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Why3
why3
Commits
8ec8429d
Commit
8ec8429d
authored
Aug 18, 2011
by
Jean-Christophe Filliâtre
Browse files
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Plain Diff
update proofs for gcd and gcd_bezout
parent
65b2b501
Changes
4
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4 changed files
with
199 additions
and
95 deletions
+199
-95
examples/programs/gcd/gcd_WP_EuclideanAlgorithm_WP_parameter_gcd_1.v
...ograms/gcd/gcd_WP_EuclideanAlgorithm_WP_parameter_gcd_1.v
+77
-31
examples/programs/gcd/why3session.xml
examples/programs/gcd/why3session.xml
+16
-24
examples/programs/gcd_bezout/gcd_bezout_WP_GcdBezout_WP_parameter_gcd_1.v
...s/gcd_bezout/gcd_bezout_WP_GcdBezout_WP_parameter_gcd_1.v
+54
-7
examples/programs/gcd_bezout/why3session.xml
examples/programs/gcd_bezout/why3session.xml
+52
-33
No files found.
examples/programs/gcd/gcd_WP_EuclideanAlgorithm_WP_parameter_gcd_1.v
View file @
8ec8429d
...
...
@@ -20,7 +20,7 @@ Definition divides(d:Z) (n:Z): Prop := exists q:Z, (n = (q * d)%Z).
Axiom
divides_refl
:
forall
(
n
:
Z
),
(
divides
n
n
).
Axiom
divides_1
:
forall
(
n
:
Z
),
(
divides
1
%
Z
n
).
Axiom
divides_1
_n
:
forall
(
n
:
Z
),
(
divides
1
%
Z
n
).
Axiom
divides_0
:
forall
(
n
:
Z
),
(
divides
n
0
%
Z
).
...
...
@@ -56,8 +56,8 @@ Axiom divides_factorl : forall (a:Z) (b:Z), (divides a (b * a)%Z).
Axiom
divides_factorr
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
a
(
a
*
b
)
%
Z
).
Axiom
divides
1
:
forall
(
x
:
Z
),
(
divides
x
1
%
Z
)
->
((
x
=
1
%
Z
)
\
/
(
x
=
(
-
1
%
Z
)
%
Z
)).
Axiom
divides
_n_1
:
forall
(
n
:
Z
),
(
divides
n
1
%
Z
)
->
((
n
=
1
%
Z
)
\
/
(
n
=
(
-
1
%
Z
)
%
Z
)).
Axiom
divides_antisym
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
a
b
)
->
((
divides
b
a
)
->
((
a
=
b
)
\
/
(
a
=
(
-
b
)
%
Z
))).
...
...
@@ -83,33 +83,11 @@ Axiom Mod_1 : forall (x:Z), ((Zmod x 1%Z) = 0%Z).
Axiom
Div_1
:
forall
(
x
:
Z
),
((
Zdiv
x
1
%
Z
)
=
x
).
Axiom
mod_divides
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
Axiom
mod_divides
_euclidean
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
(((
Zmod
a
b
)
=
0
%
Z
)
->
(
divides
b
a
)).
Axiom
divides_mod
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
Zmod
a
b
)
=
0
%
Z
)).
Parameter
gcd
:
Z
->
Z
->
Z
.
Axiom
gcd_nonneg
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
0
%
Z
<=
(
gcd
a
b
))
%
Z
.
Axiom
gcd_def1
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
(
gcd
a
b
)
a
).
Axiom
gcd_def2
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
(
gcd
a
b
)
b
).
Axiom
gcd_def3
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
x
:
Z
),
(
divides
x
a
)
->
((
divides
x
b
)
->
(
divides
x
(
gcd
a
b
))).
Axiom
Assoc
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
gcd
(
gcd
x
y
)
z
)
=
(
gcd
x
(
gcd
y
z
))).
Axiom
Comm
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
gcd
x
y
)
=
(
gcd
y
x
)).
Axiom
gcd_0
:
forall
(
a
:
Z
),
((
gcd
a
0
%
Z
)
=
a
).
Axiom
gcd_euclid
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
q
:
Z
),
((
gcd
a
b
)
=
(
gcd
a
(
b
-
(
q
*
a
)
%
Z
)
%
Z
)).
Axiom
divides_mod_euclidean
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
Zmod
a
b
)
=
0
%
Z
)).
Axiom
Div_mod1
:
forall
(
x
:
Z
)
(
y
:
Z
),
(
~
(
y
=
0
%
Z
))
->
(
x
=
((
y
*
(
ZOdiv
x
y
))
%
Z
+
(
ZOmod
x
y
))
%
Z
).
...
...
@@ -151,16 +129,84 @@ Axiom Div_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z < x)%Z /\ ((0%Z <= y)%Z /\
Axiom
Mod_mult
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
0
%
Z
<
x
)
%
Z
/
\
((
0
%
Z
<=
y
)
%
Z
/
\
(
0
%
Z
<=
z
)
%
Z
))
->
((
ZOmod
((
x
*
y
)
%
Z
+
z
)
%
Z
x
)
=
(
ZOmod
z
x
)).
Axiom
mod_divides_computer
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
(((
ZOmod
a
b
)
=
0
%
Z
)
->
(
divides
b
a
)).
Axiom
divides_mod_computer
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
ZOmod
a
b
)
=
0
%
Z
)).
Definition
even
(
n
:
Z
)
:
Prop
:=
exists
k
:
Z
,
(
n
=
(
2
%
Z
*
k
)
%
Z
).
Definition
odd
(
n
:
Z
)
:
Prop
:=
exists
k
:
Z
,
(
n
=
((
2
%
Z
*
k
)
%
Z
+
1
%
Z
)
%
Z
).
Axiom
even_or_odd
:
forall
(
n
:
Z
),
(
even
n
)
\
/
(
odd
n
).
Axiom
even_not_odd
:
forall
(
n
:
Z
),
(
even
n
)
->
~
(
odd
n
).
Axiom
odd_not_even
:
forall
(
n
:
Z
),
(
odd
n
)
->
~
(
even
n
).
Axiom
even_odd
:
forall
(
n
:
Z
),
(
even
n
)
->
(
odd
(
n
+
1
%
Z
)
%
Z
).
Axiom
odd_even
:
forall
(
n
:
Z
),
(
odd
n
)
->
(
even
(
n
+
1
%
Z
)
%
Z
).
Axiom
even_even
:
forall
(
n
:
Z
),
(
even
n
)
->
(
even
(
n
+
2
%
Z
)
%
Z
).
Axiom
odd_odd
:
forall
(
n
:
Z
),
(
odd
n
)
->
(
odd
(
n
+
2
%
Z
)
%
Z
).
Axiom
even_2k
:
forall
(
k
:
Z
),
(
even
(
2
%
Z
*
k
)
%
Z
).
Axiom
odd_2k1
:
forall
(
k
:
Z
),
(
odd
((
2
%
Z
*
k
)
%
Z
+
1
%
Z
)
%
Z
).
Axiom
even_divides
:
forall
(
a
:
Z
),
(
even
a
)
<->
(
divides
2
%
Z
a
).
Axiom
odd_divides
:
forall
(
a
:
Z
),
(
odd
a
)
<->
~
(
divides
2
%
Z
a
).
Parameter
gcd
:
Z
->
Z
->
Z
.
Axiom
gcd_nonneg
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
0
%
Z
<=
(
gcd
a
b
))
%
Z
.
Axiom
gcd_def1
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
(
gcd
a
b
)
a
).
Axiom
gcd_def2
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
(
gcd
a
b
)
b
).
Axiom
gcd_def3
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
x
:
Z
),
(
divides
x
a
)
->
((
divides
x
b
)
->
(
divides
x
(
gcd
a
b
))).
Axiom
gcd_unique
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
d
:
Z
),
(
0
%
Z
<=
d
)
%
Z
->
((
divides
d
a
)
->
((
divides
d
b
)
->
((
forall
(
x
:
Z
),
(
divides
x
a
)
->
((
divides
x
b
)
->
(
divides
x
d
)))
->
(
d
=
(
gcd
a
b
))))).
Axiom
Assoc
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
gcd
(
gcd
x
y
)
z
)
=
(
gcd
x
(
gcd
y
z
))).
Axiom
Comm
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
gcd
x
y
)
=
(
gcd
y
x
)).
Axiom
gcd_0_pos
:
forall
(
a
:
Z
),
(
0
%
Z
<=
a
)
%
Z
->
((
gcd
a
0
%
Z
)
=
a
).
Axiom
gcd_0_neg
:
forall
(
a
:
Z
),
(
a
<
0
%
Z
)
%
Z
->
((
gcd
a
0
%
Z
)
=
(
-
a
)
%
Z
).
Axiom
gcd_opp
:
forall
(
a
:
Z
)
(
b
:
Z
),
((
gcd
a
b
)
=
(
gcd
(
-
a
)
%
Z
b
)).
Axiom
gcd_euclid
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
q
:
Z
),
((
gcd
a
b
)
=
(
gcd
a
(
b
-
(
q
*
a
)
%
Z
)
%
Z
)).
Axiom
Gcd_computer_mod
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
gcd
a
(
ZOmod
a
b
))
=
(
gcd
a
b
)).
Axiom
Gcd_euclidean_mod
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
gcd
a
(
Zmod
a
b
))
=
(
gcd
a
b
)).
Axiom
gcd_mult
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
c
:
Z
),
(
0
%
Z
<=
c
)
%
Z
->
((
gcd
(
c
*
a
)
%
Z
(
c
*
b
)
%
Z
)
=
(
c
*
(
gcd
a
b
))
%
Z
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
WP_parameter_gcd
:
forall
(
u
:
Z
),
forall
(
v
:
Z
),
((
0
%
Z
<=
u
)
%
Z
/
\
(
0
%
Z
<=
v
)
%
Z
)
->
((
~
(
v
=
0
%
Z
))
->
((((
0
%
Z
<=
v
)
%
Z
/
\
((
ZOmod
u
v
)
<
v
)
%
Z
)
/
\
((
0
%
Z
<=
v
)
%
Z
/
\
(
0
%
Z
<=
(
ZOmod
u
v
))
%
Z
))
->
((
gcd
v
(
ZOmod
u
v
))
=
(
gcd
u
v
)))).
(
0
%
Z
<=
v
)
%
Z
)
->
((
~
(
v
=
0
%
Z
))
->
(((
0
%
Z
<=
v
)
%
Z
/
\
((
ZOmod
u
v
)
<
v
)
%
Z
)
/
\
((
0
%
Z
<=
v
)
%
Z
/
\
(
0
%
Z
<=
(
ZOmod
u
v
))
%
Z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intuition
.
symmetry
.
...
...
examples/programs/gcd/why3session.xml
View file @
8ec8429d
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session SYSTEM "why3session.dtd">
<why3session
name=
"examples/programs/gcd/why3session.xml"
>
<prover
id=
"alt-ergo"
name=
"Alt-Ergo"
version=
"0.93"
/>
<prover
id=
"coq"
name=
"Coq"
version=
"8.3pl2"
/>
<prover
id=
"cvc3"
name=
"CVC3"
version=
"2.2"
/>
<prover
id=
"eprover"
name=
"Eprover"
version=
"0.8 Steinthal"
/>
<prover
id=
"gappa"
name=
"Gappa"
version=
"0.15.0"
/>
<prover
id=
"simplify"
name=
"Simplify"
version=
"1.5.4"
/>
<prover
id=
"spass"
name=
"Spass"
version=
"3.7"
/>
<prover
id=
"vampire"
name=
"Vampire"
version=
"0.6"
/>
<prover
id=
"yices"
name=
"Yices"
version=
"1.0.27"
/>
<prover
id=
"z3"
name=
"Z3"
version=
"2.19"
/>
<file
name=
"../gcd.mlw"
verified=
"true"
expanded=
"true"
>
<theory
name=
"WP EuclideanAlgorithm"
verified=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd"
expl=
"correctness of parameter gcd"
sum=
"
a484c5193a5cd03b14a4f295d1e1f03e"
proved=
"true"
expanded=
"true
"
>
<goal
name=
"WP_parameter gcd"
expl=
"correctness of parameter gcd"
sum=
"
766d5a2cc24a5fd8ece90420c2f75c1f"
proved=
"true"
expanded=
"true"
shape=
"iainfix =V1c0ainfix =V0agcdV0V1ainfix =agcdV1amodV0V1agcdV0V1Aainfix >=amodV0V1c0Aainfix >=V1c0Aainfix <amodV0V1V1Aainfix <=c0V1Iainfix >=V1c0Aainfix >=V0c0FF
"
>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd.1"
expl=
"normal postcondition"
sum=
"5a586460df7b95622e77fbef9bf3b5c9"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"cvc3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<goal
name=
"WP_parameter gcd.1"
expl=
"normal postcondition"
sum=
"4465e9890b7776b88f461cfd07f6534a"
proved=
"true"
expanded=
"true"
shape=
"ainfix =V0agcdV0V1Iainfix =V1c0Iainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"alt-ergo"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"simplify"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.01"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.04"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2"
expl=
"precondition"
sum=
"b712e1e03f97fab825d3144c19e74e4b"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"cvc3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<goal
name=
"WP_parameter gcd.2"
expl=
"precondition"
sum=
"4f12dd81ef67923c17562cf06453d85e"
proved=
"true"
expanded=
"true"
shape=
"ainfix >=amodV0V1c0Aainfix >=V1c0Aainfix <amodV0V1V1Aainfix <=c0V1Iainfix =V1c0NIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.06"
/>
</proof>
<proof
prover=
"simplify"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.01"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.04"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.3"
expl=
"normal postcondition"
sum=
"
c6bdc24bc2e3c5380662a3fe96194dd7"
proved=
"true"
expanded=
"true
"
>
<goal
name=
"WP_parameter gcd.3"
expl=
"normal postcondition"
sum=
"
9d5b42228b6e27b8f3654513a09c01d0"
proved=
"true"
expanded=
"true"
shape=
"ainfix =agcdV1amodV0V1agcdV0V1Iainfix >=amodV0V1c0Aainfix >=V1c0Aainfix <amodV0V1V1Aainfix <=c0V1Iainfix =V1c0NIainfix >=V1c0Aainfix >=V0c0FF
"
>
<proof
prover=
"coq"
timelimit=
"10"
edited=
"gcd_WP_EuclideanAlgorithm_WP_parameter_gcd_1.v"
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.6
8
"
/>
<result
status=
"valid"
time=
"0.6
2
"
/>
</proof>
</goal>
</transf>
...
...
examples/programs/gcd_bezout/gcd_bezout_WP_
M
_WP_parameter_gcd_1.v
→
examples/programs/gcd_bezout/gcd_bezout_WP_
GcdBezout
_WP_parameter_gcd_1.v
View file @
8ec8429d
...
...
@@ -62,7 +62,7 @@ Definition divides(d:Z) (n:Z): Prop := exists q:Z, (n = (q * d)%Z).
Axiom
divides_refl
:
forall
(
n
:
Z
),
(
divides
n
n
).
Axiom
divides_1
:
forall
(
n
:
Z
),
(
divides
1
%
Z
n
).
Axiom
divides_1
_n
:
forall
(
n
:
Z
),
(
divides
1
%
Z
n
).
Axiom
divides_0
:
forall
(
n
:
Z
),
(
divides
n
0
%
Z
).
...
...
@@ -98,8 +98,8 @@ Axiom divides_factorl : forall (a:Z) (b:Z), (divides a (b * a)%Z).
Axiom
divides_factorr
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
a
(
a
*
b
)
%
Z
).
Axiom
divides
1
:
forall
(
x
:
Z
),
(
divides
x
1
%
Z
)
->
((
x
=
1
%
Z
)
\
/
(
x
=
(
-
1
%
Z
)
%
Z
)).
Axiom
divides
_n_1
:
forall
(
n
:
Z
),
(
divides
n
1
%
Z
)
->
((
n
=
1
%
Z
)
\
/
(
n
=
(
-
1
%
Z
)
%
Z
)).
Axiom
divides_antisym
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
divides
a
b
)
->
((
divides
b
a
)
->
((
a
=
b
)
\
/
(
a
=
(
-
b
)
%
Z
))).
...
...
@@ -123,11 +123,43 @@ Axiom Mod_11 : forall (x:Z), ((Zmod x 1%Z) = 0%Z).
Axiom
Div_11
:
forall
(
x
:
Z
),
((
Zdiv
x
1
%
Z
)
=
x
).
Axiom
mod_divides
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
Axiom
mod_divides
_euclidean
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
(((
Zmod
a
b
)
=
0
%
Z
)
->
(
divides
b
a
)).
Axiom
divides_mod
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
Zmod
a
b
)
=
0
%
Z
)).
Axiom
divides_mod_euclidean
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
Zmod
a
b
)
=
0
%
Z
)).
Axiom
mod_divides_computer
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
(((
ZOmod
a
b
)
=
0
%
Z
)
->
(
divides
b
a
)).
Axiom
divides_mod_computer
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
divides
b
a
)
->
((
ZOmod
a
b
)
=
0
%
Z
)).
Definition
even
(
n
:
Z
)
:
Prop
:=
exists
k
:
Z
,
(
n
=
(
2
%
Z
*
k
)
%
Z
).
Definition
odd
(
n
:
Z
)
:
Prop
:=
exists
k
:
Z
,
(
n
=
((
2
%
Z
*
k
)
%
Z
+
1
%
Z
)
%
Z
).
Axiom
even_or_odd
:
forall
(
n
:
Z
),
(
even
n
)
\
/
(
odd
n
).
Axiom
even_not_odd
:
forall
(
n
:
Z
),
(
even
n
)
->
~
(
odd
n
).
Axiom
odd_not_even
:
forall
(
n
:
Z
),
(
odd
n
)
->
~
(
even
n
).
Axiom
even_odd
:
forall
(
n
:
Z
),
(
even
n
)
->
(
odd
(
n
+
1
%
Z
)
%
Z
).
Axiom
odd_even
:
forall
(
n
:
Z
),
(
odd
n
)
->
(
even
(
n
+
1
%
Z
)
%
Z
).
Axiom
even_even
:
forall
(
n
:
Z
),
(
even
n
)
->
(
even
(
n
+
2
%
Z
)
%
Z
).
Axiom
odd_odd
:
forall
(
n
:
Z
),
(
odd
n
)
->
(
odd
(
n
+
2
%
Z
)
%
Z
).
Axiom
even_2k
:
forall
(
k
:
Z
),
(
even
(
2
%
Z
*
k
)
%
Z
).
Axiom
odd_2k1
:
forall
(
k
:
Z
),
(
odd
((
2
%
Z
*
k
)
%
Z
+
1
%
Z
)
%
Z
).
Axiom
even_divides
:
forall
(
a
:
Z
),
(
even
a
)
<->
(
divides
2
%
Z
a
).
Axiom
odd_divides
:
forall
(
a
:
Z
),
(
odd
a
)
<->
~
(
divides
2
%
Z
a
).
Parameter
gcd
:
Z
->
Z
->
Z
.
...
...
@@ -141,12 +173,20 @@ Axiom gcd_def2 : forall (a:Z) (b:Z), (divides (gcd a b) b).
Axiom
gcd_def3
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
x
:
Z
),
(
divides
x
a
)
->
((
divides
x
b
)
->
(
divides
x
(
gcd
a
b
))).
Axiom
gcd_unique
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
d
:
Z
),
(
0
%
Z
<=
d
)
%
Z
->
((
divides
d
a
)
->
((
divides
d
b
)
->
((
forall
(
x
:
Z
),
(
divides
x
a
)
->
((
divides
x
b
)
->
(
divides
x
d
)))
->
(
d
=
(
gcd
a
b
))))).
Axiom
Assoc
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
z
:
Z
),
((
gcd
(
gcd
x
y
)
z
)
=
(
gcd
x
(
gcd
y
z
))).
Axiom
Comm
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
gcd
x
y
)
=
(
gcd
y
x
)).
Axiom
gcd_0
:
forall
(
a
:
Z
),
((
gcd
a
0
%
Z
)
=
a
).
Axiom
gcd_0_pos
:
forall
(
a
:
Z
),
(
0
%
Z
<=
a
)
%
Z
->
((
gcd
a
0
%
Z
)
=
a
).
Axiom
gcd_0_neg
:
forall
(
a
:
Z
),
(
a
<
0
%
Z
)
%
Z
->
((
gcd
a
0
%
Z
)
=
(
-
a
)
%
Z
).
Axiom
gcd_opp
:
forall
(
a
:
Z
)
(
b
:
Z
),
((
gcd
a
b
)
=
(
gcd
(
-
a
)
%
Z
b
)).
Axiom
gcd_euclid
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
q
:
Z
),
((
gcd
a
b
)
=
(
gcd
a
(
b
-
(
q
*
a
)
%
Z
)
%
Z
)).
...
...
@@ -157,6 +197,9 @@ Axiom Gcd_computer_mod : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((gcd a
Axiom
Gcd_euclidean_mod
:
forall
(
a
:
Z
)
(
b
:
Z
),
(
~
(
b
=
0
%
Z
))
->
((
gcd
a
(
Zmod
a
b
))
=
(
gcd
a
b
)).
Axiom
gcd_mult
:
forall
(
a
:
Z
)
(
b
:
Z
)
(
c
:
Z
),
(
0
%
Z
<=
c
)
%
Z
->
((
gcd
(
c
*
a
)
%
Z
(
c
*
b
)
%
Z
)
=
(
c
*
(
gcd
a
b
))
%
Z
).
Inductive
ref
(
a
:
Type
)
:=
|
mk_ref
:
a
->
ref
a
.
Implicit
Arguments
mk_ref
.
...
...
@@ -167,6 +210,10 @@ Definition contents (a:Type)(u:(ref a)): a :=
end
.
Implicit
Arguments
contents
.
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
WP_parameter_gcd
:
forall
(
x
:
Z
),
forall
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<=
y
)
%
Z
)
->
forall
(
d
:
Z
),
forall
(
c
:
Z
),
forall
(
b
:
Z
),
forall
(
a
:
Z
),
forall
(
y1
:
Z
),
forall
(
x1
:
Z
),
((
0
%
Z
<=
x1
)
%
Z
/
\
((
0
%
Z
<=
y1
)
%
Z
/
\
(((
gcd
x1
...
...
examples/programs/gcd_bezout/why3session.xml
View file @
8ec8429d
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session SYSTEM "why3session.dtd">
<why3session
name=
"examples/programs/gcd_bezout/why3session.xml"
>
<prover
id=
"alt-ergo"
name=
"Alt-Ergo"
version=
"0.93"
/>
<prover
id=
"coq"
name=
"Coq"
version=
"8.3pl2"
/>
<prover
id=
"cvc3"
name=
"CVC3"
version=
"2.2"
/>
<prover
id=
"eprover"
name=
"Eprover"
version=
"0.8 Steinthal"
/>
<prover
id=
"gappa"
name=
"Gappa"
version=
"0.15.0"
/>
<prover
id=
"simplify"
name=
"Simplify"
version=
"1.5.4"
/>
<prover
id=
"spass"
name=
"Spass"
version=
"3.7"
/>
<prover
id=
"vampire"
name=
"Vampire"
version=
"0.6"
/>
<prover
id=
"yices"
name=
"Yices"
version=
"1.0.27"
/>
<prover
id=
"z3"
name=
"Z3"
version=
"2.19"
/>
<file
name=
"../gcd_bezout.mlw"
verified=
"true"
expanded=
"true"
>
<theory
name=
"WP GcdBezout"
verified=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd"
expl=
"correctness of parameter gcd"
sum=
"
0fd714786263b647d7092c43fe9130f8"
proved=
"true"
expanded=
"true
"
>
<goal
name=
"WP_parameter gcd"
expl=
"correctness of parameter gcd"
sum=
"
e708de1061369396b2ff1f4113f30e00"
proved=
"true"
expanded=
"true"
shape=
"iainfix >V6c0ainfix <V9V6Aainfix <=c0V6Aainfix =ainfix +ainfix *V12V0ainfix *V13V1V9Aainfix =ainfix +ainfix *V10V0ainfix *V11V1V8Aainfix =agcdV8V9agcdV0V1Aainfix >=V9c0Aainfix >=V8c0Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6Fainfix =ainfix +ainfix *V14V0ainfix *V15V1V7EAainfix =V7agcdV0V1Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFAainfix =ainfix +ainfix *c0V0ainfix *c1V1V1Aainfix =ainfix +ainfix *c1V0ainfix *c0V1V0Aainfix =agcdV0V1agcdV0V1Aainfix >=V1c0Aainfix >=V0c0Iainfix >=V1c0Aainfix >=V0c0FF
"
>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd.1"
expl=
"loop invariant init"
sum=
"17d25f35ef473e3427180dd08103cf55"
proved=
"true"
expanded=
"false"
>
<proof
prover=
"cvc3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.00"
/>
<goal
name=
"WP_parameter gcd.1"
expl=
"loop invariant init"
sum=
"44ff89a72fcf725343c0157cb7a7faf8"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *c0V0ainfix *c1V1V1Aainfix =ainfix +ainfix *c1V0ainfix *c0V1V0Aainfix =agcdV0V1agcdV0V1Aainfix >=V1c0Aainfix >=V0c0Iainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.01"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2"
expl=
"loop invariant preservation"
sum=
"8
2475812596a513e9a0cc4da9e33016d"
proved=
"true"
expanded=
"true
"
>
<goal
name=
"WP_parameter gcd.2"
expl=
"loop invariant preservation"
sum=
"8
38c6f08305500896b8e79edc0ff580f"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *V12V0ainfix *V13V1V9Aainfix =ainfix +ainfix *V10V0ainfix *V11V1V8Aainfix =agcdV8V9agcdV0V1Aainfix >=V9c0Aainfix >=V8c0Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF
"
>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd.2.1"
expl=
"correctness of parameter gcd"
sum=
"976821da4ae5ccad609c3d2287c80fe0"
proved=
"true"
expanded=
"false"
>
<goal
name=
"WP_parameter gcd.2.1"
expl=
"correctness of parameter gcd"
sum=
"b046d3dcd5fbba79c535695efd558e94"
proved=
"true"
expanded=
"true"
shape=
"ainfix >=V8c0Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
4
"
/>
<result
status=
"valid"
time=
"0.0
2
"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2.2"
expl=
"correctness of parameter gcd"
sum=
"9920d86d6b752a310cd0cacc278dd381"
proved=
"true"
expanded=
"false"
>
<goal
name=
"WP_parameter gcd.2.2"
expl=
"correctness of parameter gcd"
sum=
"080da6996a2bd84d0819ab2c8db3ce4e"
proved=
"true"
expanded=
"true"
shape=
"ainfix >=V9c0Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
6
"
/>
<result
status=
"valid"
time=
"0.0
7
"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2.3"
expl=
"correctness of parameter gcd"
sum=
"
c877b4e476cb6894559421a2c631c1fe"
proved=
"true"
expanded=
"true
"
>
<proof
prover=
"coq"
timelimit=
"10"
edited=
"gcd_bezout_WP_
M
_WP_parameter_gcd_1.v"
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.7
8
"
/>
<goal
name=
"WP_parameter gcd.2.3"
expl=
"correctness of parameter gcd"
sum=
"
6e974266fad2cd4dc4e4778c9da314d4"
proved=
"true"
expanded=
"true"
shape=
"ainfix =agcdV8V9agcdV0V1Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF
"
>
<proof
prover=
"coq"
timelimit=
"10"
edited=
"gcd_bezout_WP_
GcdBezout
_WP_parameter_gcd_1.v"
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.7
0
"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2.4"
expl=
"correctness of parameter gcd"
sum=
"0aa20dc3fd2383fb1f18175fd306a4ff"
proved=
"true"
expanded=
"false"
>
<goal
name=
"WP_parameter gcd.2.4"
expl=
"correctness of parameter gcd"
sum=
"bb890d60feb115367a6f034ea18e068e"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *V10V0ainfix *V11V1V8Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
5
"
/>
<result
status=
"valid"
time=
"0.0
2
"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.2.5"
expl=
"correctness of parameter gcd"
sum=
"a
abf483908c2b613f2697262172db1d8"
proved=
"true"
expanded=
"false
"
>
<proof
prover=
"cvc3"
timelimit=
"
3
"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
4
"
/>
<goal
name=
"WP_parameter gcd.2.5"
expl=
"correctness of parameter gcd"
sum=
"a
f06cf21d1578e0cf266716c7f811f36"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *V12V0ainfix *V13V1V9Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF
"
>
<proof
prover=
"cvc3"
timelimit=
"
10
"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
2
"
/>
</proof>
</goal>
</transf>
</goal>
<goal
name=
"WP_parameter gcd.3"
expl=
"loop variant decreases"
sum=
"b46a0daa9ac9fd8b9a204c389b2c55ee"
proved=
"true"
expanded=
"false"
>
<proof
prover=
"cvc3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.20"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.05"
/>
<goal
name=
"WP_parameter gcd.3"
expl=
"loop variant decreases"
sum=
"d4abfa67a7bde89abc856c1564c1aa3d"
proved=
"true"
expanded=
"true"
shape=
"ainfix <V9V6Aainfix <=c0V6Iainfix =ainfix +ainfix *V12V0ainfix *V13V1V9Aainfix =ainfix +ainfix *V10V0ainfix *V11V1V8Aainfix =agcdV8V9agcdV0V1Aainfix >=V9c0Aainfix >=V8c0Iainfix =V13ainfix -V4ainfix *V2adivV7V6FIainfix =V12ainfix -V5ainfix *V3adivV7V6FIainfix =V11V2FIainfix =V10V3FIainfix =V9amodV7V6FIainfix =V8V6FIainfix >V6c0Iainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.10"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.4"
expl=
"normal postcondition"
sum=
"0ea8c622bb67d0b961c102d4d481b9ac"
proved=
"true"
expanded=
"false"
>
<proof
prover=
"alt-ergo"
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.08"
/>
</proof>
<goal
name=
"WP_parameter gcd.4"
expl=
"normal postcondition"
sum=
"f519c8644e350bf6b4879e17ab8d1f45"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *V8V0ainfix *V9V1V7EAainfix =V7agcdV0V1Iainfix >V6c0NIainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_parameter gcd.4.1"
expl=
"correctness of parameter gcd"
sum=
"bb06bfcff41f64c10ab670f8e609fa11"
proved=
"true"
expanded=
"true"
shape=
"ainfix =V7agcdV0V1Iainfix >V6c0NIainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.04"
/>
</proof>
</goal>
<goal
name=
"WP_parameter gcd.4.2"
expl=
"correctness of parameter gcd"
sum=
"e7a614e3d26579b68ee56a56869bee59"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix +ainfix *V8V0ainfix *V9V1V7EIainfix >V6c0NIainfix =ainfix +ainfix *V3V0ainfix *V2V1V6Aainfix =ainfix +ainfix *V5V0ainfix *V4V1V7Aainfix =agcdV7V6agcdV0V1Aainfix >=V6c0Aainfix >=V7c0FFFFFFIainfix >=V1c0Aainfix >=V0c0FF"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
</goal>
</transf>
</goal>
</transf>
</goal>
...
...
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