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 | |
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7 | axiom graph_add_lookup: |
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8 | ∀g: graph rtl_statement. |
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9 | ∀l: label. |
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10 | ∀s: rtl_statement. |
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11 | ∀to_insert: label. |
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12 | lookup ? ? g l ≠ None ? → lookup ? ? (add ? ? g to_insert s) l ≠ None ?. |
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13 | |
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14 | definition add_graph ≝ |
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15 | λl: label. |
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16 | λstmt. |
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17 | λp. |
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18 | let rtl_if_luniverse' ≝ rtl_if_luniverse p in |
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19 | let rtl_if_runiverse' ≝ rtl_if_runiverse p in |
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20 | let rtl_if_result' ≝ rtl_if_result p in |
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21 | let rtl_if_params' ≝ rtl_if_params p in |
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22 | let rtl_if_locals' ≝ rtl_if_locals p in |
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23 | let rtl_if_stacksize' ≝ rtl_if_stacksize p in |
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24 | let rtl_if_graph' ≝ add ? ? (rtl_if_graph p) l stmt in |
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25 | let rtl_if_entry' ≝ rtl_if_entry p in |
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26 | let rtl_if_exit' ≝ rtl_if_exit p in |
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27 | mk_rtl_internal_function rtl_if_luniverse' rtl_if_runiverse' |
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28 | rtl_if_params' rtl_if_params' rtl_if_locals' |
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29 | rtl_if_stacksize' rtl_if_graph' ? ?. |
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30 | [1: cases(rtl_if_entry') |
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31 | #LABEL #HYP % |
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32 | [1: @LABEL |
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33 | |2: cases(HYP) |
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34 | generalize in match (graph_add_lookup (rtl_if_graph p) LABEL stmt l); |
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35 | /2/ |
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36 | ] |
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37 | |2: cases(rtl_if_exit') |
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38 | #LABEL #HYP % |
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39 | [1: @LABEL |
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40 | |2: cases(HYP) |
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41 | generalize in match (graph_add_lookup (rtl_if_graph p) LABEL stmt l); |
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42 | /2/ |
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43 | ] |
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44 | ] |
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45 | qed. |
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46 | |
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47 | definition fresh_label: rtl_internal_function → rtl_internal_function × label ≝ |
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48 | λdef. |
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49 | let 〈lbl, new_univ〉 ≝ fresh LabelTag (rtl_if_luniverse def) in |
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50 | let locals ≝ rtl_if_locals def in |
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51 | let rtl_if_luniverse' ≝ new_univ in |
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52 | let rtl_if_runiverse' ≝ rtl_if_runiverse def in |
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53 | let rtl_if_result' ≝ rtl_if_result def in |
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54 | let rtl_if_params' ≝ rtl_if_params def in |
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55 | let rtl_if_locals' ≝ rtl_if_locals def in |
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56 | let rtl_if_stacksize' ≝ rtl_if_stacksize def in |
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57 | let rtl_if_graph' ≝ rtl_if_graph def in |
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58 | let rtl_if_entry' ≝ rtl_if_entry def in |
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59 | let rtl_if_exit' ≝ rtl_if_exit def in |
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60 | 〈mk_rtl_internal_function |
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61 | rtl_if_luniverse' rtl_if_runiverse' rtl_if_result' |
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62 | rtl_if_params' rtl_if_locals' rtl_if_stacksize' rtl_if_graph' |
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63 | rtl_if_entry' rtl_if_exit', lbl〉. |
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64 | |
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65 | axiom register_fresh: universe RegisterTag → register. |
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66 | |
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67 | definition fresh_reg: rtl_internal_function → rtl_internal_function × register ≝ |
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68 | λdef. |
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69 | let r ≝ register_fresh (rtl_if_runiverse def) in |
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70 | let locals ≝ r :: rtl_if_locals def in |
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71 | let rtl_if_luniverse' ≝ rtl_if_luniverse def in |
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72 | let rtl_if_runiverse' ≝ rtl_if_runiverse def in |
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73 | let rtl_if_result' ≝ rtl_if_result def in |
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74 | let rtl_if_params' ≝ rtl_if_params def in |
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75 | let rtl_if_locals' ≝ locals in |
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76 | let rtl_if_stacksize' ≝ rtl_if_stacksize def in |
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77 | let rtl_if_graph' ≝ rtl_if_graph def in |
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78 | let rtl_if_entry' ≝ rtl_if_entry def in |
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79 | let rtl_if_exit' ≝ rtl_if_exit def in |
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80 | 〈mk_rtl_internal_function |
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81 | rtl_if_luniverse' rtl_if_runiverse' rtl_if_result' |
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82 | rtl_if_params' rtl_if_locals' rtl_if_stacksize' rtl_if_graph' |
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83 | rtl_if_entry' rtl_if_exit', r〉. |
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84 | |
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85 | let rec fresh_regs (def: rtl_internal_function) (n: nat) on n ≝ |
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86 | match n with |
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87 | [ O ⇒ 〈def, [ ]〉 |
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88 | | S n ⇒ |
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89 | let 〈def, res〉 ≝ fresh_regs def n in |
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90 | let 〈def, r〉 ≝ fresh_reg def in |
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91 | 〈def, r :: res〉 |
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92 | ]. |
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93 | |
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94 | axiom fresh_regs_length: |
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95 | ∀def: rtl_internal_function. |
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96 | ∀n: nat. |
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97 | |(\snd (fresh_regs def n))| = n. |
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98 | |
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99 | definition addr_regs ≝ |
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100 | λregs. |
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101 | match regs with |
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102 | [ cons hd tl ⇒ |
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103 | match tl with |
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104 | [ cons hd' tl' ⇒ Some (register × register) 〈hd, hd'〉 |
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105 | | nil ⇒ None (register × register) |
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106 | ] |
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107 | | nil ⇒ None (register × register) (* registers are not an address *) |
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108 | ]. |
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109 | |
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110 | let rec register_freshes (runiverse: universe RegisterTag) (n: nat) on n ≝ |
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111 | match n with |
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112 | [ O ⇒ [ ] |
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113 | | S n' ⇒ register_fresh runiverse :: (register_freshes runiverse n') |
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114 | ]. |
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115 | |
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116 | definition choose_rest ≝ |
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117 | λA: Type[0]. |
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118 | λleft, right: list A. |
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119 | λprfl: 0 < |left|. |
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120 | λprfr: 0 < |right|. |
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121 | match left return λx. 0 < |x| → list A with |
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122 | [ nil ⇒ λnil_prf. |
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123 | match right return λx. 0 < |x| → list A with |
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124 | [ nil ⇒ λnil_nil_absrd. ? |
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125 | | _ ⇒ λnil_cons_prf. right |
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126 | ] prfr |
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127 | | _ ⇒ λcons_prf. right |
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128 | ] prfl. |
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129 | normalize in nil_prf; |
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130 | cases(not_le_Sn_O 0) |
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131 | #HYP |
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132 | cases(HYP nil_prf) |
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133 | qed. |
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134 | |
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135 | definition complete_regs ≝ |
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136 | λdef. |
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137 | λsrcrs1. |
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138 | λsrcrs2. |
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139 | let nb_added ≝ abs ((length ? srcrs1) - (length ? srcrs2)) in |
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140 | let 〈def, added_regs〉 ≝ fresh_regs def nb_added in |
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141 | if gtb nb_added 0 then |
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142 | 〈〈srcrs1, srcrs2 @ added_regs〉, added_regs〉 |
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143 | else |
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144 | 〈〈srcrs1 @ added_regs, srcrs2〉, added_regs〉. |
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145 | |
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146 | (* obvious, but proof doesn't look easy! *) |
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147 | axiom complete_regs_length: |
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148 | ∀def. |
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149 | ∀left. |
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150 | ∀right. |
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151 | |\fst (\fst (complete_regs def left right))| = |\snd (\fst (complete_regs def left right))|. |
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152 | |
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153 | definition size_of_sig_type ≝ |
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154 | λsig. |
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155 | match sig with |
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156 | [ ASTint isize sign ⇒ |
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157 | let isize' ≝ match isize with [ I8 ⇒ 8 | I16 ⇒ 16 | I32 ⇒ 32 ] in |
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158 | isize' ÷ (nat_of_bitvector ? int_size) |
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159 | | ASTfloat _ ⇒ ? (* dpm: not implemented *) |
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160 | | ASTptr rgn ⇒ nat_of_bitvector ? ptr_size |
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161 | ]. |
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162 | cases not_implemented; |
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163 | qed. |
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164 | |
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165 | inductive register_type: Type[0] ≝ |
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166 | | register_int: register → register_type |
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167 | | register_ptr: register → register → register_type. |
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168 | |
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169 | definition local_env ≝ BitVectorTrie (list register). |
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170 | |
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171 | definition mem_local_env ≝ |
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172 | λr: register. |
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173 | match r with |
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174 | [ an_identifier w ⇒ member (list register) 16 w |
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175 | ]. |
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176 | |
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177 | definition add_local_env ≝ |
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178 | λr: register. |
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179 | match r with |
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180 | [ an_identifier w ⇒ insert (list register) 16 w |
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181 | ]. |
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182 | |
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183 | definition find_local_env ≝ |
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184 | λr: register. |
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185 | λbvt. |
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186 | match r with |
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187 | [ an_identifier w ⇒ lookup (list register) 16 w bvt [ ] |
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188 | ]. |
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189 | |
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190 | definition initialize_local_env_internal ≝ |
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191 | λruniverse. |
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192 | λlenv. |
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193 | λr_sig. |
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194 | let 〈r, sig〉 ≝ r_sig in |
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195 | let size ≝ size_of_sig_type sig in |
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196 | let rs ≝ register_freshes runiverse size in |
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197 | add_local_env r rs lenv. |
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198 | |
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199 | definition initialize_local_env ≝ |
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200 | λruniverse. |
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201 | λregisters. |
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202 | λresult. |
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203 | let registers ≝ registers @ |
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204 | match result with |
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205 | [ None ⇒ [ ] |
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206 | | Some rt ⇒ [ rt ] |
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207 | ] |
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208 | in |
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209 | foldl ? ? (initialize_local_env_internal runiverse) (Stub …) registers. |
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210 | |
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211 | definition map_list_local_env_internal ≝ |
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212 | λlenv. |
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213 | λres. |
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214 | λr. |
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215 | res @ (find_local_env r lenv). |
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216 | |
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217 | definition map_list_local_env ≝ |
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218 | λlenv. |
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219 | λregs. |
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220 | foldl ? ? (map_list_local_env_internal lenv) [ ] regs. |
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221 | |
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222 | definition make_addr ≝ |
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223 | λA. |
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224 | λlst: list A. |
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225 | λprf: 2 ≤ length A lst. |
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226 | match lst return λx. 2 ≤ |x| → A × A with |
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227 | [ nil ⇒ λlst_nil_prf. ? |
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228 | | cons hd tl ⇒ λprf. |
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229 | match tl return λx. 1 ≤ |x| → A × A with |
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230 | [ nil ⇒ λtl_nil_prf. ? |
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231 | | cons hd' tl' ⇒ λtl_cons_prf. 〈hd, hd'〉 |
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232 | ] ? |
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233 | ] prf. |
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234 | [1: normalize in lst_nil_prf; |
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235 | cases(not_le_Sn_O 1); |
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236 | #HYP cases(HYP lst_nil_prf) |
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237 | |2: normalize in prf; |
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238 | @le_S_S_to_le |
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239 | assumption |
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240 | |3: normalize in tl_nil_prf; |
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241 | cases(not_le_Sn_O 0) |
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242 | #HYP cases(HYP tl_nil_prf) |
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243 | ] |
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244 | qed. |
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245 | |
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246 | definition find_and_addr ≝ |
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247 | λr. |
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248 | λlenv. |
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249 | make_addr ? (find_local_env r lenv). |
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250 | |
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251 | definition rtl_args ≝ |
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252 | λregs_list. |
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253 | λlenv. |
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254 | flatten ? (map ? ? ( |
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255 | λr. find_local_env r lenv) regs_list). |
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256 | |
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257 | definition change_label ≝ |
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258 | λlbl. |
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259 | λstmt: rtl_statement. |
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260 | match stmt with |
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261 | [ rtl_st_skip _ ⇒ rtl_st_skip lbl |
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262 | | rtl_st_cost cost_lbl _ ⇒ rtl_st_cost cost_lbl lbl |
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263 | | rtl_st_addr r1 r2 id _ ⇒ rtl_st_addr r1 r2 id lbl |
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264 | | rtl_st_stack_addr r1 r2 _ ⇒ rtl_st_stack_addr r1 r2 lbl |
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265 | | rtl_st_int r i _ ⇒ rtl_st_int r i lbl |
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266 | | rtl_st_move r1 r2 _ ⇒ rtl_st_move r1 r2 lbl |
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267 | | rtl_st_opaccs opaccs d s1 s2 s3 _ ⇒ rtl_st_opaccs opaccs d s1 s2 s3 lbl |
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268 | | rtl_st_op1 op1 d s _ ⇒ rtl_st_op1 op1 d s lbl |
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269 | | rtl_st_op2 op2 d s1 s2 _ ⇒ rtl_st_op2 op2 d s1 s2 lbl |
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270 | | rtl_st_load d a1 a2 _ ⇒ rtl_st_load d a1 a2 lbl |
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271 | | rtl_st_store a1 a2 s _ ⇒ rtl_st_store a1 a2 s lbl |
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272 | | rtl_st_call_id f a r _ ⇒ rtl_st_call_id f a r lbl |
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273 | | rtl_st_call_ptr f1 f2 a r _ ⇒ rtl_st_call_ptr f1 f2 a r lbl |
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274 | | rtl_st_tailcall_id f a ⇒ rtl_st_tailcall_id f a |
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275 | | rtl_st_tailcall_ptr f1 f2 a ⇒ rtl_st_tailcall_ptr f1 f2 a |
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276 | | rtl_st_cond r l1 l2 ⇒ rtl_st_cond r l1 l2 |
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277 | | rtl_st_clear_carry l ⇒ rtl_st_clear_carry lbl |
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278 | | rtl_st_set_carry l ⇒ rtl_st_set_carry lbl |
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279 | | rtl_st_return ⇒ rtl_st_return |
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280 | ]. |
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281 | |
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282 | (* Ack! Should generating a fresh label ever fail? This seems to be a side-effect |
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283 | of implementing everything as a bitvector, which is bounded. If we implemented |
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284 | labels as unbounded nats then this function will never fail. |
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285 | *) |
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286 | (* Fixed with changes to label generation. |
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287 | *) |
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288 | let rec adds_graph (stmt_list: ?) (start_lbl: label) (dest_lbl: label) (def: rtl_internal_function) ≝ |
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289 | match stmt_list with |
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290 | [ nil ⇒ def |
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291 | | cons hd tl ⇒ |
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292 | match tl with |
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293 | [ nil ⇒ add_graph start_lbl (change_label dest_lbl hd) def |
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294 | | cons hd' tl' ⇒ |
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295 | let 〈new_def, tmp_lbl〉 ≝ fresh_label def in |
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296 | let stmt ≝ change_label tmp_lbl hd in |
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297 | let def ≝ add_graph start_lbl stmt new_def in |
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298 | adds_graph tl tmp_lbl dest_lbl new_def |
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299 | ] |
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300 | ]. |
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301 | |
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302 | let rec add_translates (translate_list: ?) (start_lbl: label) (dest_lbl: label) |
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303 | (def: ?) on translate_list ≝ |
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304 | match translate_list with |
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305 | [ nil ⇒ def |
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306 | | cons hd tl ⇒ |
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307 | match tl with |
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308 | [ nil ⇒ hd start_lbl dest_lbl def |
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309 | | cons hd' tl' ⇒ |
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310 | let 〈new_def, tmp_lbl〉 ≝ fresh_label def in |
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311 | let applied ≝ hd start_lbl tmp_lbl new_def in |
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312 | add_translates tl tmp_lbl dest_lbl applied |
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313 | ] |
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314 | ]. |
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315 | |
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316 | definition translate_cst_int_internal ≝ |
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317 | λdest_lbl. |
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318 | λr. |
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319 | λi. |
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320 | rtl_st_int r i dest_lbl. |
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321 | |
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322 | axiom translate_cst: |
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323 | ∀cst: constant. |
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324 | ∀destrs: list register. |
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325 | ∀start_lbl: label. |
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326 | ∀dest_lbl: label. |
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327 | ∀def: rtl_internal_function. |
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328 | rtl_internal_function. |
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329 | |
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330 | definition translate_move_internal ≝ |
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331 | λstart_lbl: label. |
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332 | λdestr: register. |
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333 | λsrcr: register. |
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334 | rtl_st_move destr srcr start_lbl. |
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335 | |
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336 | definition translate_move ≝ |
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337 | λdestrs: list register. |
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338 | λsrcrs: list register. |
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339 | λstart_lbl: label. |
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340 | match reduce_strong register register destrs srcrs with |
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341 | [ dp crl_crr len_proof ⇒ |
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342 | let commonl ≝ \fst (\fst crl_crr) in |
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343 | let commonr ≝ \fst (\snd crl_crr) in |
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344 | let restl ≝ \snd (\fst crl_crr) in |
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345 | let restr ≝ \snd (\snd crl_crr) in |
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346 | let f_common ≝ translate_move_internal start_lbl in |
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347 | let translate1 ≝ adds_graph (map2 … f_common commonl commonr ?) in |
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348 | let translate2 ≝ translate_cst (Ointconst ? (repr I8 0)) restl in (* should this be 8? *) |
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349 | add_translates [ translate1 ; translate2 ] start_lbl |
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350 | ]. |
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351 | @len_proof |
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352 | qed. |
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353 | |
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354 | let rec make |
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355 | (A: Type[0]) (elt: A) (n: nat) on n ≝ |
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356 | match n with |
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357 | [ O ⇒ [ ] |
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358 | | S n' ⇒ elt :: make A elt n' |
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359 | ]. |
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360 | |
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361 | lemma make_length: |
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362 | ∀A: Type[0]. |
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363 | ∀elt: A. |
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364 | ∀n: nat. |
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365 | n = length ? (make A elt n). |
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366 | #A #ELT #N |
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367 | elim N |
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368 | [ normalize % |
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369 | | #N #IH |
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370 | normalize <IH % |
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371 | ] |
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372 | qed. |
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373 | |
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374 | definition translate_cast_unsigned ≝ |
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375 | λdestrs. |
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376 | λstart_lbl. |
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377 | λdest_lbl. |
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378 | λdef. |
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379 | let 〈def, tmp_zero〉 ≝ fresh_reg def in |
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380 | let zeros ≝ make ? tmp_zero (length ? destrs) in |
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381 | add_translates [ |
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382 | adds_graph [ |
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383 | rtl_st_int tmp_zero (bitvector_of_nat ? 0) start_lbl |
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384 | ]; |
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385 | translate_move destrs zeros |
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386 | ] start_lbl dest_lbl def. |
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387 | |
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388 | definition translate_cast_signed ≝ |
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389 | λdestrs. |
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390 | λsrcr. |
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391 | λstart_lbl. |
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392 | λdest_lbl. |
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393 | λdef. |
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394 | let 〈def, tmp_128〉 ≝ fresh_reg def in |
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395 | let 〈def, tmp_255〉 ≝ fresh_reg def in |
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396 | let 〈def, tmpr〉 ≝ fresh_reg def in |
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397 | let 〈def, dummy〉 ≝ fresh_reg def in |
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398 | let insts ≝ [ |
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399 | rtl_st_int tmp_128 (bitvector_of_nat ? 128) start_lbl; |
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400 | rtl_st_op2 And tmpr tmp_128 srcr start_lbl; |
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401 | rtl_st_opaccs DivuModu tmpr dummy tmpr tmp_128 start_lbl; |
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402 | rtl_st_int tmp_255 (bitvector_of_nat ? 255) start_lbl; |
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403 | rtl_st_opaccs Mul tmpr dummy tmpr tmp_255 start_lbl |
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404 | ] in |
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405 | let srcrs ≝ make ? tmpr (length ? destrs) in |
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406 | add_translates [ |
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407 | adds_graph insts; |
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408 | translate_move destrs srcrs |
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409 | ] start_lbl dest_lbl def. |
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410 | |
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411 | definition translate_cast ≝ |
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412 | λsrc_size. |
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413 | λsrc_sign. |
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414 | λdest_size. |
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415 | λdestrs. |
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416 | λsrcrs. |
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417 | match |srcrs| return λx. |srcrs| = x → ? with |
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418 | [ O ⇒ λzero_prf. adds_graph [ ] |
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419 | | S n' ⇒ λsucc_prf. |
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420 | if ltb dest_size src_size then |
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421 | translate_move destrs srcrs |
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422 | else |
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423 | match reduce_strong register register destrs srcrs with |
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424 | [ dp crl len_proof ⇒ |
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425 | let commonl ≝ \fst (\fst crl) in |
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426 | let commonr ≝ \fst (\snd crl) in |
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427 | let restl ≝ \snd (\fst crl) in |
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428 | let restr ≝ \snd (\snd crl) in |
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429 | let insts_common ≝ translate_move commonl commonr in |
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430 | let sign_reg ≝ last_safe ? srcrs ? in |
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431 | let insts_sign ≝ |
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432 | match src_sign with |
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433 | [ Unsigned ⇒ translate_cast_unsigned restl |
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434 | | Signed ⇒ translate_cast_signed restl sign_reg |
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435 | ] |
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436 | in |
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437 | add_translates [ insts_common; insts_sign ] |
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438 | ] |
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439 | ] (refl ? (|srcrs|)). |
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440 | >succ_prf // |
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441 | qed. |
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442 | |
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443 | definition translate_negint ≝ |
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444 | λdestrs. |
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445 | λsrcrs. |
---|
446 | λstart_lbl. |
---|
447 | λdest_lbl. |
---|
448 | λdef. |
---|
449 | λprf: | destrs | = | srcrs |. (* assert in caml code *) |
---|
450 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
451 | let f_cmpl ≝ λdestr. λsrcr. rtl_st_op1 Cmpl destr srcr start_lbl in |
---|
452 | let insts_cmpl ≝ map2 ? ? ? f_cmpl destrs srcrs prf in |
---|
453 | let insts_init ≝ [ |
---|
454 | rtl_st_set_carry start_lbl; |
---|
455 | rtl_st_int tmpr (zero ?) start_lbl |
---|
456 | ] in |
---|
457 | let f_add ≝ λdestr. rtl_st_op2 Addc destr destr tmpr start_lbl in |
---|
458 | let insts_add ≝ map … f_add destrs in |
---|
459 | adds_graph (insts_cmpl @ insts_init @ insts_add) start_lbl dest_lbl def. |
---|
460 | |
---|
461 | definition translate_notbool ≝ |
---|
462 | λdestrs. |
---|
463 | λsrcrs. |
---|
464 | λstart_lbl. |
---|
465 | λdest_lbl. |
---|
466 | λdef. |
---|
467 | match destrs with |
---|
468 | [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def |
---|
469 | | cons destr destrs ⇒ |
---|
470 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
471 | let 〈def, tmp_srcrs〉 ≝ fresh_regs def (length ? srcrs) in |
---|
472 | let save_srcrs ≝ translate_move tmp_srcrs srcrs in |
---|
473 | let init_destr ≝ rtl_st_int destr (bitvector_of_nat ? 1) start_lbl in |
---|
474 | let f ≝ λtmp_srcr. [ |
---|
475 | rtl_st_clear_carry start_lbl; |
---|
476 | rtl_st_int tmpr (zero ?) start_lbl; |
---|
477 | rtl_st_op2 Sub tmpr tmpr tmp_srcr start_lbl; |
---|
478 | rtl_st_int tmpr (zero ?) start_lbl; |
---|
479 | rtl_st_op2 Addc tmpr tmpr tmpr start_lbl; |
---|
480 | rtl_st_op2 Xor destr destr tmpr start_lbl |
---|
481 | ] in |
---|
482 | let insts ≝ init_destr :: (flatten ? (map … f tmp_srcrs)) in |
---|
483 | let epilogue ≝ translate_cst (Ointconst I8 (bitvector_of_nat ? 0)) destrs in |
---|
484 | add_translates [ |
---|
485 | save_srcrs; adds_graph insts; epilogue |
---|
486 | ] start_lbl dest_lbl def |
---|
487 | ]. |
---|
488 | |
---|
489 | (* TODO: examine this properly. This is a change from the O'caml code due |
---|
490 | to us dropping the explicit use of a cast destination size field. We |
---|
491 | instead infer the size of the cast's destination from the context. Is |
---|
492 | this correct? |
---|
493 | *) |
---|
494 | definition translate_op1 ≝ |
---|
495 | λop1: unary_operation. |
---|
496 | λdestrs: list register. |
---|
497 | λsrcrs: list register. |
---|
498 | λprf: |destrs| = |srcrs|. |
---|
499 | λstart_lbl: label. |
---|
500 | λdest_lbl: label. |
---|
501 | λdef: rtl_internal_function. |
---|
502 | match op1 with |
---|
503 | [ Ocastint src_sign src_size ⇒ |
---|
504 | let dest_size ≝ |destrs| * 8 in |
---|
505 | let src_size ≝ bitsize_of_intsize src_size in |
---|
506 | translate_cast src_size src_sign dest_size destrs srcrs start_lbl dest_lbl def |
---|
507 | | Onegint ⇒ |
---|
508 | translate_negint destrs srcrs start_lbl dest_lbl def prf |
---|
509 | | Onotbool ⇒ |
---|
510 | translate_notbool destrs srcrs start_lbl dest_lbl def |
---|
511 | | Onotint ⇒ |
---|
512 | let f ≝ λdestr. λsrcr. rtl_st_op1 Cmpl destr srcr start_lbl in |
---|
513 | let l ≝ map2 … f destrs srcrs prf in |
---|
514 | adds_graph l start_lbl dest_lbl def |
---|
515 | | Optrofint r ⇒ |
---|
516 | translate_move destrs srcrs start_lbl dest_lbl def |
---|
517 | | Ointofptr r ⇒ |
---|
518 | translate_move destrs srcrs start_lbl dest_lbl def |
---|
519 | | Oid ⇒ |
---|
520 | translate_move destrs srcrs start_lbl dest_lbl def |
---|
521 | | _ ⇒ ? (* float operations implemented in runtime *) |
---|
522 | ]. |
---|
523 | cases not_implemented |
---|
524 | qed. |
---|
525 | |
---|
526 | definition translate_op ≝ |
---|
527 | λop. |
---|
528 | λdestrs. |
---|
529 | λsrcrs1. |
---|
530 | λsrcrs2. |
---|
531 | λstart_lbl. |
---|
532 | λdest_lbl. |
---|
533 | λdef. |
---|
534 | match reduce_strong register register srcrs1 srcrs2 with |
---|
535 | [ dp reduced first_reduced_proof ⇒ |
---|
536 | let srcrsl_common ≝ \fst (\fst reduced) in |
---|
537 | let srcrsr_common ≝ \fst (\snd reduced) in |
---|
538 | let srcrsl_rest ≝ \snd (\fst reduced) in |
---|
539 | let srcrsr_rest ≝ \snd (\snd reduced) in |
---|
540 | let srcrs_rest ≝ choose_rest ? srcrsl_rest srcrsr_rest ? ? in |
---|
541 | match reduce_strong register register destrs srcrsl_common with |
---|
542 | [ dp reduced second_reduced_proof ⇒ |
---|
543 | let destrs_common ≝ \fst (\fst reduced) in |
---|
544 | let destrs_rest ≝ \snd (\fst reduced) in |
---|
545 | match reduce_strong register register destrs_rest srcrs_rest with |
---|
546 | [ dp reduced third_reduced_proof ⇒ |
---|
547 | let destrs_cted ≝ \fst (\fst reduced) in |
---|
548 | let destrs_rest ≝ \snd (\fst reduced) in |
---|
549 | let srcrs_cted ≝ \fst (\snd reduced) in |
---|
550 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
551 | let insts_init ≝ [ |
---|
552 | rtl_st_clear_carry start_lbl; |
---|
553 | rtl_st_int tmpr (zero ?) start_lbl |
---|
554 | ] in |
---|
555 | let f_add ≝ λdestr. λsrcr1. λsrcr2. rtl_st_op2 op destr srcr1 srcr2 start_lbl in |
---|
556 | let insts_add ≝ map3 … f_add destrs_common srcrsl_common srcrsr_common ? ? in |
---|
557 | let f_add_cted ≝ λdestr. λsrcr. rtl_st_op2 op destr srcr tmpr start_lbl in |
---|
558 | let insts_add_cted ≝ map2 ? ? ? f_add_cted destrs_cted srcrs_cted ? in |
---|
559 | let f_rest ≝ λdestr. rtl_st_op2 op destr tmpr tmpr start_lbl in |
---|
560 | let insts_rest ≝ map … f_rest destrs_rest in |
---|
561 | adds_graph (insts_init @ insts_add @ insts_add_cted @ insts_rest) start_lbl dest_lbl def |
---|
562 | ] |
---|
563 | ] |
---|
564 | ]. |
---|
565 | [1: @third_reduced_proof |
---|
566 | |3: @first_reduced_proof |
---|
567 | |*: cases daemon (* TODO *) |
---|
568 | ] |
---|
569 | qed. |
---|
570 | |
---|
571 | let rec translate_mul1 |
---|
572 | (dummy: register) (tmpr: register) (destrs: list register) |
---|
573 | (srcrs1: list register) (srcr2: register) (start_lbl: label) |
---|
574 | on srcrs1 ≝ |
---|
575 | match destrs with |
---|
576 | [ nil ⇒ adds_graph [ rtl_st_skip start_lbl ] start_lbl |
---|
577 | | cons destr tl ⇒ |
---|
578 | match tl with |
---|
579 | [ nil ⇒ |
---|
580 | match srcrs1 with |
---|
581 | [ nil ⇒ |
---|
582 | adds_graph [ |
---|
583 | rtl_st_int tmpr (zero ?) start_lbl; |
---|
584 | rtl_st_op2 Addc destr destr tmpr start_lbl |
---|
585 | ] start_lbl |
---|
586 | | cons srcr1 tl' ⇒ |
---|
587 | adds_graph [ |
---|
588 | rtl_st_opaccs Mul tmpr dummy srcr2 srcr1 start_lbl; |
---|
589 | rtl_st_op2 Addc destr destr tmpr start_lbl |
---|
590 | ] start_lbl |
---|
591 | ] |
---|
592 | | cons destr2 destrs ⇒ |
---|
593 | match srcrs1 with |
---|
594 | [ nil ⇒ |
---|
595 | add_translates [ |
---|
596 | adds_graph [ |
---|
597 | rtl_st_int tmpr (zero ?) start_lbl; |
---|
598 | rtl_st_op2 Addc destr destr tmpr start_lbl; |
---|
599 | rtl_st_op2 Addc destr2 tmpr tmpr start_lbl |
---|
600 | ]; |
---|
601 | translate_cst (Ointconst I8 (zero ?)) destrs |
---|
602 | ] start_lbl |
---|
603 | | cons srcr1 srcrs1 ⇒ |
---|
604 | match destrs with |
---|
605 | [ nil ⇒ |
---|
606 | add_translates [ |
---|
607 | adds_graph [ |
---|
608 | rtl_st_int tmpr (zero ?) start_lbl; |
---|
609 | rtl_st_op2 Addc destr destr tmpr start_lbl; |
---|
610 | rtl_st_op2 Addc destr2 tmpr tmpr start_lbl |
---|
611 | ]; |
---|
612 | translate_cst (Ointconst I8 (zero ?)) destrs |
---|
613 | ] start_lbl |
---|
614 | | cons destr2 destrs ⇒ |
---|
615 | add_translates [ |
---|
616 | adds_graph [ |
---|
617 | rtl_st_opaccs Mul tmpr destr2 srcr2 srcr1 start_lbl; |
---|
618 | rtl_st_op2 Addc destr destr tmpr start_lbl |
---|
619 | ]; |
---|
620 | translate_mul1 dummy tmpr (destr2 :: destrs) srcrs1 srcr2 |
---|
621 | ] start_lbl |
---|
622 | ] |
---|
623 | ] |
---|
624 | ] |
---|
625 | ]. |
---|
626 | |
---|
627 | definition translate_muli ≝ |
---|
628 | λdummy: register. |
---|
629 | λtmpr: register. |
---|
630 | λdestrs: list register. |
---|
631 | λtmp_destrs: list register. |
---|
632 | λsrcrs1: list register. |
---|
633 | λdummy_lbl: label. |
---|
634 | λi: nat. |
---|
635 | λi_prf. |
---|
636 | λtranslates: list ?. |
---|
637 | λsrcr2i: register. |
---|
638 | let 〈tmp_destrs1, tmp_destrs2〉 ≝ split ? tmp_destrs i i_prf in |
---|
639 | let tmp_destrs2' ≝ |
---|
640 | match tmp_destrs2 with |
---|
641 | [ nil ⇒ [ ] |
---|
642 | | cons tmp_destr2 tmp_destrs2 ⇒ [ |
---|
643 | adds_graph [ |
---|
644 | rtl_st_clear_carry dummy_lbl; |
---|
645 | rtl_st_int tmp_destr2 (zero ?) dummy_lbl |
---|
646 | ]; |
---|
647 | translate_mul1 dummy tmpr tmp_destrs2 srcrs1 srcr2i; |
---|
648 | translate_cst (Ointconst I8 (zero ?)) tmp_destrs1; |
---|
649 | adds_graph [ |
---|
650 | rtl_st_clear_carry dummy_lbl |
---|
651 | ]; |
---|
652 | translate_op Addc destrs destrs tmp_destrs |
---|
653 | ] |
---|
654 | ] |
---|
655 | in |
---|
656 | translates @ tmp_destrs2'. |
---|
657 | |
---|
658 | axiom translate_mul: |
---|
659 | ∀destrs: list register. |
---|
660 | ∀srcrs1: list register. |
---|
661 | ∀srcrs2: list register. |
---|
662 | ∀start_lbl: label. |
---|
663 | ∀dest_lbl: label. |
---|
664 | ∀def: rtl_internal_function. |
---|
665 | rtl_internal_function. |
---|
666 | |
---|
667 | (* |
---|
668 | definition translate_mul ≝ |
---|
669 | λdestrs: list register. |
---|
670 | λsrcrs1: list register. |
---|
671 | λsrcrs2: list register. |
---|
672 | λstart_lbl: label. |
---|
673 | λdest_lbl: label. |
---|
674 | λdef: rtl_internal_function. |
---|
675 | let 〈def, dummy〉 ≝ fresh_reg def in |
---|
676 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
677 | let 〈def, tmp_destrs〉 ≝ fresh_regs def (|destrs|) in |
---|
678 | let 〈def, fresh_srcrs1〉 ≝ fresh_regs def (|srcrs1|) in |
---|
679 | let 〈def, fresh_srcrs2〉 ≝ fresh_regs def (|srcrs2|) in |
---|
680 | let insts_init ≝ [ |
---|
681 | translate_move fresh_srcrs1 srcrs1; |
---|
682 | translate_move fresh_srcrs2 srcrs2; |
---|
683 | translate_cst (Ointconst I8 (zero ?)) destrs |
---|
684 | ] |
---|
685 | in |
---|
686 | let f ≝ λi. translate_muli dummy tmpr destrs tmp_destrs ? fresh_srcrs1 start_lbl i ? in |
---|
687 | let insts_mul ≝ foldi ? ? [ ] srcrs2 in ?. [5: check insts_init. |
---|
688 | add_translates (insts_init @ insts_mul) start_lbl dest_lbl def. |
---|
689 | *) |
---|
690 | |
---|
691 | definition translate_divumodu8 ≝ |
---|
692 | λorder: bool. |
---|
693 | λdestrs: list register. |
---|
694 | λsrcr1: register. |
---|
695 | λsrcr2: register. |
---|
696 | λstart_lbl: label. |
---|
697 | λdest_lbl: label. |
---|
698 | λdef: rtl_internal_function. |
---|
699 | match destrs with |
---|
700 | [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def |
---|
701 | | cons destr destrs ⇒ |
---|
702 | let 〈def, dummy〉 ≝ fresh_reg def in |
---|
703 | let 〈destr1, destr2〉 ≝ match order with [ true ⇒ 〈destr, dummy〉 | _ ⇒ 〈dummy, destr〉 ] in |
---|
704 | let inst_div ≝ adds_graph [ |
---|
705 | rtl_st_opaccs DivuModu destr1 destr2 srcr1 srcr2 start_lbl |
---|
706 | ] |
---|
707 | in |
---|
708 | let insts_rest ≝ translate_cst (Ointconst I8 (zero ?)) destrs in |
---|
709 | add_translates [ inst_div; insts_rest ] start_lbl dest_lbl def |
---|
710 | ]. |
---|
711 | |
---|
712 | definition translate_ne: ? → ? → ? → ? → ? → ? → rtl_internal_function ≝ |
---|
713 | λdestrs: list register. |
---|
714 | λsrcrs1: list register. |
---|
715 | λsrcrs2: list register. |
---|
716 | λstart_lbl: label. |
---|
717 | λdest_lbl: label. |
---|
718 | λdef: rtl_internal_function. |
---|
719 | match destrs with |
---|
720 | [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def |
---|
721 | | cons destr destrs ⇒ |
---|
722 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
723 | let 〈def, tmp_zero〉 ≝ fresh_reg def in |
---|
724 | let 〈def, tmp_srcrs1〉 ≝ fresh_regs def (|srcrs1|) in |
---|
725 | let save_srcrs1 ≝ translate_move tmp_srcrs1 srcrs1 in |
---|
726 | let 〈def, tmp_srcrs2〉 ≝ fresh_regs def (|srcrs2|) in |
---|
727 | let save_srcrs2 ≝ translate_move tmp_srcrs2 srcrs2 in |
---|
728 | match reduce_strong register register tmp_srcrs1 tmp_srcrs2 with |
---|
729 | [ dp crl their_proof ⇒ |
---|
730 | let commonl ≝ \fst (\fst crl) in |
---|
731 | let commonr ≝ \fst (\snd crl) in |
---|
732 | let restl ≝ \snd (\snd crl) in |
---|
733 | let restr ≝ \snd (\snd crl) in |
---|
734 | let rest ≝ choose_rest ? restl restr ? ? in |
---|
735 | let inits ≝ [ |
---|
736 | rtl_st_int destr (zero ?) start_lbl; |
---|
737 | rtl_st_int tmp_zero (zero ?) start_lbl |
---|
738 | ] |
---|
739 | in |
---|
740 | let f_common ≝ λtmp_srcr1. λtmp_srcr2. [ |
---|
741 | rtl_st_clear_carry start_lbl; |
---|
742 | rtl_st_op2 Sub tmpr tmp_srcr1 tmp_srcr2 start_lbl; |
---|
743 | rtl_st_op2 Or destr destr tmpr start_lbl |
---|
744 | ] |
---|
745 | in |
---|
746 | let insts_common ≝ flatten ? (map2 … f_common commonl commonr ?) in |
---|
747 | let f_rest ≝ λtmp_srcr. [ |
---|
748 | rtl_st_clear_carry start_lbl; |
---|
749 | rtl_st_op2 Sub tmpr tmp_zero tmp_srcr start_lbl; |
---|
750 | rtl_st_op2 Or destr destr tmpr start_lbl |
---|
751 | ] |
---|
752 | in |
---|
753 | let insts_rest ≝ flatten ? (map ? ? f_rest rest) in |
---|
754 | let insts ≝ inits @ insts_common @ insts_rest in |
---|
755 | let epilogue ≝ translate_cst (Ointconst I8 (zero ?)) destrs in |
---|
756 | add_translates [ |
---|
757 | save_srcrs1; save_srcrs2; adds_graph insts; epilogue |
---|
758 | ] start_lbl dest_lbl def |
---|
759 | ] |
---|
760 | ]. |
---|
761 | [1: @their_proof |
---|
762 | |*: cases not_implemented (* choose rest requires non-emptiness proofs but |
---|
763 | these appear impossible to obtain, similar proof |
---|
764 | to above *) |
---|
765 | ] |
---|
766 | qed. |
---|
767 | |
---|
768 | definition translate_eq_reg ≝ |
---|
769 | λtmp_zero: register. |
---|
770 | λtmp_one: register. |
---|
771 | λtmpr1: register. |
---|
772 | λtmpr2: register. |
---|
773 | λdestr: register. |
---|
774 | λdummy_lbl: label. |
---|
775 | λsrcr12: register × register. |
---|
776 | let 〈srcr1, srcr2〉 ≝ srcr12 in |
---|
777 | [ rtl_st_clear_carry dummy_lbl; |
---|
778 | rtl_st_op2 Sub tmpr1 srcr1 srcr2 dummy_lbl; |
---|
779 | rtl_st_op2 Addc tmpr1 tmp_zero tmp_zero dummy_lbl; |
---|
780 | rtl_st_op2 Sub tmpr2 srcr2 srcr1 dummy_lbl; |
---|
781 | rtl_st_op2 Addc tmpr2 tmp_zero tmp_zero dummy_lbl; |
---|
782 | rtl_st_op2 Or tmpr1 tmpr1 tmpr2 dummy_lbl; |
---|
783 | rtl_st_op2 Xor tmpr1 tmpr1 tmp_one dummy_lbl; |
---|
784 | rtl_st_op2 And destr destr tmpr1 dummy_lbl ]. |
---|
785 | |
---|
786 | definition translate_eq_list ≝ |
---|
787 | λtmp_zero: register. |
---|
788 | λtmp_one: register. |
---|
789 | λtmpr1: register. |
---|
790 | λtmpr2: register. |
---|
791 | λdestr: register. |
---|
792 | λleq: list (register × register). |
---|
793 | λdummy_lbl: label. |
---|
794 | let f ≝ translate_eq_reg tmp_zero tmp_one tmpr1 tmpr2 destr dummy_lbl in |
---|
795 | (rtl_st_int destr (bitvector_of_nat ? 1) dummy_lbl) :: |
---|
796 | flatten ? (map ? ? f leq). |
---|
797 | |
---|
798 | definition translate_atom ≝ |
---|
799 | λtmp_zero: register. |
---|
800 | λtmp_one: register. |
---|
801 | λtmpr1: register. |
---|
802 | λtmpr2: register. |
---|
803 | λtmpr3: register. |
---|
804 | λdestr: register. |
---|
805 | λdummy_lbl: label. |
---|
806 | λleq: list (register × register). |
---|
807 | λsrcr1: register. |
---|
808 | λsrcr2: register. |
---|
809 | translate_eq_list tmp_zero tmp_one tmpr1 tmpr2 tmpr3 leq dummy_lbl @ |
---|
810 | [ rtl_st_clear_carry dummy_lbl; |
---|
811 | rtl_st_op2 Sub tmpr1 srcr1 srcr2 dummy_lbl; |
---|
812 | rtl_st_op2 Addc tmpr1 tmp_zero tmp_zero dummy_lbl; |
---|
813 | rtl_st_op2 And tmpr3 tmpr3 tmpr1 dummy_lbl; |
---|
814 | rtl_st_op2 Or destr destr tmpr3 dummy_lbl ]. |
---|
815 | |
---|
816 | definition translate_lt_main ≝ |
---|
817 | λtmp_zero: register. |
---|
818 | λtmp_one: register. |
---|
819 | λtmpr1: register. |
---|
820 | λtmpr2: register. |
---|
821 | λtmpr3: register. |
---|
822 | λdestr: register. |
---|
823 | λdummy_lbl: label. |
---|
824 | λsrcrs1: list register. |
---|
825 | λsrcrs2: list register. |
---|
826 | λproof: |srcrs1| = |srcrs2|. |
---|
827 | let f ≝ λinsts_leq. λsrcr1. λsrcr2. |
---|
828 | let 〈insts, leq〉 ≝ insts_leq in |
---|
829 | let added_insts ≝ translate_atom tmp_zero tmp_one tmpr1 tmpr2 tmpr3 destr dummy_lbl leq srcr1 srcr2 in |
---|
830 | 〈insts @ added_insts, leq @ [〈srcr1, srcr2〉]〉 |
---|
831 | in |
---|
832 | \fst (fold_left2 ? ? ? f 〈[ ], [ ]〉 srcrs1 srcrs2 proof). |
---|
833 | |
---|
834 | definition fresh_regs_strong: |
---|
835 | rtl_internal_function → ∀n: nat. Σfresh: rtl_internal_function × (list register). |\snd fresh| = n ≝ |
---|
836 | λdef. |
---|
837 | λn. |
---|
838 | fresh_regs def n. |
---|
839 | @fresh_regs_length |
---|
840 | qed. |
---|
841 | |
---|
842 | definition complete_regs_strong: |
---|
843 | rtl_internal_function → list register → list register → Σcomplete: (list register) × (list register) × (list register). |\fst (\fst complete)| = |\snd (\fst complete)| ≝ |
---|
844 | λdef. |
---|
845 | λleft. |
---|
846 | λright. |
---|
847 | complete_regs def left right. |
---|
848 | @complete_regs_length |
---|
849 | qed. |
---|
850 | |
---|
851 | definition translate_lt ≝ |
---|
852 | λdestrs: list register. |
---|
853 | λprf_destrs: lt 0 (|destrs|). |
---|
854 | λsrcrs1: list register. |
---|
855 | λsrcrs2: list register. |
---|
856 | λstart_lbl: label. |
---|
857 | λdest_lbl: label. |
---|
858 | λdef: rtl_internal_function. |
---|
859 | match destrs with |
---|
860 | [ nil ⇒ add_graph start_lbl (rtl_st_skip dest_lbl) def |
---|
861 | | _ ⇒ |
---|
862 | match fresh_regs_strong def (|destrs|) with |
---|
863 | [ dp def_tmp_destrs tmp_destrs_proof ⇒ |
---|
864 | let def ≝ \fst def_tmp_destrs in |
---|
865 | let tmp_destrs ≝ \snd def_tmp_destrs in |
---|
866 | let tmp_destr ≝ hd_safe ? tmp_destrs ? in |
---|
867 | let 〈def, tmp_zero〉 ≝ fresh_reg def in |
---|
868 | let 〈def, tmp_one〉 ≝ fresh_reg def in |
---|
869 | let 〈def, tmpr1〉 ≝ fresh_reg def in |
---|
870 | let 〈def, tmpr2〉 ≝ fresh_reg def in |
---|
871 | let 〈def, tmpr3〉 ≝ fresh_reg def in |
---|
872 | match complete_regs_strong def srcrs1 srcrs2 with |
---|
873 | [ dp srcrs12_added srcrs12_proof ⇒ |
---|
874 | let srcrs1 ≝ \fst (\fst srcrs12_added) in |
---|
875 | let srcrs2 ≝ \snd (\fst srcrs12_added) in |
---|
876 | let added ≝ \snd srcrs12_added in |
---|
877 | let srcrs1' ≝ rev ? srcrs1 in |
---|
878 | let srcrs2' ≝ rev ? srcrs2 in |
---|
879 | let insts_init ≝ [ |
---|
880 | translate_cst (Ointconst I8 (zero ?)) tmp_destrs; |
---|
881 | translate_cst (Ointconst I8 (zero ?)) added; |
---|
882 | adds_graph [ |
---|
883 | rtl_st_int tmp_zero (zero ?) start_lbl; |
---|
884 | rtl_st_int tmp_one (bitvector_of_nat ? 1) start_lbl |
---|
885 | ] |
---|
886 | ] |
---|
887 | in |
---|
888 | let insts_main ≝ |
---|
889 | translate_lt_main tmp_zero tmp_one tmpr1 tmpr2 tmpr3 tmp_destr start_lbl srcrs1' srcrs2' ? in |
---|
890 | let insts_main ≝ [ adds_graph insts_main ] in |
---|
891 | let insts_exit ≝ [ translate_move destrs tmp_destrs ] in |
---|
892 | add_translates (insts_init @ insts_main @ insts_exit) start_lbl dest_lbl def |
---|
893 | ] |
---|
894 | ] |
---|
895 | ]. |
---|
896 | [2: >tmp_destrs_proof @prf_destrs |
---|
897 | |1: normalize nodelta |
---|
898 | generalize in match srcrs12_proof |
---|
899 | #HYP >rev_length >rev_length @HYP |
---|
900 | ] |
---|
901 | qed. |
---|
902 | |
---|
903 | definition add_128_to_last ≝ |
---|
904 | λtmp_128: register. |
---|
905 | λrs. |
---|
906 | λprf: 0 < |rs|. |
---|
907 | λstart_lbl: label. |
---|
908 | match rs with |
---|
909 | [ nil ⇒ adds_graph [ ] start_lbl |
---|
910 | | _ ⇒ |
---|
911 | let r ≝ last_safe ? rs prf in |
---|
912 | adds_graph [ |
---|
913 | rtl_st_op2 Add r r tmp_128 start_lbl |
---|
914 | ] start_lbl |
---|
915 | ]. |
---|
916 | |
---|
917 | definition translate_lts ≝ |
---|
918 | λdestrs: list register. |
---|
919 | λdestrs_prf: lt 0 (|destrs|). |
---|
920 | λsrcrs1: list register. |
---|
921 | λsrcrs2: list register. |
---|
922 | λsrcrs1_lt_prf: 0 < |srcrs1|. |
---|
923 | λsrcrs2_lt_prf: 0 < |srcrs2|. |
---|
924 | λstart_lbl: label. |
---|
925 | λdest_lbl: label. |
---|
926 | λdef: rtl_internal_function. |
---|
927 | match fresh_regs_strong def (|srcrs1|) with |
---|
928 | [ dp def_tmp_srcrs1 srcrs1_prf ⇒ |
---|
929 | let def ≝ \fst def_tmp_srcrs1 in |
---|
930 | let tmp_srcrs1 ≝ \snd def_tmp_srcrs1 in |
---|
931 | match fresh_regs_strong def (|srcrs2|) with |
---|
932 | [ dp def_tmp_srcrs2 srcrs2_prf ⇒ |
---|
933 | let def ≝ \fst def_tmp_srcrs2 in |
---|
934 | let tmp_srcrs2 ≝ \snd def_tmp_srcrs2 in |
---|
935 | let 〈def, tmp_128〉 ≝ fresh_reg def in |
---|
936 | add_translates [ |
---|
937 | translate_move tmp_srcrs1 srcrs1; |
---|
938 | translate_move tmp_srcrs2 srcrs2; |
---|
939 | adds_graph [ |
---|
940 | rtl_st_int tmp_128 (bitvector_of_nat ? 128) start_lbl |
---|
941 | ]; |
---|
942 | add_128_to_last tmp_128 tmp_srcrs1 ?; |
---|
943 | add_128_to_last tmp_128 tmp_srcrs2 ?; |
---|
944 | translate_lt destrs destrs_prf tmp_srcrs1 tmp_srcrs2 |
---|
945 | ] start_lbl dest_lbl def |
---|
946 | ] |
---|
947 | ]. |
---|
948 | [1: >srcrs1_prf @srcrs1_lt_prf |
---|
949 | |2: >srcrs2_prf @srcrs2_lt_prf |
---|
950 | ] |
---|
951 | qed. |
---|
952 | |
---|
953 | definition translate_op2 ≝ |
---|
954 | λop2. |
---|
955 | λdestrs: list register. |
---|
956 | λdestrs_prf: lt 0 (|destrs|). |
---|
957 | λsrcrs1: list register. |
---|
958 | λsrcrs2: list register. |
---|
959 | λsrcrs1_prf: lt 0 (|srcrs1|). (* can this be relaxed? i think it can |
---|
960 | but we need more dep. typ. in modu/divu |
---|
961 | cases *) |
---|
962 | λsrcrs2_prf: lt 0 (|srcrs2|). |
---|
963 | λstart_lbl: label. |
---|
964 | λdest_lbl: label. |
---|
965 | λdef: rtl_internal_function. |
---|
966 | match op2 with |
---|
967 | [ Oadd ⇒ |
---|
968 | translate_op Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
969 | | Oaddp ⇒ |
---|
970 | translate_op Addc destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
971 | | Osub ⇒ |
---|
972 | translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
973 | | Osubpi ⇒ |
---|
974 | translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
975 | | Osubpp sz ⇒ |
---|
976 | translate_op Sub destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
977 | | Omul ⇒ |
---|
978 | translate_mul destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
979 | | Odivu ⇒ |
---|
980 | match srcrs1 return λx. 0 < |x| → rtl_internal_function with |
---|
981 | [ cons hd tl ⇒ λcons_prf. |
---|
982 | match tl with |
---|
983 | [ nil ⇒ translate_divumodu8 true destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def |
---|
984 | | _ ⇒ ? (* not implemented *) |
---|
985 | ] |
---|
986 | | nil ⇒ λnil_absrd. ? |
---|
987 | ] srcrs1_prf |
---|
988 | | Omodu ⇒ |
---|
989 | match srcrs1 return λx. 0 < |x| → rtl_internal_function with |
---|
990 | [ cons hd tl ⇒ λcons_prf. |
---|
991 | match tl with |
---|
992 | [ nil ⇒ translate_divumodu8 false destrs hd (hd_safe register srcrs2 ?) start_lbl dest_lbl def |
---|
993 | | _ ⇒ ? (* not implemented *) |
---|
994 | ] |
---|
995 | | nil ⇒ λnil_absrd. ? |
---|
996 | ] srcrs1_prf |
---|
997 | | Oand ⇒ |
---|
998 | translate_op And destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
999 | | Oor ⇒ |
---|
1000 | translate_op Or destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1001 | | Oxor ⇒ |
---|
1002 | translate_op Xor destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1003 | | Ocmp c ⇒ |
---|
1004 | match c with |
---|
1005 | [ Ceq ⇒ |
---|
1006 | add_translates [ |
---|
1007 | translate_ne destrs srcrs1 srcrs2; |
---|
1008 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1009 | ] start_lbl dest_lbl def |
---|
1010 | | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1011 | | Clt ⇒ translate_lts destrs destrs_prf srcrs1 srcrs2 ? ? start_lbl dest_lbl def |
---|
1012 | | Cgt ⇒ translate_lts destrs destrs_prf srcrs2 srcrs1 ? ? start_lbl dest_lbl def |
---|
1013 | | Cle ⇒ |
---|
1014 | add_translates [ |
---|
1015 | translate_lts destrs destrs_prf srcrs2 srcrs1 ? ?; |
---|
1016 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1017 | ] start_lbl dest_lbl def |
---|
1018 | | Cge ⇒ |
---|
1019 | add_translates [ |
---|
1020 | translate_lts destrs destrs_prf srcrs1 srcrs2 ? ?; |
---|
1021 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1022 | ] start_lbl dest_lbl def |
---|
1023 | ] |
---|
1024 | | Ocmpu c ⇒ |
---|
1025 | match c with |
---|
1026 | [ Ceq ⇒ |
---|
1027 | add_translates [ |
---|
1028 | translate_ne destrs srcrs1 srcrs2; |
---|
1029 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1030 | ] start_lbl dest_lbl def |
---|
1031 | | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1032 | | Clt ⇒ translate_lt destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1033 | | Cgt ⇒ translate_lt destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def |
---|
1034 | | Cle ⇒ |
---|
1035 | add_translates [ |
---|
1036 | translate_lt destrs destrs_prf srcrs2 srcrs1; |
---|
1037 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1038 | ] start_lbl dest_lbl def |
---|
1039 | | Cge ⇒ |
---|
1040 | add_translates [ |
---|
1041 | translate_lt destrs destrs_prf srcrs1 srcrs2; |
---|
1042 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1043 | ] start_lbl dest_lbl def |
---|
1044 | ] |
---|
1045 | | Ocmpp c ⇒ |
---|
1046 | match c with |
---|
1047 | [ Ceq ⇒ |
---|
1048 | add_translates [ |
---|
1049 | translate_ne destrs srcrs1 srcrs2; |
---|
1050 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1051 | ] start_lbl dest_lbl def |
---|
1052 | | Cne ⇒ translate_ne destrs srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1053 | | Clt ⇒ translate_lt destrs destrs_prf srcrs1 srcrs2 start_lbl dest_lbl def |
---|
1054 | | Cgt ⇒ translate_lt destrs destrs_prf srcrs2 srcrs1 start_lbl dest_lbl def |
---|
1055 | | Cle ⇒ |
---|
1056 | add_translates [ |
---|
1057 | translate_lt destrs destrs_prf srcrs2 srcrs1; |
---|
1058 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1059 | ] start_lbl dest_lbl def |
---|
1060 | | Cge ⇒ |
---|
1061 | add_translates [ |
---|
1062 | translate_lt destrs destrs_prf srcrs1 srcrs2; |
---|
1063 | translate_op1 Onotbool destrs destrs (refl ? (|destrs|)) |
---|
1064 | ] start_lbl dest_lbl def |
---|
1065 | ] |
---|
1066 | | _ ⇒ ? (* assert false, implemented in run time or float op *) |
---|
1067 | ]. |
---|
1068 | [ 2,6: normalize in nil_absrd; |
---|
1069 | cases(not_le_Sn_O 0) |
---|
1070 | #HYP cases(HYP nil_absrd) |
---|
1071 | | 3,7,12,17,13,15,18,19,20,21,14,16: |
---|
1072 | assumption |
---|
1073 | | *: cases not_implemented (* yes, really *) |
---|
1074 | ] |
---|
1075 | qed. |
---|
1076 | |
---|
1077 | definition translate_cond ≝ |
---|
1078 | λsrcrs: list register. |
---|
1079 | λstart_lbl: label. |
---|
1080 | λlbl_true: label. |
---|
1081 | λlbl_false: label. |
---|
1082 | λdef: rtl_internal_function. |
---|
1083 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
1084 | let 〈def, tmp_lbl〉 ≝ fresh_label def in |
---|
1085 | let init ≝ rtl_st_int tmpr (zero ?) start_lbl in |
---|
1086 | let f ≝ λsrcr. rtl_st_op2 Or tmpr tmpr srcr start_lbl in |
---|
1087 | let def ≝ adds_graph (init :: (map … f srcrs)) start_lbl tmp_lbl def in |
---|
1088 | add_graph tmp_lbl (rtl_st_cond tmpr lbl_true lbl_false) def. |
---|
1089 | |
---|
1090 | definition translate_load ≝ |
---|
1091 | λaddr. |
---|
1092 | λaddr_prf: 2 ≤ |addr|. |
---|
1093 | λdestrs: list register. |
---|
1094 | λstart_lbl: label. |
---|
1095 | λdest_lbl: label. |
---|
1096 | λdef: rtl_internal_function. |
---|
1097 | match fresh_regs_strong def (|addr|) with |
---|
1098 | [ dp def_save_addr save_addr_prf ⇒ |
---|
1099 | let def ≝ \fst def_save_addr in |
---|
1100 | let save_addr ≝ \snd def_save_addr in |
---|
1101 | match fresh_regs_strong def (|addr|) with |
---|
1102 | [ dp def_tmp_addr tmp_addr_prf ⇒ |
---|
1103 | let def ≝ \fst def_tmp_addr in |
---|
1104 | let tmp_addr ≝ \snd def_tmp_addr in |
---|
1105 | let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr ? tmp_addr ? in |
---|
1106 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
1107 | let insts_save_addr ≝ translate_move save_addr addr in |
---|
1108 | let f ≝ λtranslates_off. λr. |
---|
1109 | let 〈translates, off〉 ≝ translates_off in |
---|
1110 | let translates ≝ translates @ [ |
---|
1111 | adds_graph [ |
---|
1112 | rtl_st_int tmpr off start_lbl |
---|
1113 | ]; |
---|
1114 | translate_op2 Oaddp tmp_addr ? save_addr [tmpr] ? ?; |
---|
1115 | adds_graph [ |
---|
1116 | rtl_st_load r tmp_addr1 tmp_addr2 dest_lbl |
---|
1117 | ] |
---|
1118 | ] |
---|
1119 | in |
---|
1120 | let 〈carry, result〉 ≝ half_add ? off int_size in |
---|
1121 | 〈translates, result〉 |
---|
1122 | in |
---|
1123 | let 〈translates, ignore〉 ≝ foldl ? ? f 〈[ ], (zero ?)〉 destrs in |
---|
1124 | add_translates (insts_save_addr :: translates) start_lbl dest_lbl def |
---|
1125 | ] |
---|
1126 | ]. |
---|
1127 | [1: normalize // |
---|
1128 | |2: >save_addr_prf normalize |
---|
1129 | @(transitive_le 1 2 (|addr|)) // |
---|
1130 | |3: >tmp_addr_prf normalize |
---|
1131 | @(transitive_le 1 2 (|addr|)) // |
---|
1132 | |*: >tmp_addr_prf assumption |
---|
1133 | ] |
---|
1134 | qed. |
---|
1135 | |
---|
1136 | definition translate_store ≝ |
---|
1137 | λaddr. |
---|
1138 | λaddr_prf: 2 ≤ |addr|. |
---|
1139 | λsrcrs: list register. |
---|
1140 | λstart_lbl: label. |
---|
1141 | λdest_lbl: label. |
---|
1142 | λdef: rtl_internal_function. |
---|
1143 | match fresh_regs_strong def (|addr|) with |
---|
1144 | [ dp def_tmp_addr tmp_addr_prf ⇒ |
---|
1145 | let def ≝ \fst def_tmp_addr in |
---|
1146 | let tmp_addr ≝ \snd def_tmp_addr in |
---|
1147 | let 〈tmp_addr1, tmp_addr2〉 ≝ make_addr ? tmp_addr ? in |
---|
1148 | let 〈def, tmpr〉 ≝ fresh_reg def in |
---|
1149 | let f ≝ λtranslate_off. λsrcr. |
---|
1150 | let 〈translates, off〉 ≝ translate_off in |
---|
1151 | let translates ≝ translates @ [ |
---|
1152 | adds_graph [ |
---|
1153 | rtl_st_int tmpr off start_lbl |
---|
1154 | ]; |
---|
1155 | translate_op2 Oaddp tmp_addr ? addr [tmpr] ? ?; |
---|
1156 | adds_graph [ |
---|
1157 | rtl_st_store tmp_addr1 tmp_addr2 srcr dest_lbl |
---|
1158 | ] |
---|
1159 | ] |
---|
1160 | in |
---|
1161 | let 〈carry, result〉 ≝ half_add ? off int_size in |
---|
1162 | 〈translates, result〉 |
---|
1163 | in |
---|
1164 | let 〈translates, ignore〉 ≝ foldl ? ? f 〈[ ], zero ?〉 srcrs in |
---|
1165 | add_translates translates start_lbl dest_lbl def |
---|
1166 | ]. |
---|
1167 | [1: normalize // |
---|
1168 | |2: >tmp_addr_prf normalize |
---|
1169 | @(transitive_le 1 2 (|addr|)) // |
---|
1170 | |3: >tmp_addr_prf normalize |
---|
1171 | @(transitive_le 1 2 (|addr|)) // |
---|
1172 | |*: >tmp_addr_prf assumption |
---|
1173 | ] |
---|
1174 | qed. |
---|
1175 | |
---|
1176 | definition translate_stmt ≝ |
---|
1177 | λlenv. |
---|
1178 | λlbl: label. |
---|
1179 | λstmt. |
---|
1180 | λdef: rtl_internal_function. |
---|
1181 | match stmt with |
---|
1182 | [ St_skip lbl' ⇒ add_graph lbl (rtl_st_skip lbl') def |
---|
1183 | | St_cost cost_lbl lbl' ⇒ add_graph lbl (rtl_st_cost cost_lbl lbl') def |
---|
1184 | | St_const destr cst lbl' ⇒ |
---|
1185 | translate_cst cst (find_local_env destr lenv) lbl lbl' def |
---|
1186 | | St_op1 op1 destr srcr lbl' ⇒ |
---|
1187 | translate_op1 op1 (find_local_env destr lenv) (find_local_env srcr lenv) ? lbl lbl' def |
---|
1188 | | St_op2 op2 destr srcr1 srcr2 lbl' ⇒ |
---|
1189 | translate_op2 op2 (find_local_env destr lenv) ? (find_local_env srcr1 lenv) (find_local_env srcr2 lenv) ? ? lbl lbl' def |
---|
1190 | | St_load ignore addr destr lbl' ⇒ |
---|
1191 | translate_load (find_local_env addr lenv) ? (find_local_env destr lenv) lbl lbl' def |
---|
1192 | | St_store ignore addr srcr lbl' ⇒ |
---|
1193 | translate_store (find_local_env addr lenv) ? (find_local_env srcr lenv) lbl lbl' def |
---|
1194 | | St_call_id f args retr lbl' ⇒ |
---|
1195 | match retr with |
---|
1196 | [ Some retr ⇒ |
---|
1197 | add_graph lbl (rtl_st_call_id f (rtl_args args lenv) (find_local_env retr lenv) lbl') def |
---|
1198 | | None ⇒ add_graph lbl (rtl_st_call_id f (rtl_args args lenv) [ ] lbl') def |
---|
1199 | ] |
---|
1200 | | St_call_ptr f args retr lbl' ⇒ |
---|
1201 | match retr with |
---|
1202 | [ None ⇒ |
---|
1203 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1204 | add_graph lbl (rtl_st_call_ptr f1 f2 (rtl_args args lenv) [ ] lbl') def |
---|
1205 | | Some retr ⇒ |
---|
1206 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1207 | add_graph lbl (rtl_st_call_ptr f1 f2 (rtl_args args lenv) (find_local_env retr lenv) lbl') def |
---|
1208 | ] |
---|
1209 | | St_tailcall_id f args ⇒ |
---|
1210 | add_graph lbl (rtl_st_tailcall_id f (rtl_args args lenv)) def |
---|
1211 | | St_tailcall_ptr f args ⇒ |
---|
1212 | let 〈f1, f2〉 ≝ find_and_addr f lenv ? in |
---|
1213 | add_graph lbl (rtl_st_tailcall_ptr f1 f2 (rtl_args args lenv)) def |
---|
1214 | | St_cond r lbl_true lbl_false ⇒ |
---|
1215 | translate_cond (find_local_env r lenv) lbl lbl_true lbl_false def |
---|
1216 | | St_jumptable r l ⇒ ? (* assert false: not implemented yet *) |
---|
1217 | | St_return ⇒ add_graph lbl rtl_st_return def |
---|
1218 | ]. |
---|
1219 | [10: cases not_implemented (* jtable case *) |
---|
1220 | |*: cases daemon (* TODO: proofs regarding lengths of lists, examine! *) |
---|
1221 | ] |
---|
1222 | qed. |
---|
1223 | |
---|
1224 | axiom graph_add: |
---|
1225 | ∀g: graph rtl_statement. |
---|
1226 | ∀l: label. |
---|
1227 | ∀s: rtl_statement. |
---|
1228 | lookup ? ? (add ? ? g l s) l ≠ None ?. |
---|
1229 | |
---|
1230 | (* XXX: we use a "hack" here to circumvent the proof obligations required to |
---|
1231 | create res, an empty internal_function record. we insert into our graph |
---|
1232 | the start and end nodes to ensure that they are in, and place dummy |
---|
1233 | skip instructions at these nodes. *) |
---|
1234 | definition translate_internal ≝ |
---|
1235 | λdef. |
---|
1236 | let runiverse ≝ f_reggen def in |
---|
1237 | let lenv ≝ initialize_local_env runiverse |
---|
1238 | (f_params def @ f_locals def) (f_result def) in |
---|
1239 | let params ≝ map_list_local_env lenv (map ? ? \fst (f_params def)) in |
---|
1240 | let locals ≝ map_list_local_env lenv (map ? ? \fst (f_locals def)) in |
---|
1241 | let result ≝ |
---|
1242 | match (f_result def) with |
---|
1243 | [ None ⇒ [ ] |
---|
1244 | | Some r_typ ⇒ |
---|
1245 | let 〈r, typ〉 ≝ r_typ in |
---|
1246 | find_local_env r lenv |
---|
1247 | ] |
---|
1248 | in |
---|
1249 | let luniverse' ≝ f_labgen def in |
---|
1250 | let runiverse' ≝ runiverse in |
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1251 | let result' ≝ result in |
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1252 | let params' ≝ params in |
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1253 | let locals' ≝ locals in |
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1254 | let stack_size' ≝ f_stacksize def in |
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1255 | let entry' ≝ f_entry def in |
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1256 | let exit' ≝ f_exit def in |
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1257 | let graph' ≝ add ? ? (empty_map ? ?) entry' (rtl_st_skip entry') in |
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1258 | let graph' ≝ add ? ? graph' exit' (rtl_st_skip exit') in |
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1259 | let res ≝ |
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1260 | mk_rtl_internal_function luniverse' runiverse' result' params' |
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1261 | locals' stack_size' graph' ? ? in |
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1262 | fold ? ? ? (translate_stmt lenv) (f_graph def) res. |
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1263 | [1: % |
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1264 | [1: @entry' |
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1265 | |2: normalize nodelta; |
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1266 | @(graph_add ? (f_entry def) ?) |
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1267 | @graph_add_lookup |
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1268 | ] |
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1269 | |2: % |
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1270 | [1: @exit' |
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1271 | |2: /2/ |
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1272 | ] |
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1273 | ] |
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1274 | qed. |
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1275 | |
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1276 | definition translate_fun_def ≝ |
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1277 | λfdef: fundef internal_function. |
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1278 | match fdef with |
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1279 | [ Internal f ⇒ Internal ? (translate_internal f) |
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1280 | | External e ⇒ External ? e |
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1281 | ]. |
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1282 | |
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1283 | definition init_data_to_nat ≝ |
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1284 | λi: init_data. |
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1285 | match i with |
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1286 | [ Init_int8 i8 ⇒ 8 |
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1287 | | Init_int16 i16 ⇒ 16 |
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1288 | | Init_int32 i32 ⇒ 32 |
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1289 | | Init_float32 f32 ⇒ 32 |
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1290 | | Init_float64 f64 ⇒ 64 |
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1291 | | Init_space s ⇒ s |
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1292 | | Init_null r ⇒ 0 |
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1293 | | Init_addrof r i o ⇒ 8 (* XXX: not at all sure about this one *) |
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1294 | ]. |
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1295 | |
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1296 | definition translate ≝ |
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1297 | λp: program (fundef internal_function) unit. |
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1298 | let f_funct ≝ λid_fun_def. |
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1299 | let 〈id, fun_def〉 ≝ id_fun_def in |
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1300 | 〈id, translate_fun_def fun_def〉 |
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1301 | in |
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1302 | let vars' ≝ map ? ? (λx. |
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1303 | let 〈id_init_ignore, ignore〉 ≝ x in |
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1304 | let 〈id_init, ignore〉 ≝ id_init_ignore in |
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1305 | let 〈id, init〉 ≝ id_init in |
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1306 | let init_total ≝ foldr ? ? plus 0 (map ? ? init_data_to_nat init) in |
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1307 | 〈id, init_total〉) (prog_vars ? ? p) |
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1308 | in |
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1309 | let functs' ≝ map ? ? f_funct (prog_funct ? ? p) in |
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1310 | let main' ≝ prog_main ? ? p in |
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1311 | mk_rtl_program vars' functs' (Some ? main'). |
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