1 | include "ASM/Util.ma". |
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2 | include "utilities/BitVectorTrieSet.ma". |
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3 | include "LIN/LIN.ma". |
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4 | |
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5 | let rec association (i: ident) (l: list (ident × nat)) |
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6 | on l: member i (eq_identifier ?) (map ? ? (fst ? ?) l) → nat ≝ |
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7 | match l return λl. member i (eq_identifier ?) (map ? ? (fst ? ?) l) → nat with |
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8 | [ nil ⇒ λabsd. ? |
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9 | | cons hd tl ⇒ |
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10 | λprf: member i (eq_identifier ?) (map ? ? (fst ? ?) (cons ? hd tl)). |
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11 | (match eq_identifier ? (\fst hd) i return λb. eq_identifier ? (\fst hd) i = b → nat with |
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12 | [ true ⇒ λeq_prf. \snd hd |
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13 | | false ⇒ λeq_prf. association i tl ? |
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14 | ]) (refl ? (eq_identifier ? (\fst hd) i)) |
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15 | ]. |
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16 | [ cases absd |
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17 | | cases prf |
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18 | [ > eq_prf |
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19 | # H |
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20 | cases H |
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21 | | # H |
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22 | assumption |
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23 | ] |
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24 | ] |
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25 | qed. |
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26 | |
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27 | definition statement_labels ≝ |
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28 | λg: list ident. |
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29 | λs: lin_statement g. |
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30 | let 〈label, instr〉 ≝ s in |
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31 | let generated ≝ |
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32 | match instr with |
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33 | [ joint_st_sequential instr' _ ⇒ |
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34 | match instr' with |
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35 | [ joint_instr_cost_label lbl ⇒ set_insert ? (word_of_identifier ? lbl) (set_empty ?) |
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36 | | joint_instr_cond acc_a_reg lbl ⇒ set_insert ? (word_of_identifier ? lbl) (set_empty ?) |
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37 | | _ ⇒ set_empty ? |
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38 | ] |
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39 | | joint_st_return ⇒ set_empty ? |
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40 | | joint_st_goto lbl ⇒ set_insert ? (word_of_identifier ? lbl) (set_empty ?) ] |
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41 | in |
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42 | match label with |
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43 | [ None ⇒ generated |
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44 | | Some lbl ⇒ set_insert ? (word_of_identifier ? lbl) generated |
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45 | ]. |
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46 | |
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47 | definition function_labels_internal ≝ |
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48 | λglobals: list ident. |
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49 | λlabels: BitVectorTrieSet ?. |
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50 | λstatement: lin_statement globals. |
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51 | set_union ? labels (statement_labels globals statement). |
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52 | |
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53 | |
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54 | (* dpm: A = Identifier *) |
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55 | definition function_labels: ∀A. ∀globals. ∀f. BitVectorTrieSet ? ≝ |
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56 | λA: Type[0]. |
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57 | λglobals: list ident. |
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58 | λf: A × (lin_function globals). |
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59 | let 〈ignore, fun_def〉 ≝ f in |
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60 | match fun_def return λ_. BitVectorTrieSet ? with |
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61 | [ Internal stmts ⇒ |
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62 | foldl ? ? (function_labels_internal globals) (set_empty ?) (pi1 … (joint_if_code ?? stmts)) |
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63 | | External _ ⇒ set_empty ? |
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64 | ]. |
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65 | |
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66 | definition program_labels_internal: ∀A. ∀globals. ∀labels. ∀funct. BitVectorTrieSet ? ≝ |
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67 | λA: Type[0]. |
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68 | λglobals: list ident. |
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69 | λlabels: BitVectorTrieSet ?. |
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70 | λfunct: A × (lin_function globals). |
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71 | set_union ? labels (function_labels ? globals funct). |
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72 | |
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73 | (* CSC: here we are silently throwing away the region information *) |
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74 | definition program_labels ≝ |
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75 | λprogram: lin_program. |
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76 | foldl … (program_labels_internal … (map … (λx. fst … (fst … x)) (prog_vars … program))) |
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77 | (set_empty …) (prog_funct … program). |
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78 | |
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79 | definition data_of_int ≝ λbv. DATA bv. |
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80 | definition data16_of_int ≝ λbv. DATA16 (bitvector_of_nat 16 bv). |
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81 | definition accumulator_address ≝ DIRECT (bitvector_of_nat 8 224). |
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82 | |
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83 | definition translate_statements ≝ |
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84 | λglobals: list (ident × nat). |
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85 | λglobals_old: list ident. |
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86 | λprf: ∀i: ident. member i (eq_identifier ?) globals_old → member i (eq_identifier ?) (map ? ? (fst ? ?) globals). |
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87 | λstatement: pre_lin_statement globals_old. |
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88 | match statement with |
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89 | [ joint_st_goto lbl ⇒ Jmp (word_of_identifier ? lbl) |
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90 | | joint_st_return ⇒ Instruction (RET ?) |
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91 | | joint_st_sequential instr _ ⇒ |
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92 | match instr with |
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93 | [ joint_instr_extension ext ⇒ ⊥ |
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94 | | joint_instr_comment comment ⇒ Comment comment |
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95 | | joint_instr_cost_label lbl ⇒ Cost (word_of_identifier ? lbl) |
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96 | | joint_instr_pop _ ⇒ Instruction (POP ? accumulator_address) |
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97 | | joint_instr_push _ ⇒ Instruction (PUSH ? accumulator_address) |
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98 | | joint_instr_clear_carry ⇒ Instruction (CLR ? CARRY) |
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99 | | joint_instr_call_id f _ _ ⇒ Call (word_of_identifier ? f) |
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100 | | joint_instr_opaccs accs _ _ ⇒ |
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101 | match accs with |
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102 | [ Mul ⇒ Instruction (MUL ? ACC_A ACC_B) |
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103 | | DivuModu ⇒ Instruction (DIV ? ACC_A ACC_B) |
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104 | ] |
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105 | | joint_instr_op1 op1 _ _ ⇒ |
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106 | match op1 with |
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107 | [ Cmpl ⇒ Instruction (CPL ? ACC_A) |
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108 | | Inc ⇒ Instruction (INC ? ACC_A) |
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109 | ] |
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110 | | joint_instr_op2 op2 _ _ reg ⇒ |
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111 | match op2 with |
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112 | [ Add ⇒ |
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113 | let reg' ≝ register_address reg in |
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114 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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115 | direct; |
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116 | registr ]] x) → ? with |
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117 | [ ACC_A ⇒ λacc_a: True. |
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118 | Instruction (ADD ? ACC_A accumulator_address) |
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119 | | DIRECT d ⇒ λdirect1: True. |
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120 | Instruction (ADD ? ACC_A (DIRECT d)) |
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121 | | REGISTER r ⇒ λregister1: True. |
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122 | Instruction (ADD ? ACC_A (REGISTER r)) |
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123 | | _ ⇒ λother: False. ⊥ |
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124 | ] (subaddressing_modein … reg') |
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125 | | Addc ⇒ |
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126 | let reg' ≝ register_address reg in |
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127 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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128 | direct; |
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129 | registr ]] x) → ? with |
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130 | [ ACC_A ⇒ λacc_a: True. |
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131 | Instruction (ADDC ? ACC_A accumulator_address) |
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132 | | DIRECT d ⇒ λdirect2: True. |
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133 | Instruction (ADDC ? ACC_A (DIRECT d)) |
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134 | | REGISTER r ⇒ λregister2: True. |
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135 | Instruction (ADDC ? ACC_A (REGISTER r)) |
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136 | | _ ⇒ λother: False. ⊥ |
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137 | ] (subaddressing_modein … reg') |
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138 | | Sub ⇒ |
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139 | let reg' ≝ register_address reg in |
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140 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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141 | direct; |
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142 | registr ]] x) → ? with |
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143 | [ ACC_A ⇒ λacc_a: True. |
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144 | Instruction (SUBB ? ACC_A accumulator_address) |
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145 | | DIRECT d ⇒ λdirect3: True. |
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146 | Instruction (SUBB ? ACC_A (DIRECT d)) |
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147 | | REGISTER r ⇒ λregister3: True. |
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148 | Instruction (SUBB ? ACC_A (REGISTER r)) |
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149 | | _ ⇒ λother: False. ⊥ |
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150 | ] (subaddressing_modein … reg') |
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151 | | And ⇒ |
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152 | let reg' ≝ register_address reg in |
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153 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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154 | direct; |
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155 | registr ]] x) → ? with |
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156 | [ ACC_A ⇒ λacc_a: True. |
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157 | Instruction (NOP ?) |
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158 | | DIRECT d ⇒ λdirect4: True. |
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159 | Instruction (ANL ? (inl ? ? (inl ? ? 〈ACC_A, DIRECT d〉))) |
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160 | | REGISTER r ⇒ λregister4: True. |
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161 | Instruction (ANL ? (inl ? ? (inl ? ? 〈ACC_A, REGISTER r〉))) |
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162 | | _ ⇒ λother: False. ⊥ |
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163 | ] (subaddressing_modein … reg') |
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164 | | Or ⇒ |
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165 | let reg' ≝ register_address reg in |
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166 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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167 | direct; |
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168 | registr ]] x) → ? with |
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169 | [ ACC_A ⇒ λacc_a: True. |
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170 | Instruction (NOP ?) |
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171 | | DIRECT d ⇒ λdirect5: True. |
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172 | Instruction (ORL ? (inl ? ? (inl ? ? 〈ACC_A, DIRECT d〉))) |
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173 | | REGISTER r ⇒ λregister5: True. |
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174 | Instruction (ORL ? (inl ? ? (inl ? ? 〈ACC_A, REGISTER r〉))) |
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175 | | _ ⇒ λother: False. ⊥ |
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176 | ] (subaddressing_modein … reg') |
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177 | | Xor ⇒ |
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178 | let reg' ≝ register_address reg in |
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179 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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180 | direct; |
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181 | registr ]] x) → ? with |
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182 | [ ACC_A ⇒ λacc_a: True. |
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183 | Instruction (XRL ? (inr ? ? 〈accumulator_address, ACC_A〉)) |
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184 | | DIRECT d ⇒ λdirect6: True. |
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185 | Instruction (XRL ? (inl ? ? 〈ACC_A, DIRECT d〉)) |
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186 | | REGISTER r ⇒ λregister6: True. |
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187 | Instruction (XRL ? (inl ? ? 〈ACC_A, REGISTER r〉)) |
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188 | | _ ⇒ λother: False. ⊥ |
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189 | ] (subaddressing_modein … reg') |
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190 | ] |
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191 | | joint_instr_int reg byte ⇒ |
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192 | let reg' ≝ register_address reg in |
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193 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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194 | direct; |
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195 | registr ]] x) → ? with |
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196 | [ REGISTER r ⇒ λregister7: True. |
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197 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? (inr ? ? 〈REGISTER r, (data_of_int byte)〉)))))) |
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198 | | ACC_A ⇒ λacc: True. |
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199 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? 〈ACC_A, (data_of_int byte)〉)))))) |
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200 | | DIRECT d ⇒ λdirect7: True. |
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201 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inr ? ? 〈DIRECT d, (data_of_int byte)〉))))) |
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202 | | _ ⇒ λother: False. ⊥ |
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203 | ] (subaddressing_modein … reg') |
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204 | | joint_instr_move regs ⇒ |
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205 | match regs with |
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206 | [ from_acc reg ⇒ |
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207 | let reg' ≝ register_address reg in |
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208 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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209 | direct; |
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210 | registr ]] x) → ? with |
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211 | [ REGISTER r ⇒ λregister8: True. |
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212 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? (inr ? ? 〈REGISTER r, ACC_A〉)))))) |
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213 | | ACC_A ⇒ λacc: True. |
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214 | Instruction (NOP ?) |
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215 | | DIRECT d ⇒ λdirect8: True. |
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216 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inr ? ? 〈DIRECT d, ACC_A〉))))) |
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217 | | _ ⇒ λother: False. ⊥ |
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218 | ] (subaddressing_modein … reg') |
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219 | | to_acc reg ⇒ |
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220 | let reg' ≝ register_address reg in |
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221 | match reg' return λx. bool_to_Prop (is_in … [[ acc_a; |
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222 | direct; |
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223 | registr ]] x) → ? with |
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224 | [ REGISTER r ⇒ λregister9: True. |
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225 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? 〈ACC_A, REGISTER r〉)))))) |
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226 | | DIRECT d ⇒ λdirect9: True. |
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227 | Instruction (MOV ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? (inl ? ? 〈ACC_A, DIRECT d〉)))))) |
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228 | | ACC_A ⇒ λacc_a: True. |
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229 | Instruction (NOP ?) |
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230 | | _ ⇒ λother: False. ⊥ |
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231 | ] (subaddressing_modein … reg')] |
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232 | | joint_instr_load _ _ _ ⇒ Instruction (MOVX ? (inl ? ? 〈ACC_A, EXT_INDIRECT_DPTR〉)) |
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233 | | joint_instr_store _ _ _ ⇒ Instruction (MOVX ? (inr ? ? 〈EXT_INDIRECT_DPTR, ACC_A〉)) |
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234 | | joint_instr_address addr proof _ _ ⇒ |
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235 | let look ≝ association addr globals (prf ? proof) in |
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236 | Instruction (MOV ? (inl ? ? (inl ? ? (inr ? ? (〈DPTR, (data16_of_int look)〉))))) |
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237 | | joint_instr_cond _ lbl ⇒ |
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238 | (* dpm: this should be handled in translate_code! *) |
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239 | Instruction (JNZ ? (word_of_identifier ? lbl)) |
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240 | | joint_instr_set_carry ⇒ |
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241 | Instruction (SETB ? CARRY) |
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242 | ] |
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243 | ]. |
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244 | try assumption |
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245 | try @ I |
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246 | qed. |
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247 | |
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248 | (*CSC: XXXXXXXXXXX looks bad: what invariant is needed here? *) |
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249 | definition ident_of_label: label → ident ≝ |
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250 | λl. an_identifier … (word_of_identifier … l). |
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251 | |
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252 | definition build_translated_statement ≝ |
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253 | λglobals. |
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254 | λglobals_old. |
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255 | λprf. |
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256 | λstatement: lin_statement globals_old. |
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257 | 〈option_map … ident_of_label (\fst statement), translate_statements globals globals_old prf (\snd statement)〉. |
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258 | |
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259 | definition translate_code ≝ |
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260 | λglobals: list (ident × nat). |
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261 | λglobals_old: list ident. |
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262 | λprf. |
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263 | λcode: list (lin_statement globals_old). |
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264 | map ? ? (build_translated_statement globals globals_old prf) code. |
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265 | |
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266 | lemma translate_code_preserves_WellFormedP: |
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267 | ∀globals, globals_old, prf, code. |
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268 | well_formed_P ? ? code → well_formed_P ? ? (translate_code globals globals_old prf code). |
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269 | #G #GO #P #C |
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270 | elim C |
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271 | [ normalize |
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272 | // |
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273 | | #G2 #G02 #IH (* CSC: understand here *) |
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274 | whd in ⊢ (% → %) whd in match build_translated_statement normalize nodelta |
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275 | cases (\fst G2) normalize // ] |
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276 | qed. |
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277 | |
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278 | definition translate_fun_def ≝ |
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279 | λglobals: list (ident × nat). |
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280 | λglobals_old. |
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281 | λprf. |
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282 | λid_def. |
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283 | let 〈id, def〉 ≝ id_def in |
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284 | match def with |
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285 | [ Internal int ⇒ |
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286 | let code_proof ≝ joint_if_code … (lin_params globals_old) int in |
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287 | match translate_code globals globals_old prf (pi1 ?? code_proof) return λtranscode. well_formed_P ? ? transcode → list labelled_instruction with |
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288 | [ nil ⇒ λprf2. ⊥ |
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289 | | cons hd tl ⇒ λ_. |
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290 | let rest ≝ 〈Some ? id, \snd hd〉 :: tl in |
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291 | map ? ? ( |
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292 | λr. |
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293 | match fst ? ? r with |
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294 | [ Some id' ⇒ 〈Some ? (word_of_identifier ? id'), snd ? ? r〉 |
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295 | | None ⇒ 〈None ?, \snd r〉 |
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296 | ]) rest |
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297 | ] (translate_code_preserves_WellFormedP globals globals_old prf (pi1 ?? code_proof) (pi2 ?? code_proof)) |
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298 | | External _ ⇒ [ ] |
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299 | ]. |
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300 | @prf2 |
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301 | qed. |
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302 | |
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303 | definition translate_functs ≝ |
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304 | λglobals. |
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305 | λglobals_old. |
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306 | λprf. |
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307 | λexit_label. |
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308 | λmain. |
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309 | λfuncts. |
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310 | let preamble ≝ [ 〈None ?, Call main〉 ; 〈Some ? exit_label, Jmp exit_label〉 ] in |
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311 | preamble @ |
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312 | (flatten ? (map ? ? (translate_fun_def globals globals_old prf) functs)). |
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313 | |
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314 | (*CSC: region silently thrown away here *) |
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315 | definition globals_addr ≝ |
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316 | λl. |
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317 | let globals_addr_internal ≝ |
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318 | λres_offset. |
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319 | λx_size: ident × region × nat. |
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320 | let 〈res, offset〉 ≝ res_offset in |
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321 | let 〈x, region, size〉 ≝ x_size in |
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322 | 〈〈x, offset〉 :: res, offset + size〉 |
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323 | in |
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324 | \fst (foldl ? ? globals_addr_internal 〈[ ], 0〉 l). |
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325 | |
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326 | (* dpm: plays the role of the string "_exit" in the O'caml source *) |
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327 | axiom identifier_prefix: Word. |
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328 | (*CSC: XXXXXXX wrong anyway since labels from different functions can now |
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329 | clash with each other and with names of functions *) |
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330 | axiom fresh_prefix: BitVectorTrieSet 16 → Word → Word. |
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331 | |
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332 | (* dpm: fresh prefix stuff needs unifying with brian *) |
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333 | definition translate : lin_program → pseudo_assembly_program ≝ |
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334 | λp. |
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335 | let prog_lbls ≝ program_labels … p in |
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336 | let exit_label ≝ fresh_prefix prog_lbls identifier_prefix in |
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337 | let global_addr ≝ globals_addr (prog_vars … p) in |
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338 | let global_addr' ≝ map ?? (λx_off. let 〈x,off〉 ≝ x_off in 〈word_of_identifier ? x,off〉) global_addr in |
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339 | 〈global_addr', translate_functs global_addr (prog_var_names … p) ? exit_label (word_of_identifier … (prog_main … p)) (prog_funct … p)〉. |
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340 | #i normalize nodelta -global_addr' global_addr exit_label prog_lbls; |
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341 | normalize in match prog_var_names normalize nodelta |
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342 | elim (prog_vars … p) // |
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343 | #hd #tl #IH whd in ⊢ (% → %) |
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344 | whd in match globals_addr normalize nodelta |
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345 | whd in match (foldl ???? (hd::tl)) normalize nodelta |
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346 | cases hd * #id #reg #size normalize nodelta |
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347 | cases daemon (*CSC: provable using a pair of lemmas over foldl *) |
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348 | qed. |
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