1 | (* *********************************************************************) |
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2 | (* *) |
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3 | (* The Compcert verified compiler *) |
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4 | (* *) |
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5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
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6 | (* *) |
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7 | (* Copyright Institut National de Recherche en Informatique et en *) |
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8 | (* Automatique. All rights reserved. This file is distributed *) |
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9 | (* under the terms of the GNU General Public License as published by *) |
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10 | (* the Free Software Foundation, either version 2 of the License, or *) |
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11 | (* (at your option) any later version. This file is also distributed *) |
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12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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13 | (* *) |
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14 | (* *********************************************************************) |
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15 | |
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16 | (* * This file defines a number of data types and operations used in |
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17 | the abstract syntax trees of many of the intermediate languages. *) |
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18 | |
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19 | include "basics/types.ma". |
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20 | include "common/Integers.ma". |
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21 | include "common/Floats.ma". |
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22 | include "ASM/Arithmetic.ma". |
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23 | include "common/Identifiers.ma". |
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24 | |
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25 | |
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26 | (* * * Syntactic elements *) |
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27 | |
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28 | (* Global variables and functions are represented by identifiers with the |
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29 | tag for symbols. Note that Clight also uses them for locals due to |
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30 | the ambiguous syntax. *) |
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31 | |
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32 | axiom SymbolTag : String. |
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33 | |
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34 | definition ident ≝ identifier SymbolTag. |
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35 | |
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36 | definition ident_eq : ∀x,y:ident. (x=y) + (x≠y) ≝ identifier_eq ?. |
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37 | |
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38 | definition ident_of_nat : nat → ident ≝ identifier_of_nat ?. |
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39 | |
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40 | (* dpm: not needed |
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41 | inductive quantity: Type[0] ≝ |
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42 | | q_int: Byte → quantity |
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43 | | q_offset: quantity |
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44 | | q_ptr: quantity. |
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45 | |
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46 | inductive abstract_size: Type[0] ≝ |
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47 | | size_q: quantity → abstract_size |
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48 | | size_prod: list abstract_size → abstract_size |
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49 | | size_sum: list abstract_size → abstract_size |
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50 | | size_array: nat → abstract_size → abstract_size. |
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51 | *) |
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52 | |
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53 | |
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54 | (* Memory spaces |
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55 | |
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56 | For full 8051 memory spaces support we have internal memory pointers, |
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57 | PData pointers which are 8 bit pointers to the first page of XData, and |
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58 | a tagged Any pointer for any of the spaces. |
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59 | |
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60 | We only support the 16 bit XData and Code pointers for now. Some commented |
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61 | out code is still present to suggest how to add the rest, which includes |
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62 | having pointers and pointer types contain a region field to indicate what |
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63 | kind of pointer they are (in addition to the region in the block which |
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64 | indicates where the pointer points to). |
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65 | |
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66 | *) |
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67 | |
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68 | inductive region : Type[0] ≝ |
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69 | (* | Any : region |
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70 | | Data : region |
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71 | | IData : region |
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72 | | PData : region*) |
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73 | | XData : region |
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74 | | Code : region. |
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75 | |
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76 | definition eq_region : region → region → bool ≝ |
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77 | λr1,r2. |
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78 | match r1 with |
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79 | [ (*Any ⇒ match r2 with [ Any ⇒ true | _ ⇒ false ] |
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80 | | Data ⇒ match r2 with [ Data ⇒ true | _ ⇒ false ] |
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81 | | IData ⇒ match r2 with [ IData ⇒ true | _ ⇒ false ] |
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82 | | PData ⇒ match r2 with [ PData ⇒ true | _ ⇒ false ] |
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83 | |*) XData ⇒ match r2 with [ XData ⇒ true | _ ⇒ false ] |
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84 | | Code ⇒ match r2 with [ Code ⇒ true | _ ⇒ false ] |
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85 | ]. |
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86 | |
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87 | lemma eq_region_elim : ∀P:bool → Type[0]. ∀r1,r2. |
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88 | (r1 = r2 → P true) → (r1 ≠ r2 → P false) → |
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89 | P (eq_region r1 r2). |
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90 | #P #r1 #r2 cases r1; cases r2; #Ptrue #Pfalse |
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91 | try ( @Ptrue // ) |
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92 | @Pfalse % #E destruct |
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93 | qed. |
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94 | |
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95 | lemma reflexive_eq_region: ∀r. eq_region r r = true. |
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96 | * // |
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97 | qed. |
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98 | |
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99 | definition eq_region_dec : ∀r1,r2:region. (r1=r2)+(r1≠r2). |
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100 | #r1 #r2 @(eq_region_elim ? r1 r2) /2/; qed. |
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101 | |
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102 | (* |
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103 | (* Carefully defined to be convertably nonzero *) |
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104 | definition size_pointer : region → nat ≝ |
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105 | λsp. S match sp with [ Data ⇒ 0 | IData ⇒ 0 | PData ⇒ 0 | XData ⇒ 1 | Code ⇒ 1 | Any ⇒ 2 ]. |
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106 | *) |
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107 | definition size_pointer : nat ≝ 2. |
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108 | |
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109 | (* We maintain some reasonable type information through the front end of the |
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110 | compiler. *) |
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111 | |
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112 | inductive signedness : Type[0] ≝ |
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113 | | Signed: signedness |
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114 | | Unsigned: signedness. |
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115 | |
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116 | inductive intsize : Type[0] ≝ |
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117 | | I8: intsize |
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118 | | I16: intsize |
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119 | | I32: intsize. |
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120 | |
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121 | (* * Float types come in two sizes: 32 bits (single precision) |
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122 | and 64-bit (double precision). *) |
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123 | |
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124 | inductive floatsize : Type[0] ≝ |
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125 | | F32: floatsize |
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126 | | F64: floatsize. |
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127 | |
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128 | inductive typ : Type[0] ≝ |
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129 | | ASTint : intsize → signedness → typ |
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130 | | ASTptr : (*region →*) typ. |
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131 | (* | ASTfloat : floatsize → typ. *) |
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132 | |
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133 | (* XXX aliases *) |
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134 | definition SigType ≝ typ. |
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135 | definition SigType_Int ≝ ASTint I32 Unsigned. |
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136 | (* |
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137 | | SigType_Float: SigType |
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138 | *) |
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139 | definition SigType_Ptr ≝ ASTptr (*Any*). |
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140 | |
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141 | (* Define these carefully so that we always know that the result is nonzero, |
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142 | and can be used in dependent types of the form (S n). |
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143 | (At the time of writing this is only used for bitsize_of_intsize.) *) |
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144 | |
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145 | definition pred_size_intsize : intsize → nat ≝ |
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146 | λsz. match sz with [ I8 ⇒ 0 | I16 ⇒ 1 | I32 ⇒ 3]. |
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147 | |
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148 | definition size_intsize : intsize → nat ≝ |
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149 | λsz. S (pred_size_intsize sz). |
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150 | |
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151 | definition bitsize_of_intsize : intsize → nat ≝ |
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152 | λsz. size_intsize sz * 8. |
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153 | |
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154 | definition eq_intsize : intsize → intsize → bool ≝ |
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155 | λsz1,sz2. |
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156 | match sz1 with |
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157 | [ I8 ⇒ match sz2 with [ I8 ⇒ true | _ ⇒ false ] |
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158 | | I16 ⇒ match sz2 with [ I16 ⇒ true | _ ⇒ false ] |
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159 | | I32 ⇒ match sz2 with [ I32 ⇒ true | _ ⇒ false ] |
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160 | ]. |
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161 | |
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162 | lemma eq_intsize_elim : ∀sz1,sz2. ∀P:bool → Type[0]. |
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163 | (sz1 ≠ sz2 → P false) → (sz1 = sz2 → P true) → P (eq_intsize sz1 sz2). |
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164 | * * #P #Hne #Heq whd in ⊢ (?%); try (@Heq @refl) @Hne % #E destruct |
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165 | qed. |
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166 | |
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167 | lemma eq_intsize_true : ∀sz. eq_intsize sz sz = true. |
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168 | * @refl |
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169 | qed. |
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170 | |
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171 | lemma eq_intsize_false : ∀sz,sz'. sz ≠ sz' → eq_intsize sz sz' = false. |
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172 | * * * #NE try @refl @False_ind @NE @refl |
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173 | qed. |
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174 | |
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175 | definition signedness_check : ∀sg1,sg2:signedness. ∀P:signedness → signedness → Type[0]. |
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176 | P sg1 sg1 → P sg1 sg2 → P sg1 sg2 ≝ |
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177 | λsg1,sg2,P. |
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178 | match sg1 return λsg1. P sg1 sg1 → P sg1 sg2 → P sg1 sg2 with |
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179 | [ Signed ⇒ λx. match sg2 return λsg2. P ? sg2 → P Signed sg2 with [ Signed ⇒ λd. x | _ ⇒ λd. d ] |
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180 | | Unsigned ⇒ λx. match sg2 return λsg2. P ? sg2 → P Unsigned sg2 with [ Unsigned ⇒ λd. x | _ ⇒ λd. d ] |
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181 | ]. |
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182 | |
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183 | let rec inttyp_eq_elim' (sz1,sz2:intsize) (sg1,sg2:signedness) (P:intsize → signedness → intsize → signedness → Type[0]) on sz1 : |
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184 | P sz1 sg1 sz1 sg1 → P sz1 sg1 sz2 sg2 → P sz1 sg1 sz2 sg2 ≝ |
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185 | match sz1 return λsz. P sz sg1 sz sg1 → P sz sg1 sz2 sg2 → P sz sg1 sz2 sg2 with |
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186 | [ I8 ⇒ λx. match sz2 return λsz. P ?? sz ? → P I8 ? sz ? with [ I8 ⇒ signedness_check sg1 sg2 (λs1,s2. P ? s1 ? s2) x | _ ⇒ λd. d ] |
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187 | | I16 ⇒ λx. match sz2 return λsz. P I16 sg1 sz sg2 → P I16 sg1 sz sg2 with [ I16 ⇒ signedness_check sg1 sg2 (λs1,s2. P ? s1 ? s2) x | _ ⇒ λd. d ] |
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188 | | I32 ⇒ λx. match sz2 return λsz. P I32 sg1 sz sg2 → P I32 sg1 sz sg2 with [ I32 ⇒ signedness_check sg1 sg2 (λs1,s2. P ? s1 ? s2) x | _ ⇒ λd. d ] |
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189 | ]. |
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190 | |
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191 | let rec intsize_eq_elim' (sz1,sz2:intsize) (P:intsize → intsize → Type[0]) on sz1 : |
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192 | P sz1 sz1 → P sz1 sz2 → P sz1 sz2 ≝ |
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193 | match sz1 return λsz. P sz sz → P sz sz2 → P sz sz2 with |
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194 | [ I8 ⇒ λx. match sz2 return λsz. P ? sz → P I8 sz with [ I8 ⇒ λd. x | _ ⇒ λd. d ] |
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195 | | I16 ⇒ λx. match sz2 return λsz. P ? sz → P I16 sz with [ I16 ⇒ λd. x | _ ⇒ λd. d ] |
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196 | | I32 ⇒ λx. match sz2 return λsz. P ? sz → P I32 sz with [ I32 ⇒ λd. x | _ ⇒ λd. d ] |
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197 | ]. |
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198 | |
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199 | (* [intsize_eq_elim ? sz1 sz2 ? n (λn.e1) e2] checks if [sz1] equals [sz2] and |
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200 | if it is returns [e1] where the type of [n] has its dependency on [sz1] |
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201 | changed to [sz2], and if not returns [e2]. *) |
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202 | let rec intsize_eq_elim (A:Type[0]) (sz1,sz2:intsize) (P:intsize → Type[0]) on sz1 : |
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203 | P sz1 → (P sz2 → A) → A → A ≝ |
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204 | match sz1 return λsz. P sz → (P sz2 → A) → A → A with |
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205 | [ I8 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I8 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
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206 | | I16 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I16 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
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207 | | I32 ⇒ λx. match sz2 return λsz. (P sz → A) → A → A with [ I32 ⇒ λf,d. f x | _ ⇒ λf,d. d ] |
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208 | ]. |
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209 | |
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210 | lemma intsize_eq_elim_true : ∀A,sz,P,p,f,d. |
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211 | intsize_eq_elim A sz sz P p f d = f p. |
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212 | #A * // |
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213 | qed. |
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214 | |
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215 | lemma intsize_eq_elim_elim : ∀A,sz1,sz2,P,p,f,d. ∀Q:A → Type[0]. |
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216 | (sz1 ≠ sz2 → Q d) → (∀E:sz1 = sz2. match sym_eq ??? E return λx.λ_.P x → Type[0] with [ refl ⇒ λp. Q (f p) ] p ) → Q (intsize_eq_elim A sz1 sz2 P p f d). |
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217 | #A * * #P #p #f #d #Q #Hne #Heq |
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218 | [ 1,5,9: whd in ⊢ (?%); @(Heq (refl ??)) |
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219 | | *: whd in ⊢ (?%); @Hne % #E destruct |
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220 | ] qed. |
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221 | |
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222 | lemma intsize_eq_elim_false : ∀A,sz,sz',P,p,f,d. |
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223 | sz ≠ sz' → |
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224 | intsize_eq_elim A sz sz' P p f d = d. |
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225 | #A * * // #P #p #f #d * #H cases (H (refl ??)) |
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226 | qed. |
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227 | |
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228 | (* A type for the integers that appear in the semantics. *) |
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229 | definition bvint : intsize → Type[0] ≝ λsz. BitVector (bitsize_of_intsize sz). |
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230 | |
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231 | definition repr : ∀sz:intsize. nat → bvint sz ≝ |
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232 | λsz,x. bitvector_of_nat (bitsize_of_intsize sz) x. |
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233 | |
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234 | definition size_floatsize : floatsize → nat ≝ |
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235 | λsz. S match sz with [ F32 ⇒ 3 | F64 ⇒ 7 ]. |
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236 | |
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237 | let rec floatsize_eq_elim (sz1,sz2:floatsize) (P:floatsize → floatsize → Type[0]) on sz1 : |
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238 | P sz1 sz1 → P sz1 sz2 → P sz1 sz2 ≝ |
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239 | match sz1 return λsz. P sz sz → P sz sz2 → P sz sz2 with |
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240 | [ F32 ⇒ λx. match sz2 return λsz. P ? sz → P F32 sz with [ F32 ⇒ λd. x | _ ⇒ λd. d ] |
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241 | | F64 ⇒ λx. match sz2 return λsz. P ? sz → P F64 sz with [ F64 ⇒ λd. x | _ ⇒ λd. d ] |
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242 | ]. |
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243 | |
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244 | |
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245 | definition typesize : typ → nat ≝ λty. |
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246 | match ty with |
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247 | [ ASTint sz _ ⇒ size_intsize sz |
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248 | | ASTptr ⇒ size_pointer ]. |
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249 | (* | ASTfloat sz ⇒ size_floatsize sz ].*) |
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250 | |
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251 | lemma typesize_pos: ∀ty. typesize ty > 0. |
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252 | *; try *; try * /2 by le_n/ qed. |
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253 | |
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254 | lemma typ_eq: ∀t1,t2: typ. (t1=t2) + (t1≠t2). |
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255 | * *; try *; try *; try *; try *; try (%1 @refl) %2 @nmk #H destruct |
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256 | qed. |
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257 | |
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258 | lemma opt_typ_eq: ∀t1,t2: option typ. (t1=t2) + (t1≠t2). |
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259 | #t1 #t2 cases t1 cases t2 |
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260 | [ %1 @refl |
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261 | | 2,3: #ty %2 % #H destruct |
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262 | | #ty1 #ty2 elim (typ_eq ty1 ty2) #E [ %1 >E @refl | %2 % #E' destruct cases E /2/ |
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263 | ] |
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264 | qed. |
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265 | |
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266 | (* * Additionally, function definitions and function calls are annotated |
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267 | by function signatures indicating the number and types of arguments, |
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268 | as well as the type of the returned value if any. These signatures |
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269 | are used in particular to determine appropriate calling conventions |
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270 | for the function. *) |
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271 | |
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272 | record signature : Type[0] ≝ { |
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273 | sig_args: list typ; |
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274 | sig_res: option typ |
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275 | }. |
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276 | |
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277 | (* XXX aliases *) |
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278 | definition Signature ≝ signature. |
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279 | definition signature_args ≝ sig_args. |
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280 | definition signature_return ≝ sig_res. |
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281 | |
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282 | definition proj_sig_res : signature → typ ≝ λs. |
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283 | match sig_res s with |
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284 | [ None ⇒ ASTint I32 Unsigned |
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285 | | Some t ⇒ t |
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286 | ]. |
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287 | |
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288 | (* * Initialization data for global variables. *) |
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289 | |
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290 | inductive init_data: Type[0] ≝ |
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291 | | Init_int8: bvint I8 → init_data |
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292 | | Init_int16: bvint I16 → init_data |
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293 | | Init_int32: bvint I32 → init_data |
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294 | | Init_float32: float → init_data |
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295 | | Init_float64: float → init_data |
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296 | | Init_space: nat → init_data |
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297 | | Init_null: (*region →*) init_data |
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298 | | Init_addrof: (*region →*) ident → nat → init_data. (*r address of symbol + offset *) |
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299 | |
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300 | (* * Whole programs consist of: |
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301 | - a collection of function definitions (name and description); |
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302 | - the name of the ``main'' function that serves as entry point in the program; |
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303 | - a collection of global variable declarations, consisting of |
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304 | a name, initialization data, and additional information. |
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305 | |
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306 | The type of function descriptions and that of additional information |
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307 | for variables vary among the various intermediate languages and are |
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308 | taken as parameters to the [program] type. The other parts of whole |
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309 | programs are common to all languages. *) |
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310 | |
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311 | record program (F: list ident → Type[0]) (V: Type[0]) : Type[0] := { |
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312 | prog_vars: list (ident × region × V); |
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313 | prog_funct: list (ident × (F (map … (λx. \fst (\fst x)) prog_vars))); |
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314 | prog_main: ident |
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315 | }. |
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316 | |
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317 | |
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318 | definition prog_funct_names ≝ λF,V. λp: program F V. |
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319 | map ?? \fst (prog_funct … p). |
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320 | |
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321 | definition prog_var_names ≝ λF,V. λp: program F V. |
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322 | map ?? (λx. \fst (\fst x)) (prog_vars … p). |
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323 | |
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324 | (* * * Generic transformations over programs *) |
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325 | |
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326 | (* * We now define a general iterator over programs that applies a given |
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327 | code transformation function to all function descriptions and leaves |
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328 | the other parts of the program unchanged. *) |
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329 | (* |
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330 | Section TRANSF_PROGRAM. |
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331 | |
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332 | Variable A B V: Type. |
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333 | Variable transf: A -> B. |
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334 | *) |
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335 | |
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336 | definition transf_program : ∀A,B. (A → B) → list (ident × A) → list (ident × B) ≝ |
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337 | λA,B,transf,l. |
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338 | map ?? (λid_fn. 〈fst ?? id_fn, transf (snd ?? id_fn)〉) l. |
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339 | |
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340 | (* In principle we could allow the transformation to be specialised to a |
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341 | particular set of variable names, but that makes it much harder to state |
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342 | and prove properties. *) |
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343 | |
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344 | definition transform_program : ∀A,B,V. ∀p:program A V. (∀varnames. A varnames → B varnames) → program B V ≝ |
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345 | λA,B,V,p,transf. |
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346 | mk_program B V |
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347 | (prog_vars A V p) |
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348 | (transf_program ?? (transf ?) (prog_funct A V p)) |
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349 | (prog_main A V p). |
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350 | |
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351 | (* Versions of the above that thread a fresh name generator through the |
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352 | transformation. *) |
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353 | |
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354 | definition transf_program_gen : ∀tag,A,B. universe tag → |
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355 | (universe tag → A → B × (universe tag)) → |
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356 | list (ident × A) → |
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357 | list (ident × B) × (universe tag) ≝ |
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358 | λtag,A,B,gen,transf,l. |
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359 | foldr ?? (λid_fn,bs_gen. let 〈fn',gen'〉 ≝ transf (\snd bs_gen) (\snd id_fn) in |
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360 | 〈〈\fst id_fn, fn'〉::(\fst bs_gen), gen'〉) 〈[ ], gen〉 l. |
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361 | |
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362 | lemma transf_program_gen_step : ∀tag,A,B,gen,transf,id,fn,tl. |
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363 | ∃gen'. |
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364 | \fst (transf_program_gen tag A B gen transf (〈id,fn〉::tl)) = |
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365 | 〈id, \fst (transf gen' fn)〉::(\fst (transf_program_gen tag A B gen transf tl)). |
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366 | #tag #A #B #gen #transf #id #fn #tl |
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367 | whd in ⊢ (??(λ_.??(???%)?)); |
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368 | change with (transf_program_gen ??????) in match (foldr ?????); |
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369 | cases (transf_program_gen ??????) #tl' #gen' |
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370 | %{gen'} cases (transf gen' fn) // |
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371 | qed. |
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372 | |
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373 | definition transform_program_gen : ∀tag,A,B,V. universe tag → ∀p:program A V. |
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374 | (∀varnames. universe tag → A varnames → B varnames × (universe tag)) → |
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375 | program B V × (universe tag) ≝ |
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376 | λtag,A,B,V,gen,p,trans. |
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377 | let fsg ≝ transf_program_gen tag ?? gen (trans ?) (prog_funct A V p) in |
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378 | 〈mk_program B V |
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379 | (prog_vars A V p) |
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380 | (\fst fsg) |
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381 | (prog_main A V p), \snd fsg〉. |
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382 | |
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383 | |
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384 | (* |
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385 | lemma transform_program_function: |
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386 | ∀A,B,V,transf,p,i,tf. |
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387 | in_list ? 〈i, tf〉 (prog_funct ?? (transform_program A B V transf p)) → |
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388 | ∃f. in_list ? 〈i, f〉 (prog_funct ?? p) ∧ transf f = tf. |
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389 | normalize; #A #B #V #transf #p #i #tf #H elim (list_in_map_inv ????? H); |
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390 | #x elim x; #i' #tf' *; #e #H destruct; %{tf'} /2/; |
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391 | qed. |
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392 | *) |
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393 | (* |
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394 | End TRANSF_PROGRAM. |
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395 | |
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396 | (** The following is a variant of [transform_program] where the |
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397 | code transformation function can fail and therefore returns an |
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398 | option type. *) |
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399 | |
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400 | Open Local Scope error_monad_scope. |
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401 | Open Local Scope string_scope. |
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402 | |
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403 | Section MAP_PARTIAL. |
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404 | |
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405 | Variable A B C: Type. |
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406 | Variable prefix_errmsg: A -> errmsg. |
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407 | Variable f: B -> res C. |
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408 | *) |
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409 | definition map_partial : ∀A,B,C:Type[0]. (B → res C) → |
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410 | list (A × B) → res (list (A × C)) ≝ |
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411 | λA,B,C,f. m_list_map ??? (λab. let 〈a,b〉 ≝ ab in do c ← f b; OK ? 〈a,c〉). |
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412 | |
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413 | lemma map_partial_preserves_first: |
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414 | ∀A,B,C:Type[0]. ∀f: B → res C. ∀l: list (A × B). ∀l': list (A × C). |
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415 | map_partial … f l = OK ? l' → |
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416 | map … \fst l = map … \fst l'. |
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417 | #A #B #C #f #l elim l |
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418 | [ #l' #E normalize in E; destruct % |
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419 | | * #a #b #tl #IH #l' normalize in ⊢ (??%? → ?); cases (f b) normalize in ⊢ (? → ??%? → ?); |
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420 | [2: #err #E destruct |
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421 | | #c change with (match map_partial … f tl with [ OK x ⇒ ? | Error y ⇒ ?] = ? → ?) |
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422 | cases (map_partial … f tl) in IH ⊢ %; |
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423 | #x #IH normalize in ⊢ (??%? → ?); #ABS destruct normalize |
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424 | >(IH x) // ]] |
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425 | qed. |
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426 | |
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427 | lemma map_partial_All2 : ∀A,B,C,f. ∀P:A×B → A×C → Prop. |
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428 | (∀i,x,y. f x = OK ? y → P 〈i,x〉 〈i,y〉) → |
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429 | ∀l,l'. |
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430 | map_partial A B C f l = OK ? l' → |
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431 | All2 (A×B) (A×C) P l l'. |
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432 | #A #B #C #f #P #H #l elim l |
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433 | [ * [ // | #h #t #E normalize in E; destruct ] |
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434 | | * #a #b #tl #IH #l' #M |
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435 | cases (bind_inversion ????? M) -M * #a' #c * #AC #M |
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436 | cases (bind_inversion ????? M) -M #tl' * #TL #M |
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437 | cases (bind_inversion ????? AC) -AC #c' * #C #C' |
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438 | normalize in C C' M; destruct % |
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439 | [ @H @C |
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440 | | @IH @TL |
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441 | ] |
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442 | ] qed. |
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443 | |
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444 | (* |
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445 | Fixpoint map_partial (l: list (A * B)) : res (list (A * C)) := |
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446 | match l with |
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447 | | nil => OK nil |
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448 | | (a, b) :: rem => |
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449 | match f b with |
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450 | | Error msg => Error (prefix_errmsg a ++ msg)%list |
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451 | | OK c => |
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452 | do rem' <- map_partial rem; |
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453 | OK ((a, c) :: rem') |
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454 | end |
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455 | end. |
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456 | |
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457 | Remark In_map_partial: |
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458 | forall l l' a c, |
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459 | map_partial l = OK l' -> |
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460 | In (a, c) l' -> |
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461 | exists b, In (a, b) l /\ f b = OK c. |
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462 | Proof. |
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463 | induction l; simpl. |
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464 | intros. inv H. elim H0. |
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465 | intros until c. destruct a as [a1 b1]. |
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466 | caseEq (f b1); try congruence. |
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467 | intro c1; intros. monadInv H0. |
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468 | elim H1; intro. inv H0. exists b1; auto. |
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469 | exploit IHl; eauto. intros [b [P Q]]. exists b; auto. |
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470 | Qed. |
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471 | |
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472 | Remark map_partial_forall2: |
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473 | forall l l', |
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474 | map_partial l = OK l' -> |
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475 | list_forall2 |
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476 | (fun (a_b: A * B) (a_c: A * C) => |
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477 | fst a_b = fst a_c /\ f (snd a_b) = OK (snd a_c)) |
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478 | l l'. |
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479 | Proof. |
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480 | induction l; simpl. |
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481 | intros. inv H. constructor. |
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482 | intro l'. destruct a as [a b]. |
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483 | caseEq (f b). 2: congruence. intro c; intros. monadInv H0. |
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484 | constructor. simpl. auto. auto. |
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485 | Qed. |
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486 | |
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487 | End MAP_PARTIAL. |
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488 | |
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489 | Remark map_partial_total: |
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490 | forall (A B C: Type) (prefix: A -> errmsg) (f: B -> C) (l: list (A * B)), |
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491 | map_partial prefix (fun b => OK (f b)) l = |
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492 | OK (List.map (fun a_b => (fst a_b, f (snd a_b))) l). |
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493 | Proof. |
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494 | induction l; simpl. |
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495 | auto. |
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496 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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497 | Qed. |
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498 | |
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499 | Remark map_partial_identity: |
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500 | forall (A B: Type) (prefix: A -> errmsg) (l: list (A * B)),cmp |
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501 | map_partial prefix (fun b => OK b) l = OK l. |
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502 | Proof. |
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503 | induction l; simpl. |
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504 | auto. |
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505 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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506 | Qed. |
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507 | |
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508 | Section TRANSF_PARTIAL_PROGRAM. |
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509 | |
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510 | Variable A B V: Type. |
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511 | Variable transf_partial: A -> res B. |
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512 | |
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513 | Definition prefix_funct_name (id: ident) : errmsg := |
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514 | MSG "In function " :: CTX id :: MSG ": " :: nil. |
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515 | *) |
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516 | definition transform_partial_program : ∀A,B,V. ∀p:program A V. (∀varnames. A varnames → res (B varnames)) → res (program B V) ≝ |
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517 | λA,B,V,p,transf_partial. |
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518 | do fl ← map_partial … (transf_partial ?) (prog_funct … p); |
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519 | OK (program B V) (mk_program … (prog_vars … p) fl (prog_main ?? p)). |
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520 | |
---|
521 | (* |
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522 | Lemma transform_partial_program_function: |
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523 | forall p tp i tf, |
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524 | transform_partial_program p = OK tp -> |
---|
525 | In (i, tf) tp.(prog_funct) -> |
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526 | exists f, In (i, f) p.(prog_funct) /\ transf_partial f = OK tf. |
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527 | Proof. |
---|
528 | intros. monadInv H. simpl in H0. |
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529 | eapply In_map_partial; eauto. |
---|
530 | Qed. |
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531 | |
---|
532 | Lemma transform_partial_program_main: |
---|
533 | forall p tp, |
---|
534 | transform_partial_program p = OK tp -> |
---|
535 | tp.(prog_main) = p.(prog_main). |
---|
536 | Proof. |
---|
537 | intros. monadInv H. reflexivity. |
---|
538 | Qed. |
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539 | |
---|
540 | Lemma transform_partial_program_vars: |
---|
541 | forall p tp, |
---|
542 | transform_partial_program p = OK tp -> |
---|
543 | tp.(prog_vars) = p.(prog_vars). |
---|
544 | Proof. |
---|
545 | intros. monadInv H. reflexivity. |
---|
546 | Qed. |
---|
547 | |
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548 | End TRANSF_PARTIAL_PROGRAM. |
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549 | |
---|
550 | (** The following is a variant of [transform_program_partial] where |
---|
551 | both the program functions and the additional variable information |
---|
552 | are transformed by functions that can fail. *) |
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553 | |
---|
554 | Section TRANSF_PARTIAL_PROGRAM2. |
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555 | |
---|
556 | Variable A B V W: Type. |
---|
557 | Variable transf_partial_function: A -> res B. |
---|
558 | Variable transf_partial_variable: V -> res W. |
---|
559 | |
---|
560 | Definition prefix_var_name (id_init: ident * list init_data) : errmsg := |
---|
561 | MSG "In global variable " :: CTX (fst id_init) :: MSG ": " :: nil. |
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562 | *) |
---|
563 | |
---|
564 | (* CSC: ad hoc lemma, move away? *) |
---|
565 | lemma map_fst: |
---|
566 | ∀A,B,C,C':Type[0].∀l:list (A × B × C).∀l':list (A × B × C'). |
---|
567 | map … \fst l = map … \fst l' → |
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568 | map … (λx. \fst (\fst x)) l = map … (λx. \fst (\fst x)) l'. |
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569 | #A #B #C #C' #l elim l |
---|
570 | [ #l' elim l' // #he #tl #IH #ABS normalize in ABS; destruct |
---|
571 | | #he1 #tl1 #IH #l' cases l' [ #ABS normalize in ABS; destruct ] |
---|
572 | #he2 #tl2 #EQ whd in EQ:(??%%) ⊢ (??%%); >(IH tl2) destruct normalize in e1 ⊢ %; >e0 // |
---|
573 | >e0 in e1; normalize #H @H ] |
---|
574 | qed. |
---|
575 | |
---|
576 | definition transform_partial_program2 : |
---|
577 | ∀A,B,V,W. ∀p: program A V. |
---|
578 | (∀varnames. A varnames → res (B varnames)) |
---|
579 | → (V → res W) → res (program B W) ≝ |
---|
580 | λA,B,V,W,p, transf_partial_function, transf_partial_variable. |
---|
581 | do fl ← map_partial … (*prefix_funct_name*) (transf_partial_function ?) (prog_funct ?? p); ?. |
---|
582 | (*CSC: interactive mode because of dependent types *) |
---|
583 | generalize in match (map_partial_preserves_first … transf_partial_variable (prog_vars … p)); |
---|
584 | cases (map_partial … transf_partial_variable (prog_vars … p)) |
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585 | [ #vl #EQ |
---|
586 | @(OK (program B W) (mk_program … vl … (prog_main … p))) |
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587 | <(map_fst … (EQ vl (refl …))) @fl |
---|
588 | | #err #_ @(Error … err)] |
---|
589 | qed. |
---|
590 | |
---|
591 | lemma transform_partial_program2_preserves_var_names : ∀A,B,V,W,p,tf,tv,p'. |
---|
592 | transform_partial_program2 A B V W p tf tv = OK ? p' → |
---|
593 | prog_var_names … p = prog_var_names … p'. |
---|
594 | #A #B #V #W * #vars #fns #main #tf #tv * #vars' #fns' #main' |
---|
595 | #T cases (bind_inversion ????? T) -T #vars1 * #Evars1 |
---|
596 | generalize in match (map_partial_preserves_first ?????); #MPPF |
---|
597 | lapply (refl ? (map_partial ??? tv vars)) |
---|
598 | cases (map_partial ?????) in ⊢ (???% → ?); |
---|
599 | [ 2: #m #M >M in MPPF ⊢ %; #MPPF #E normalize in E; destruct ] |
---|
600 | #vs #VS >VS in MPPF ⊢ %; #MPPF |
---|
601 | whd in ⊢ (??%% → ?); |
---|
602 | generalize in match (map_fst ???????); #MF |
---|
603 | #E destruct |
---|
604 | whd in ⊢ (??%%); @MF |
---|
605 | qed. |
---|
606 | |
---|
607 | |
---|
608 | (* |
---|
609 | Lemma transform_partial_program2_function: |
---|
610 | forall p tp i tf, |
---|
611 | transform_partial_program2 p = OK tp -> |
---|
612 | In (i, tf) tp.(prog_funct) -> |
---|
613 | exists f, In (i, f) p.(prog_funct) /\ transf_partial_function f = OK tf. |
---|
614 | Proof. |
---|
615 | intros. monadInv H. |
---|
616 | eapply In_map_partial; eauto. |
---|
617 | Qed. |
---|
618 | |
---|
619 | Lemma transform_partial_program2_variable: |
---|
620 | forall p tp i tv, |
---|
621 | transform_partial_program2 p = OK tp -> |
---|
622 | In (i, tv) tp.(prog_vars) -> |
---|
623 | exists v, In (i, v) p.(prog_vars) /\ transf_partial_variable v = OK tv. |
---|
624 | Proof. |
---|
625 | intros. monadInv H. |
---|
626 | eapply In_map_partial; eauto. |
---|
627 | Qed. |
---|
628 | |
---|
629 | Lemma transform_partial_program2_main: |
---|
630 | forall p tp, |
---|
631 | transform_partial_program2 p = OK tp -> |
---|
632 | tp.(prog_main) = p.(prog_main). |
---|
633 | Proof. |
---|
634 | intros. monadInv H. reflexivity. |
---|
635 | Qed. |
---|
636 | |
---|
637 | End TRANSF_PARTIAL_PROGRAM2. |
---|
638 | |
---|
639 | (** The following is a relational presentation of |
---|
640 | [transform_program_partial2]. Given relations between function |
---|
641 | definitions and between variable information, it defines a relation |
---|
642 | between programs stating that the two programs have the same shape |
---|
643 | (same global names, etc) and that identically-named function definitions |
---|
644 | are variable information are related. *) |
---|
645 | |
---|
646 | Section MATCH_PROGRAM. |
---|
647 | |
---|
648 | Variable A B V W: Type. |
---|
649 | Variable match_fundef: A -> B -> Prop. |
---|
650 | Variable match_varinfo: V -> W -> Prop. |
---|
651 | |
---|
652 | *) |
---|
653 | |
---|
654 | record matching : Type[1] ≝ { |
---|
655 | m_A : list ident → Type[0]; m_B : list ident → Type[0]; (* function types *) |
---|
656 | m_V : Type[0]; m_W : Type[0]; (* variable types *) |
---|
657 | match_fundef : ∀vs. m_A vs → m_B vs → Prop; |
---|
658 | match_varinfo : m_V → m_W → Prop |
---|
659 | }. |
---|
660 | |
---|
661 | (* When defining a matching between function entries, quietly enforce equality |
---|
662 | of the list of global variables (vs and vs'). *) |
---|
663 | |
---|
664 | inductive match_funct_entry (M:matching) : ∀vs,vs'. ident × (m_A M vs) → ident × (m_B M vs') → Prop ≝ |
---|
665 | | mfe_intro : ∀vs,id,f1,f2. match_fundef M vs f1 f2 → match_funct_entry M vs vs 〈id,f1〉 〈id,f2〉. |
---|
666 | |
---|
667 | (* but we'll need some care to usefully invert it *) |
---|
668 | |
---|
669 | definition mfe_cast_fn_type : ∀M,vs,vs'. ∀E:vs'=vs. m_B M vs' → m_B M vs ≝ |
---|
670 | λM,vs,vs',E. ?. |
---|
671 | >E #m @m |
---|
672 | qed. |
---|
673 | |
---|
674 | let rec match_funct_entry_inv (M:matching) |
---|
675 | (P:∀vs,id,f,id',f'. Prop) |
---|
676 | (H:∀vs,id,f,f'. match_fundef M vs f f' → P vs id f id f') |
---|
677 | vs id f id' f' |
---|
678 | (MFE:match_funct_entry M vs vs 〈id,f〉 〈id',f'〉) on MFE : P vs id f id' f' ≝ |
---|
679 | match MFE return λvs,vs',idf,idf',MFE. ∀E:vs'=vs. P vs (\fst idf) (\snd idf) (\fst idf') (mfe_cast_fn_type … E (\snd idf')) with |
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680 | [ mfe_intro vs id f1 f2 MF ⇒ ? |
---|
681 | ] (refl ??). |
---|
682 | #E >(K ?? E) @H @MF |
---|
683 | qed. |
---|
684 | |
---|
685 | (* and continue as before *) |
---|
686 | |
---|
687 | inductive match_var_entry (M:matching) : ident × region × (m_V M) → ident × region × (m_W M) → Prop ≝ |
---|
688 | | mve_intro : ∀id,r,v1,v2. match_varinfo M v1 v2 → match_var_entry M 〈id,r,v1〉 〈id,r,v2〉. |
---|
689 | |
---|
690 | lemma matching_vars : ∀M.∀p1:program (m_A M) (m_V M).∀p2:program (m_B M) (m_W M). |
---|
691 | All2 … (match_var_entry M) (prog_vars … p1) (prog_vars … p2) → |
---|
692 | prog_var_names … p1 = prog_var_names … p2. |
---|
693 | #M * #vs1 #mn1 #fn1 * #vs2 #mn2 #fn2 |
---|
694 | normalize generalize in match vs2; |
---|
695 | elim vs1 |
---|
696 | [ * [ // | #h #t * ] |
---|
697 | | * * #id1 #r1 #v1 #tl1 #IH * [ * ] |
---|
698 | * * #id2 #r2 #v2 #tl2 * * |
---|
699 | #id #r #v1' #v2' #_ #H |
---|
700 | whd in ⊢ (??%%); >(IH … H) % |
---|
701 | ] qed. |
---|
702 | |
---|
703 | record match_program (M:matching) (p1: program (m_A M) (m_V M)) (p2: program (m_B M) (m_W M)) : Prop ≝ { |
---|
704 | mp_vars : All2 … (match_var_entry M) (prog_vars … p1) (prog_vars … p2); |
---|
705 | mp_funct : All2 ?? … (match_funct_entry M (prog_var_names … p1) (prog_var_names … p2)) (prog_funct … p1) (prog_funct ??… p2); |
---|
706 | mp_main : prog_main … p1 = prog_main … p2 |
---|
707 | }. |
---|
708 | |
---|
709 | (* |
---|
710 | End MATCH_PROGRAM. |
---|
711 | *) |
---|
712 | |
---|
713 | (* Now show that all the transformations are instances of match_program. *) |
---|
714 | |
---|
715 | lemma transform_partial_program2_match: |
---|
716 | ∀A,B,V,W. |
---|
717 | ∀transf_partial_function: ∀vs. A vs -> res (B vs). |
---|
718 | ∀transf_partial_variable: V -> res W. |
---|
719 | ∀p: program A V. ∀tp: program B W. |
---|
720 | transform_partial_program2 … p transf_partial_function transf_partial_variable = OK ? tp → |
---|
721 | match_program (mk_matching A B V W |
---|
722 | (λvs,fd,tfd. transf_partial_function vs … fd = OK ? tfd) |
---|
723 | (λinfo,tinfo. transf_partial_variable info = OK ? tinfo)) |
---|
724 | p tp. |
---|
725 | #A #B #V #W #transf_partial_function #transf_partial_variable |
---|
726 | * #vars #main #functs * #vars' #main' #functs' |
---|
727 | #T cases (bind_inversion ????? T) -T #fl * #Efl |
---|
728 | generalize in match (map_partial_preserves_first ?????); #MPPF |
---|
729 | lapply (refl ? (map_partial ??? transf_partial_variable vars)) |
---|
730 | cases (map_partial ?????) in ⊢ (???% → ?); |
---|
731 | [ 2: #m #M >M in MPPF ⊢ %; #MPPF #E normalize in E; destruct ] |
---|
732 | #vs #VS >VS in MPPF ⊢ %; #MPPF |
---|
733 | whd in ⊢ (??%% → ?); |
---|
734 | generalize in match (map_fst ???????); #MF |
---|
735 | #E destruct |
---|
736 | % |
---|
737 | [ @(map_partial_All2 … VS) * /2/ |
---|
738 | | whd in match (prog_var_names ???); |
---|
739 | whd in match (prog_var_names ???); |
---|
740 | <MF @(map_partial_All2 … Efl) #id #f1 #f2 /2/ |
---|
741 | | // |
---|
742 | ] qed. |
---|
743 | |
---|
744 | lemma transform_partial_program_match: |
---|
745 | ∀A,B,V. |
---|
746 | ∀trans_partial_function: ∀vs. A vs → res (B vs). |
---|
747 | ∀p: program A V. ∀tp: program B V. |
---|
748 | transform_partial_program … p trans_partial_function = OK ? tp → |
---|
749 | match_program (mk_matching A B V V |
---|
750 | (λvs,fd,tfd. trans_partial_function vs fd = OK ? tfd) |
---|
751 | (λv,w. v = w)) |
---|
752 | p tp. |
---|
753 | #A #B #V #tpf |
---|
754 | * #vars #fns #main * #vars' #fns' #main' |
---|
755 | #TPP cases (bind_inversion ????? TPP) -TPP #fns'' * #MAP |
---|
756 | #E normalize in E; destruct |
---|
757 | % |
---|
758 | [ elim vars' // * * #id #r #v #tl #H % /2/ |
---|
759 | | @(map_partial_All2 … MAP) #i #f #f' #TPF % @TPF |
---|
760 | | // |
---|
761 | ] qed. |
---|
762 | |
---|
763 | lemma transform_program_match: |
---|
764 | ∀A,B,V. |
---|
765 | ∀trans_function: ∀vs. A vs → B vs. |
---|
766 | ∀p: program A V. |
---|
767 | match_program (mk_matching A B V V |
---|
768 | (λvs,fd,tfd. trans_function vs fd = tfd) |
---|
769 | (λv,w. v = w)) |
---|
770 | p (transform_program … p trans_function). |
---|
771 | #A #B #V #tf |
---|
772 | * #vars #fns #main |
---|
773 | % |
---|
774 | [ normalize elim vars // * * #id #r #v #tl #H % /2/ |
---|
775 | | whd in match (prog_var_names ???); |
---|
776 | whd in match (prog_vars ???); |
---|
777 | elim fns |
---|
778 | [ // |
---|
779 | | * #id #f #tl #IH % // % // |
---|
780 | ] |
---|
781 | | // |
---|
782 | ] qed. |
---|
783 | |
---|
784 | lemma transform_program_gen_match: |
---|
785 | ∀A,B,V,tag,gen. |
---|
786 | ∀trans_function: ∀vs. universe tag → A vs → B vs × (universe tag). |
---|
787 | ∀p: program A V. |
---|
788 | match_program (mk_matching A B V V |
---|
789 | (λvs,fd,tfd. ∃g,g'. trans_function vs g fd = 〈tfd,g'〉) |
---|
790 | (λv,w. v = w)) |
---|
791 | p (\fst (transform_program_gen … gen p trans_function)). |
---|
792 | #A #B #V #tag #gen #tf |
---|
793 | * #vars #fns #main |
---|
794 | % |
---|
795 | [ normalize elim vars // * * #id #r #v #tl #H % /2/ |
---|
796 | | whd in match (prog_var_names ???); |
---|
797 | whd in match (prog_vars ???); |
---|
798 | whd in match (transform_program_gen ???????); |
---|
799 | generalize in match gen; |
---|
800 | elim fns |
---|
801 | [ // |
---|
802 | | * #id #f #tl #IH #g |
---|
803 | cases (transf_program_gen_step tag (A ?) (B ?) g (tf ?) id f tl) |
---|
804 | #g' #E >E % /4/ |
---|
805 | ] |
---|
806 | | // |
---|
807 | ] qed. |
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808 | |
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809 | (* * * External functions *) |
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810 | |
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811 | (* * For most languages, the functions composing the program are either |
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812 | internal functions, defined within the language, or external functions |
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813 | (a.k.a. system calls) that emit an event when applied. We define |
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814 | a type for such functions and some generic transformation functions. *) |
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815 | |
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816 | record external_function : Type[0] ≝ { |
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817 | ef_id: ident; |
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818 | ef_sig: signature |
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819 | }. |
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820 | |
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821 | definition ExternalFunction ≝ external_function. |
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822 | definition external_function_tag ≝ ef_id. |
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823 | definition external_function_sig ≝ ef_sig. |
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824 | |
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825 | inductive fundef (F: Type[0]): Type[0] ≝ |
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826 | | Internal: F → fundef F |
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827 | | External: external_function → fundef F. |
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828 | |
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829 | (* Implicit Arguments External [F]. *) |
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830 | (* |
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831 | Section TRANSF_FUNDEF. |
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832 | |
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833 | Variable A B: Type. |
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834 | Variable transf: A -> B. |
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835 | *) |
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836 | definition transf_fundef : ∀A,B. (A→B) → fundef A → fundef B ≝ |
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837 | λA,B,transf,fd. |
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838 | match fd with |
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839 | [ Internal f ⇒ Internal ? (transf f) |
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840 | | External ef ⇒ External ? ef |
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841 | ]. |
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842 | |
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843 | (* |
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844 | End TRANSF_FUNDEF. |
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845 | |
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846 | Section TRANSF_PARTIAL_FUNDEF. |
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847 | |
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848 | Variable A B: Type. |
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849 | Variable transf_partial: A -> res B. |
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850 | *) |
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851 | |
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852 | definition transf_partial_fundef : ∀A,B. (A → res B) → fundef A → res (fundef B) ≝ |
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853 | λA,B,transf_partial,fd. |
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854 | match fd with |
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855 | [ Internal f ⇒ do f' ← transf_partial f; OK ? (Internal ? f') |
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856 | | External ef ⇒ OK ? (External ? ef) |
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857 | ]. |
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858 | (* |
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859 | End TRANSF_PARTIAL_FUNDEF. |
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860 | *) |
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861 | |
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862 | |
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863 | |
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864 | (* Partially merged stuff derived from the prototype cerco compiler. *) |
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865 | |
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866 | (* |
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867 | definition bool_to_Prop ≝ |
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868 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
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869 | |
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870 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
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871 | *) |
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872 | |
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873 | (* dpm: should go to standard library *) |
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874 | let rec member (i: ident) (eq_i: ident → ident → bool) |
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875 | (g: list ident) on g: Prop ≝ |
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876 | match g with |
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877 | [ nil ⇒ False |
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878 | | cons hd tl ⇒ |
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879 | bool_to_Prop (eq_i hd i) ∨ member i eq_i tl |
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880 | ]. |
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