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|
(**************************************************************************)
(* *)
(* This file is part of Frama-C. *)
(* *)
(* Copyright (C) 2007-2010 *)
(* CEA (Commissariat l'nergie atomique et aux nergies *)
(* alternatives) *)
(* INRIA (Institut National de Recherche en Informatique et en *)
(* Automatique) *)
(* *)
(* you can redistribute it and/or modify it under the terms of the GNU *)
(* Lesser General Public License as published by the Free Software *)
(* Foundation, version 2.1. *)
(* *)
(* It is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU Lesser General Public License for more details. *)
(* *)
(* See the GNU Lesser General Public License version v2.1 *)
(* for more details (enclosed in the file licenses/LGPLv2.1). *)
(* *)
(**************************************************************************)
(* Integer modulo definitions as in compcert integers*)
logic modul : int
axiom modul_pos : modul > 0
logic eqmod: int ,int -> prop
axiom eqmod_def :
forall x,y:int.
exists k:int. x =k*modul + y
axiom eqmod_refl :
forall x:int.
eqmod(x,x)
axiom eqmod_refl2 :
forall x,y:int.
x=y ->
eqmod(x,y)
axiom eqmod_sym :
forall x,y:int.
eqmod(x,y) ->
eqmod(y,x)
axiom eqmod_trans:
forall x,y,z:int.
eqmod(x,y) ->
eqmod(y,z) ->
eqmod(x,z)
axiom eqmod_small_eq :
forall x,y:int.
eqmod(x,y) ->
0<= x < modul ->
0<= y < modul ->
x=y
axiom eqmod_mod_eq:
forall x,y:int.
eqmod(x,y) ->
x % modul = y% modul
axiom eqmod_mod:
forall x:int.
eqmod(x,x %modul)
axiom eqmod_neg:
forall x,y:int.
eqmod(x,y) ->
eqmod(-x,-y)
axiom eqmod_add:
forall a,b,c,d:int.
eqmod(a,b) ->
eqmod(c,d) ->
eqmod(a+c,b+d)
axiom eqmod_sub:
forall a,b,c,d:int.
eqmod(a,b) ->
eqmod(c,d) ->
eqmod(a-c,b-d)
axiom eqmod_mult:
forall a,b,c,d:int.
eqmod(a,b) ->
eqmod(c,d) ->
eqmod(a*c,b*d)
axiom mod_in_range :
forall x:int.
0 <= x%modul < modul
(* integer definition *)
type Int
logic intval : Int -> int (* Z representation of the int *)
logic intrange : Int -> prop (* range property of the int vs a modul *)
logic unsigned : Int -> int
logic signed : Int -> int
logic repr : int -> Int
logic Int_eq : Int equality
logic half_modul : int
logic max_unsigned : int
logic max_signed : int
logic min_signed : int
axiom half_modul_value : half_modul = modul /2
axiom max_unsigned_value : max_signed = modul - 1
axiom max_signed_value : max_signed = half_modul - 1
axiom min_signed_value : min_signed =- half_modul
axiom unsigned_def :
forall i:Int.
unsigned(i) = intval(i)
axiom signed_lt :
forall i:Int.
intval(i) < half_modul ->
signed(i) = unsigned(i)
axiom signed_ge :
forall i:Int.
intval(i) >= half_modul ->
signed(i) = unsigned(i) - modul
axiom mod_in_range :
forall x:Int. 0 <= (intval(x))%modul < modul
axiom repr_intval :
forall x:int. intval(repr(x))=x
axiom repr_intrange :
forall x:int. intrange(repr(x))
logic zero : Int
logic one : Int
logic mone : Int
axiom zero_def : zero = repr(0)
axiom one_def : one = repr(1)
axiom mone_def : mone = repr(-1)
(** coercions between int and Int **)
axiom eqm_unsigned_repr :
forall z:int. eqmod(z,unsigned(repr(z)))
axiom eqm_unsigned_repr_l :
forall a,b:int.
eqmod(a,b) ->
eqmod(unsigned(repr(a)),b)
axiom eqm_unsigned_repr_r :
forall a,b:int.
eqmod(a,b) ->
eqmod(a,unsigned(repr(b)))
axiom eqm_signed_unsigned :
forall x:Int.
eqmod(signed(x),unsigned(x))
axiom unsigned_range:
forall i:Int. 0 <= unsigned(i) < max_unsigned
axiom unsigned_range_2:
forall i:Int. 0 <= unsigned(i) <= max_unsigned
(** boolean intepretation **)
axiom is_true :
forall i: Int. i <> zero
axiom is_false :
forall i:Int. i = zero
(** arithmetics operations **)
(** comparisons definition **)
logic cmpi_eq : Int,Int -> Int
logic cmpi_lt : Int,Int -> Int
logic cmpi_ltu : Int,Int -> Int
axiom cmpi_eq_def_true:
forall x,y:Int.
unsigned(x) = unsigned(y) ->
cmpi_eq(x,y) = one
axiom cmpi_eq_def_false:
forall x,y:Int.
unsigned(x) <> unsigned(y) ->
cmpi_eq(x,y) = zero
axiom cmpi_lt_def_true :
forall x,y:Int.
signed(x) < signed(y) ->
cmpi_lt(x,y) = one
axiom cmpi_lt_def_false :
forall x,y:Int.
signed(x) >= signed(y) ->
cmpi_lt(x,y) = zero
axiom cmpi_ltu_def_true :
forall x,y:Int.
unsigned(x) < unsigned(y) ->
cmpi_lt(x,y) = one
axiom cmpi_ltu_def_false :
forall x,y:Int.
unsigned(x) >= unsigned(y) ->
cmpi_lt(x,y) = zero
(** negation of a Int definition and properties **)
logic i_neg: Int -> Int
axiom neg_def :
forall i:Int. i_neg(i) = repr(-unsigned(i))
axiom neg_repr :
forall i:int . i_neg(repr(i)) = repr(-i)
axiom neg_zero :
i_neg(zero) = zero
axiom neg_involutive :
forall i:Int. i_neg (i_neg(i)) = i
(** addition of int definition and properties **)
logic i_add : Int,Int -> Int
axiom i_add_unsigned :
forall x,y:Int.
i_add(x,y) = repr (unsigned(x)+unsigned(y))
axiom add_signed :
forall x,y:Int.
i_add(x,y) = repr(signed(x)+signed(y))
axiom add_commut :
forall x,y:Int.
i_add(x,y) = i_add(y,x)
axiom add_permut :
forall x,y,z:Int.
i_add(x,i_add(y,z)) = i_add(y,i_add(x,z))
axiom add_neg_zero :
forall x:Int. i_add(x,i_neg(x)) = zero
axiom neg_add_distr :
forall x,y:Int.
i_neg(i_add(x,y)) = i_add (i_neg(x),i_neg(y))
(** subtraction of Int definition and properties **)
logic i_sub : Int,Int -> Int
axiom i_sub_def :
forall x,y:Int.
i_sub(x,y) = repr (unsigned(x)-unsigned(y))
axiom sub_zero_l :
forall x:Int. i_sub(x,zero) = x
axiom sub_zero_r :
forall x:Int. i_sub(zero,x) = i_neg(x)
axiom sub_add_opp :
forall x,y:Int. i_sub(x,y) = i_add(x,i_neg(y))
axiom sub_idem :
forall x:Int. i_sub(x,x) = zero
axiom sub_add_l :
forall x,y,z:Int. i_sub (i_add(x,y),z) = i_add(i_sub(x,z),y)
axiom sub_add_r:
forall x,y,z:Int. i_sub(x,i_add(y,z)) = i_add(i_sub(x,z),i_neg(y))
axiom sub_shifted:
forall x,y,z:Int. i_sub(i_add(x,z),i_add(y,z)) =i_sub(x,y)
(** multiplication of Int definition and properties **)
logic i_mul : Int,Int -> Int
axiom i_mul_def :
forall x,y:Int.
i_mul(x,y) = repr (unsigned(x)*unsigned(y))
axiom mul_commut:
forall x,y:Int. i_mul(x,y) = i_mul(y,x)
axiom mul_zero:
forall x:Int. i_mul(x,zero) = zero
axiom mul_one:
forall x:Int. i_mul(x,one) = x
axiom mul_assoc:
forall x,y,z:Int. i_mul(i_mul(x,y),z) = i_mul(x,i_mul(y,z))
axiom mul_add_distr_l :
forall x,y,z:Int. i_mul(i_add(x,y),z) = i_add(i_mul(x,z),i_mul(y,z))
axiom mul_add_distr_r :
forall x,y,z:Int. i_mul(x,i_add(y,z)) = i_add(i_mul(x,y),i_mul(x,z))
axiom neg_mul_distr_l:
forall x,y:Int. i_neg(i_mul(x,y)) = i_mul (i_neg(x),y)
axiom neg_mul_distr_l:
forall x,y:Int. i_neg(i_mul(x,y)) = i_mul (x,i_neg(y))
(** divisions on signed Int definition and properties **)
logic divs:Int,Int -> Int
axiom divs_neg_neg :
forall x,y:Int.
signed(x) < 0 ->
signed(y) < 0 ->
divs(x,y) = repr (-signed(x)/(-signed(y)))
axiom divs_neg_pos :
forall x,y:Int.
signed(x) < 0 ->
signed(y) >= 0 ->
divs(x,y) = repr (-((-signed(x))/signed(y)))
axiom divs_pos_neg :
forall x,y:Int.
signed(x)>= 0 ->
signed(y) < 0 ->
divs(x,y) = repr(-(signed(x)/(-signed(y))))
axiom divs_pos_pos :
forall x,y:Int.
signed(x)>= 0 ->
signed(y)>= 0 ->
divs(x,y) = repr (signed(x)/signed(y))
(** divisions of unsigned Int definition and properties **)
logic divu : Int,Int -> Int
axiom divu_def :
forall x,y:Int. divu(x,y) = repr (unsigned(x)/unsigned(y))
(** Modulo of signed Int definition and properties **)
logic mods :Int,Int -> Int
axiom mods_neg_neg:
forall x,y:Int.
signed(x) < 0 ->
signed(y) < 0 ->
mods(x,y) = repr (signed(x)-(((-signed(x))/(-signed(y)))*signed(y)))
axiom divs_neg_pos :
forall x,y:Int.
signed(x) < 0 ->
signed(y) >= 0 ->
mods(x,y) = repr (signed(x) +(((-signed(x))/signed(y))*signed(y)))
axiom divs_pos_neg :
forall x,y:Int.
signed(x)>= 0 ->
signed(y) < 0 ->
mods(x,y) = repr(signed(x)+((signed(x)/(-signed(y)))*signed(y)))
axiom divs_pos_pos :
forall x,y:Int.
signed(x)>= 0 ->
signed(y)>= 0 ->
divs(x,y) = repr(signed(x)-((signed(x)/signed(y))*signed(y)))
(** Modulo of unsigned Int definition and properties **)
logic modu:Int,Int ->Int
axiom modu_def:
forall x,y:Int. modu(x,y)=repr (unsigned(x)%unsigned(y))
|
