcrs2: list register. |
---|
752 | λstart_lbl: label. |
---|
753 | λdest_lbl: label. |
---|
754 | λdef: rtl_internal_function globals. |
---|
755 | match destrs with |
---|
756 | [ nil ⇒ add_graph rtl_params1 globals start_lbl (GOTO … dest_lbl) def |
---|
757 | | _ ⇒ |
---|
758 | match fresh_regs_strong globals def (|destrs|) with |
---|
759 | [ dp def_tmp_destrs tmp_destrs_proof ⇒ |
---|
760 | let def ≝ \fst def_tmp_destrs in |
---|
761 | let tmp_destrs ≝ \snd def_tmp_destrs in |
---|
762 | let tmp_destr ≝ hd_safe ? tmp_destrs ? in |
---|
763 | let 〈def, tmp_zero〉 ≝ fresh_reg rtl_params1 globals def in |
---|
764 | let 〈def, tmp_one〉 ≝ fresh_reg rtl_params1 globals def in |
---|
765 | let 〈def, tmpr1〉 ≝ fresh_reg rtl_params1 globals def in |
---|
766 | let 〈def, tmpr2〉 ≝ fresh_reg rtl_params1 globals def in |
---|
767 | let 〈def, tmpr3〉 ≝ fresh_reg rtl_params1 globals def in |
---|
768 | match complete_regs_strong globals def srcrs1 srcrs2 with |
---|
769 | [ dp srcrs12_added srcrs12_proof ⇒ |
---|
770 | let srcrs1 ≝ \fst (\fst srcrs12_added) in |
---|
771 | let srcrs2 ≝ \snd (\fst srcrs12_added) in |
---|
772 | let added ≝ \snd srcrs12_added in |
---|
773 | let srcrs1' ≝ rev … srcrs1 in |
---|
774 | let srcrs2' ≝ rev … srcrs2 in |
---|
775 | let insts_init ≝ [ |
---|
776 | translate_cst globals (Ointconst I8 (zero ?)) tmp_destrs; |
---|
777 | translate_cst globals (Ointconst I8 (zero ?)) added; |
---|
778 | adds_graph rtl_params1 globals [ |
---|
779 | sequential rtl_params_ globals (INT rtl_params_ globals tmp_zero (zero …)); |
---|
780 | sequential rtl_params_ globals (INT rtl_params_ globals tmp_one (bitvector_of_nat … 1)) |
---|
781 | ] |
---|
782 | ] |
---|
783 | in |
---|
784 | let insts_main ≝ |
---|
785 | translate_lt_main globals tmp_zero tmp_one tmpr1 tmpr2 tmpr3 tmp_destr start_lbl srcrs1' srcrs2' ? in |
---|
786 | let insts_main ≝ [ adds_graph rtl_params1 globals insts_main ] in |
---|
787 | let insts_exit ≝ [ translate_move globals destrs tmp_destrs ] in |
---|
788 | add_translates rtl_params1 globals (insts_init @ insts_main @ insts_exit) start_lbl dest_lbl def |
---|
789 | ] |
---|
790 | ] |
---|
791 | ]. |
---|
792 | [2: >tmp_destrs_proof @prf_destrs |
---|
793 | |1: normalize nodelta |
---|
794 | generalize in match srcrs12_proof |
---|
795 | #HYP >rev_length >rev_length @HYP |
---|
796 | ] |
---|
797 | qed. |
---|
798 | |
---|
799 | definition add_128_to_last ≝ |
---|
800 | λglobals: list ident. |
---|
801 | λtmp_128: register. |
---|
802 | λrs. |
---|
803 | λprf: 0 < |rs|. |
---|
804 | λstart_lbl: label. |
---|
805 | match rs with |
---|
806 | [ nil ⇒ adds_graph rtl_params1 globals [ ] start_lbl |
---|
807 | | _ ⇒ |
---|
808 | let r ≝ last_safe … rs prf in |
---|
809 | adds_graph rtl_params1 globals [ |
---|
810 | sequential rtl_params_ globals (OP2 rtl_params_ globals Add r r tmp_128) |
---|
811 | ] start_lbl |
---|
812 | ]. |
---|
813 | |
---|
814 | definition translate_lts ≝ |
---|
815 | λglobals: list ident. |
---|
816 | λdestrs: list register. |
---|
817 | λdestrs_prf: 0 < |destrs|. |
---|
818 | λsrcrs1: list register. |
---|
819 | λsrcrs2: list register. |
---|
820 | λsrcrs1_lt_prf: 0 < |srcrs1|. |
---|
821 | λsrcrs2_lt_prf: 0 < |srcrs2|. |
---|
822 | λstart_lbl: label. |
---|
823 | λdest_lbl: label. |
---|
824 | λdef: rtl_internal_function globals. |
---|
825 | match fresh_regs_strong globals def (|srcrs1|) with |
---|
826 | [ dp def_tmp_srcrs1 srcrs1_prf ⇒ |
---|
827 | let def ≝ \fst def_tmp_srcrs1 in |
---|
828 | let tmp_srcrs1 ≝ \snd def_tmp_srcrs1 in |
---|
829 | match fresh_regs_strong globals def (|srcrs2|) with |
---|
830 | [ dp def_tmp_srcrs2 srcrs2_prf ⇒ |
---|
831 | let def ≝ \fst def_tmp_srcrs2 in |
---|
832 | let tmp_srcrs2 ≝ \snd def_tmp_srcrs2 in |
---|
833 | let 〈def, tmp_128〉 ≝ fresh_reg rtl_params0 globals def in |
---|
834 | add_translates rtl_params1 globals [ |
---|
835 | translate_move globals tmp_srcrs1 srcrs1; |
---|
836 | translate_move globals tmp_srcrs2 srcrs2; |
---|
837 | adds_graph rtl_params1 globals [ |
---|
838 | sequential rtl_params_ globals (INT rtl_params_ globals tmp_128 (bitvector_of_nat ? 128)) |
---|
839 | ]; |
---|
840 | add_128_to_last globals tmp_128 tmp_srcrs1 ?; |
---|
841 | add_128_to_last globals tmp_128 tmp_srcrs2 ?; |
---|
842 | translate_lt globals destrs destrs_prf tmp_srcrs1 tmp_srcrs2 |
---|
843 | ] start_lbl dest_lbl def |
---|
844 | ] |
---|
845 | ]. |
---|
846 | [1: >srcrs1_prf @srcrs1_lt_prf |
---|
847 | |2: >srcrs2_prf @srcrs2_lt_prf |
---|
848 | ] |
---|
849 | qed. |
---|
850 | |
---|
851 | definition translate_op2 ≝ |
---|
852 | λglobals: list ident. |
---|
853 | λop2. |
---|
854 | λdestrs: list register. |
---|
855 | λdestrs_prf: 0 < |destrs|. |
---|
856 | λsrcrs1: list register. |
---|
857 | λsrcrs2: list register. |
---|
858 | λsrcrs1_prf: 0 < |srcrs1|. (* XXX: can this be relaxed? i think it can |
---|
859 | but we need more dep. typ. in modu/divu |
---|
860 | cases *) |
---|
861 | λsrcrs2_prf: |srcrs2| = |destrs|. |
---|
862 | λstart_lbl: label. |
---|
863 | λdest_lbl: label. |
---|
864 | λdef: rtl_internal_function globals. |
---|
865 | match op2 with |
---|
866 | [ Oadd ⇒ |
---|
867 | translate_op globals Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
868 | | Oaddp ⇒ |
---|
869 | translate_op globals Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
870 | | Osub ⇒ |
---|
871 | translate_op globals Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
872 | | Osubpi ⇒ |
---|
873 | translate_op globals Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
874 | | Osubpp sz ⇒ |
---|
875 | translate_op globals Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
876 | | Omul ⇒ |
---|
877 | translate_mul globals destrs srcrs1 srcrs2 ? start_lbl dest_lbl def |
---|
878 | | Odivu ⇒ |
---|
879 | match srcrs1 return λx. 0 < |x| → rtl_internal_function globals with |
---|
880 | [ cons hd tl ⇒ λcons_prf. |
---|
881 | match tl with |
---|
882 | [ nil ⇒ translate_divumodu8 globals true destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def |
---|
883 | | _ ⇒ ? (* not implemented *) |
---|
884 | ] |
---|
885 | | nil ⇒ λnil_absrd. ? |
---|
886 | ] srcrs1_prf |
---|
887 | | Omodu ⇒ |
---|
888 | match srcrs1 return λx. 0 < |x| → rtl_internal_function globals with |
---|
889 | [ cons hd tl ⇒ λcons_prf. |
---|
890 | match tl with |
---|
891 | [ nil ⇒ translate_divumodu8 globals false destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def |
---|
892 | | _ ⇒ ? (* not implemented *) |
---|
893 | ] |
---|
894 | | nil ⇒ λnil_absrd. ? |
---|
895 | ] srcrs1_prf |
---|
896 | | Oand ⇒ |
---|
897 | translate_op globals And destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
898 | | Oor ⇒ |
---|
899 | translate_op globals Or destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
900 | | Oxor ⇒ |
---|
901 | translate_op globals Xor destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
902 | | Ocmp c ⇒ |
---|
903 | match c with |
---|
904 | [ Ceq ⇒ |
---|
905 | add_translates rtl_params1 globals [ |
---|
906 | translate_ne globals destrs srcrs1 srcrs2; |
---|
907 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
908 | ] start_lbl dest_lbl def |
---|
909 | | Cne ⇒ translate_ne globals destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
910 | | Clt ⇒ translate_lts globals destrs destrs_prf srcrs1 srcrs2 ? ? start_lbl dest_lbl def |
---|
911 | | Cgt ⇒ translate_lts globals destrs destrs_prf srcrs2 srcrs1 ? ? start_lbl dest_lbl def |
---|
912 | | Cle ⇒ |
---|
913 | add_translates rtl_params1 globals [ |
---|
914 | translate_lts globals destrs destrs_prf srcrs2 srcrs1 ? ?; |
---|
915 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
916 | ] start_lbl dest_lbl def |
---|
917 | | Cge ⇒ |
---|
918 | add_translates rtl_params1 globals [ |
---|
919 | translate_lts globals destrs destrs_prf srcrs1 srcrs2 ? ?; |
---|
920 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
921 | ] start_lbl dest_lbl def |
---|
922 | ] |
---|
923 | | Ocmpu c ⇒ |
---|
924 | match c with |
---|
925 | [ Ceq ⇒ |
---|
926 | add_translates rtl_params1 globals [ |
---|
927 | translate_ne globals destrs srcrs1 srcrs2; |
---|
928 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
929 | ] start_lbl dest_lbl def |
---|
930 | | Cne ⇒ translate_ne globals destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
931 | | Clt ⇒ translate_lt globals destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def |
---|
932 | | Cgt ⇒ translate_lt globals destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def |
---|
933 | | Cle ⇒ |
---|
934 | add_translates rtl_params1 globals [ |
---|
935 | translate_lt globals destrs destrs_prf srcrs2 srcrs1; |
---|
936 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
937 | ] start_lbl dest_lbl def |
---|
938 | | Cge ⇒ |
---|
939 | add_translates rtl_params1 globals [ |
---|
940 | translate_lt globals destrs destrs_prf srcrs1 srcrs2; |
---|
941 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
942 | ] start_lbl dest_lbl def |
---|
943 | ] |
---|
944 | | Ocmpp c ⇒ |
---|
945 | match c with |
---|
946 | [ Ceq ⇒ |
---|
947 | add_translates rtl_params1 globals [ |
---|
948 | translate_ne globals destrs srcrs1 srcrs2; |
---|
949 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
950 | ] start_lbl dest_lbl def |
---|
951 | | Cne ⇒ translate_ne globals destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
952 | | Clt ⇒ translate_lt globals destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def |
---|
953 | | Cgt ⇒ translate_lt globals destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def |
---|
954 | | Cle ⇒ |
---|
955 | add_translates rtl_params1 globals [ |
---|
956 | translate_lt globals destrs destrs_prf srcrs2 srcrs1; |
---|
957 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
958 | ] start_lbl dest_lbl def |
---|
959 | | Cge ⇒ |
---|
960 | add_translates rtl_params1 globals [ |
---|
961 | translate_lt globals destrs destrs_prf srcrs1 srcrs2; |
---|
962 | translate_op1 globals Onotbool destrs destrs (refl ? (|destrs|)) |
---|
963 | ] start_lbl dest_lbl def |
---|
964 | ] |
---|
965 | | _ ⇒ ? (* assert false, implemented in run time or float op *) |
---|
966 | ]. |
---|
967 | [ 3,7: normalize in nil_absrd; |
---|
968 | cases(not_le_Sn_O 0) |
---|
969 | #HYP cases(HYP nil_absrd) |
---|
970 | | 4,8,19,20,21,22,17,18,24: |
---|
971 | >srcrs2_prf |
---|
972 | assumption |
---|
973 | | 1,23: |
---|
974 | assumption |
---|
975 | | *: cases not_implemented (* yes, really *) |
---|
976 | ] |
---|
977 | qed. |
---|
978 | |
---|
979 | definition translate_cond: ∀globals: list ident. list register → label → label → label → rtl_internal_function globals → rtl_internal_function globals ≝ |
---|
980 | λglobals: list ident. |
---|
981 | λsrcrs: list register. |
---|
982 | λstart_lbl: label. |
---|
983 | λlbl_true: label. |
---|
984 | λlbl_false: label. |
---|
985 | λdef: rtl_internal_function globals. |
---|
986 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
---|
987 | let 〈tmp_lbl, def〉 ≝ fresh_label rtl_params1 globals def in |
---|
988 | let init ≝ sequential rtl_params_ globals (INT rtl_params_ globals tmpr (zero …)) in |
---|
989 | let f ≝ λsrcr. sequential rtl_params_ globals (OP2 rtl_params_ globals Or tmpr tmpr srcr) in |
---|
990 | let def ≝ adds_graph rtl_params1 globals (init :: (map ? ? f srcrs)) start_lbl tmp_lbl def in |
---|
991 | add_graph rtl_params1 globals tmp_lbl (sequential rtl_params_ globals (COND rtl_params_ globals tmpr lbl_true) lbl_false) def. |
---|
992 | |
---|
993 | definition translate_load ≝ |
---|
994 | λglobals: list ident. |
---|
995 | λaddr. |
---|
996 | λaddr_prf: 2 ≤ |addr|. |
---|
997 | λdestrs: list register. |
---|
998 | λstart_lbl: label. |
---|
999 | λdest_lbl: label. |
---|
1000 | λdef: rtl_internal_function globals. |
---|
1001 | match fresh_regs_strong rtl_params0 globals def (|addr|) with |
---|
1002 | [ dp def_save_addr save_addr_prf ⇒ |
---|
1003 | let def ≝ \fst def_save_addr in |
---|
1004 | let save_addr ≝ \snd def_save_addr in |
---|
1005 | match fresh_regs_strong rtl_params0 globals def (|addr|) with |
---|
1006 | [ dp def_tmp_addr tmp_addr_prf ⇒ |
---|
1007 | let def ≝ \fst def_tmp_addr in |
---|
1008 | let tmp_addr ≝ \snd def_tmp_addr in |
---|
1009 | let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr … tmp_addr ? in |
---|
1010 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
---|
1011 | let insts_save_addr ≝ translate_move globals save_addr addr in |
---|
1012 | let f ≝ λtranslates_off. λr. |
---|
1013 | let 〈translates, off〉 ≝ translates_off in |
---|
1014 | let translates ≝ translates @ [ |
---|
1015 | adds_graph rtl_params1 globals [ |
---|
1016 | sequential rtl_params_ globals (INT rtl_params_ globals tmpr off) |
---|
1017 | ]; |
---|
1018 | translate_op2 globals Oaddp tmp_addr ? save_addr [tmpr] ? ?; |
---|
1019 | adds_graph rtl_params1 globals [ |
---|
1020 | sequential rtl_params_ globals (LOAD rtl_params_ globals r tmp_addr1 tmp_addr2) |
---|
1021 | ] |
---|
1022 | ] |
---|
1023 | in |
---|
1024 | let 〈carry, result〉 ≝ half_add … off int_size in |
---|
1025 | 〈translates, result〉 |
---|
1026 | in |
---|
1027 | let 〈translates, ignore〉 ≝ foldl … f 〈[ ], (zero ?)〉 destrs in |
---|
1028 | add_translates rtl_params1 globals (insts_save_addr :: translates) start_lbl dest_lbl def |
---|
1029 | ] |
---|
1030 | ]. |
---|
1031 | [1: normalize // |
---|
1032 | |2: >save_addr_prf normalize |
---|
1033 | @(transitive_le 1 2 (|addr|)) // |
---|
1034 | |3: >tmp_addr_prf normalize |
---|
1035 | @(transitive_le 1 2 (|addr|)) // |
---|
1036 | |*: >tmp_addr_prf assumption |
---|
1037 | ] |
---|
1038 | qed. |
---|
1039 | |
---|
1040 | definition translate_store ≝ |
---|
1041 | λglobals: list ident. |
---|
1042 | λaddr. |
---|
1043 | λaddr_prf: 2 ≤ |addr|. |
---|
1044 | λsrcrs: list register. |
---|
1045 | λstart_lbl: label. |
---|
1046 | λdest_lbl: label. |
---|
1047 | λdef: rtl_internal_function globals. |
---|
1048 | match fresh_regs_strong rtl_params0 globals def (|addr|) with |
---|
1049 | [ dp def_tmp_addr tmp_addr_prf ⇒ |
---|
1050 | let def ≝ \fst def_tmp_addr in |
---|
1051 | let tmp_addr ≝ \snd def_tmp_addr in |
---|
1052 | let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr … tmp_addr ? in |
---|
1053 | let 〈def, tmpr〉 ≝ fresh_reg rtl_params0 globals def in |
---|
1054 | let f ≝ λtranslate_off. λsrcr. |
---|
1055 | let 〈translates, off〉 ≝ translate_off in |
---|
1056 | let translates ≝ translates @ [ |
---|
1057 | adds_graph rtl_params1 globals [ |
---|
1058 | sequential rtl_params_ globals (INT rtl_params_ globals tmpr off) |
---|
1059 | ]; |
---|
1060 | translate_op2 globals Oaddp tmp_addr … addr [tmpr] ? ?; |
---|
1061 | adds_graph rtl_params1 globals [ |
---|
1062 | sequential rtl_params_ globals (STORE rtl_params_ globals tmp_addr1 tmp_addr2 srcr) |
---|
1063 | ] |
---|
1064 | ] |
---|
1065 | in |
---|
1066 | let 〈carry, result〉 ≝ half_add … off int_size in |
---|
1067 | 〈translates, result〉 |
---|
1068 | in |
---|
1069 | let 〈translates, ignore〉 ≝ foldl … f 〈[ ], zero …〉 srcrs in |
---|
1070 | add_translates rtl_params1 globals translates start_lbl dest_lbl def |
---|
1071 | ]. |
---|
1072 | [1: normalize // |
---|
1073 | |2: >tmp_addr_prf normalize |
---|
1074 | @(transitive_le 1 2 (|addr|)) // |
---|
1075 | |3: >tmp_addr_prf normalize |
---|
1076 | @(transitive_le 1 2 (|addr|)) // |
---|
1077 | |*: >tmp_addr_prf assumption |
---|
1078 | ] |
---|
1079 | qed. |
---|
1080 | |
---|
1081 | axiom fake_label: label. |
---|
1082 | |
---|
1083 | (* XXX: following conversation with CSC about the mix up in extension statements |
---|
1084 | and extension instructions in RTL, use fake_label in calls to |
---|
1085 | tailcall_* instructions. *) |
---|
1086 | definition translate_stmt ≝ |
---|
1087 | λglobals: list ident. |
---|
1088 | λlenv: local_env. |
---|
1089 | λlbl: label. |
---|
1090 | λstmt. |
---|
1091 | λdef: rtl_internal_function globals. |
---|
1092 | match stmt with |
---|
1093 | [ St_skip lbl' ⇒ add_graph … lbl (GOTO … lbl') def |
---|
1094 | | St_cost cost_lbl lbl' ⇒ add_graph rtl_params1 globals lbl (sequential rtl_params_ globals (COST_LABEL rtl_params_ globals cost_lbl) lbl') def |
---|
1095 | | St_const destr cst lbl' ⇒ |
---|
1096 | translate_cst globals cst (find_local_env destr lenv) lbl lbl' def |
---|
1097 | | St_op1 op1 destr srcr lbl' ⇒ |
---|
1098 | translate_op1 globals op1 (find_local_env destr lenv) (find_local_env srcr lenv) … lbl lbl' def |
---|
1099 | | St_op2 op2 destr srcr1 srcr2 lbl' ⇒ |
---|
1100 | translate_op2 globals op2 (find_local_env destr lenv) ? (find_local_env srcr1 lenv) (find_local_env srcr2 lenv) ? ? lbl lbl' def |
---|
1101 | (* XXX: should we be ignoring this? *) |
---|
1102 | | St_load ignore addr destr lbl' ⇒ |
---|
1103 | translate_load globals (find_local_env addr lenv) ? (find_local_env destr lenv) lbl lbl' def |
---|
1104 | (* XXX: should we be ignoring this? *) |
---|
1105 | | St_store ignore addr srcr lbl' ⇒ |
---|
1106 | translate_store globals (find_local_env addr lenv) ? (find_local_env srcr lenv) lbl lbl' def |
---|
1107 | | St_call_id f args retr lbl' ⇒ |
---|
1108 | match retr with |
---|
1109 | [ Some retr ⇒ |
---|
1110 | add_graph rtl_params1 globals lbl (sequential … (CALL_ID rtl_params1 globals f (rtl_args args lenv) (find_local_env retr lenv)) lbl') def |
---|
1111 | | None ⇒ add_graph rtl_params1 globals lbl (sequential … (CALL_ID rtl_params1 globals f (rtl_args args lenv) [ ]) lbl') def |
---|
1112 | ] |
---|
1113 | | St_call_ptr f args retr lbl' ⇒ |
---|
1114 | match retr with |
---|
1115 | [ None ⇒ |
---|
1116 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1117 | add_graph rtl_params1 globals lbl (sequential … (extension … (rtl_st_ext_call_ptr f1 f2 (rtl_args args lenv) [ ])) lbl') def |
---|
1118 | | Some retr ⇒ |
---|
1119 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1120 | add_graph rtl_params1 globals lbl (sequential … (extension … (rtl_st_ext_call_ptr f1 f2 (rtl_args args lenv) (find_local_env retr lenv))) lbl') def |
---|
1121 | ] |
---|
1122 | | St_tailcall_id f args ⇒ |
---|
1123 | add_graph rtl_params1 globals lbl (sequential rtl_params_ globals (extension rtl_params_ globals (rtl_st_ext_tailcall_id f (rtl_args args lenv))) fake_label) def |
---|
1124 | | St_tailcall_ptr f args ⇒ |
---|
1125 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1126 | add_graph rtl_params1 globals lbl (sequential rtl_params_ globals (extension rtl_params1 globals (rtl_st_ext_tailcall_ptr f1 f2 (rtl_args args lenv))) fake_label) def |
---|
1127 | | St_cond r lbl_true lbl_false ⇒ |
---|
1128 | translate_cond globals (find_local_env r lenv) lbl lbl_true lbl_false def |
---|
1129 | | St_jumptable r l ⇒ ? (* assert false: not implemented yet *) |
---|
1130 | | St_return ⇒ add_graph rtl_params1 globals lbl (RETURN …) def |
---|
1131 | ]. |
---|
1132 | [10: cases not_implemented (* jtable case *) |
---|
1133 | |*: cases daemon (* TODO: proofs regarding lengths of lists, examine! *) |
---|
1134 | ] |
---|
1135 | qed. |
---|
1136 | |
---|
1137 | (* XXX: we use a "hack" here to circumvent the proof obligations required to |
---|
1138 | create res, an empty internal_function record. we insert into our graph |
---|
1139 | the start and end nodes to ensure that they are in, and place dummy |
---|
1140 | skip instructions at these nodes. *) |
---|
1141 | definition translate_internal ≝ |
---|
1142 | λglobals: list ident. |
---|
1143 | λdef. |
---|
1144 | let runiverse ≝ f_reggen def in |
---|
1145 | let 〈lenv,runiverse〉 ≝ initialize_local_env runiverse |
---|
1146 | (f_params def @ f_locals def) (f_result def) in |
---|
1147 | let params ≝ map_list_local_env lenv (map … \fst (f_params def)) in |
---|
1148 | let locals ≝ map_list_local_env lenv (map … \fst (f_locals def)) in |
---|
1149 | let result ≝ |
---|
1150 | match (f_result def) with |
---|
1151 | [ None ⇒ [ ] |
---|
1152 | | Some r_typ ⇒ |
---|
1153 | let 〈r, typ〉 ≝ r_typ in |
---|
1154 | find_local_env r lenv |
---|
1155 | ] |
---|
1156 | in |
---|
1157 | let luniverse' ≝ f_labgen def in |
---|
1158 | let runiverse' ≝ runiverse in |
---|
1159 | let result' ≝ result in |
---|
1160 | let params' ≝ params in |
---|
1161 | let locals' ≝ locals in |
---|
1162 | let stack_size' ≝ f_stacksize def in |
---|
1163 | let entry' ≝ f_entry def in |
---|
1164 | let exit' ≝ f_exit def in |
---|
1165 | let graph' ≝ add LabelTag ? (empty_map LabelTag ?) entry' (GOTO rtl_params_ globals entry') in |
---|
1166 | let graph' ≝ add LabelTag ? graph' exit' (GOTO rtl_params_ globals exit') in |
---|
1167 | let res ≝ |
---|
1168 | mk_joint_internal_function … (rtl_params globals) luniverse' runiverse' result' params' |
---|
1169 | locals' stack_size' graph' ? ? in |
---|
1170 | foldi … (translate_stmt globals … lenv) (f_graph def) res. |
---|
1171 | [ % [@entry' | @graph_add_lookup @graph_add] |
---|
1172 | | % [@exit' | @graph_add]] |
---|
1173 | qed. |
---|
1174 | |
---|
1175 | (*CSC: here we are not doing any alignment on variables, like it is done in OCaml |
---|
1176 | because of CompCert heritage *) |
---|
1177 | definition rtlabs_to_rtl: RTLabs_program → rtl_program ≝ |
---|
1178 | λp. transform_program … p (transf_fundef … (translate_internal (prog_var_names …))). |
---|