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 | (* * Representation of observable events and execution traces. *) |
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17 | (* |
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18 | Require Import Coqlib. |
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19 | *) |
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20 | (*include "AST.ma".*) |
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21 | (*include "Integers.ma".*) |
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22 | (*include "Floats.ma".*) |
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23 | include "Values.ma". |
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24 | include "basics/list.ma". |
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25 | include "extralib.ma". |
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26 | include "CostLabel.ma". |
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27 | |
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28 | (* * The observable behaviour of programs is stated in terms of |
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29 | input/output events, which can also be thought of as system calls |
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30 | to the operating system. An event is generated each time an |
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31 | external function (see module AST) is invoked. The event records |
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32 | the name of the external function, the arguments to the function |
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33 | invocation provided by the program, and the return value provided by |
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34 | the outside world (e.g. the operating system). Arguments and values |
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35 | are either integers or floating-point numbers. We currently do not |
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36 | allow pointers to be exchanged between the program and the outside |
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37 | world. *) |
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38 | |
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39 | inductive eventval: Type[0] := |
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40 | | EVint: int -> eventval |
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41 | | EVfloat: float -> eventval. |
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42 | |
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43 | inductive event : Type[0] ≝ |
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44 | | EVcost: costlabel → event |
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45 | | EVextcall: ∀ev_name: ident. ∀ev_args: list eventval. ∀ev_res: eventval. event. |
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46 | |
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47 | (* * The dynamic semantics for programs collect traces of events. |
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48 | Traces are of two kinds: finite (type [trace]) or infinite (type [traceinf]). *) |
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49 | |
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50 | definition trace := list event. |
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51 | |
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52 | definition E0 : trace := nil ?. |
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53 | |
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54 | definition Echarge : costlabel → trace ≝ |
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55 | λlabel. EVcost label :: (nil ?). |
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56 | |
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57 | definition Eextcall : ident → list eventval → eventval → trace ≝ |
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58 | λname: ident. λargs: list eventval. λres: eventval. |
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59 | EVextcall name args res :: (nil ?). |
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60 | |
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61 | definition Eapp : trace → trace → trace ≝ λt1,t2. t1 @ t2. |
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62 | |
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63 | coinductive traceinf : Type[0] := |
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64 | | Econsinf: event -> traceinf -> traceinf. |
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65 | |
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66 | let rec Eappinf (t: trace) (T: traceinf) on t : traceinf := |
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67 | match t with |
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68 | [ nil => T |
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69 | | cons ev t' => Econsinf ev (Eappinf t' T) |
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70 | ]. |
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71 | |
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72 | (* * Concatenation of traces is written [**] in the finite case |
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73 | or [***] in the infinite case. *) |
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74 | |
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75 | notation "hvbox(l1 break ⧺ l2)" |
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76 | right associative with precedence 47 |
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77 | for @{'doubleplus $l1 $l2 }. |
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78 | |
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79 | notation "hvbox(l1 break ⧻ l2)" |
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80 | right associative with precedence 47 |
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81 | for @{'tripleplus $l1 $l2 }. |
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82 | |
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83 | interpretation "trace concatenation" 'doubleplus l1 l2 = (Eapp l1 l2). |
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84 | interpretation "infinite trace concatenation" 'tripleplus l1 l2 = (Eappinf l1 l2). |
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85 | (* |
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86 | Infix "**" := Eapp (at level 60, right associativity). |
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87 | Infix "***" := Eappinf (at level 60, right associativity). |
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88 | *) |
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89 | lemma E0_left: ∀t. E0 ⧺ t = t. |
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90 | //; qed. |
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91 | |
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92 | lemma E0_right: ∀t. t ⧺ E0 = t. |
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93 | @append_nil qed. |
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94 | |
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95 | lemma Eapp_assoc: ∀t1,t2,t3. (t1 ⧺ t2) ⧺ t3 = t1 ⧺ (t2 ⧺ t3). |
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96 | @associative_append qed. |
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97 | |
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98 | lemma Eapp_E0_inv: ∀t1,t2. t1 ⧺ t2 = E0 → t1 = E0 ∧ t2 = E0. |
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99 | @nil_append_nil_both qed. |
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100 | |
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101 | lemma E0_left_inf: ∀T. E0 ⧻ T = T. |
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102 | //; qed. |
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103 | |
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104 | lemma Eappinf_assoc: ∀t1,t2,T. (t1 ⧺ t2) ⧻ T = t1 ⧻ (t2 ⧻ T). |
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105 | #t1 elim t1; #t2 #T normalize; //; qed. |
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106 | |
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107 | (* |
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108 | Hint Rewrite E0_left E0_right Eapp_assoc |
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109 | E0_left_inf Eappinf_assoc: trace_rewrite. |
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110 | |
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111 | Opaque trace E0 Eextcall Eapp Eappinf. |
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112 | |
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113 | (** The following [traceEq] tactic proves equalities between traces |
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114 | or infinite traces. *) |
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115 | |
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116 | Ltac substTraceHyp := |
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117 | match goal with |
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118 | | [ H: (@eq trace ?x ?y) |- _ ] => |
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119 | subst x || clear H |
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120 | end. |
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121 | |
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122 | Ltac decomposeTraceEq := |
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123 | match goal with |
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124 | | [ |- (_ ** _) = (_ ** _) ] => |
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125 | apply (f_equal2 Eapp); auto; decomposeTraceEq |
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126 | | _ => |
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127 | auto |
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128 | end. |
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129 | |
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130 | Ltac traceEq := |
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131 | repeat substTraceHyp; autorewrite with trace_rewrite; decomposeTraceEq. |
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132 | *) |
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133 | |
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134 | (* Ported from CompCert 1.7.1 >>> *) |
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135 | |
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136 | (* * An alternate presentation of infinite traces as |
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137 | infinite concatenations of nonempty finite traces. *) |
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138 | |
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139 | coinductive traceinf': Type[0] ≝ |
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140 | | Econsinf': ∀t: trace. ∀T: traceinf'. t ≠ E0 → traceinf'. |
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141 | |
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142 | definition split_traceinf' : ∀t:trace. traceinf' → t ≠ E0 → event × traceinf' ≝ |
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143 | λt,T. |
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144 | match t return λt0.t0 ≠ E0 → ? with |
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145 | [ nil ⇒ ? |
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146 | | cons e t' ⇒ λ_. |
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147 | (match t' return λt0. t' = t0 → ? with |
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148 | [ nil ⇒ λ_.〈e, T〉 |
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149 | | cons e' t'' ⇒ λE.〈e, Econsinf' t' T ?〉 |
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150 | ]) (refl ? t') |
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151 | ]. |
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152 | [ *; #NE @False_rect_Type0 @NE @refl |
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153 | | % #E' >E' in E #E whd in E:(??%%); destruct (E); |
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154 | ] qed. |
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155 | |
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156 | let corec traceinf_of_traceinf' (T': traceinf') : traceinf ≝ |
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157 | match T' with |
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158 | [ Econsinf' t T'' NOTEMPTY ⇒ |
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159 | match split_traceinf' t T'' NOTEMPTY with [ pair e tl ⇒ |
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160 | Econsinf e (traceinf_of_traceinf' tl) ] |
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161 | ]. |
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162 | |
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163 | lemma unroll_traceinf': |
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164 | ∀T. T = match T with [ Econsinf' t T' NE ⇒ Econsinf' t T' NE ]. |
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165 | #T cases T; //; qed. |
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166 | |
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167 | lemma unroll_traceinf: |
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168 | ∀T. T = match T with [ Econsinf t T' ⇒ Econsinf t T' ]. |
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169 | #T cases T #ev #tr @refl (* XXX //; *) |
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170 | qed. |
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171 | |
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172 | lemma traceinf_traceinfp_app: |
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173 | ∀t,T,NE. |
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174 | traceinf_of_traceinf' (Econsinf' t T NE) = t ⧻ traceinf_of_traceinf' T. |
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175 | #t elim t; |
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176 | [ #T #NE cases NE; #NE' @False_ind @NE' @refl |
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177 | | #h #t' cases t'; [ 2: #h' #t'' ] #IH #T #NE |
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178 | >(unroll_traceinf (traceinf_of_traceinf' ?)) |
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179 | whd in ⊢ (??%?); //; >(IH …) @refl |
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180 | ] qed. |
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181 | |
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182 | (* <<< *) |
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183 | |
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184 | (* * The predicate [event_match ef vargs t vres] expresses that |
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185 | the event [t] is generated when invoking external function [ef] |
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186 | with arguments [vargs], and obtaining [vres] as a return value |
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187 | from the operating system. *) |
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188 | |
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189 | inductive eventval_match: eventval -> typ -> val -> Prop := |
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190 | | ev_match_int: |
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191 | ∀i. eventval_match (EVint i) ASTint (Vint i) |
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192 | | ev_match_float: |
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193 | ∀f. eventval_match (EVfloat f) ASTfloat (Vfloat f). |
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194 | |
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195 | inductive eventval_list_match: list eventval -> list typ -> list val -> Prop := |
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196 | | evl_match_nil: |
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197 | eventval_list_match (nil ?) (nil ?) (nil ?) |
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198 | | evl_match_cons: |
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199 | ∀ev1,evl,ty1,tyl,v1,vl. |
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200 | eventval_match ev1 ty1 v1 -> |
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201 | eventval_list_match evl tyl vl -> |
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202 | eventval_list_match (ev1::evl) (ty1::tyl) (v1::vl). |
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203 | |
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204 | inductive event_match: |
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205 | external_function -> list val -> trace -> val -> Prop := |
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206 | event_match_intro: |
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207 | ∀ef,vargs,vres,eargs,eres. |
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208 | eventval_list_match eargs (sig_args (ef_sig ef)) vargs -> |
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209 | eventval_match eres (proj_sig_res (ef_sig ef)) vres -> |
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210 | event_match ef vargs (Eextcall (ef_id ef) eargs eres) vres. |
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211 | (* |
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212 | (** The following section shows that [event_match] is stable under |
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213 | relocation of pointer values, as performed by memory injections |
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214 | (see module [Mem]). *) |
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215 | |
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216 | Require Import Mem. |
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217 | |
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218 | Section EVENT_MATCH_INJECT. |
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219 | |
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220 | Variable f: meminj. |
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221 | |
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222 | Remark eventval_match_inject: |
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223 | forall ev ty v1, eventval_match ev ty v1 -> |
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224 | forall v2, val_inject f v1 v2 -> |
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225 | eventval_match ev ty v2. |
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226 | Proof. |
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227 | induction 1; intros; inversion H; constructor. |
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228 | Qed. |
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229 | |
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230 | Remark eventval_list_match_inject: |
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231 | forall evl tyl vl1, eventval_list_match evl tyl vl1 -> |
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232 | forall vl2, val_list_inject f vl1 vl2 -> |
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233 | eventval_list_match evl tyl vl2. |
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234 | Proof. |
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235 | induction 1; intros. |
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236 | inversion H; constructor. |
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237 | inversion H1; constructor. |
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238 | eapply eventval_match_inject; eauto. |
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239 | eauto. |
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240 | Qed. |
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241 | |
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242 | Lemma event_match_inject: |
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243 | forall ef args1 t res args2, |
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244 | event_match ef args1 t res -> |
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245 | val_list_inject f args1 args2 -> |
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246 | event_match ef args2 t res /\ val_inject f res res. |
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247 | Proof. |
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248 | intros. inversion H; subst. |
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249 | split. constructor. eapply eventval_list_match_inject; eauto. auto. |
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250 | inversion H2; constructor. |
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251 | Qed. |
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252 | |
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253 | End EVENT_MATCH_INJECT. |
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254 | |
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255 | (** The following section shows that [event_match] is stable under |
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256 | replacement of [Vundef] values by more defined values. *) |
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257 | |
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258 | Section EVENT_MATCH_LESSDEF. |
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259 | |
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260 | Remark eventval_match_lessdef: |
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261 | forall ev ty v1, eventval_match ev ty v1 -> |
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262 | forall v2, Val.lessdef v1 v2 -> |
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263 | eventval_match ev ty v2. |
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264 | Proof. |
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265 | induction 1; intros; inv H; constructor. |
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266 | Qed. |
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267 | |
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268 | Remark eventval_list_match_moredef: |
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269 | forall evl tyl vl1, eventval_list_match evl tyl vl1 -> |
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270 | forall vl2, Val.lessdef_list vl1 vl2 -> |
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271 | eventval_list_match evl tyl vl2. |
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272 | Proof. |
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273 | induction 1; intros. |
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274 | inversion H; constructor. |
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275 | inversion H1; constructor. |
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276 | eapply eventval_match_lessdef; eauto. |
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277 | eauto. |
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278 | Qed. |
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279 | |
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280 | Lemma event_match_lessdef: |
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281 | forall ef args1 t res1 args2, |
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282 | event_match ef args1 t res1 -> |
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283 | Val.lessdef_list args1 args2 -> |
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284 | exists res2, event_match ef args2 t res2 /\ Val.lessdef res1 res2. |
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285 | Proof. |
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286 | intros. inversion H; subst. exists res1; split. |
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287 | constructor. eapply eventval_list_match_moredef; eauto. auto. |
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288 | auto. |
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289 | Qed. |
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290 | |
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291 | End EVENT_MATCH_LESSDEF. |
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292 | |
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293 | (** Bisimilarity between infinite traces. *) |
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294 | |
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295 | CoInductive traceinf_sim: traceinf -> traceinf -> Prop := |
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296 | | traceinf_sim_cons: forall e T1 T2, |
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297 | traceinf_sim T1 T2 -> |
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298 | traceinf_sim (Econsinf e T1) (Econsinf e T2). |
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299 | |
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300 | Lemma traceinf_sim_refl: |
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301 | forall T, traceinf_sim T T. |
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302 | Proof. |
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303 | cofix COINDHYP; intros. |
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304 | destruct T. constructor. apply COINDHYP. |
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305 | Qed. |
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306 | |
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307 | Lemma traceinf_sim_sym: |
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308 | forall T1 T2, traceinf_sim T1 T2 -> traceinf_sim T2 T1. |
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309 | Proof. |
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310 | cofix COINDHYP; intros. inv H; constructor; auto. |
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311 | Qed. |
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312 | |
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313 | Lemma traceinf_sim_trans: |
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314 | forall T1 T2 T3, |
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315 | traceinf_sim T1 T2 -> traceinf_sim T2 T3 -> traceinf_sim T1 T3. |
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316 | Proof. |
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317 | cofix COINDHYP;intros. inv H; inv H0; constructor; eauto. |
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318 | Qed. |
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319 | |
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320 | (** The "is prefix of" relation between a finite and an infinite trace. *) |
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321 | |
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322 | Inductive traceinf_prefix: trace -> traceinf -> Prop := |
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323 | | traceinf_prefix_nil: forall T, |
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324 | traceinf_prefix nil T |
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325 | | traceinf_prefix_cons: forall e t1 T2, |
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326 | traceinf_prefix t1 T2 -> |
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327 | traceinf_prefix (e :: t1) (Econsinf e T2). |
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328 | |
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329 | (* |
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330 | Lemma traceinf_prefix_compat: |
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331 | forall T1 T2 T1' T2', |
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332 | traceinf_prefix T1 T2 -> traceinf_sim T1 T1' -> traceinf_sim T2 T2' -> |
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333 | traceinf_prefix T1' T2'. |
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334 | Proof. |
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335 | cofix COINDHYP; intros. |
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336 | inv H; inv H0; inv H1; constructor; eapply COINDHYP; eauto. |
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337 | Qed. |
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338 | |
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339 | Transparent trace E0 Eapp Eappinf. |
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340 | *) |
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341 | |
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342 | Lemma traceinf_prefix_app: |
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343 | forall t1 T2 t, |
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344 | traceinf_prefix t1 T2 -> |
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345 | traceinf_prefix (t ** t1) (t *** T2). |
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346 | Proof. |
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347 | induction t; simpl; intros. auto. |
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348 | change ((a :: t) ** t1) with (a :: (t ** t1)). |
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349 | change ((a :: t) *** T2) with (Econsinf a (t *** T2)). |
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350 | constructor. auto. |
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351 | Qed. |
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352 | |
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353 | *) |
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