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1
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(**************************************************************************)
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2
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(* *)
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3
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(* The Why3 Verification Platform / The Why3 Development Team *)
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4
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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5
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(* *)
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6
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(* This software is distributed under the terms of the GNU Lesser *)
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7
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(* General Public License version 2.1, with the special exception *)
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(* on linking described in file LICENSE. *)
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(* *)
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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(* énergies alternatives). *)
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(* *)
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(**************************************************************************)
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(* this is a prelude for Alt-Ergo*)
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16
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(* this is a prelude for Alt-Ergo integer arithmetic *)
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(** The theory BuiltIn_ must be appended to this file*)
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(** The theory Bool_ must be appended to this file*)
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(**************************************************************************)
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20
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(* *)
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21
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(* The Why3 Verification Platform / The Why3 Development Team *)
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22
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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(* *)
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(* This software is distributed under the terms of the GNU Lesser *)
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25
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(* General Public License version 2.1, with the special exception *)
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26
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(* on linking described in file LICENSE. *)
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(* *)
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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29
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(* énergies alternatives). *)
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(* *)
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31
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(**************************************************************************)
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33
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(* this is a prelude for Alt-Ergo*)
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(** The theory BuiltIn_ must be appended to this file*)
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(** The theory Bool_ must be appended to this file*)
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36
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(** The theory int_Int_ must be appended to this file*)
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logic abs_int : int -> int
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axiom abs_def : (forall x:int. ((0 <= x) -> (abs_int(x) = x)))
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axiom abs_def1 : (forall x:int. ((not (0 <= x)) -> (abs_int(x) = (-x))))
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42
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axiom Abs_le :
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(forall x:int. forall y:int. ((abs_int(x) <= y) -> ((-y) <= x)))
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45
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axiom Abs_le1 : (forall x:int. forall y:int. ((abs_int(x) <= y) -> (x <= y)))
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axiom Abs_le2 :
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(forall x:int. forall y:int. ((((-y) <= x) and (x <= y)) ->
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(abs_int(x) <= y)))
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51
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axiom Abs_pos : (forall x:int. (0 <= abs_int(x)))
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53
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54
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(**************************************************************************)
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55
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(* *)
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56
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(* The Why3 Verification Platform / The Why3 Development Team *)
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57
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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58
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(* *)
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59
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(* This software is distributed under the terms of the GNU Lesser *)
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60
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(* General Public License version 2.1, with the special exception *)
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61
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(* on linking described in file LICENSE. *)
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62
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(* *)
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63
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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64
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(* énergies alternatives). *)
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65
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(* *)
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66
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(**************************************************************************)
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67
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68
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(* this is a prelude for Alt-Ergo*)
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69
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logic safe_comp_div: int, int -> int
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axiom safe_comp_div_def: forall x, y:int. x >= 0 and y > 0 -> safe_comp_div(x,y) = x / y
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logic safe_comp_mod: int, int -> int
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axiom safe_comp_mod_def: forall x, y:int. x >= 0 and y > 0 -> safe_comp_mod(x,y) = x % y
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(** The theory BuiltIn_ must be appended to this file*)
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(** The theory Bool_ must be appended to this file*)
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(** The theory int_Int_ must be appended to this file*)
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(** The theory int_Abs_ must be appended to this file*)
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axiom Div_bound :
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(forall x:int. forall y:int. (((0 <= x) and (0 < y)) ->
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(0 <= safe_comp_div(x,y))))
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axiom Div_bound1 :
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(forall x:int. forall y:int. (((0 <= x) and (0 < y)) ->
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(safe_comp_div(x,y) <= x)))
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axiom Div_1 : (forall x:int. (safe_comp_div(x,1) = x))
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axiom Mod_1 : (forall x:int. (safe_comp_mod(x,1) = 0))
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axiom Div_inf :
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(forall x:int. forall y:int. (((0 <= x) and (x < y)) ->
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(safe_comp_div(x,y) = 0)))
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92
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axiom Mod_inf :
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(forall x:int. forall y:int. (((0 <= x) and (x < y)) ->
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(safe_comp_mod(x,y) = x)))
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96
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97
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axiom Div_mult :
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(forall x:int. forall y:int. forall z:int [safe_comp_div(((x * y) + z),x)].
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(((0 < x) and ((0 <= y) and (0 <= z))) ->
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(safe_comp_div(((x * y) + z),x) = (y + safe_comp_div(z,x)))))
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axiom Mod_mult :
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103
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(forall x:int. forall y:int. forall z:int [safe_comp_mod(((x * y) + z),x)].
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(((0 < x) and ((0 <= y) and (0 <= z))) ->
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(safe_comp_mod(((x * y) + z),x) = safe_comp_mod(z,x))))
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106
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107
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(**************************************************************************)
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108
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(* *)
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109
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(* The Why3 Verification Platform / The Why3 Development Team *)
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110
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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111
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(* *)
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112
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(* This software is distributed under the terms of the GNU Lesser *)
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113
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(* General Public License version 2.1, with the special exception *)
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114
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(* on linking described in file LICENSE. *)
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115
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(* *)
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116
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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117
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(* énergies alternatives). *)
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118
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(* *)
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119
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(**************************************************************************)
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120
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121
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(* this is a prelude for Alt-Ergo*)
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122
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(* this is a prelude for Alt-Ergo real arithmetic *)
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123
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(** The theory BuiltIn_ must be appended to this file*)
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124
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(** The theory Bool_ must be appended to this file*)
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125
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axiom add_div :
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(forall x:real. forall y:real. forall z:real. ((not (z = 0.0)) ->
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(((x + y) / z) = ((x / z) + (y / z)))))
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128
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129
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axiom sub_div :
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(forall x:real. forall y:real. forall z:real. ((not (z = 0.0)) ->
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(((x - y) / z) = ((x / z) - (y / z)))))
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132
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133
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axiom neg_div :
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(forall x:real. forall y:real. ((not (y = 0.0)) ->
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135
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(((-x) / y) = (-(x / y)))))
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136
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137
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axiom assoc_mul_div :
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138
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(forall x:real. forall y:real. forall z:real. ((not (z = 0.0)) ->
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139
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(((x * y) / z) = (x * (y / z)))))
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140
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141
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axiom assoc_div_mul :
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142
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(forall x:real. forall y:real. forall z:real. (((not (y = 0.0)) and
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(not (z = 0.0))) -> (((x / y) / z) = (x / (y * z)))))
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144
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145
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axiom assoc_div_div :
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146
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(forall x:real. forall y:real. forall z:real. (((not (y = 0.0)) and
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147
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(not (z = 0.0))) -> ((x / (y / z)) = ((x * z) / y))))
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148
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149
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(**************************************************************************)
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150
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(* *)
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151
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(* The Why3 Verification Platform / The Why3 Development Team *)
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152
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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153
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(* *)
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154
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(* This software is distributed under the terms of the GNU Lesser *)
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155
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(* General Public License version 2.1, with the special exception *)
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156
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(* on linking described in file LICENSE. *)
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157
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(* *)
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158
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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159
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(* énergies alternatives). *)
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160
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(* *)
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161
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(**************************************************************************)
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162
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163
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(* this is a prelude for Alt-Ergo*)
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164
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(** The theory BuiltIn_ must be appended to this file*)
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165
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(** The theory Bool_ must be appended to this file*)
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166
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(** The theory real_Real_ must be appended to this file*)
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167
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(**************************************************************************)
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168
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(* *)
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169
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(* The Why3 Verification Platform / The Why3 Development Team *)
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170
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(* Copyright 2010-2013 -- INRIA - CNRS - Paris-Sud University *)
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171
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(* *)
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172
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(* This software is distributed under the terms of the GNU Lesser *)
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173
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(* General Public License version 2.1, with the special exception *)
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174
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(* on linking described in file LICENSE. *)
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175
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(* *)
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176
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(* File modified by CEA (Commissariat à l'énergie atomique et aux *)
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177
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(* énergies alternatives). *)
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178
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(* *)
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179
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(**************************************************************************)
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180
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181
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(* this is a prelude for Alt-Ergo*)
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182
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(** The theory BuiltIn_ must be appended to this file*)
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183
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(** The theory Bool_ must be appended to this file*)
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184
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(** The theory int_Int_ must be appended to this file*)
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185
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(** The theory real_Real_ must be appended to this file*)
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186
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logic from_int : int -> real
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187
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188
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axiom Zero : (from_int(0) = 0.0)
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189
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190
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axiom One : (from_int(1) = 1.0)
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191
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192
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axiom Add :
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193
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(forall x:int. forall y:int.
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194
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(from_int((x + y)) = (from_int(x) + from_int(y))))
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195
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196
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axiom Sub :
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197
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(forall x:int. forall y:int.
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198
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(from_int((x - y)) = (from_int(x) - from_int(y))))
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199
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200
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axiom Mul :
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201
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(forall x:int. forall y:int.
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202
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(from_int((x * y)) = (from_int(x) * from_int(y))))
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203
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204
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axiom Neg : (forall x:int. (from_int((-x)) = (-from_int(x))))
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205
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206
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(**************************************************************************)
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207
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(* *)
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208
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(* This file is part of WP plug-in of Frama-C. *)
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209
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(* *)
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210
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(* Copyright (C) 2007-2014 *)
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211
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(* CEA (Commissariat a l'energie atomique et aux energies *)
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212
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(* alternatives) *)
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213
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(* *)
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214
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(* you can redistribute it and/or modify it under the terms of the GNU *)
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215
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(* Lesser General Public License as published by the Free Software *)
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216
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(* Foundation, version 2.1. *)
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217
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(* *)
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218
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(* It is distributed in the hope that it will be useful, *)
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219
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(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
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220
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
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221
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(* GNU Lesser General Public License for more details. *)
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222
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(* *)
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223
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(* See the GNU Lesser General Public License version 2.1 *)
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224
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(* for more details (enclosed in the file licenses/LGPLv2.1). *)
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225
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(* *)
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226
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(**************************************************************************)
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227
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228
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(* this is a prelude for Alt-Ergo*)
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229
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(** The theory BuiltIn_ must be appended to this file*)
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230
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(** The theory Bool_ must be appended to this file*)
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231
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(** The theory bool_Bool_ must be appended to this file*)
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232
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(** The theory int_Int_ must be appended to this file*)
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233
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(** The theory int_Abs_ must be appended to this file*)
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234
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(** The theory int_ComputerDivision_ must be appended to this file*)
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235
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(** The theory real_Real_ must be appended to this file*)
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236
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(** The theory real_RealInfix_ must be appended to this file*)
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237
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(** The theory real_FromInt_ must be appended to this file*)
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238
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logic ite : bool, 'a, 'a -> 'a
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239
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240
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axiom ite1 :
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241
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(forall p:bool. forall x:'a. forall y:'a [ite(p, x, y)]. (((p = true) and
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242
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(ite(p, x, y) = x)) or ((p = false) and (ite(p, x, y) = y))))
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243
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244
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logic eqb : 'a, 'a -> bool
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245
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246
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axiom eqb1 : (forall x:'a. forall y:'a. ((eqb(x, y) = true) -> (x = y)))
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247
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248
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axiom eqb2 : (forall x:'a. forall y:'a. ((x = y) -> (eqb(x, y) = true)))
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249
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250
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logic neqb : 'a, 'a -> bool
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251
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252
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axiom neqb1 :
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253
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(forall x:'a. forall y:'a. ((neqb(x, y) = true) -> (not (x = y))))
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254
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255
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axiom neqb2 :
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256
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(forall x:'a. forall y:'a. ((not (x = y)) -> (neqb(x, y) = true)))
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257
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258
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logic zlt : int, int -> bool
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259
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260
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logic zleq : int, int -> bool
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261
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262
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axiom zlt1 : (forall x:int. forall y:int. ((zlt(x, y) = true) -> (x < y)))
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263
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264
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axiom zlt2 : (forall x:int. forall y:int. ((x < y) -> (zlt(x, y) = true)))
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265
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266
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axiom zleq1 : (forall x:int. forall y:int. ((zleq(x, y) = true) -> (x <= y)))
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267
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268
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axiom zleq2 : (forall x:int. forall y:int. ((x <= y) -> (zleq(x, y) = true)))
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269
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270
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logic rlt : real, real -> bool
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271
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272
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logic rleq : real, real -> bool
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273
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274
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axiom rlt1 : (forall x:real. forall y:real. ((rlt(x, y) = true) -> (x < y)))
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275
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276
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axiom rlt2 : (forall x:real. forall y:real. ((x < y) -> (rlt(x, y) = true)))
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277
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278
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axiom rleq1 :
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279
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(forall x:real. forall y:real. ((rleq(x, y) = true) -> (x <= y)))
|
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280
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281
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axiom rleq2 :
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282
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(forall x:real. forall y:real. ((x <= y) -> (rleq(x, y) = true)))
|
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283
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284
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logic truncate : real -> int
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285
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286
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function real_of_int(x: int) : real = from_int(x)
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287
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288
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axiom truncate_of_int : (forall x:int. (truncate(real_of_int(x)) = x))
|
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289
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290
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axiom c_euclidian :
|
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291
|
(forall n:int. forall d:int [safe_comp_div(n,d), safe_comp_mod(n,d)].
|
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292
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((not (d = 0)) -> (n = ((safe_comp_div(n,d) * d) + safe_comp_mod(n,d)))))
|
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293
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294
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axiom cdiv_cases :
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295
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(forall n:int. forall d:int [safe_comp_div(n,d)]. ((0 <= n) -> ((0 < d) ->
|
|
296
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(safe_comp_div(n,d) = (n / d)))))
|
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297
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298
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axiom cdiv_cases1 :
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299
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(forall n:int. forall d:int [safe_comp_div(n,d)]. ((n <= 0) -> ((0 < d) ->
|
|
300
|
(safe_comp_div(n,d) = (-((-n) / d))))))
|
|
301
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302
|
axiom cdiv_cases2 :
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303
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(forall n:int. forall d:int [safe_comp_div(n,d)]. ((0 <= n) -> ((d < 0) ->
|
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304
|
(safe_comp_div(n,d) = (-(n / (-d)))))))
|
|
305
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306
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axiom cdiv_cases3 :
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307
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(forall n:int. forall d:int [safe_comp_div(n,d)]. ((n <= 0) -> ((d < 0) ->
|
|
308
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(safe_comp_div(n,d) = ((-n) / (-d))))))
|
|
309
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310
|
axiom cmod_cases :
|
|
311
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(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((0 < d) ->
|
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312
|
(safe_comp_mod(n,d) = (n % d)))))
|
|
313
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314
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axiom cmod_cases1 :
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315
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(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((0 < d) ->
|
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316
|
(safe_comp_mod(n,d) = (-((-n) % d))))))
|
|
317
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|
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318
|
axiom cmod_cases2 :
|
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319
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(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((d < 0) ->
|
|
320
|
(safe_comp_mod(n,d) = (n % (-d))))))
|
|
321
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322
|
axiom cmod_cases3 :
|
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323
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(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((d < 0) ->
|
|
324
|
(safe_comp_mod(n,d) = (-((-n) % (-d)))))))
|
|
325
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326
|
axiom cmod_remainder :
|
|
327
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((0 < d) ->
|
|
328
|
(0 <= safe_comp_mod(n,d)))))
|
|
329
|
|
|
330
|
axiom cmod_remainder1 :
|
|
331
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((0 < d) ->
|
|
332
|
(safe_comp_mod(n,d) < d))))
|
|
333
|
|
|
334
|
axiom cmod_remainder2 :
|
|
335
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((0 < d) ->
|
|
336
|
((-d) < safe_comp_mod(n,d)))))
|
|
337
|
|
|
338
|
axiom cmod_remainder3 :
|
|
339
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((0 < d) ->
|
|
340
|
(safe_comp_mod(n,d) <= 0))))
|
|
341
|
|
|
342
|
axiom cmod_remainder4 :
|
|
343
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((d < 0) ->
|
|
344
|
(0 <= safe_comp_mod(n,d)))))
|
|
345
|
|
|
346
|
axiom cmod_remainder5 :
|
|
347
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((0 <= n) -> ((d < 0) ->
|
|
348
|
(safe_comp_mod(n,d) < (-d)))))
|
|
349
|
|
|
350
|
axiom cmod_remainder6 :
|
|
351
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((d < 0) ->
|
|
352
|
(d < safe_comp_mod(n,d)))))
|
|
353
|
|
|
354
|
axiom cmod_remainder7 :
|
|
355
|
(forall n:int. forall d:int [safe_comp_mod(n,d)]. ((n <= 0) -> ((d < 0) ->
|
|
356
|
(safe_comp_mod(n,d) <= 0))))
|
|
357
|
|
|
358
|
axiom cdiv_neutral :
|
|
359
|
(forall a:int [safe_comp_div(a,1)]. (safe_comp_div(a,1) = a))
|
|
360
|
|
|
361
|
axiom cdiv_inv :
|
|
362
|
(forall a:int [safe_comp_div(a,a)]. ((not (a = 0)) ->
|
|
363
|
(safe_comp_div(a,a) = 1)))
|
|
364
|
|
|
365
|
(**************************************************************************)
|
|
366
|
(* *)
|
|
367
|
(* This file is part of WP plug-in of Frama-C. *)
|
|
368
|
(* *)
|
|
369
|
(* Copyright (C) 2007-2014 *)
|
|
370
|
(* CEA (Commissariat a l'energie atomique et aux energies *)
|
|
371
|
(* alternatives) *)
|
|
372
|
(* *)
|
|
373
|
(* you can redistribute it and/or modify it under the terms of the GNU *)
|
|
374
|
(* Lesser General Public License as published by the Free Software *)
|
|
375
|
(* Foundation, version 2.1. *)
|
|
376
|
(* *)
|
|
377
|
(* It is distributed in the hope that it will be useful, *)
|
|
378
|
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
|
|
379
|
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
|
|
380
|
(* GNU Lesser General Public License for more details. *)
|
|
381
|
(* *)
|
|
382
|
(* See the GNU Lesser General Public License version 2.1 *)
|
|
383
|
(* for more details (enclosed in the file licenses/LGPLv2.1). *)
|
|
384
|
(* *)
|
|
385
|
(**************************************************************************)
|
|
386
|
|
|
387
|
(* this is a prelude for Alt-Ergo*)
|
|
388
|
(** The theory BuiltIn_ must be appended to this file*)
|
|
389
|
(** The theory Bool_ must be appended to this file*)
|
|
390
|
(** The theory bool_Bool_ must be appended to this file*)
|
|
391
|
(** The theory int_Int_ must be appended to this file*)
|
|
392
|
logic is_uint8 : int -> prop
|
|
393
|
|
|
394
|
axiom is_uint8_def : (forall x:int [is_uint8(x)]. (is_uint8(x) -> (0 <= x)))
|
|
395
|
|
|
396
|
axiom is_uint8_def1 :
|
|
397
|
(forall x:int [is_uint8(x)]. (is_uint8(x) -> (x < 256)))
|
|
398
|
|
|
399
|
axiom is_uint8_def2 :
|
|
400
|
(forall x:int [is_uint8(x)]. (((0 <= x) and (x < 256)) -> is_uint8(x)))
|
|
401
|
|
|
402
|
logic is_sint8 : int -> prop
|
|
403
|
|
|
404
|
axiom is_sint8_def :
|
|
405
|
(forall x:int [is_sint8(x)]. (is_sint8(x) -> ((-128) <= x)))
|
|
406
|
|
|
407
|
axiom is_sint8_def1 :
|
|
408
|
(forall x:int [is_sint8(x)]. (is_sint8(x) -> (x < 128)))
|
|
409
|
|
|
410
|
axiom is_sint8_def2 :
|
|
411
|
(forall x:int [is_sint8(x)]. ((((-128) <= x) and (x < 128)) ->
|
|
412
|
is_sint8(x)))
|
|
413
|
|
|
414
|
logic is_uint16 : int -> prop
|
|
415
|
|
|
416
|
axiom is_uint16_def :
|
|
417
|
(forall x:int [is_uint16(x)]. (is_uint16(x) -> (0 <= x)))
|
|
418
|
|
|
419
|
axiom is_uint16_def1 :
|
|
420
|
(forall x:int [is_uint16(x)]. (is_uint16(x) -> (x < 65536)))
|
|
421
|
|
|
422
|
axiom is_uint16_def2 :
|
|
423
|
(forall x:int [is_uint16(x)]. (((0 <= x) and (x < 65536)) ->
|
|
424
|
is_uint16(x)))
|
|
425
|
|
|
426
|
predicate is_sint16(x: int) = (((-32768) <= x) and (x < 32768))
|
|
427
|
|
|
428
|
logic is_uint32 : int -> prop
|
|
429
|
|
|
430
|
axiom is_uint32_def :
|
|
431
|
(forall x:int [is_uint32(x)]. (is_uint32(x) -> (0 <= x)))
|
|
432
|
|
|
433
|
axiom is_uint32_def1 :
|
|
434
|
(forall x:int [is_uint32(x)]. (is_uint32(x) -> (x < 4294967296)))
|
|
435
|
|
|
436
|
axiom is_uint32_def2 :
|
|
437
|
(forall x:int [is_uint32(x)]. (((0 <= x) and (x < 4294967296)) ->
|
|
438
|
is_uint32(x)))
|
|
439
|
|
|
440
|
logic is_sint32 : int -> prop
|
|
441
|
|
|
442
|
axiom is_sint32_def :
|
|
443
|
(forall x:int [is_sint32(x)]. (is_sint32(x) -> ((-2147483648) <= x)))
|
|
444
|
|
|
445
|
axiom is_sint32_def1 :
|
|
446
|
(forall x:int [is_sint32(x)]. (is_sint32(x) -> (x < 2147483648)))
|
|
447
|
|
|
448
|
axiom is_sint32_def2 :
|
|
449
|
(forall x:int [is_sint32(x)]. ((((-2147483648) <= x) and
|
|
450
|
(x < 2147483648)) -> is_sint32(x)))
|
|
451
|
|
|
452
|
logic is_uint64 : int -> prop
|
|
453
|
|
|
454
|
axiom is_uint64_def :
|
|
455
|
(forall x:int [is_uint64(x)]. (is_uint64(x) -> (0 <= x)))
|
|
456
|
|
|
457
|
axiom is_uint64_def1 :
|
|
458
|
(forall x:int [is_uint64(x)]. (is_uint64(x) ->
|
|
459
|
(x < 18446744073709551616)))
|
|
460
|
|
|
461
|
axiom is_uint64_def2 :
|
|
462
|
(forall x:int [is_uint64(x)]. (((0 <= x) and
|
|
463
|
(x < 18446744073709551616)) -> is_uint64(x)))
|
|
464
|
|
|
465
|
logic is_sint64 : int -> prop
|
|
466
|
|
|
467
|
axiom is_sint64_def :
|
|
468
|
(forall x:int [is_sint64(x)]. (is_sint64(x) ->
|
|
469
|
((-9223372036854775808) <= x)))
|
|
470
|
|
|
471
|
axiom is_sint64_def1 :
|
|
472
|
(forall x:int [is_sint64(x)]. (is_sint64(x) -> (x < 9223372036854775808)))
|
|
473
|
|
|
474
|
axiom is_sint64_def2 :
|
|
475
|
(forall x:int [is_sint64(x)]. ((((-9223372036854775808) <= x) and
|
|
476
|
(x < 9223372036854775808)) -> is_sint64(x)))
|
|
477
|
|
|
478
|
logic to_uint8 : int -> int
|
|
479
|
|
|
480
|
logic to_sint8 : int -> int
|
|
481
|
|
|
482
|
logic to_uint16 : int -> int
|
|
483
|
|
|
484
|
logic to_sint16 : int -> int
|
|
485
|
|
|
486
|
logic to_uint32 : int -> int
|
|
487
|
|
|
488
|
logic to_sint32 : int -> int
|
|
489
|
|
|
490
|
logic to_uint64 : int -> int
|
|
491
|
|
|
492
|
logic to_sint64 : int -> int
|
|
493
|
|
|
494
|
axiom is_to_uint8 :
|
|
495
|
(forall x:int [is_uint8(to_uint8(x))]. is_uint8(to_uint8(x)))
|
|
496
|
|
|
497
|
axiom is_to_sint8 :
|
|
498
|
(forall x:int [is_sint8(to_sint8(x))]. is_sint8(to_sint8(x)))
|
|
499
|
|
|
500
|
axiom is_to_uint16 :
|
|
501
|
(forall x:int [is_uint16(to_uint16(x))]. is_uint16(to_uint16(x)))
|
|
502
|
|
|
503
|
axiom is_to_sint16 :
|
|
504
|
(forall x:int [is_sint16(to_sint16(x))]. is_sint16(to_sint16(x)))
|
|
505
|
|
|
506
|
axiom is_to_uint32 :
|
|
507
|
(forall x:int [is_uint32(to_uint32(x))]. is_uint32(to_uint32(x)))
|
|
508
|
|
|
509
|
axiom is_to_sint32 :
|
|
510
|
(forall x:int [is_sint32(to_sint32(x))]. is_sint32(to_sint32(x)))
|
|
511
|
|
|
512
|
axiom is_to_uint64 :
|
|
513
|
(forall x:int [is_uint64(to_uint64(x))]. is_uint64(to_uint64(x)))
|
|
514
|
|
|
515
|
axiom is_to_sint64 :
|
|
516
|
(forall x:int [is_sint64(to_sint64(x))]. is_sint64(to_sint64(x)))
|
|
517
|
|
|
518
|
axiom id_uint8 :
|
|
519
|
(forall x:int [to_uint8(x)]. (((0 <= x) and (x < 256)) ->
|
|
520
|
(to_uint8(x) = x)))
|
|
521
|
|
|
522
|
axiom id_sint8 :
|
|
523
|
(forall x:int [to_sint8(x)]. ((((-128) <= x) and (x < 128)) ->
|
|
524
|
(to_sint8(x) = x)))
|
|
525
|
|
|
526
|
axiom id_uint16 :
|
|
527
|
(forall x:int [to_uint16(x)]. (((0 <= x) and (x < 65536)) ->
|
|
528
|
(to_uint16(x) = x)))
|
|
529
|
|
|
530
|
axiom id_sint16 :
|
|
531
|
(forall x:int [to_sint16(x)]. ((((-32768) <= x) and (x < 32768)) ->
|
|
532
|
(to_sint16(x) = x)))
|
|
533
|
|
|
534
|
axiom id_uint32 :
|
|
535
|
(forall x:int [to_uint32(x)]. (((0 <= x) and (x < 4294967296)) ->
|
|
536
|
(to_uint32(x) = x)))
|
|
537
|
|
|
538
|
axiom id_sint32 :
|
|
539
|
(forall x:int [to_sint32(x)]. ((((-2147483648) <= x) and
|
|
540
|
(x < 2147483648)) -> (to_sint32(x) = x)))
|
|
541
|
|
|
542
|
axiom id_uint64 :
|
|
543
|
(forall x:int [to_uint64(x)]. (((0 <= x) and
|
|
544
|
(x < 18446744073709551616)) -> (to_uint64(x) = x)))
|
|
545
|
|
|
546
|
axiom id_sint64 :
|
|
547
|
(forall x:int [to_sint64(x)]. ((((-9223372036854775808) <= x) and
|
|
548
|
(x < 9223372036854775808)) -> (to_sint64(x) = x)))
|
|
549
|
|
|
550
|
logic lnot : int -> int
|
|
551
|
|
|
552
|
logic ac land : int, int -> int
|
|
553
|
|
|
554
|
logic ac lxor : int, int -> int
|
|
555
|
|
|
556
|
logic ac lor : int, int -> int
|
|
557
|
|
|
558
|
logic lsl : int, int -> int
|
|
559
|
|
|
560
|
logic lsr : int, int -> int
|
|
561
|
|
|
562
|
logic bit_testb : int, int -> bool
|
|
563
|
|
|
564
|
logic bit_test : int, int -> prop
|
|
565
|
|
|
566
|
(* ---------------------------------------------------------- *)
|
|
567
|
(* --- Global Definitions --- *)
|
|
568
|
(* ---------------------------------------------------------- *)
|
|
569
|
|
|
570
|
predicate P_trans_totoA() = true
|
|
571
|
|
|
572
|
predicate P_clock_d() = true
|
|
573
|
|
|
574
|
predicate P_trans_d
|
|
575
|
(x:int,
|
|
576
|
d:int) =
|
|
577
|
((0 = d) and (x <= 0)) or ((1 = d) and (0 < x))
|
|
578
|
|
|
579
|
predicate P_trans_totoB
|
|
580
|
(x:int,
|
|
581
|
d:int) =
|
|
582
|
P_trans_totoA and (P_clock_d -> P_trans_d(x, d))
|
|
583
|
|
|
584
|
predicate P_clock_toto_2(d:int) = 1 = d
|
|
585
|
|
|
586
|
predicate P_trans_toto_2(x:int, toto_2_0:int) = (0 = x) <-> (0 <> toto_2_0)
|
|
587
|
|
|
588
|
predicate P_trans_totoC
|
|
589
|
(x:int,
|
|
590
|
d:int,
|
|
591
|
toto_2_0:int) =
|
|
592
|
P_trans_totoB(x, d) and
|
|
593
|
(P_clock_toto_2(d) -> P_trans_toto_2(x, toto_2_0))
|
|
594
|
|
|
595
|
predicate P_clock_c(d:int) = 1 = d
|
|
596
|
|
|
597
|
predicate P_trans_c
|
|
598
|
(toto_2_0:int,
|
|
599
|
c:int) =
|
|
600
|
((0 = c) and (0 <> toto_2_0)) or ((0 = toto_2_0) and (1 = c))
|
|
601
|
|
|
602
|
predicate P_trans_totoD
|
|
603
|
(x:int,
|
|
604
|
d:int,
|
|
605
|
c:int) =
|
|
606
|
exists i : int. is_uint32(i) and P_trans_totoC(x, d, i) and
|
|
607
|
(P_clock_c(d) -> P_trans_c(i, c))
|
|
608
|
|
|
609
|
predicate P_clock_b2(d:int, c:int) = (0 = c) and P_clock_c(d)
|
|
610
|
|
|
611
|
predicate P_trans_b2(b2_0:int) = 2 = b2_0
|
|
612
|
|
|
613
|
predicate P_trans_totoE
|
|
614
|
(x:int,
|
|
615
|
d:int,
|
|
616
|
c:int,
|
|
617
|
b2_0:int) =
|
|
618
|
P_trans_totoD(x, d, c) and (P_clock_b2(d, c) -> P_trans_b2(b2_0))
|
|
619
|
|
|
620
|
predicate P_clock_b1(d:int, c:int) = (1 = c) and P_clock_c(d)
|
|
621
|
|
|
622
|
predicate P_trans_b1(b1_0:int) = 1 = b1_0
|
|
623
|
|
|
624
|
predicate P_trans_totoF
|
|
625
|
(x:int,
|
|
626
|
d:int,
|
|
627
|
c:int,
|
|
628
|
b2_0:int,
|
|
629
|
b1_0:int) =
|
|
630
|
P_trans_totoE(x, d, c, b2_0) and (P_clock_b1(d, c) -> P_trans_b1(b1_0))
|
|
631
|
|
|
632
|
predicate P_clock_z(d:int) = P_clock_c(d)
|
|
633
|
|
|
634
|
predicate P_trans_z
|
|
635
|
(c:int,
|
|
636
|
b1_0:int,
|
|
637
|
b2_0:int,
|
|
638
|
z:int) =
|
|
639
|
((1 = c) -> (b1_0 = z)) and ((0 = c) -> (b2_0 = z))
|
|
640
|
|
|
641
|
predicate P_trans_totoG
|
|
642
|
(x:int,
|
|
643
|
d:int,
|
|
644
|
z:int) =
|
|
645
|
exists i_2,i_1,i : int. is_sint32(i) and is_sint32(i_1) and
|
|
646
|
is_uint32(i_2) and P_trans_totoF(x, d, i_2, i_1, i) and
|
|
647
|
(P_clock_z(d) -> P_trans_z(i_2, i, i_1, z))
|
|
648
|
|
|
649
|
predicate P_clock_b3(d:int) = 0 = d
|
|
650
|
|
|
651
|
predicate P_trans_b3(b3_0:int) = 3 = b3_0
|
|
652
|
|
|
653
|
predicate P_trans_totoH
|
|
654
|
(x:int,
|
|
655
|
d:int,
|
|
656
|
z:int,
|
|
657
|
b3_0:int) =
|
|
658
|
P_trans_totoG(x, d, z) and (P_clock_b3(d) -> P_trans_b3(b3_0))
|
|
659
|
|
|
660
|
predicate P_clock_y() = P_clock_d
|
|
661
|
|
|
662
|
predicate P_trans_y
|
|
663
|
(d:int,
|
|
664
|
b3_0:int,
|
|
665
|
z:int,
|
|
666
|
y:int) =
|
|
667
|
((0 = d) -> (b3_0 = y)) and ((1 = d) -> (y = z))
|
|
668
|
|
|
669
|
predicate P_trans_totoI
|
|
670
|
(x:int,
|
|
671
|
y:int) =
|
|
672
|
exists i_2,i_1,i : int. is_sint32(i_1) and is_sint32(i_2) and
|
|
673
|
is_uint32(i) and P_trans_totoH(x, i, i_2, i_1) and
|
|
674
|
(P_clock_y -> P_trans_y(i, i_1, i_2, y))
|
|
675
|
|
|
676
|
(**************************************************************************)
|
|
677
|
(* *)
|
|
678
|
(* This file is part of WP plug-in of Frama-C. *)
|
|
679
|
(* *)
|
|
680
|
(* Copyright (C) 2007-2014 *)
|
|
681
|
(* CEA (Commissariat a l'energie atomique et aux energies *)
|
|
682
|
(* alternatives) *)
|
|
683
|
(* *)
|
|
684
|
(* you can redistribute it and/or modify it under the terms of the GNU *)
|
|
685
|
(* Lesser General Public License as published by the Free Software *)
|
|
686
|
(* Foundation, version 2.1. *)
|
|
687
|
(* *)
|
|
688
|
(* It is distributed in the hope that it will be useful, *)
|
|
689
|
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
|
|
690
|
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
|
|
691
|
(* GNU Lesser General Public License for more details. *)
|
|
692
|
(* *)
|
|
693
|
(* See the GNU Lesser General Public License version 2.1 *)
|
|
694
|
(* for more details (enclosed in the file licenses/LGPLv2.1). *)
|
|
695
|
(* *)
|
|
696
|
(**************************************************************************)
|
|
697
|
|
|
698
|
(* this is a prelude for Alt-Ergo*)
|
|
699
|
(** The theory BuiltIn_ must be appended to this file*)
|
|
700
|
(** The theory Bool_ must be appended to this file*)
|
|
701
|
(** The theory bool_Bool_ must be appended to this file*)
|
|
702
|
(** The theory int_Int_ must be appended to this file*)
|
|
703
|
(** The theory map_Map_ must be appended to this file*)
|
|
704
|
type addr = { base : int; offset : int
|
|
705
|
}
|
|
706
|
|
|
707
|
logic addr_le : addr, addr -> prop
|
|
708
|
|
|
709
|
logic addr_lt : addr, addr -> prop
|
|
710
|
|
|
711
|
logic addr_le_bool : addr, addr -> bool
|
|
712
|
|
|
713
|
logic addr_lt_bool : addr, addr -> bool
|
|
714
|
|
|
715
|
axiom addr_le_def :
|
|
716
|
(forall p:addr. forall q:addr [addr_le(p, q)]. (((p).base = (q).base) ->
|
|
717
|
(addr_le(p, q) -> ((p).offset <= (q).offset))))
|
|
718
|
|
|
719
|
axiom addr_le_def1 :
|
|
720
|
(forall p:addr. forall q:addr [addr_le(p, q)]. (((p).base = (q).base) ->
|
|
721
|
(((p).offset <= (q).offset) -> addr_le(p, q))))
|
|
722
|
|
|
723
|
axiom addr_lt_def :
|
|
724
|
(forall p:addr. forall q:addr [addr_lt(p, q)]. (((p).base = (q).base) ->
|
|
725
|
(addr_lt(p, q) -> ((p).offset < (q).offset))))
|
|
726
|
|
|
727
|
axiom addr_lt_def1 :
|
|
728
|
(forall p:addr. forall q:addr [addr_lt(p, q)]. (((p).base = (q).base) ->
|
|
729
|
(((p).offset < (q).offset) -> addr_lt(p, q))))
|
|
730
|
|
|
731
|
axiom addr_le_bool_def :
|
|
732
|
(forall p:addr. forall q:addr [addr_le_bool(p, q)]. (addr_le(p, q) ->
|
|
733
|
(addr_le_bool(p, q) = true)))
|
|
734
|
|
|
735
|
axiom addr_le_bool_def1 :
|
|
736
|
(forall p:addr. forall q:addr [addr_le_bool(p, q)]. ((addr_le_bool(p,
|
|
737
|
q) = true) -> addr_le(p, q)))
|
|
738
|
|
|
739
|
axiom addr_lt_bool_def :
|
|
740
|
(forall p:addr. forall q:addr [addr_lt_bool(p, q)]. (addr_lt(p, q) ->
|
|
741
|
(addr_lt_bool(p, q) = true)))
|
|
742
|
|
|
743
|
axiom addr_lt_bool_def1 :
|
|
744
|
(forall p:addr. forall q:addr [addr_lt_bool(p, q)]. ((addr_lt_bool(p,
|
|
745
|
q) = true) -> addr_lt(p, q)))
|
|
746
|
|
|
747
|
function null() : addr = { base = 0; offset = 0 }
|
|
748
|
|
|
749
|
function global(b: int) : addr = { base = b; offset = 0 }
|
|
750
|
|
|
751
|
function shift(p: addr, k: int) : addr = { base = (p).base; offset =
|
|
752
|
((p).offset + k) }
|
|
753
|
|
|
754
|
predicate included(p: addr, a: int, q: addr, b: int) = ((0 < a) ->
|
|
755
|
((0 <= b) and (((p).base = (q).base) and (((q).offset <= (p).offset) and
|
|
756
|
(((p).offset + a) <= ((q).offset + b))))))
|
|
757
|
|
|
758
|
predicate separated(p: addr, a: int, q: addr, b: int) = ((a <= 0) or
|
|
759
|
((b <= 0) or ((not ((p).base = (q).base)) or
|
|
760
|
((((q).offset + b) <= (p).offset) or (((p).offset + a) <= (q).offset)))))
|
|
761
|
|
|
762
|
predicate eqmem(m1: (addr,'a) farray, m2: (addr,'a) farray, p: addr,
|
|
763
|
a1: int) =
|
|
764
|
(forall q:addr [(m1[p])| (m2[q])]. (included(q, 1, p, a1) ->
|
|
765
|
((m1[q]) = (m2[q]))))
|
|
766
|
|
|
767
|
predicate havoc(m1: (addr,'a) farray, m2: (addr,'a) farray, p: addr,
|
|
768
|
a1: int) =
|
|
769
|
(forall q:addr [(m1[p])| (m2[q])]. (separated(q, 1, p, a1) ->
|
|
770
|
((m1[q]) = (m2[q]))))
|
|
771
|
|
|
772
|
predicate valid_rd(m: (int,int) farray, p: addr, n: int) = ((0 < n) ->
|
|
773
|
((0 <= (p).offset) and (((p).offset + n) <= (m[(p).base]))))
|
|
774
|
|
|
775
|
predicate valid_rw(m: (int,int) farray, p: addr, n: int) = ((0 < n) ->
|
|
776
|
((0 < (p).base) and ((0 <= (p).offset) and
|
|
777
|
(((p).offset + n) <= (m[(p).base])))))
|
|
778
|
|
|
779
|
axiom valid_rw_rd :
|
|
780
|
(forall m:(int,int) farray.
|
|
781
|
(forall p:addr. (forall n:int. (valid_rw(m, p, n) -> valid_rd(m, p, n)))))
|
|
782
|
|
|
783
|
axiom valid_string :
|
|
784
|
(forall m:(int,int) farray.
|
|
785
|
(forall p:addr. (((p).base < 0) -> (((0 <= (p).offset) and
|
|
786
|
((p).offset < (m[(p).base]))) -> valid_rd(m, p, 1)))))
|
|
787
|
|
|
788
|
axiom valid_string1 :
|
|
789
|
(forall m:(int,int) farray.
|
|
790
|
(forall p:addr. (((p).base < 0) -> (((0 <= (p).offset) and
|
|
791
|
((p).offset < (m[(p).base]))) -> (not valid_rw(m, p, 1))))))
|
|
792
|
|
|
793
|
axiom separated_1 :
|
|
794
|
(forall p:addr. forall q:addr.
|
|
795
|
(forall a:int. forall b:int. forall i:int. forall j:int [separated(p, a, q,
|
|
796
|
b), { base = (p).base; offset = i }, { base = (q).base; offset = j }].
|
|
797
|
(separated(p, a, q, b) -> ((((p).offset <= i) and
|
|
798
|
(i < ((p).offset + a))) -> ((((q).offset <= j) and
|
|
799
|
(j < ((q).offset + b))) -> (not ({ base = (p).base; offset = i } = {
|
|
800
|
base = (q).base; offset = j })))))))
|
|
801
|
|
|
802
|
logic region : int -> int
|
|
803
|
|
|
804
|
logic linked : (int,int) farray -> prop
|
|
805
|
|
|
806
|
logic sconst : (addr,int) farray -> prop
|
|
807
|
|
|
808
|
predicate framed(m: (addr,addr) farray) =
|
|
809
|
(forall p:addr [(m[p])]. (region(((m[p])).base) <= 0))
|
|
810
|
|
|
811
|
axiom separated_included :
|
|
812
|
(forall p:addr. forall q:addr.
|
|
813
|
(forall a:int. forall b:int [separated(p, a, q, b), included(p, a, q, b)].
|
|
814
|
((0 < a) -> ((0 < b) -> (separated(p, a, q, b) -> (not included(p, a, q,
|
|
815
|
b)))))))
|
|
816
|
|
|
817
|
axiom included_trans :
|
|
818
|
(forall p:addr. forall q:addr. forall r:addr.
|
|
819
|
(forall a:int. forall b:int. forall c:int [included(p, a, q, b),
|
|
820
|
included(q, b, r, c)]. (included(p, a, q, b) -> (included(q, b, r, c) ->
|
|
821
|
included(p, a, r, c)))))
|
|
822
|
|
|
823
|
axiom separated_trans :
|
|
824
|
(forall p:addr. forall q:addr. forall r:addr.
|
|
825
|
(forall a:int. forall b:int. forall c:int [included(p, a, q, b),
|
|
826
|
separated(q, b, r, c)]. (included(p, a, q, b) -> (separated(q, b, r, c) ->
|
|
827
|
separated(p, a, r, c)))))
|
|
828
|
|
|
829
|
axiom separated_sym :
|
|
830
|
(forall p:addr. forall q:addr.
|
|
831
|
(forall a:int. forall b:int [separated(p, a, q, b)]. (separated(p, a, q,
|
|
832
|
b) -> separated(q, b, p, a))))
|
|
833
|
|
|
834
|
axiom separated_sym1 :
|
|
835
|
(forall p:addr. forall q:addr.
|
|
836
|
(forall a:int. forall b:int [separated(p, a, q, b)]. (separated(q, b, p,
|
|
837
|
a) -> separated(p, a, q, b))))
|
|
838
|
|
|
839
|
axiom eqmem_included :
|
|
840
|
(forall m1:(addr,'a) farray. forall m2:(addr,'a) farray.
|
|
841
|
(forall p:addr. forall q:addr.
|
|
842
|
(forall a1:int. forall b:int [eqmem(m1, m2, p, a1), eqmem(m1, m2, q, b)].
|
|
843
|
(included(p, a1, q, b) -> (eqmem(m1, m2, q, b) -> eqmem(m1, m2, p, a1))))))
|
|
844
|
|
|
845
|
axiom eqmem_sym :
|
|
846
|
(forall m1:(addr,'a) farray. forall m2:(addr,'a) farray.
|
|
847
|
(forall p:addr.
|
|
848
|
(forall a1:int. (eqmem(m1, m2, p, a1) -> eqmem(m2, m1, p, a1)))))
|
|
849
|
|
|
850
|
axiom havoc_sym :
|
|
851
|
(forall m1:(addr,'a) farray. forall m2:(addr,'a) farray.
|
|
852
|
(forall p:addr.
|
|
853
|
(forall a1:int. (havoc(m1, m2, p, a1) -> havoc(m2, m1, p, a1)))))
|
|
854
|
|
|
855
|
logic cast : addr -> int
|
|
856
|
|
|
857
|
axiom cast_injective :
|
|
858
|
(forall p:addr. forall q:addr [cast(p), cast(q)]. ((cast(p) = cast(q)) ->
|
|
859
|
(p = q)))
|
|
860
|
|
|
861
|
logic hardware : int -> int
|
|
862
|
|
|
863
|
axiom hardnull : (hardware(0) = 0)
|
|
864
|
|
|
865
|
(* ---------------------------------------------------------- *)
|
|
866
|
(* --- Assertion (file clocks7/clocks7_7.c, line 155) --- *)
|
|
867
|
(* ---------------------------------------------------------- *)
|
|
868
|
|
|
869
|
goal toto_step_assert_4:
|
|
870
|
forall i : int.
|
|
871
|
forall t_4,t_3,t_2,t_1,t : (addr,int) farray.
|
|
872
|
forall a : addr.
|
|
873
|
P_trans_totoA ->
|
|
874
|
(0 < i) ->
|
|
875
|
P_clock_toto_2(1) ->
|
|
876
|
is_sint32(i) ->
|
|
877
|
is_sint32(t[a]) ->
|
|
878
|
is_sint32(t_1[a]) ->
|
|
879
|
is_sint32(t_2[a]) ->
|
|
880
|
is_sint32(t_3[a]) ->
|
|
881
|
is_sint32(t_4[a]) ->
|
|
882
|
P_trans_totoB(i, 1) ->
|
|
883
|
(0 = (ite((eqb(0, i)), 1, 0))) ->
|
|
884
|
(region(a.base) <= 0) ->
|
|
885
|
(P_clock_d -> P_trans_d(i, 1)) ->
|
|
886
|
P_trans_toto_2(i, 0)
|
|
887
|
|