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Why3
why3
Commits
f691a29d
Commit
f691a29d
authored
Feb 17, 2013
by
MARCHE Claude
Browse files
bitvectors: more proofs
parent
2347a6e6
Changes
7
Expand all
Hide whitespace changes
Inline
Side-by-side
examples/bitvectors/bitvector/bitvector_BitVector_nth_from_int2c_plus_pow2_1.v
View file @
f691a29d
(
*
This
file
is
generated
by
Why3
'
s
Coq
8.4
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
BuiltIn
.
Require
Import
ZOdiv
.
Require
BuiltIn
.
Require
bool
.
Bool
.
Require
int
.
Int
.
Require
int
.
Abs
.
Require
int
.
Computer
Division
.
Require
int
.
Euclidean
Division
.
Parameter
pow2
:
Z
->
Z
.
...
...
@@ -150,15 +149,26 @@ Axiom pow2_62 : ((pow2 62%Z) = 4611686018427387904%Z).
Axiom
pow2_63
:
((
pow2
63
%
Z
)
=
9223372036854775808
%
Z
).
Axiom
Div_
pow
:
forall
(
x
:
Z
)
(
i
:
Z
),
(((
pow2
(
i
-
1
%
Z
)
%
Z
)
<=
x
)
%
Z
/
\
(
x
<
(
pow2
i
))
%
Z
)
->
((
ZOdiv
x
(
pow2
(
i
-
1
%
Z
)
%
Z
)
)
=
1
%
Z
).
Axiom
Div_
double
:
forall
(
x
:
Z
)
(
y
:
Z
),
(((
0
%
Z
<
y
)
%
Z
/
\
(
y
<=
x
)
%
Z
)
/
\
(
x
<
(
2
%
Z
*
y
)
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
y
)
=
1
%
Z
).
Axiom
Div_pow
2
:
forall
(
x
:
Z
)
(
i
:
Z
),
(
((
-
(
pow2
i
))
%
Z
<
=
x
)
%
Z
/
\
(
x
<
(
-
(
pow2
(
i
-
1
%
Z
)
%
Z
)
)
%
Z
)
%
Z
)
->
((
ZO
div
x
(
pow2
(
i
-
1
%
Z
)
%
Z
))
=
(
-
2
%
Z
)
%
Z
).
Axiom
Div_pow
:
forall
(
x
:
Z
)
(
i
:
Z
),
(
0
%
Z
<
i
)
%
Z
->
(
((
(
pow2
(
i
-
1
%
Z
)
%
Z
)
<=
x
)
%
Z
/
\
(
x
<
(
pow2
i
)
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
(
pow2
(
i
-
1
%
Z
)
%
Z
))
=
1
%
Z
)).
Axiom
Mod_pow2_gen
:
forall
(
x
:
Z
)
(
i
:
Z
)
(
k
:
Z
),
((
0
%
Z
<=
k
)
%
Z
/
\
(
k
<
i
)
%
Z
)
->
((
ZOmod
(
ZOdiv
(
x
+
(
pow2
i
))
%
Z
(
pow2
k
))
2
%
Z
)
=
(
ZOmod
(
ZOdiv
x
(
pow2
k
))
2
%
Z
)).
Axiom
Div_double_neg
:
forall
(
x
:
Z
)
(
y
:
Z
),
(((((
-
2
%
Z
)
%
Z
*
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<
(
-
y
)
%
Z
)
%
Z
)
/
\
((
-
y
)
%
Z
<
0
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
y
)
=
(
-
2
%
Z
)
%
Z
).
Axiom
Div_pow2
:
forall
(
x
:
Z
)
(
i
:
Z
),
(
0
%
Z
<
i
)
%
Z
->
((((
-
(
pow2
i
))
%
Z
<=
x
)
%
Z
/
\
(
x
<
(
-
(
pow2
(
i
-
1
%
Z
)
%
Z
))
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
(
pow2
(
i
-
1
%
Z
)
%
Z
))
=
(
-
2
%
Z
)
%
Z
)).
Axiom
Mod_pow2_gen
:
forall
(
x
:
Z
)
(
i
:
Z
)
(
k
:
Z
),
(
0
%
Z
<=
x
)
%
Z
->
(((
0
%
Z
<=
k
)
%
Z
/
\
(
k
<
i
)
%
Z
)
->
((
int
.
EuclideanDivision
.
mod1
(
int
.
EuclideanDivision
.
div
(
x
+
(
pow2
i
))
%
Z
(
pow2
k
))
2
%
Z
)
=
(
int
.
EuclideanDivision
.
mod1
(
int
.
EuclideanDivision
.
div
x
(
pow2
k
))
2
%
Z
))).
Parameter
size
:
Z
.
...
...
@@ -287,18 +297,19 @@ Axiom to_nat_sub_footprint : forall (b1:bv) (b2:bv) (j:Z) (i:Z),
Parameter
from_int
:
Z
->
bv
.
Axiom
nth_from_int_high_even
:
forall
(
n
:
Z
)
(
i
:
Z
),
(((
i
<
size
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
/
\
((
ZO
mod
(
ZOdiv
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
i
)
=
false
).
(
0
%
Z
<=
i
)
%
Z
)
/
\
((
int
.
EuclideanDivision
.
mod
1
(
int
.
EuclideanDivision
.
div
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
i
)
=
false
).
Axiom
nth_from_int_high_odd
:
forall
(
n
:
Z
)
(
i
:
Z
),
(((
i
<
size
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
/
\
~
((
ZOmod
(
ZOdiv
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
i
)
=
true
).
(
0
%
Z
<=
i
)
%
Z
)
/
\
~
((
int
.
EuclideanDivision
.
mod1
(
int
.
EuclideanDivision
.
div
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
i
)
=
true
).
Axiom
nth_from_int_low_even
:
forall
(
n
:
Z
),
((
ZO
mod
n
2
%
Z
)
=
0
%
Z
)
->
((
nth
(
from_int
n
)
0
%
Z
)
=
false
).
Axiom
nth_from_int_low_even
:
forall
(
n
:
Z
),
((
int
.
EuclideanDivision
.
mod
1
n
2
%
Z
)
=
0
%
Z
)
->
((
nth
(
from_int
n
)
0
%
Z
)
=
false
).
Axiom
nth_from_int_low_odd
:
forall
(
n
:
Z
),
(
~
((
ZO
mod
n
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
0
%
Z
)
=
true
).
Axiom
nth_from_int_low_odd
:
forall
(
n
:
Z
),
(
~
((
int
.
EuclideanDivision
.
mod
1
n
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int
n
)
0
%
Z
)
=
true
).
Axiom
nth_from_int_0
:
forall
(
i
:
Z
),
((
i
<
size
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
->
((
nth
(
from_int
0
%
Z
)
i
)
=
false
).
...
...
@@ -313,19 +324,20 @@ Axiom nth_sign_negative : forall (n:Z), (n < 0%Z)%Z -> ((nth (from_int2c n)
Axiom
nth_from_int2c_high_even
:
forall
(
n
:
Z
)
(
i
:
Z
),
(((
i
<
(
size
-
1
%
Z
)
%
Z
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
/
\
((
ZO
mod
(
ZOdiv
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
i
)
=
false
).
((
int
.
EuclideanDivision
.
mod
1
(
int
.
EuclideanDivision
.
div
n
(
pow2
i
)
)
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
i
)
=
false
).
Axiom
nth_from_int2c_high_odd
:
forall
(
n
:
Z
)
(
i
:
Z
),
(((
i
<
(
size
-
1
%
Z
)
%
Z
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
/
\
~
((
ZO
mod
(
ZOdiv
n
(
pow2
i
))
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
i
)
=
true
).
~
((
int
.
EuclideanDivision
.
mod
1
(
int
.
EuclideanDivision
.
div
n
(
pow2
i
)
)
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
i
)
=
true
).
Axiom
nth_from_int2c_low_even
:
forall
(
n
:
Z
),
((
ZO
mod
n
2
%
Z
)
=
0
%
Z
)
->
((
nth
(
from_int2c
n
)
0
%
Z
)
=
false
).
Axiom
nth_from_int2c_low_even
:
forall
(
n
:
Z
),
((
int
.
EuclideanDivision
.
mod
1
n
2
%
Z
)
=
0
%
Z
)
->
((
nth
(
from_int2c
n
)
0
%
Z
)
=
false
).
Axiom
nth_from_int2c_low_odd
:
forall
(
n
:
Z
),
(
~
((
ZOmod
n
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
0
%
Z
)
=
true
).
Axiom
nth_from_int2c_low_odd
:
forall
(
n
:
Z
),
(
~
((
int
.
EuclideanDivision
.
mod1
n
2
%
Z
)
=
0
%
Z
))
->
((
nth
(
from_int2c
n
)
0
%
Z
)
=
true
).
Axiom
nth_from_int2c_0
:
forall
(
i
:
Z
),
((
i
<
size
)
%
Z
/
\
(
0
%
Z
<=
i
)
%
Z
)
->
((
nth
(
from_int2c
0
%
Z
)
i
)
=
false
).
...
...
@@ -341,8 +353,8 @@ Theorem nth_from_int2c_plus_pow2 : forall (x:Z) (k:Z) (i:Z),
(
*
intros
x
k
i
((
h1
,
h2
),
h3
).
*
)
intros
x
k
i
(
h1
&
h2
).
assert
(
h
:
ZO
mod
(
ZO
div
x
(
pow2
k
))
2
=
0
\
/
ZO
mod
(
ZO
div
x
(
pow2
k
))
2
<>
0
)
assert
(
h
:
int
.
EuclideanDivision
.
mod
1
(
int
.
EuclideanDivision
.
div
x
(
pow2
k
))
2
=
0
\
/
int
.
EuclideanDivision
.
mod
1
(
int
.
EuclideanDivision
.
div
x
(
pow2
k
))
2
<>
0
)
by
omega
.
destruct
h
.
rewrite
nth_from_int2c_high_even
;
intuition
.
...
...
examples/bitvectors/bitvector/why3session.xml
View file @
f691a29d
This diff is collapsed.
Click to expand it.
examples/bitvectors/double/why3session.xml
View file @
f691a29d
...
...
@@ -50,7 +50,7 @@
name=
"nth_one1"
locfile=
"../double.why"
loclnum=
"73"
loccnumb=
"8"
loccnume=
"16"
sum=
"
64b742f26230c95a6f017247e361064e
"
sum=
"
52d8a3fb72c7fbd67b6242f14de4ec07
"
proved=
"true"
expanded=
"true"
shape=
"ainfix =anthaoneV0aFalseIainfix <=V0c51Aainfix <=c0V0F"
>
...
...
@@ -75,7 +75,7 @@
name=
"nth_one2"
locfile=
"../double.why"
loclnum=
"74"
loccnumb=
"8"
loccnume=
"16"
sum=
"
84a85185c5147202068cebe22848d18f
"
sum=
"
5d9e03517d2cc0bf19926e772b281575
"
proved=
"true"
expanded=
"true"
shape=
"ainfix =anthaoneV0aTrueIainfix <=V0c61Aainfix <=c52V0F"
>
...
...
@@ -100,7 +100,7 @@
name=
"nth_one3"
locfile=
"../double.why"
loclnum=
"75"
loccnumb=
"8"
loccnume=
"16"
sum=
"
3fa54a156946495d247aa0b4e5d2308b
"
sum=
"
f5f820393566394effd1eecae3fa8ef8
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =anthaoneV0aFalseIainfix <=V0c63Aainfix <=c62V0F"
>
...
...
@@ -117,7 +117,7 @@
name=
"sign_one"
locfile=
"../double.why"
loclnum=
"77"
loccnumb=
"8"
loccnume=
"16"
sum=
"
be9ca7041b727df4ca7493d2af0cd4c6
"
sum=
"
c88fd3753e8153b90619ca7be5ed1117
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =asignaoneaFalse"
>
...
...
@@ -166,7 +166,7 @@
name=
"exp_one"
locfile=
"../double.why"
loclnum=
"78"
loccnumb=
"8"
loccnume=
"15"
sum=
"
a95859f4aee7c4d050a5a4db5dd9c1fd
"
sum=
"
62a94560121b0bf05e66471064063151
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =aexpaonec1023"
>
...
...
@@ -192,7 +192,7 @@
name=
"mantissa_one"
locfile=
"../double.why"
loclnum=
"79"
loccnumb=
"8"
loccnume=
"20"
sum=
"
1e2b71c93f02723b910f5b0520c73873
"
sum=
"
8f958039e1df42ec82b32367fcd739c7
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =amantissaaonec0"
>
...
...
@@ -225,7 +225,7 @@
name=
"double_value_of_1"
locfile=
"../double.why"
loclnum=
"81"
loccnumb=
"8"
loccnume=
"25"
sum=
"
ad540400343cee2ffb0b8a07d70165c
f"
sum=
"
8058d2bf7c95995fe09bb6908781d19
f"
proved=
"true"
expanded=
"false"
shape=
"ainfix =adouble_of_bv64aonec1.0"
>
...
...
examples/bitvectors/double_of_int/why3session.xml
View file @
f691a29d
This diff is collapsed.
Click to expand it.
examples/bitvectors/neg_as_xor/why3session.xml
View file @
f691a29d
...
...
@@ -35,7 +35,7 @@
name=
"Nth_j"
locfile=
"../neg_as_xor.why"
loclnum=
"13"
loccnumb=
"8"
loccnume=
"13"
sum=
"
34b383120a6c9582c26cf0005c710a92
"
sum=
"
6c8374d08ad76160b6097f0c92f918f8
"
proved=
"true"
expanded=
"true"
shape=
"ainfix =anthajV0aFalseIainfix <=V0c62Aainfix <=c0V0F"
>
...
...
@@ -60,7 +60,7 @@
name=
"sign_of_j"
locfile=
"../neg_as_xor.why"
loclnum=
"15"
loccnumb=
"8"
loccnume=
"17"
sum=
"
8874135905e94507fde3c62c691db58
1"
sum=
"
1760b20625d63815ec0aa452d6ad34d
1"
proved=
"true"
expanded=
"false"
shape=
"ainfix =asignajaTrue"
>
...
...
@@ -77,7 +77,7 @@
name=
"mantissa_of_j"
locfile=
"../neg_as_xor.why"
loclnum=
"16"
loccnumb=
"8"
loccnume=
"21"
sum=
"
7a9b27b0536d17d854cf27f8234d761b
"
sum=
"
a3e5fd7d3642b40f5cc15957027b57c3
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =amantissaajc0"
>
...
...
@@ -103,14 +103,14 @@
memlimit=
"1000"
obsolete=
"false"
archived=
"false"
>
<result
status=
"valid"
time=
"
3.10
"
/>
<result
status=
"valid"
time=
"
2.68
"
/>
</proof>
</goal>
<goal
name=
"exp_of_j"
locfile=
"../neg_as_xor.why"
loclnum=
"17"
loccnumb=
"8"
loccnume=
"16"
sum=
"
d30d723f5c789d43e3644685e0f09ad3
"
sum=
"
3b531125475692b4d418a663a72c1838
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =aexpajc0"
>
...
...
@@ -143,7 +143,7 @@
name=
"int_of_bv"
locfile=
"../neg_as_xor.why"
loclnum=
"18"
loccnumb=
"8"
loccnume=
"17"
sum=
"
a5352ad948c731bc7e1b1d044870bb95
"
sum=
"
65a13df3701b2fbbbaf06857bd5fece6
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =adouble_of_bv64ajc0.0"
>
...
...
@@ -176,7 +176,7 @@
name=
"MainResultBits"
locfile=
"../neg_as_xor.why"
loclnum=
"20"
loccnumb=
"8"
loccnume=
"22"
sum=
"
52f645e44bf665a7bc0953d0ab22cbec
"
sum=
"
be8ed15fd950211d237c83c4aa76424a
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =anthabw_xorV0ajV1anthV0V1Iainfix <V1c63Aainfix <=c0V1FF"
>
...
...
@@ -193,7 +193,7 @@
name=
"MainResultSign"
locfile=
"../neg_as_xor.why"
loclnum=
"23"
loccnumb=
"8"
loccnume=
"22"
sum=
"
84408f58ca669e966ac3dd0e8e26c501
"
sum=
"
634a2fb3ae42a2e5996ab58a4428682c
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =anthabw_xorV0ajc63anotbanthV0c63F"
>
...
...
@@ -210,7 +210,7 @@
name=
"Sign_of_xor_j"
locfile=
"../neg_as_xor.why"
loclnum=
"25"
loccnumb=
"8"
loccnume=
"21"
sum=
"
8e35e7b3978720fcffa4654f4d40a8bb
"
sum=
"
083892c9a58aff7b918bc3d5aca51d8e
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =asignabw_xorV0ajanotbasignV0F"
>
...
...
@@ -243,7 +243,7 @@
name=
"Exp_of_xor_j"
locfile=
"../neg_as_xor.why"
loclnum=
"27"
loccnumb=
"8"
loccnume=
"20"
sum=
"
bd2a20ad78883a9b246d819dfc050046
"
sum=
"
e23bdb76d6bcfd9e06cfe30f4eea16e7
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =aexpabw_xorV0ajaexpV0F"
>
...
...
@@ -276,7 +276,7 @@
name=
"Mantissa_of_xor_j"
locfile=
"../neg_as_xor.why"
loclnum=
"29"
loccnumb=
"8"
loccnume=
"25"
sum=
"
ce1ceaaaf0b5d079ac350e484e3a5051
"
sum=
"
6294006381c02dc3f9556ae437087baa
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =amantissaabw_xorV0ajamantissaV0F"
>
...
...
@@ -309,7 +309,7 @@
name=
"MainResultZero"
locfile=
"../neg_as_xor.why"
loclnum=
"31"
loccnumb=
"8"
loccnume=
"22"
sum=
"
e71da47a4d26e5f9bb78680af2110e9d
"
sum=
"
f6fbb4d35e0473dc1e92b447b37a3189
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =adouble_of_bv64abw_xorV0ajaprefix -.adouble_of_bv64V0Iainfix =amantissaV0c0Aainfix =c0aexpV0F"
>
...
...
@@ -342,7 +342,7 @@
name=
"sign_neg"
locfile=
"../neg_as_xor.why"
loclnum=
"34"
loccnumb=
"8"
loccnume=
"16"
sum=
"
1a8d522bc703c9be8a1629374fd9d762
"
sum=
"
e4ee54c231d60fa6ef6184307c4a30c3
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =asign_valueanotbasignV0aprefix -.asign_valueasignV0F"
>
...
...
@@ -359,7 +359,7 @@
name=
"MainResult"
locfile=
"../neg_as_xor.why"
loclnum=
"36"
loccnumb=
"8"
loccnume=
"18"
sum=
"
172f68d15ea46e505257c40efb1e8b07
"
sum=
"
6826df4299dee21a669e748d7738a8c4
"
proved=
"true"
expanded=
"true"
shape=
"ainfix =adouble_of_bv64abw_xorV0ajaprefix -.adouble_of_bv64V0Iainfix <aexpV0c2047Aainfix <c0aexpV0F"
>
...
...
examples/bitvectors/power2/power2_Pow2int_Mod_pow2_gen_1.v
0 → 100644
View file @
f691a29d
(
*
This
file
is
generated
by
Why3
'
s
Coq
8.4
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
BuiltIn
.
Require
BuiltIn
.
Require
int
.
Int
.
Require
int
.
Abs
.
Require
int
.
EuclideanDivision
.
Parameter
pow2
:
Z
->
Z
.
Axiom
Power_0
:
((
pow2
0
%
Z
)
=
1
%
Z
).
Axiom
Power_s
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((
pow2
(
n
+
1
%
Z
)
%
Z
)
=
(
2
%
Z
*
(
pow2
n
))
%
Z
).
Axiom
Power_1
:
((
pow2
1
%
Z
)
=
2
%
Z
).
Axiom
Power_sum
:
forall
(
n
:
Z
)
(
m
:
Z
),
((
0
%
Z
<=
n
)
%
Z
/
\
(
0
%
Z
<=
m
)
%
Z
)
->
((
pow2
(
n
+
m
)
%
Z
)
=
((
pow2
n
)
*
(
pow2
m
))
%
Z
).
Axiom
pow2pos
:
forall
(
i
:
Z
),
(
0
%
Z
<=
i
)
%
Z
->
(
0
%
Z
<
(
pow2
i
))
%
Z
.
Axiom
pow2_0
:
((
pow2
0
%
Z
)
=
1
%
Z
).
Axiom
pow2_1
:
((
pow2
1
%
Z
)
=
2
%
Z
).
Axiom
pow2_2
:
((
pow2
2
%
Z
)
=
4
%
Z
).
Axiom
pow2_3
:
((
pow2
3
%
Z
)
=
8
%
Z
).
Axiom
pow2_4
:
((
pow2
4
%
Z
)
=
16
%
Z
).
Axiom
pow2_5
:
((
pow2
5
%
Z
)
=
32
%
Z
).
Axiom
pow2_6
:
((
pow2
6
%
Z
)
=
64
%
Z
).
Axiom
pow2_7
:
((
pow2
7
%
Z
)
=
128
%
Z
).
Axiom
pow2_8
:
((
pow2
8
%
Z
)
=
256
%
Z
).
Axiom
pow2_9
:
((
pow2
9
%
Z
)
=
512
%
Z
).
Axiom
pow2_10
:
((
pow2
10
%
Z
)
=
1024
%
Z
).
Axiom
pow2_11
:
((
pow2
11
%
Z
)
=
2048
%
Z
).
Axiom
pow2_12
:
((
pow2
12
%
Z
)
=
4096
%
Z
).
Axiom
pow2_13
:
((
pow2
13
%
Z
)
=
8192
%
Z
).
Axiom
pow2_14
:
((
pow2
14
%
Z
)
=
16384
%
Z
).
Axiom
pow2_15
:
((
pow2
15
%
Z
)
=
32768
%
Z
).
Axiom
pow2_16
:
((
pow2
16
%
Z
)
=
65536
%
Z
).
Axiom
pow2_17
:
((
pow2
17
%
Z
)
=
131072
%
Z
).
Axiom
pow2_18
:
((
pow2
18
%
Z
)
=
262144
%
Z
).
Axiom
pow2_19
:
((
pow2
19
%
Z
)
=
524288
%
Z
).
Axiom
pow2_20
:
((
pow2
20
%
Z
)
=
1048576
%
Z
).
Axiom
pow2_21
:
((
pow2
21
%
Z
)
=
2097152
%
Z
).
Axiom
pow2_22
:
((
pow2
22
%
Z
)
=
4194304
%
Z
).
Axiom
pow2_23
:
((
pow2
23
%
Z
)
=
8388608
%
Z
).
Axiom
pow2_24
:
((
pow2
24
%
Z
)
=
16777216
%
Z
).
Axiom
pow2_25
:
((
pow2
25
%
Z
)
=
33554432
%
Z
).
Axiom
pow2_26
:
((
pow2
26
%
Z
)
=
67108864
%
Z
).
Axiom
pow2_27
:
((
pow2
27
%
Z
)
=
134217728
%
Z
).
Axiom
pow2_28
:
((
pow2
28
%
Z
)
=
268435456
%
Z
).
Axiom
pow2_29
:
((
pow2
29
%
Z
)
=
536870912
%
Z
).
Axiom
pow2_30
:
((
pow2
30
%
Z
)
=
1073741824
%
Z
).
Axiom
pow2_31
:
((
pow2
31
%
Z
)
=
2147483648
%
Z
).
Axiom
pow2_32
:
((
pow2
32
%
Z
)
=
4294967296
%
Z
).
Axiom
pow2_33
:
((
pow2
33
%
Z
)
=
8589934592
%
Z
).
Axiom
pow2_34
:
((
pow2
34
%
Z
)
=
17179869184
%
Z
).
Axiom
pow2_35
:
((
pow2
35
%
Z
)
=
34359738368
%
Z
).
Axiom
pow2_36
:
((
pow2
36
%
Z
)
=
68719476736
%
Z
).
Axiom
pow2_37
:
((
pow2
37
%
Z
)
=
137438953472
%
Z
).
Axiom
pow2_38
:
((
pow2
38
%
Z
)
=
274877906944
%
Z
).
Axiom
pow2_39
:
((
pow2
39
%
Z
)
=
549755813888
%
Z
).
Axiom
pow2_40
:
((
pow2
40
%
Z
)
=
1099511627776
%
Z
).
Axiom
pow2_41
:
((
pow2
41
%
Z
)
=
2199023255552
%
Z
).
Axiom
pow2_42
:
((
pow2
42
%
Z
)
=
4398046511104
%
Z
).
Axiom
pow2_43
:
((
pow2
43
%
Z
)
=
8796093022208
%
Z
).
Axiom
pow2_44
:
((
pow2
44
%
Z
)
=
17592186044416
%
Z
).
Axiom
pow2_45
:
((
pow2
45
%
Z
)
=
35184372088832
%
Z
).
Axiom
pow2_46
:
((
pow2
46
%
Z
)
=
70368744177664
%
Z
).
Axiom
pow2_47
:
((
pow2
47
%
Z
)
=
140737488355328
%
Z
).
Axiom
pow2_48
:
((
pow2
48
%
Z
)
=
281474976710656
%
Z
).
Axiom
pow2_49
:
((
pow2
49
%
Z
)
=
562949953421312
%
Z
).
Axiom
pow2_50
:
((
pow2
50
%
Z
)
=
1125899906842624
%
Z
).
Axiom
pow2_51
:
((
pow2
51
%
Z
)
=
2251799813685248
%
Z
).
Axiom
pow2_52
:
((
pow2
52
%
Z
)
=
4503599627370496
%
Z
).
Axiom
pow2_53
:
((
pow2
53
%
Z
)
=
9007199254740992
%
Z
).
Axiom
pow2_54
:
((
pow2
54
%
Z
)
=
18014398509481984
%
Z
).
Axiom
pow2_55
:
((
pow2
55
%
Z
)
=
36028797018963968
%
Z
).
Axiom
pow2_56
:
((
pow2
56
%
Z
)
=
72057594037927936
%
Z
).
Axiom
pow2_57
:
((
pow2
57
%
Z
)
=
144115188075855872
%
Z
).
Axiom
pow2_58
:
((
pow2
58
%
Z
)
=
288230376151711744
%
Z
).
Axiom
pow2_59
:
((
pow2
59
%
Z
)
=
576460752303423488
%
Z
).
Axiom
pow2_60
:
((
pow2
60
%
Z
)
=
1152921504606846976
%
Z
).
Axiom
pow2_61
:
((
pow2
61
%
Z
)
=
2305843009213693952
%
Z
).
Axiom
pow2_62
:
((
pow2
62
%
Z
)
=
4611686018427387904
%
Z
).
Axiom
pow2_63
:
((
pow2
63
%
Z
)
=
9223372036854775808
%
Z
).
Axiom
Div_double
:
forall
(
x
:
Z
)
(
y
:
Z
),
(((
0
%
Z
<
y
)
%
Z
/
\
(
y
<=
x
)
%
Z
)
/
\
(
x
<
(
2
%
Z
*
y
)
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
y
)
=
1
%
Z
).
Axiom
Div_pow
:
forall
(
x
:
Z
)
(
i
:
Z
),
(
0
%
Z
<
i
)
%
Z
->
((((
pow2
(
i
-
1
%
Z
)
%
Z
)
<=
x
)
%
Z
/
\
(
x
<
(
pow2
i
))
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
(
pow2
(
i
-
1
%
Z
)
%
Z
))
=
1
%
Z
)).
Axiom
Div_double_neg
:
forall
(
x
:
Z
)
(
y
:
Z
),
(((((
-
2
%
Z
)
%
Z
*
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<
(
-
y
)
%
Z
)
%
Z
)
/
\
((
-
y
)
%
Z
<
0
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
y
)
=
(
-
2
%
Z
)
%
Z
).
Axiom
Div_pow2
:
forall
(
x
:
Z
)
(
i
:
Z
),
(
0
%
Z
<
i
)
%
Z
->
((((
-
(
pow2
i
))
%
Z
<=
x
)
%
Z
/
\
(
x
<
(
-
(
pow2
(
i
-
1
%
Z
)
%
Z
))
%
Z
)
%
Z
)
->
((
int
.
EuclideanDivision
.
div
x
(
pow2
(
i
-
1
%
Z
)
%
Z
))
=
(
-
2
%
Z
)
%
Z
)).
Open
Scope
Z_scope
.
Require
Import
Why3
.
Ltac
ae
:=
why3
"alt-ergo"
timelimit
3.
(
*
Why3
goal
*
)
Theorem
Mod_pow2_gen
:
forall
(
x
:
Z
)
(
i
:
Z
)
(
k
:
Z
),
((
0
%
Z
<=
k
)
%
Z
/
\
(
k
<
i
)
%
Z
)
->
((
int
.
EuclideanDivision
.
mod1
(
int
.
EuclideanDivision
.
div
(
x
+
(
pow2
i
))
%
Z
(
pow2
k
))
2
%
Z
)
=
(
int
.
EuclideanDivision
.
mod1
(
int
.
EuclideanDivision
.
div
x
(
pow2
k
))
2
%
Z
)).
(
*
intros
x
i
k
(
h1
,
h2
).
*
)
intros
x
i
k
(
h1
,
h2
).
replace
(
x
+
pow2
i
)
with
(
pow2
k
*
pow2
(
i
-
k
)
+
x
).
2
:
ae
.
rewrite
int
.
EuclideanDivision
.
Div_mult
.
2
:
ae
.
replace
(
pow2
(
i
-
k
))
with
(
2
*
pow2
(
i
-
k
-
1
)).
2
:
ae
.
ae
.
Qed
.
examples/bitvectors/power2/why3session.xml
View file @
f691a29d
...
...
@@ -37,14 +37,6 @@
id=
"8"
name=
"Z3"
version=
"3.2"
/>
<prover
id=
"9"
name=
"Z3"
version=
"4.2"
/>
<prover
id=
"10"
name=
"Z3"
version=
"4.3.1"
/>
<file
name=
"../power2.why"
verified=
"false"
...
...
@@ -3232,7 +3224,7 @@
name=
"Div_double"
locfile=
"../power2.why"
loclnum=
"113"
loccnumb=
"8"
loccnume=
"18"
sum=
"
54ca81fd8dd2cc7b0616dbb62374c1dc
"
sum=
"
6f01a44c48ae3d43c3c4f83b1d870e98
"
proved=
"false"
expanded=
"true"
shape=
"ainfix =adivV0V1c1Iainfix <V0ainfix *c2V1Aainfix <=V1V0Aainfix <c0V1F"
>
...
...
@@ -3241,9 +3233,9 @@
name=
"Div_pow"
locfile=
"../power2.why"
loclnum=
"116"
loccnumb=
"8"
loccnume=
"15"
sum=
"
38721ee10b8925bf9da7fdd92bfcadad
"
sum=
"
94c0d831a4dcda27b2e02ffd748efbcf
"
proved=
"true"
expanded=
"
tru
e"
expanded=
"
fals
e"
shape=
"ainfix =adivV0apow2ainfix -V1c1c1Iainfix <V0apow2V1Aainfix <=apow2ainfix -V1c1V0Iainfix >V1c0F"
>
<proof
prover=
"0"
...
...
@@ -3259,7 +3251,7 @@
memlimit=
"1000"
obsolete=
"false"
archived=
"false"
>
<result
status=
"valid"
time=
"0.2
6
"
/>
<result
status=
"valid"
time=
"0.2
7
"
/>
</proof>
<proof
prover=
"2"
...
...
@@ -3290,7 +3282,7 @@
name=
"Div_double_neg"
locfile=
"../power2.why"
loclnum=
"119"
loccnumb=
"8"
loccnume=
"22"
sum=
"
973e29204b30d40a7bab41c85eb8f9ff
"
sum=
"
abb5d03ab24579b6ee213167c0a3133c
"
proved=
"false"
expanded=
"true"
shape=
"ainfix =adivV0V1aprefix -c2Iainfix <aprefix -V1c0Aainfix <V0aprefix -V1Aainfix <=ainfix *aprefix -c2V1V0F"
>
...
...
@@ -3299,9 +3291,9 @@
name=
"Div_pow2"
locfile=
"../power2.why"
loclnum=
"122"
loccnumb=
"8"
loccnume=
"16"
sum=
"
cf96a17cbeab4ecc0a75f7
fc5f
c09ad3
"
sum=
"
2c89c
fc
9
5f
a863dc50eff7a69b9fa7c8
"
proved=
"true"
expanded=
"
tru
e"
expanded=
"
fals
e"
shape=
"ainfix =adivV0apow2ainfix -V1c1aprefix -c2Iainfix <V0aprefix -apow2ainfix -V1c1Aainfix <=aprefix -apow2V1V0Iainfix >V1c0F"
>
<proof
prover=
"0"
...
...
@@ -3309,7 +3301,7 @@
memlimit=
"1000"
obsolete=
"false"
archived=
"false"
>
<result
status=
"valid"
time=
"0.1
3
"
/>
<result
status=
"valid"
time=
"0.
2
1"
/>
</proof>
<proof
prover=
"1"
...
...
@@ -3317,7 +3309,7 @@
memlimit=
"1000"
obsolete=
"false"
archived=
"false"
>
<result
status=
"valid"
time=
"0.
08
"
/>
<result
status=
"valid"
time=
"0.
11
"
/>
</proof>
<proof
prover=
"2"
...
...
@@ -3348,81 +3340,18 @@
