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 "extralib.ma". |
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21 | include "Integers.ma". |
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22 | include "Floats.ma". |
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23 | include "binary/positive.ma". |
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24 | |
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25 | (* * * Syntactic elements *) |
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26 | |
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27 | (* * Identifiers (names of local variables, of global symbols and functions, |
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28 | etc) are represented by the type [positive] of positive integers. *) |
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29 | |
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30 | definition ident ≝ Pos. |
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31 | |
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32 | definition ident_eq : ∀x,y:ident. (x=y) + (x≠y). |
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33 | #x #y lapply (pos_compare_to_Prop x y); cases (pos_compare x y); |
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34 | [ #H %2 /2/; | #H %1 //; | #H %2 /2/ ] qed. |
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35 | |
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36 | (* |
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37 | (* XXX: we use nats for now, but if in future we use binary like compcert |
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38 | then the maps will be easier to define. *) |
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39 | definition ident ≝ nat. |
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40 | definition ident_eq : ∀x,y:ident. (x=y) + (x≠y). |
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41 | #x elim x; |
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42 | [ #y cases y; /3/; |
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43 | | #x' #IH #y cases y; |
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44 | [ %{2} % #H destruct |
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45 | | #y' elim (IH y'); |
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46 | [ #e destruct; /2/ |
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47 | | #ne %{2} /2/; |
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48 | ] |
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49 | ] |
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50 | ] qed. |
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51 | *) |
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52 | (* * The intermediate languages are weakly typed, using only two types: |
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53 | [Tint] for integers and pointers, and [Tfloat] for floating-point |
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54 | numbers. *) |
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55 | |
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56 | inductive typ : Type[0] ≝ |
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57 | | ASTint : typ |
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58 | | ASTfloat : typ. |
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59 | |
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60 | definition typesize : typ → Z ≝ λty. |
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61 | match ty return λ_.Z with [ ASTint ⇒ 4 | ASTfloat ⇒ 8 ]. |
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62 | |
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63 | lemma typesize_pos: ∀ty. typesize ty > 0. |
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64 | #ty cases ty; //; qed. |
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65 | |
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66 | lemma typ_eq: ∀t1,t2: typ. (t1=t2) + (t1≠t2). |
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67 | #t1 #t2 cases t1;cases t2;/2/; %2 @nmk #H destruct; qed. |
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68 | |
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69 | lemma opt_typ_eq: ∀t1,t2: option typ. (t1=t2) + (t1≠t2). |
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70 | #t1 #t2 cases t1;cases t2;/2/; |
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71 | [ 1,2: #ty %2 @nmk #H destruct; |
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72 | | #ty1 #ty2 elim (typ_eq ty1 ty2); /2/; #neq %2 @nmk |
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73 | #H destruct; /2/; |
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74 | ] qed. |
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75 | |
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76 | (* * Additionally, function definitions and function calls are annotated |
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77 | by function signatures indicating the number and types of arguments, |
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78 | as well as the type of the returned value if any. These signatures |
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79 | are used in particular to determine appropriate calling conventions |
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80 | for the function. *) |
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81 | |
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82 | record signature : Type[0] ≝ { |
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83 | sig_args: list typ; |
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84 | sig_res: option typ |
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85 | }. |
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86 | |
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87 | definition proj_sig_res : signature → typ ≝ λs. |
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88 | match sig_res s with |
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89 | [ None ⇒ ASTint |
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90 | | Some t ⇒ t |
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91 | ]. |
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92 | |
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93 | (* Memory spaces *) |
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94 | |
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95 | inductive region : Type[0] ≝ |
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96 | | Any : region |
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97 | | Data : region |
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98 | | IData : region |
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99 | | PData : region |
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100 | | XData : region |
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101 | | Code : region. |
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102 | |
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103 | definition eq_region : region → region → bool ≝ |
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104 | λr1,r2. |
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105 | match r1 with |
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106 | [ Any ⇒ match r2 with [ Any ⇒ true | _ ⇒ false ] |
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107 | | Data ⇒ match r2 with [ Data ⇒ true | _ ⇒ false ] |
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108 | | IData ⇒ match r2 with [ IData ⇒ true | _ ⇒ false ] |
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109 | | PData ⇒ match r2 with [ PData ⇒ true | _ ⇒ false ] |
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110 | | XData ⇒ match r2 with [ XData ⇒ true | _ ⇒ false ] |
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111 | | Code ⇒ match r2 with [ Code ⇒ true | _ ⇒ false ] |
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112 | ]. |
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113 | |
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114 | lemma eq_region_elim : ∀P:bool → Type[0]. ∀r1,r2. |
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115 | (r1 = r2 → P true) → (r1 ≠ r2 → P false) → |
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116 | P (eq_region r1 r2). |
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117 | #P #r1 #r2 cases r1; cases r2; #Ptrue #Pfalse |
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118 | try ( @Ptrue // ) |
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119 | @Pfalse % #E destruct; |
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120 | qed. |
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121 | |
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122 | definition eq_region_dec : ∀r1,r2:region. (r1=r2)+(r1≠r2). |
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123 | #r1 #r2 @(eq_region_elim ? r1 r2) /2/; qed. |
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124 | |
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125 | (* * Memory accesses (load and store instructions) are annotated by |
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126 | a ``memory chunk'' indicating the type, size and signedness of the |
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127 | chunk of memory being accessed. *) |
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128 | |
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129 | inductive memory_chunk : Type[0] ≝ |
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130 | | Mint8signed : memory_chunk (*r 8-bit signed integer *) |
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131 | | Mint8unsigned : memory_chunk (*r 8-bit unsigned integer *) |
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132 | | Mint16signed : memory_chunk (*r 16-bit signed integer *) |
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133 | | Mint16unsigned : memory_chunk (*r 16-bit unsigned integer *) |
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134 | | Mint32 : memory_chunk (*r 32-bit integer, or pointer *) |
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135 | | Mfloat32 : memory_chunk (*r 32-bit single-precision float *) |
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136 | | Mfloat64 : memory_chunk (*r 64-bit double-precision float *) |
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137 | | Mpointer : region → memory_chunk. (* pointer addressing given region *) |
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138 | |
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139 | (* * Initialization data for global variables. *) |
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140 | |
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141 | inductive init_data: Type[0] ≝ |
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142 | | Init_int8: int → init_data |
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143 | | Init_int16: int → init_data |
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144 | | Init_int32: int → init_data |
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145 | | Init_float32: float → init_data |
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146 | | Init_float64: float → init_data |
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147 | | Init_space: Z → init_data |
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148 | | Init_null: region → init_data |
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149 | | Init_addrof: region → ident → int → init_data. (*r address of symbol + offset *) |
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150 | |
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151 | (* * Whole programs consist of: |
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152 | - a collection of function definitions (name and description); |
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153 | - the name of the ``main'' function that serves as entry point in the program; |
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154 | - a collection of global variable declarations, consisting of |
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155 | a name, initialization data, and additional information. |
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156 | |
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157 | The type of function descriptions and that of additional information |
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158 | for variables vary among the various intermediate languages and are |
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159 | taken as parameters to the [program] type. The other parts of whole |
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160 | programs are common to all languages. *) |
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161 | |
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162 | record program (F,V: Type[0]) : Type[0] := { |
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163 | prog_funct: list (ident × F); |
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164 | prog_main: ident; |
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165 | prog_vars: list (ident × (list init_data) × region × V) |
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166 | }. |
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167 | |
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168 | definition prog_funct_names ≝ λF,V: Type[0]. λp: program F V. |
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169 | map ?? (fst ident F) (prog_funct ?? p). |
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170 | |
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171 | definition prog_var_names ≝ λF,V: Type[0]. λp: program F V. |
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172 | map ?? (λx: ident × (list init_data) × region × V. fst ?? (fst ?? (fst ?? x))) (prog_vars ?? p). |
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173 | (* |
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174 | (** * Generic transformations over programs *) |
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175 | |
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176 | (** We now define a general iterator over programs that applies a given |
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177 | code transformation function to all function descriptions and leaves |
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178 | the other parts of the program unchanged. *) |
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179 | |
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180 | Section TRANSF_PROGRAM. |
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181 | |
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182 | Variable A B V: Type. |
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183 | Variable transf: A -> B. |
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184 | *) |
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185 | |
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186 | definition transf_program : ∀A,B. (A → B) → list (ident × A) → list (ident × B) ≝ |
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187 | λA,B,transf,l. |
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188 | map ?? (λid_fn. 〈fst ?? id_fn, transf (snd ?? id_fn)〉) l. |
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189 | |
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190 | definition transform_program : ∀A,B,V. (A → B) → program A V → program B V ≝ |
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191 | λA,B,V,transf,p. |
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192 | mk_program B V |
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193 | (transf_program ?? transf (prog_funct A V p)) |
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194 | (prog_main A V p) |
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195 | (prog_vars A V p). |
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196 | (* |
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197 | lemma transform_program_function: |
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198 | ∀A,B,V,transf,p,i,tf. |
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199 | in_list ? 〈i, tf〉 (prog_funct ?? (transform_program A B V transf p)) → |
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200 | ∃f. in_list ? 〈i, f〉 (prog_funct ?? p) ∧ transf f = tf. |
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201 | normalize; #A #B #V #transf #p #i #tf #H elim (list_in_map_inv ????? H); |
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202 | #x elim x; #i' #tf' *; #e #H destruct; %{tf'} /2/; |
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203 | qed. |
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204 | *) |
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205 | (* |
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206 | End TRANSF_PROGRAM. |
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207 | |
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208 | (** The following is a variant of [transform_program] where the |
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209 | code transformation function can fail and therefore returns an |
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210 | option type. *) |
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211 | |
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212 | Open Local Scope error_monad_scope. |
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213 | Open Local Scope string_scope. |
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214 | |
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215 | Section MAP_PARTIAL. |
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216 | |
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217 | Variable A B C: Type. |
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218 | Variable prefix_errmsg: A -> errmsg. |
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219 | Variable f: B -> res C. |
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220 | |
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221 | Fixpoint map_partial (l: list (A * B)) : res (list (A * C)) := |
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222 | match l with |
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223 | | nil => OK nil |
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224 | | (a, b) :: rem => |
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225 | match f b with |
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226 | | Error msg => Error (prefix_errmsg a ++ msg)%list |
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227 | | OK c => |
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228 | do rem' <- map_partial rem; |
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229 | OK ((a, c) :: rem') |
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230 | end |
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231 | end. |
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232 | |
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233 | Remark In_map_partial: |
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234 | forall l l' a c, |
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235 | map_partial l = OK l' -> |
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236 | In (a, c) l' -> |
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237 | exists b, In (a, b) l /\ f b = OK c. |
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238 | Proof. |
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239 | induction l; simpl. |
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240 | intros. inv H. elim H0. |
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241 | intros until c. destruct a as [a1 b1]. |
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242 | caseEq (f b1); try congruence. |
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243 | intro c1; intros. monadInv H0. |
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244 | elim H1; intro. inv H0. exists b1; auto. |
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245 | exploit IHl; eauto. intros [b [P Q]]. exists b; auto. |
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246 | Qed. |
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247 | |
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248 | Remark map_partial_forall2: |
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249 | forall l l', |
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250 | map_partial l = OK l' -> |
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251 | list_forall2 |
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252 | (fun (a_b: A * B) (a_c: A * C) => |
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253 | fst a_b = fst a_c /\ f (snd a_b) = OK (snd a_c)) |
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254 | l l'. |
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255 | Proof. |
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256 | induction l; simpl. |
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257 | intros. inv H. constructor. |
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258 | intro l'. destruct a as [a b]. |
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259 | caseEq (f b). 2: congruence. intro c; intros. monadInv H0. |
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260 | constructor. simpl. auto. auto. |
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261 | Qed. |
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262 | |
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263 | End MAP_PARTIAL. |
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264 | |
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265 | Remark map_partial_total: |
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266 | forall (A B C: Type) (prefix: A -> errmsg) (f: B -> C) (l: list (A * B)), |
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267 | map_partial prefix (fun b => OK (f b)) l = |
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268 | OK (List.map (fun a_b => (fst a_b, f (snd a_b))) l). |
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269 | Proof. |
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270 | induction l; simpl. |
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271 | auto. |
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272 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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273 | Qed. |
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274 | |
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275 | Remark map_partial_identity: |
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276 | forall (A B: Type) (prefix: A -> errmsg) (l: list (A * B)), |
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277 | map_partial prefix (fun b => OK b) l = OK l. |
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278 | Proof. |
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279 | induction l; simpl. |
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280 | auto. |
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281 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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282 | Qed. |
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283 | |
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284 | Section TRANSF_PARTIAL_PROGRAM. |
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285 | |
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286 | Variable A B V: Type. |
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287 | Variable transf_partial: A -> res B. |
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288 | |
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289 | Definition prefix_funct_name (id: ident) : errmsg := |
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290 | MSG "In function " :: CTX id :: MSG ": " :: nil. |
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291 | |
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292 | Definition transform_partial_program (p: program A V) : res (program B V) := |
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293 | do fl <- map_partial prefix_funct_name transf_partial p.(prog_funct); |
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294 | OK (mkprogram fl p.(prog_main) p.(prog_vars)). |
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295 | |
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296 | Lemma transform_partial_program_function: |
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297 | forall p tp i tf, |
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298 | transform_partial_program p = OK tp -> |
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299 | In (i, tf) tp.(prog_funct) -> |
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300 | exists f, In (i, f) p.(prog_funct) /\ transf_partial f = OK tf. |
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301 | Proof. |
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302 | intros. monadInv H. simpl in H0. |
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303 | eapply In_map_partial; eauto. |
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304 | Qed. |
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305 | |
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306 | Lemma transform_partial_program_main: |
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307 | forall p tp, |
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308 | transform_partial_program p = OK tp -> |
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309 | tp.(prog_main) = p.(prog_main). |
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310 | Proof. |
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311 | intros. monadInv H. reflexivity. |
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312 | Qed. |
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313 | |
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314 | Lemma transform_partial_program_vars: |
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315 | forall p tp, |
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316 | transform_partial_program p = OK tp -> |
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317 | tp.(prog_vars) = p.(prog_vars). |
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318 | Proof. |
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319 | intros. monadInv H. reflexivity. |
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320 | Qed. |
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321 | |
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322 | End TRANSF_PARTIAL_PROGRAM. |
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323 | |
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324 | (** The following is a variant of [transform_program_partial] where |
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325 | both the program functions and the additional variable information |
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326 | are transformed by functions that can fail. *) |
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327 | |
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328 | Section TRANSF_PARTIAL_PROGRAM2. |
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329 | |
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330 | Variable A B V W: Type. |
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331 | Variable transf_partial_function: A -> res B. |
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332 | Variable transf_partial_variable: V -> res W. |
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333 | |
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334 | Definition prefix_var_name (id_init: ident * list init_data) : errmsg := |
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335 | MSG "In global variable " :: CTX (fst id_init) :: MSG ": " :: nil. |
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336 | |
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337 | Definition transform_partial_program2 (p: program A V) : res (program B W) := |
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338 | do fl <- map_partial prefix_funct_name transf_partial_function p.(prog_funct); |
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339 | do vl <- map_partial prefix_var_name transf_partial_variable p.(prog_vars); |
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340 | OK (mkprogram fl p.(prog_main) vl). |
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341 | |
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342 | Lemma transform_partial_program2_function: |
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343 | forall p tp i tf, |
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344 | transform_partial_program2 p = OK tp -> |
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345 | In (i, tf) tp.(prog_funct) -> |
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346 | exists f, In (i, f) p.(prog_funct) /\ transf_partial_function f = OK tf. |
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347 | Proof. |
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348 | intros. monadInv H. |
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349 | eapply In_map_partial; eauto. |
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350 | Qed. |
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351 | |
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352 | Lemma transform_partial_program2_variable: |
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353 | forall p tp i tv, |
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354 | transform_partial_program2 p = OK tp -> |
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355 | In (i, tv) tp.(prog_vars) -> |
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356 | exists v, In (i, v) p.(prog_vars) /\ transf_partial_variable v = OK tv. |
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357 | Proof. |
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358 | intros. monadInv H. |
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359 | eapply In_map_partial; eauto. |
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360 | Qed. |
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361 | |
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362 | Lemma transform_partial_program2_main: |
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363 | forall p tp, |
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364 | transform_partial_program2 p = OK tp -> |
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365 | tp.(prog_main) = p.(prog_main). |
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366 | Proof. |
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367 | intros. monadInv H. reflexivity. |
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368 | Qed. |
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369 | |
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370 | End TRANSF_PARTIAL_PROGRAM2. |
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371 | |
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372 | (** The following is a relational presentation of |
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373 | [transform_program_partial2]. Given relations between function |
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374 | definitions and between variable information, it defines a relation |
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375 | between programs stating that the two programs have the same shape |
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376 | (same global names, etc) and that identically-named function definitions |
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377 | are variable information are related. *) |
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378 | |
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379 | Section MATCH_PROGRAM. |
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380 | |
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381 | Variable A B V W: Type. |
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382 | Variable match_fundef: A -> B -> Prop. |
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383 | Variable match_varinfo: V -> W -> Prop. |
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384 | |
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385 | Definition match_funct_entry (x1: ident * A) (x2: ident * B) := |
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386 | match x1, x2 with |
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387 | | (id1, fn1), (id2, fn2) => id1 = id2 /\ match_fundef fn1 fn2 |
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388 | end. |
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389 | |
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390 | Definition match_var_entry (x1: ident * list init_data * V) (x2: ident * list init_data * W) := |
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391 | match x1, x2 with |
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392 | | (id1, init1, info1), (id2, init2, info2) => id1 = id2 /\ init1 = init2 /\ match_varinfo info1 info2 |
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393 | end. |
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394 | |
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395 | Definition match_program (p1: program A V) (p2: program B W) : Prop := |
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396 | list_forall2 match_funct_entry p1.(prog_funct) p2.(prog_funct) |
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397 | /\ p1.(prog_main) = p2.(prog_main) |
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398 | /\ list_forall2 match_var_entry p1.(prog_vars) p2.(prog_vars). |
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399 | |
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400 | End MATCH_PROGRAM. |
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401 | |
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402 | Remark transform_partial_program2_match: |
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403 | forall (A B V W: Type) |
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404 | (transf_partial_function: A -> res B) |
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405 | (transf_partial_variable: V -> res W) |
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406 | (p: program A V) (tp: program B W), |
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407 | transform_partial_program2 transf_partial_function transf_partial_variable p = OK tp -> |
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408 | match_program |
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409 | (fun fd tfd => transf_partial_function fd = OK tfd) |
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410 | (fun info tinfo => transf_partial_variable info = OK tinfo) |
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411 | p tp. |
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412 | Proof. |
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413 | intros. monadInv H. split. |
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414 | apply list_forall2_imply with |
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415 | (fun (ab: ident * A) (ac: ident * B) => |
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416 | fst ab = fst ac /\ transf_partial_function (snd ab) = OK (snd ac)). |
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417 | eapply map_partial_forall2. eauto. |
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418 | intros. destruct v1; destruct v2; simpl in *. auto. |
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419 | split. auto. |
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420 | apply list_forall2_imply with |
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421 | (fun (ab: ident * list init_data * V) (ac: ident * list init_data * W) => |
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422 | fst ab = fst ac /\ transf_partial_variable (snd ab) = OK (snd ac)). |
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423 | eapply map_partial_forall2. eauto. |
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424 | intros. destruct v1; destruct v2; simpl in *. destruct p0; destruct p1. intuition congruence. |
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425 | Qed. |
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426 | *) |
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427 | (* * * External functions *) |
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428 | |
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429 | (* * For most languages, the functions composing the program are either |
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430 | internal functions, defined within the language, or external functions |
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431 | (a.k.a. system calls) that emit an event when applied. We define |
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432 | a type for such functions and some generic transformation functions. *) |
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433 | |
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434 | record external_function : Type[0] ≝ { |
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435 | ef_id: ident; |
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436 | ef_sig: signature |
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437 | }. |
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438 | (* |
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439 | inductive fundef (F: Type): Type ≝ |
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440 | | Internal: F → fundef F |
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441 | | External: external_function → fundef F. |
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442 | |
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443 | (* Implicit Arguments External [F]. *) |
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444 | (* |
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445 | Section TRANSF_FUNDEF. |
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446 | |
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447 | Variable A B: Type. |
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448 | Variable transf: A -> B. |
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449 | *) |
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450 | definition transf_fundef : ∀A,B. (A→B) → fundef A → fundef B ≝ |
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451 | λA,B,transf,fd. |
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452 | match fd with |
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453 | [ Internal f ⇒ Internal ? (transf f) |
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454 | | External ef ⇒ External ? ef |
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455 | ]. |
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456 | *) |
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457 | (* |
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458 | End TRANSF_FUNDEF. |
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459 | |
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460 | Section TRANSF_PARTIAL_FUNDEF. |
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461 | |
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462 | Variable A B: Type. |
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463 | Variable transf_partial: A -> res B. |
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464 | |
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465 | Definition transf_partial_fundef (fd: fundef A): res (fundef B) := |
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466 | match fd with |
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467 | | Internal f => do f' <- transf_partial f; OK (Internal f') |
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468 | | External ef => OK (External ef) |
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469 | end. |
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470 | |
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471 | End TRANSF_PARTIAL_FUNDEF. |
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472 | *) |
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