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Why3
why3
Commits
72b7667e
Commit
72b7667e
authored
Dec 15, 2015
by
Jean-Christophe Filliâtre
Browse files
optimal replay: fixed (and simplified) proof
parent
97466b34
Changes
4
Hide whitespace changes
Inline
Side-by-side
examples/optimal_replay.mlw
View file @
72b7667e
...
...
@@ -57,7 +57,7 @@ module OptimalReplay
0 < d[k] = d[g[k]] + 1 /\
forall k': int. g[k] < k' < k -> d[g[k]] < d[k'] }
(* could be deduced from above, but avoids induction *)
invariant { forall k: int. 0 <= k < i ->
path
d[k] k }
invariant { forall k: int. 0 <= k < i ->
distance
d[k] k }
let j = ref (i-1) in
while g[!j] >= f i do
invariant { f i <= !j < i /\ !count + d[!j] <= i-1 }
...
...
examples/optimal_replay/distance_Distance_WP_parameter_distance_1.v
deleted
100644 → 0
View file @
97466b34
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
BuiltIn
.
Require
BuiltIn
.
Require
int
.
Int
.
Require
map
.
Map
.
(
*
Why3
assumption
*
)
Definition
unit
:=
unit
.
Axiom
qtmark
:
Type
.
Parameter
qtmark_WhyType
:
WhyType
qtmark
.
Existing
Instance
qtmark_WhyType
.
(
*
Why3
assumption
*
)
Inductive
ref
(
a
:
Type
)
:=
|
mk_ref
:
a
->
ref
a
.
Axiom
ref_WhyType
:
forall
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
,
WhyType
(
ref
a
).
Existing
Instance
ref_WhyType
.
Implicit
Arguments
mk_ref
[[
a
]].
(
*
Why3
assumption
*
)
Definition
contents
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
v
:
(
ref
a
))
:
a
:=
match
v
with
|
(
mk_ref
x
)
=>
x
end
.
(
*
Why3
assumption
*
)
Inductive
array
(
a
:
Type
)
:=
|
mk_array
:
Z
->
(
map
.
Map
.
map
Z
a
)
->
array
a
.
Axiom
array_WhyType
:
forall
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
,
WhyType
(
array
a
).
Existing
Instance
array_WhyType
.
Implicit
Arguments
mk_array
[[
a
]].
(
*
Why3
assumption
*
)
Definition
elts
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
v
:
(
array
a
))
:
(
map
.
Map
.
map
Z
a
)
:=
match
v
with
|
(
mk_array
x
x1
)
=>
x1
end
.
(
*
Why3
assumption
*
)
Definition
length
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
v
:
(
array
a
))
:
Z
:=
match
v
with
|
(
mk_array
x
x1
)
=>
x
end
.
(
*
Why3
assumption
*
)
Definition
get
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
i
:
Z
)
:
a
:=
(
map
.
Map
.
get
(
elts
a1
)
i
).
(
*
Why3
assumption
*
)
Definition
set
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
i
:
Z
)
(
v
:
a
)
:
(
array
a
)
:=
(
mk_array
(
length
a1
)
(
map
.
Map
.
set
(
elts
a1
)
i
v
)).
Parameter
n
:
Z
.
Axiom
n_nonneg
:
(
0
%
Z
<
n
)
%
Z
.
Parameter
f
:
Z
->
Z
.
Axiom
f_prop
:
forall
(
k
:
Z
),
((
0
%
Z
<
k
)
%
Z
/
\
(
k
<
n
)
%
Z
)
->
((
0
%
Z
<=
(
f
k
))
%
Z
/
\
((
f
k
)
<
k
)
%
Z
).
(
*
Why3
assumption
*
)
Inductive
path
:
Z
->
Z
->
Prop
:=
|
path0
:
(
path
0
%
Z
0
%
Z
)
|
paths
:
forall
(
i
:
Z
),
((
0
%
Z
<=
i
)
%
Z
/
\
(
i
<
n
)
%
Z
)
->
forall
(
d
:
Z
)
(
j
:
Z
),
(
path
d
j
)
->
((((
f
i
)
<=
j
)
%
Z
/
\
(
j
<
i
)
%
Z
)
->
(
path
(
d
+
1
%
Z
)
%
Z
i
)).
(
*
Why3
assumption
*
)
Definition
distance
(
d
:
Z
)
(
i
:
Z
)
:
Prop
:=
(
path
d
i
)
/
\
forall
(
d
'
:
Z
),
(
path
d
'
i
)
->
(
d
<=
d
'
)
%
Z
.
Require
Import
Why3
.
Ltac
ae
:=
why3
"Alt-Ergo,0.95.2,"
timelimit
30.
Ltac
z3
:=
why3
"Z3,4.3.1,"
timelimit
30.
(
*
Why3
goal
*
)
Theorem
WP_parameter_distance
:
let
o
:=
n
in
((
0
%
Z
<=
o
)
%
Z
->
forall
(
g
:
Z
)
(
g1
:
(
map
.
Map
.
map
Z
Z
)),
(((
0
%
Z
<
g
)
%
Z
\
/
(
0
%
Z
=
g
))
/
\
((
g
=
o
)
/
\
forall
(
i
:
Z
),
(((
0
%
Z
<
i
)
%
Z
\
/
(
0
%
Z
=
i
))
/
\
(
i
<
o
)
%
Z
)
->
((
map
.
Map
.
get
g1
i
)
=
0
%
Z
)))
->
((
0
%
Z
<
g
)
%
Z
->
forall
(
g2
:
(
map
.
Map
.
map
Z
Z
)),
(((
0
%
Z
<
g
)
%
Z
\
/
(
0
%
Z
=
g
))
/
\
(
g2
=
(
map
.
Map
.
set
g1
0
%
Z
(
-
1
%
Z
)
%
Z
)))
->
let
o1
:=
n
in
((
0
%
Z
<=
o1
)
%
Z
->
forall
(
d
:
Z
)
(
d1
:
(
map
.
Map
.
map
Z
Z
)),
(((
0
%
Z
<
d
)
%
Z
\
/
(
0
%
Z
=
d
))
/
\
((
d
=
o1
)
/
\
forall
(
i
:
Z
),
(((
0
%
Z
<
i
)
%
Z
\
/
(
0
%
Z
=
i
))
/
\
(
i
<
o1
)
%
Z
)
->
((
map
.
Map
.
get
d1
i
)
=
0
%
Z
)))
->
let
o2
:=
(
n
-
1
%
Z
)
%
Z
in
(((
1
%
Z
<
o2
)
%
Z
\
/
(
1
%
Z
=
o2
))
->
forall
(
count
:
Z
)
(
d2
:
(
map
.
Map
.
map
Z
Z
))
(
g3
:
(
map
.
Map
.
map
Z
Z
)),
((((
map
.
Map
.
get
d2
0
%
Z
)
=
0
%
Z
)
/
\
(((
map
.
Map
.
get
g3
0
%
Z
)
=
(
-
1
%
Z
)
%
Z
)
/
\
(((
count
+
(
map
.
Map
.
get
d2
((
o2
+
1
%
Z
)
%
Z
-
1
%
Z
)
%
Z
))
%
Z
<
((
o2
+
1
%
Z
)
%
Z
-
1
%
Z
)
%
Z
)
%
Z
\
/
((
count
+
(
map
.
Map
.
get
d2
((
o2
+
1
%
Z
)
%
Z
-
1
%
Z
)
%
Z
))
%
Z
=
((
o2
+
1
%
Z
)
%
Z
-
1
%
Z
)
%
Z
))))
/
\
((
forall
(
k
:
Z
),
((
0
%
Z
<
k
)
%
Z
/
\
(
k
<
(
o2
+
1
%
Z
)
%
Z
)
%
Z
)
->
((((
map
.
Map
.
get
g3
(
map
.
Map
.
get
g3
k
))
<
(
f
k
))
%
Z
/
\
((((
f
k
)
<
(
map
.
Map
.
get
g3
k
))
%
Z
\
/
((
f
k
)
=
(
map
.
Map
.
get
g3
k
)))
/
\
((
map
.
Map
.
get
g3
k
)
<
k
)
%
Z
))
/
\
(((
0
%
Z
<
(
map
.
Map
.
get
d2
k
))
%
Z
/
\
((
map
.
Map
.
get
d2
k
)
=
((
map
.
Map
.
get
d2
(
map
.
Map
.
get
g3
k
))
+
1
%
Z
)
%
Z
))
/
\
forall
(
k
'
:
Z
),
(((
map
.
Map
.
get
g3
k
)
<
k
'
)
%
Z
/
\
(
k
'
<
k
)
%
Z
)
->
((
map
.
Map
.
get
d2
(
map
.
Map
.
get
g3
k
))
<
(
map
.
Map
.
get
d2
k
'
))
%
Z
)))
/
\
forall
(
k
:
Z
),
(((
0
%
Z
<
k
)
%
Z
\
/
(
0
%
Z
=
k
))
/
\
(
k
<
(
o2
+
1
%
Z
)
%
Z
)
%
Z
)
->
(
path
(
map
.
Map
.
get
d2
k
)
k
)))
->
((
count
<
n
)
%
Z
->
forall
(
k
:
Z
),
(((
0
%
Z
<
k
)
%
Z
\
/
(
0
%
Z
=
k
))
/
\
(
k
<
n
)
%
Z
)
->
forall
(
d
'
:
Z
),
(
path
d
'
k
)
->
((
map
.
Map
.
get
d2
k
)
<=
d
'
)
%
Z
))))).
(
*
Why3
intros
o
h1
g
g1
(
h2
,(
h3
,
h4
))
h5
g2
(
h6
,
h7
)
o1
h8
d
d1
(
h9
,(
h10
,
h11
))
o2
h12
count
d2
g3
((
h13
,(
h14
,
h15
)),(
h16
,
h17
))
h18
k
(
h19
,
h20
)
d
'
h21
.
*
)
(
*
intros
o
_
_
h3
g
_
o1
h4
h5
o2
h6
count
d
g1
((
h7
,(
h8
,
h9
)),(
h10
,
h11
))
h12
k
(
h13
,
h14
)
d
'
h15
.
*
)
intros
o
h1
g
g1
(
h2
,(
h3
,
h4
))
h5
g2
(
h6
,
h7
)
o1
h8
d
d1
(
h9
,(
h10
,
h11
))
o2
h12
count
d2
g3
((
h13
,(
h14
,
h15
)),(
h16
,
h17
))
h18
k
(
h19
,
h20
)
d
'
h21
.
subst
o
o1
o2
.
(
*
clear
h4
h5
h6
.
*
)
replace
(
n
-
1
+
1
)
%
Z
with
n
in
*
by
omega
.
(
*
clear
count
h9
h12
.
*
)
generalize
h19
h20
d
'
h21
.
pattern
k
.
apply
Z_lt_induction
;
[
clear
k
h19
h20
d
'
h21
|
omega
].
intros
k
IH
hk1
hk2
d
'
hd
'
.
assert
(
case
:
(
Map
.
get
d2
k
<=
d
'
\
/
d
'
<
Map
.
get
d2
k
)
%
Z
)
by
omega
.
destruct
case
;
auto
.
destruct
hk1
.
(
*
0
<
k
*
)
inversion
hd
'
.
omega
.
subst
i
.
assert
(
h
:
(
0
<
k
<
n
)
%
Z
)
by
omega
.
generalize
(
h16
k
h
).
intros
h16k
.
assert
(
case
:
(
j
<
Map
.
get
g1
k
\
/
j
=
Map
.
get
g1
k
\
/
j
>
Map
.
get
g1
k
)
%
Z
)
by
omega
.
destruct
case
.
(
*
j
<
g
[
k
]
*
)
z3
.
destruct
H5
.
(
*
j
=
g
[
k
]
*
)
z3
.
(
*
j
>
g
[
k
]
*
)
z3
.
(
*
k
=
0
*
)
ae
.
Qed
.
examples/optimal_replay/why3session.xml
View file @
72b7667e
...
...
@@ -4,12 +4,11 @@
<why3session
shape_version=
"4"
>
<prover
id=
"0"
name=
"Alt-Ergo"
version=
"0.99.1"
timelimit=
"5"
memlimit=
"1000"
/>
<prover
id=
"1"
name=
"CVC3"
version=
"2.4.1"
timelimit=
"5"
memlimit=
"1000"
/>
<prover
id=
"2"
name=
"Coq"
version=
"8.4pl4"
timelimit=
"5"
memlimit=
"1000"
/>
<prover
id=
"3"
name=
"CVC4"
version=
"1.4"
timelimit=
"5"
memlimit=
"1000"
/>
<prover
id=
"4"
name=
"Z3"
version=
"4.3.1"
timelimit=
"5"
memlimit=
"1000"
/>
<prover
id=
"5"
name=
"Z3"
version=
"3.2"
timelimit=
"5"
memlimit=
"1000"
/>
<file
name=
"../optimal_replay.mlw"
expanded=
"true"
>
<theory
name=
"OptimalReplay"
sum=
"
4b8a258f2e1b3ac0543c6a100e50f6f8
"
expanded=
"true"
>
<theory
name=
"OptimalReplay"
sum=
"
af50008bb1221478dad330d6726d16c3
"
expanded=
"true"
>
<goal
name=
"WP_parameter distance"
expl=
"VC for distance"
expanded=
"true"
>
<transf
name=
"split_goal_wp"
expanded=
"true"
>
<goal
name=
"WP_parameter distance.1"
expl=
"1. array creation size"
>
...
...
@@ -25,7 +24,7 @@
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.01"
steps=
"11"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.5"
expl=
"5. assertion"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"
41
"
/></proof>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"
30
"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.6"
expl=
"6. loop invariant init"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.01"
steps=
"12"
/></proof>
...
...
@@ -34,7 +33,7 @@
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"12"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.8"
expl=
"8. loop invariant init"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"
1
4"
/></proof>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"
2
4"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.9"
expl=
"9. loop invariant init"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"17"
/></proof>
...
...
@@ -59,10 +58,10 @@
</goal>
<goal
name=
"WP_parameter distance.15"
expl=
"15. loop invariant preservation"
>
<proof
prover=
"1"
timelimit=
"17"
><result
status=
"valid"
time=
"0.04"
/></proof>
<proof
prover=
"5"
timelimit=
"17"
><result
status=
"valid"
time=
"0.
4
3"
/></proof>
<proof
prover=
"5"
timelimit=
"17"
><result
status=
"valid"
time=
"0.3
9
"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.16"
expl=
"16. loop variant decrease"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.
36
"
steps=
"
434
"
/></proof>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.
11
"
steps=
"
177
"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.17"
expl=
"17. type invariant"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"22"
/></proof>
...
...
@@ -84,25 +83,25 @@
<proof
prover=
"3"
><result
status=
"valid"
time=
"0.05"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.23"
expl=
"23. loop invariant preservation"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.06"
steps=
"56"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.24"
expl=
"24. assertion"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"37"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.25"
expl=
"25. assertion"
>
<transf
name=
"inline_goal"
>
<goal
name=
"WP_parameter distance.2
5
.1"
expl=
"1.
as
sertion"
>
<goal
name=
"WP_parameter distance.2
3
.1"
expl=
"1.
loop invariant pre
ser
va
tion"
>
<transf
name=
"split_goal_wp"
>
<goal
name=
"WP_parameter distance.2
5
.1.1"
expl=
"1.
as
sertion"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.
0
1"
steps=
"
12
"
/></proof>
<goal
name=
"WP_parameter distance.2
3
.1.1"
expl=
"1.
loop invariant pre
ser
va
tion"
>
<proof
prover=
"0"
timelimit=
"36"
><result
status=
"valid"
time=
"0.1
2
"
steps=
"
79
"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.2
5
.1.2"
expl=
"2.
as
sertion"
>
<proof
prover=
"
2
"
edited=
"distance_Distance_WP_parameter_distance_1.v
"
><result
status=
"valid"
time=
"
1
0.1
0
"
/></proof>
<goal
name=
"WP_parameter distance.2
3
.1.2"
expl=
"2.
loop invariant pre
ser
va
tion"
>
<proof
prover=
"
3
"
timelimit=
"36
"
><result
status=
"valid"
time=
"0.1
7
"
/></proof>
</goal>
</transf>
</goal>
</transf>
</goal>
<goal
name=
"WP_parameter distance.24"
expl=
"24. assertion"
>
<proof
prover=
"0"
><result
status=
"valid"
time=
"0.02"
steps=
"38"
/></proof>
</goal>
<goal
name=
"WP_parameter distance.25"
expl=
"25. assertion"
>
<proof
prover=
"0"
timelimit=
"36"
><result
status=
"valid"
time=
"0.00"
steps=
"18"
/></proof>
</goal>
</transf>
</goal>
</theory>
...
...
examples/optimal_replay/why3shapes.gz
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