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/BitVector.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 | (* XXX backend is currently using untagged words as identifiers. *) |
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41 | definition Identifier ≝ Word. |
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42 | |
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43 | definition Immediate ≝ nat. (* XXX is this the best place to put this? *) |
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44 | |
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45 | (* Memory spaces *) |
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46 | |
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47 | inductive region : Type[0] ≝ |
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48 | | Any : region |
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49 | | Data : region |
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50 | | IData : region |
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51 | | PData : region |
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52 | | XData : region |
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53 | | Code : region. |
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54 | |
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55 | definition eq_region : region → region → bool ≝ |
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56 | λr1,r2. |
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57 | match r1 with |
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58 | [ Any ⇒ match r2 with [ Any ⇒ true | _ ⇒ false ] |
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59 | | Data ⇒ match r2 with [ Data ⇒ true | _ ⇒ false ] |
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60 | | IData ⇒ match r2 with [ IData ⇒ true | _ ⇒ false ] |
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61 | | PData ⇒ match r2 with [ PData ⇒ true | _ ⇒ false ] |
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62 | | XData ⇒ match r2 with [ XData ⇒ true | _ ⇒ false ] |
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63 | | Code ⇒ match r2 with [ Code ⇒ true | _ ⇒ false ] |
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64 | ]. |
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65 | |
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66 | lemma eq_region_elim : ∀P:bool → Type[0]. ∀r1,r2. |
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67 | (r1 = r2 → P true) → (r1 ≠ r2 → P false) → |
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68 | P (eq_region r1 r2). |
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69 | #P #r1 #r2 cases r1; cases r2; #Ptrue #Pfalse |
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70 | try ( @Ptrue // ) |
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71 | @Pfalse % #E destruct; |
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72 | qed. |
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73 | |
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74 | definition eq_region_dec : ∀r1,r2:region. (r1=r2)+(r1≠r2). |
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75 | #r1 #r2 @(eq_region_elim ? r1 r2) /2/; qed. |
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76 | |
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77 | definition size_pointer : region → nat ≝ |
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78 | λsp. match sp with [ Data ⇒ 1 | IData ⇒ 1 | PData ⇒ 1 | XData ⇒ 2 | Code ⇒ 2 | Any ⇒ 3 ]. |
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79 | |
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80 | (* * The intermediate languages are weakly typed, using only three types: |
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81 | [ASTint] for integers, [ASTptr] for pointers, and [ASTfloat] for floating-point |
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82 | numbers. *) |
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83 | |
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84 | inductive typ : Type[0] ≝ |
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85 | | ASTint : typ |
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86 | | ASTptr : region → typ |
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87 | | ASTfloat : typ. |
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88 | |
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89 | (* XXX aliases *) |
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90 | definition SigType ≝ typ. |
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91 | definition SigType_Int ≝ ASTint. |
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92 | (* |
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93 | | SigType_Float: SigType |
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94 | *) |
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95 | definition SigType_Ptr ≝ ASTptr Any. |
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96 | |
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97 | |
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98 | (* FIXME: ASTint size is not 1 for Clight32 *) |
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99 | definition typesize : typ → nat ≝ λty. |
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100 | match ty with [ ASTint ⇒ 1 | ASTptr r ⇒ size_pointer r | ASTfloat ⇒ 8 ]. |
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101 | |
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102 | lemma typesize_pos: ∀ty. typesize ty > 0. |
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103 | #ty cases ty [ 2: * ] /2/ qed. |
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104 | |
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105 | lemma typ_eq: ∀t1,t2: typ. (t1=t2) + (t1≠t2). |
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106 | #t1 #t2 cases t1 try *; cases t2 try *; /2/ %2 @nmk #H destruct; qed. |
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107 | |
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108 | lemma opt_typ_eq: ∀t1,t2: option typ. (t1=t2) + (t1≠t2). |
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109 | #t1 #t2 cases t1 cases t2 |
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110 | [ %1 @refl |
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111 | | 2,3: #ty %2 % #H destruct |
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112 | | #ty1 #ty2 elim (typ_eq ty1 ty2) #E [ %1 >E @refl | %2 % #E' destruct cases E /2/ |
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113 | ] |
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114 | qed. |
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115 | |
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116 | (* * Additionally, function definitions and function calls are annotated |
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117 | by function signatures indicating the number and types of arguments, |
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118 | as well as the type of the returned value if any. These signatures |
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119 | are used in particular to determine appropriate calling conventions |
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120 | for the function. *) |
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121 | |
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122 | record signature : Type[0] ≝ { |
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123 | sig_args: list typ; |
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124 | sig_res: option typ |
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125 | }. |
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126 | |
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127 | (* XXX aliases *) |
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128 | definition Signature ≝ signature. |
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129 | definition signature_args ≝ sig_args. |
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130 | definition signature_return ≝ sig_res. |
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131 | |
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132 | definition proj_sig_res : signature → typ ≝ λs. |
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133 | match sig_res s with |
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134 | [ None ⇒ ASTint |
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135 | | Some t ⇒ t |
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136 | ]. |
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137 | |
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138 | (* * Memory accesses (load and store instructions) are annotated by |
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139 | a ``memory chunk'' indicating the type, size and signedness of the |
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140 | chunk of memory being accessed. *) |
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141 | |
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142 | inductive memory_chunk : Type[0] ≝ |
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143 | | Mint8signed : memory_chunk (*r 8-bit signed integer *) |
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144 | | Mint8unsigned : memory_chunk (*r 8-bit unsigned integer *) |
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145 | | Mint16signed : memory_chunk (*r 16-bit signed integer *) |
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146 | | Mint16unsigned : memory_chunk (*r 16-bit unsigned integer *) |
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147 | | Mint32 : memory_chunk (*r 32-bit integer, or pointer *) |
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148 | | Mfloat32 : memory_chunk (*r 32-bit single-precision float *) |
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149 | | Mfloat64 : memory_chunk (*r 64-bit double-precision float *) |
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150 | | Mpointer : region → memory_chunk. (* pointer addressing given region *) |
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151 | |
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152 | (* * Initialization data for global variables. *) |
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153 | |
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154 | inductive init_data: Type[0] ≝ |
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155 | | Init_int8: int → init_data |
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156 | | Init_int16: int → init_data |
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157 | | Init_int32: int → init_data |
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158 | | Init_float32: float → init_data |
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159 | | Init_float64: float → init_data |
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160 | | Init_space: nat → init_data |
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161 | | Init_null: region → init_data |
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162 | | Init_addrof: region → ident → int → init_data. (*r address of symbol + offset *) |
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163 | |
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164 | (* * Whole programs consist of: |
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165 | - a collection of function definitions (name and description); |
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166 | - the name of the ``main'' function that serves as entry point in the program; |
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167 | - a collection of global variable declarations, consisting of |
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168 | a name, initialization data, and additional information. |
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169 | |
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170 | The type of function descriptions and that of additional information |
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171 | for variables vary among the various intermediate languages and are |
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172 | taken as parameters to the [program] type. The other parts of whole |
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173 | programs are common to all languages. *) |
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174 | |
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175 | record program (F,V: Type[0]) : Type[0] := { |
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176 | prog_funct: list (ident × F); |
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177 | prog_main: ident; |
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178 | prog_vars: list (ident × (list init_data) × region × V) |
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179 | }. |
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180 | |
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181 | definition prog_funct_names ≝ λF,V: Type[0]. λp: program F V. |
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182 | map ?? (fst ident F) (prog_funct ?? p). |
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183 | |
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184 | definition prog_var_names ≝ λF,V: Type[0]. λp: program F V. |
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185 | map ?? (λx: ident × (list init_data) × region × V. fst ?? (fst ?? (fst ?? x))) (prog_vars ?? p). |
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186 | |
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187 | (* * * Generic transformations over programs *) |
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188 | |
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189 | (* * We now define a general iterator over programs that applies a given |
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190 | code transformation function to all function descriptions and leaves |
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191 | the other parts of the program unchanged. *) |
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192 | (* |
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193 | Section TRANSF_PROGRAM. |
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194 | |
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195 | Variable A B V: Type. |
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196 | Variable transf: A -> B. |
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197 | *) |
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198 | |
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199 | definition transf_program : ∀A,B. (A → B) → list (ident × A) → list (ident × B) ≝ |
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200 | λA,B,transf,l. |
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201 | map ?? (λid_fn. 〈fst ?? id_fn, transf (snd ?? id_fn)〉) l. |
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202 | |
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203 | definition transform_program : ∀A,B,V. (A → B) → program A V → program B V ≝ |
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204 | λA,B,V,transf,p. |
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205 | mk_program B V |
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206 | (transf_program ?? transf (prog_funct A V p)) |
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207 | (prog_main A V p) |
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208 | (prog_vars A V p). |
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209 | (* |
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210 | lemma transform_program_function: |
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211 | ∀A,B,V,transf,p,i,tf. |
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212 | in_list ? 〈i, tf〉 (prog_funct ?? (transform_program A B V transf p)) → |
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213 | ∃f. in_list ? 〈i, f〉 (prog_funct ?? p) ∧ transf f = tf. |
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214 | normalize; #A #B #V #transf #p #i #tf #H elim (list_in_map_inv ????? H); |
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215 | #x elim x; #i' #tf' *; #e #H destruct; %{tf'} /2/; |
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216 | qed. |
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217 | *) |
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218 | (* |
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219 | End TRANSF_PROGRAM. |
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220 | |
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221 | (** The following is a variant of [transform_program] where the |
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222 | code transformation function can fail and therefore returns an |
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223 | option type. *) |
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224 | |
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225 | Open Local Scope error_monad_scope. |
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226 | Open Local Scope string_scope. |
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227 | |
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228 | Section MAP_PARTIAL. |
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229 | |
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230 | Variable A B C: Type. |
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231 | Variable prefix_errmsg: A -> errmsg. |
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232 | Variable f: B -> res C. |
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233 | *) |
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234 | definition map_partial : ∀A,B,C:Type[0]. (B → res C) → |
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235 | list (A × B) → res (list (A × C)) ≝ |
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236 | λA,B,C,f. mmap ?? (λab. let 〈a,b〉 ≝ ab in do c ← f b; OK ? 〈a,c〉). |
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237 | (* |
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238 | Fixpoint map_partial (l: list (A * B)) : res (list (A * C)) := |
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239 | match l with |
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240 | | nil => OK nil |
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241 | | (a, b) :: rem => |
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242 | match f b with |
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243 | | Error msg => Error (prefix_errmsg a ++ msg)%list |
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244 | | OK c => |
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245 | do rem' <- map_partial rem; |
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246 | OK ((a, c) :: rem') |
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247 | end |
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248 | end. |
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249 | |
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250 | Remark In_map_partial: |
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251 | forall l l' a c, |
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252 | map_partial l = OK l' -> |
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253 | In (a, c) l' -> |
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254 | exists b, In (a, b) l /\ f b = OK c. |
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255 | Proof. |
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256 | induction l; simpl. |
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257 | intros. inv H. elim H0. |
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258 | intros until c. destruct a as [a1 b1]. |
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259 | caseEq (f b1); try congruence. |
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260 | intro c1; intros. monadInv H0. |
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261 | elim H1; intro. inv H0. exists b1; auto. |
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262 | exploit IHl; eauto. intros [b [P Q]]. exists b; auto. |
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263 | Qed. |
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264 | |
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265 | Remark map_partial_forall2: |
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266 | forall l l', |
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267 | map_partial l = OK l' -> |
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268 | list_forall2 |
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269 | (fun (a_b: A * B) (a_c: A * C) => |
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270 | fst a_b = fst a_c /\ f (snd a_b) = OK (snd a_c)) |
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271 | l l'. |
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272 | Proof. |
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273 | induction l; simpl. |
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274 | intros. inv H. constructor. |
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275 | intro l'. destruct a as [a b]. |
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276 | caseEq (f b). 2: congruence. intro c; intros. monadInv H0. |
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277 | constructor. simpl. auto. auto. |
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278 | Qed. |
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279 | |
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280 | End MAP_PARTIAL. |
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281 | |
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282 | Remark map_partial_total: |
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283 | forall (A B C: Type) (prefix: A -> errmsg) (f: B -> C) (l: list (A * B)), |
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284 | map_partial prefix (fun b => OK (f b)) l = |
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285 | OK (List.map (fun a_b => (fst a_b, f (snd a_b))) l). |
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286 | Proof. |
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287 | induction l; simpl. |
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288 | auto. |
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289 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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290 | Qed. |
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291 | |
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292 | Remark map_partial_identity: |
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293 | forall (A B: Type) (prefix: A -> errmsg) (l: list (A * B)), |
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294 | map_partial prefix (fun b => OK b) l = OK l. |
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295 | Proof. |
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296 | induction l; simpl. |
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297 | auto. |
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298 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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299 | Qed. |
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300 | |
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301 | Section TRANSF_PARTIAL_PROGRAM. |
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302 | |
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303 | Variable A B V: Type. |
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304 | Variable transf_partial: A -> res B. |
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305 | |
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306 | Definition prefix_funct_name (id: ident) : errmsg := |
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307 | MSG "In function " :: CTX id :: MSG ": " :: nil. |
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308 | *) |
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309 | definition transform_partial_program : ∀A,B,V. (A → res B) → program A V → res (program B V) ≝ |
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310 | λA,B,V,transf_partial,p. |
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311 | do fl ← map_partial … (*prefix_funct_name*) transf_partial (prog_funct ?? p); |
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312 | OK ? (mk_program … fl (prog_main ?? p) (prog_vars ?? p)). |
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313 | (* |
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314 | Lemma transform_partial_program_function: |
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315 | forall p tp i tf, |
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316 | transform_partial_program p = OK tp -> |
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317 | In (i, tf) tp.(prog_funct) -> |
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318 | exists f, In (i, f) p.(prog_funct) /\ transf_partial f = OK tf. |
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319 | Proof. |
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320 | intros. monadInv H. simpl in H0. |
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321 | eapply In_map_partial; eauto. |
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322 | Qed. |
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323 | |
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324 | Lemma transform_partial_program_main: |
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325 | forall p tp, |
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326 | transform_partial_program p = OK tp -> |
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327 | tp.(prog_main) = p.(prog_main). |
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328 | Proof. |
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329 | intros. monadInv H. reflexivity. |
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330 | Qed. |
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331 | |
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332 | Lemma transform_partial_program_vars: |
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333 | forall p tp, |
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334 | transform_partial_program p = OK tp -> |
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335 | tp.(prog_vars) = p.(prog_vars). |
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336 | Proof. |
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337 | intros. monadInv H. reflexivity. |
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338 | Qed. |
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339 | |
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340 | End TRANSF_PARTIAL_PROGRAM. |
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341 | |
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342 | (** The following is a variant of [transform_program_partial] where |
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343 | both the program functions and the additional variable information |
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344 | are transformed by functions that can fail. *) |
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345 | |
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346 | Section TRANSF_PARTIAL_PROGRAM2. |
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347 | |
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348 | Variable A B V W: Type. |
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349 | Variable transf_partial_function: A -> res B. |
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350 | Variable transf_partial_variable: V -> res W. |
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351 | |
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352 | Definition prefix_var_name (id_init: ident * list init_data) : errmsg := |
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353 | MSG "In global variable " :: CTX (fst id_init) :: MSG ": " :: nil. |
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354 | *) |
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355 | definition transform_partial_program2 : ∀A,B,V,W. (A → res B) → (V → res W) → |
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356 | program A V → res (program B W) ≝ |
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357 | λA,B,V,W, transf_partial_function, transf_partial_variable, p. |
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358 | do fl ← map_partial … (*prefix_funct_name*) transf_partial_function (prog_funct ?? p); |
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359 | do vl ← map_partial … (*prefix_var_name*) transf_partial_variable (prog_vars ?? p); |
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360 | OK ? (mk_program … fl (prog_main ?? p) vl). |
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361 | (* |
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362 | Lemma transform_partial_program2_function: |
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363 | forall p tp i tf, |
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364 | transform_partial_program2 p = OK tp -> |
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365 | In (i, tf) tp.(prog_funct) -> |
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366 | exists f, In (i, f) p.(prog_funct) /\ transf_partial_function f = OK tf. |
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367 | Proof. |
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368 | intros. monadInv H. |
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369 | eapply In_map_partial; eauto. |
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370 | Qed. |
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371 | |
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372 | Lemma transform_partial_program2_variable: |
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373 | forall p tp i tv, |
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374 | transform_partial_program2 p = OK tp -> |
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375 | In (i, tv) tp.(prog_vars) -> |
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376 | exists v, In (i, v) p.(prog_vars) /\ transf_partial_variable v = OK tv. |
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377 | Proof. |
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378 | intros. monadInv H. |
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379 | eapply In_map_partial; eauto. |
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380 | Qed. |
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381 | |
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382 | Lemma transform_partial_program2_main: |
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383 | forall p tp, |
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384 | transform_partial_program2 p = OK tp -> |
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385 | tp.(prog_main) = p.(prog_main). |
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386 | Proof. |
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387 | intros. monadInv H. reflexivity. |
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388 | Qed. |
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389 | |
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390 | End TRANSF_PARTIAL_PROGRAM2. |
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391 | |
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392 | (** The following is a relational presentation of |
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393 | [transform_program_partial2]. Given relations between function |
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394 | definitions and between variable information, it defines a relation |
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395 | between programs stating that the two programs have the same shape |
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396 | (same global names, etc) and that identically-named function definitions |
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397 | are variable information are related. *) |
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398 | |
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399 | Section MATCH_PROGRAM. |
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400 | |
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401 | Variable A B V W: Type. |
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402 | Variable match_fundef: A -> B -> Prop. |
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403 | Variable match_varinfo: V -> W -> Prop. |
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404 | |
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405 | Definition match_funct_entry (x1: ident * A) (x2: ident * B) := |
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406 | match x1, x2 with |
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407 | | (id1, fn1), (id2, fn2) => id1 = id2 /\ match_fundef fn1 fn2 |
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408 | end. |
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409 | |
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410 | Definition match_var_entry (x1: ident * list init_data * V) (x2: ident * list init_data * W) := |
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411 | match x1, x2 with |
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412 | | (id1, init1, info1), (id2, init2, info2) => id1 = id2 /\ init1 = init2 /\ match_varinfo info1 info2 |
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413 | end. |
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414 | |
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415 | Definition match_program (p1: program A V) (p2: program B W) : Prop := |
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416 | list_forall2 match_funct_entry p1.(prog_funct) p2.(prog_funct) |
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417 | /\ p1.(prog_main) = p2.(prog_main) |
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418 | /\ list_forall2 match_var_entry p1.(prog_vars) p2.(prog_vars). |
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419 | |
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420 | End MATCH_PROGRAM. |
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421 | |
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422 | Remark transform_partial_program2_match: |
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423 | forall (A B V W: Type) |
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424 | (transf_partial_function: A -> res B) |
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425 | (transf_partial_variable: V -> res W) |
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426 | (p: program A V) (tp: program B W), |
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427 | transform_partial_program2 transf_partial_function transf_partial_variable p = OK tp -> |
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428 | match_program |
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429 | (fun fd tfd => transf_partial_function fd = OK tfd) |
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430 | (fun info tinfo => transf_partial_variable info = OK tinfo) |
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431 | p tp. |
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432 | Proof. |
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433 | intros. monadInv H. split. |
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434 | apply list_forall2_imply with |
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435 | (fun (ab: ident * A) (ac: ident * B) => |
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436 | fst ab = fst ac /\ transf_partial_function (snd ab) = OK (snd ac)). |
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437 | eapply map_partial_forall2. eauto. |
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438 | intros. destruct v1; destruct v2; simpl in *. auto. |
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439 | split. auto. |
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440 | apply list_forall2_imply with |
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441 | (fun (ab: ident * list init_data * V) (ac: ident * list init_data * W) => |
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442 | fst ab = fst ac /\ transf_partial_variable (snd ab) = OK (snd ac)). |
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443 | eapply map_partial_forall2. eauto. |
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444 | intros. destruct v1; destruct v2; simpl in *. destruct p0; destruct p1. intuition congruence. |
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445 | Qed. |
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446 | *) |
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447 | (* * * External functions *) |
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448 | |
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449 | (* * For most languages, the functions composing the program are either |
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450 | internal functions, defined within the language, or external functions |
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451 | (a.k.a. system calls) that emit an event when applied. We define |
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452 | a type for such functions and some generic transformation functions. *) |
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453 | |
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454 | record external_function : Type[0] ≝ { |
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455 | ef_id: ident; |
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456 | ef_sig: signature |
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457 | }. |
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458 | |
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459 | definition ExternalFunction ≝ external_function. |
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460 | definition external_function_tag ≝ ef_id. |
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461 | definition external_function_sig ≝ ef_sig. |
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462 | |
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463 | inductive fundef (F: Type[0]): Type[0] ≝ |
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464 | | Internal: F → fundef F |
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465 | | External: external_function → fundef F. |
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466 | |
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467 | (* Implicit Arguments External [F]. *) |
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468 | (* |
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469 | Section TRANSF_FUNDEF. |
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470 | |
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471 | Variable A B: Type. |
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472 | Variable transf: A -> B. |
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473 | *) |
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474 | definition transf_fundef : ∀A,B. (A→B) → fundef A → fundef B ≝ |
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475 | λA,B,transf,fd. |
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476 | match fd with |
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477 | [ Internal f ⇒ Internal ? (transf f) |
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478 | | External ef ⇒ External ? ef |
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479 | ]. |
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480 | |
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481 | (* |
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482 | End TRANSF_FUNDEF. |
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483 | |
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484 | Section TRANSF_PARTIAL_FUNDEF. |
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485 | |
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486 | Variable A B: Type. |
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487 | Variable transf_partial: A -> res B. |
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488 | *) |
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489 | |
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490 | definition transf_partial_fundef : ∀A,B. (A → res B) → fundef A → res (fundef B) ≝ |
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491 | λA,B,transf_partial,fd. |
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492 | match fd with |
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493 | [ Internal f ⇒ do f' ← transf_partial f; OK ? (Internal ? f') |
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494 | | External ef ⇒ OK ? (External ? ef) |
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495 | ]. |
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496 | (* |
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497 | End TRANSF_PARTIAL_FUNDEF. |
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498 | *) |
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499 | |
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500 | |
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501 | |
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502 | (* Partially merged stuff derived from the prototype cerco compiler. *) |
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503 | |
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504 | definition bool_to_Prop ≝ |
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505 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
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506 | |
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507 | (* dpm: should go to standard library *) |
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508 | let rec member (i: ident) (eq_i: ident → ident → bool) |
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509 | (g: list ident) on g: Prop ≝ |
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510 | match g with |
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511 | [ nil ⇒ False |
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512 | | cons hd tl ⇒ |
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513 | bool_to_Prop (eq_i hd i) ∨ member i eq_i tl |
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514 | ]. |
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515 | |
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516 | (* TODO: merge in comparison definitions from Integers.ma. *) |
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517 | inductive Compare: Type[0] ≝ |
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518 | Compare_Equal: Compare |
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519 | | Compare_Less: Compare |
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520 | | Compare_Greater: Compare |
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521 | | Compare_LessEqual: Compare |
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522 | | Compare_GreaterEqual: Compare. |
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523 | |
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524 | (* XXX unused definitions |
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525 | inductive Cast: Type[0] ≝ |
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526 | Cast_Int: Byte → Cast |
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527 | | Cast_AddrSymbol: Identifier → Cast |
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528 | | Cast_StackOffset: Immediate → Cast. |
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529 | |
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530 | inductive Data: Type[0] ≝ |
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531 | Data_Reserve: nat → Data |
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532 | | Data_Int8: Byte → Data |
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533 | | Data_Int16: Word → Data. |
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534 | |
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535 | inductive DataSize: Type[0] ≝ |
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536 | DataSize_Byte: DataSize |
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537 | | DataSize_HalfWord: DataSize |
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538 | | DataSize_Word: DataSize. |
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539 | *) |
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540 | |
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541 | definition Trace ≝ list Identifier. |
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