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Why3
why3
Commits
c6ede020
Commit
c6ede020
authored
Jan 23, 2013
by
MARCHE Claude
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insertion sort examples: added generic versions
parent
bb609f33
Changes
5
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5 changed files
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1488 additions
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75 deletions
+1488
-75
examples/programs/insertion_sort.mlw
examples/programs/insertion_sort.mlw
+48
-0
examples/programs/insertion_sort/insertion_sort_InsertionSortGen_WP_parameter_insertion_sort_1.v
...ion_sort_InsertionSortGen_WP_parameter_insertion_sort_1.v
+220
-0
examples/programs/insertion_sort/why3session.xml
examples/programs/insertion_sort/why3session.xml
+517
-44
examples/programs/insertion_sort_dumb.mlw
examples/programs/insertion_sort_dumb.mlw
+53
-1
examples/programs/insertion_sort_dumb/why3session.xml
examples/programs/insertion_sort_dumb/why3session.xml
+650
-30
No files found.
examples/programs/insertion_sort.mlw
View file @
c6ede020
...
...
@@ -35,3 +35,51 @@ module InsertionSort
done
end
module InsertionSortGen
use import int.Int
use import ref.Ref
use import array.Array
use import array.ArrayPermut
use import array.ArrayEq
clone import map.MapSorted as M
axiom le_refl: forall x:elt. le x x
axiom le_asym: forall x y:elt. not (le x y) -> le y x
axiom le_trans: forall x y z:elt. le x y /\ le y z -> le x z
predicate sorted_sub (a : array elt) (l u : int) =
M.sorted_sub a.elts l u
predicate sorted (a : array elt) =
M.sorted_sub a.elts 0 a.length
let insertion_sort (a: array elt) =
ensures { sorted a /\ permut (old a) a }
'L:
for i = 1 to length a - 1 do
(* a[0..i[ is sorted; now insert a[i] *)
invariant { sorted_sub a 0 i /\ permut (at a 'L) a }
let v = a[i] in
let j = ref i in
while !j > 0 && not (le a[!j - 1] v) do
invariant { 0 <= !j <= i }
invariant { permut (at a 'L) a[!j <- v] }
invariant { forall k1 k2: int.
0 <= k1 <= k2 <= i -> k1 <> !j -> k2 <> !j -> le a[k1] a[k2] }
invariant { forall k: int. !j+1 <= k <= i -> le v a[k] }
variant { !j }
'L1:
a[!j] <- a[!j - 1];
assert { exchange (at a 'L1)[!j <- v] a[!j-1 <- v] (!j - 1) !j };
j := !j - 1
done;
assert { forall k: int. 0 <= k < !j -> le a[k] v };
a[!j] <- v
done
end
examples/programs/insertion_sort/insertion_sort_InsertionSortGen_WP_parameter_insertion_sort_1.v
0 → 100644
View file @
c6ede020
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
BuiltIn
.
Require
BuiltIn
.
Require
int
.
Int
.
(
*
Why3
assumption
*
)
Definition
unit
:=
unit
.
(
*
Why3
assumption
*
)
Inductive
ref
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
:=
|
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
]
[
a_WT
]].
(
*
Why3
assumption
*
)
Definition
contents
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
v
:
(
ref
a
))
:
a
:=
match
v
with
|
(
mk_ref
x
)
=>
x
end
.
Axiom
map
:
forall
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
(
b
:
Type
)
{
b_WT
:
WhyType
b
}
,
Type
.
Parameter
map_WhyType
:
forall
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
(
b
:
Type
)
{
b_WT
:
WhyType
b
}
,
WhyType
(
map
a
b
).
Existing
Instance
map_WhyType
.
Parameter
get
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
(
map
a
b
)
->
a
->
b
.
Parameter
set
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
(
map
a
b
)
->
a
->
b
->
(
map
a
b
).
Axiom
Select_eq
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
forall
(
m
:
(
map
a
b
)),
forall
(
a1
:
a
)
(
a2
:
a
),
forall
(
b1
:
b
),
(
a1
=
a2
)
->
((
get
(
set
m
a1
b1
)
a2
)
=
b1
).
Axiom
Select_neq
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
forall
(
m
:
(
map
a
b
)),
forall
(
a1
:
a
)
(
a2
:
a
),
forall
(
b1
:
b
),
(
~
(
a1
=
a2
))
->
((
get
(
set
m
a1
b1
)
a2
)
=
(
get
m
a2
)).
Parameter
const
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
b
->
(
map
a
b
).
Axiom
Const
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
{
b
:
Type
}
{
b_WT
:
WhyType
b
}
,
forall
(
b1
:
b
)
(
a1
:
a
),
((
get
(
const
b1
:
(
map
a
b
))
a1
)
=
b1
).
(
*
Why3
assumption
*
)
Inductive
array
(
a
:
Type
)
{
a_WT
:
WhyType
a
}
:=
|
mk_array
:
Z
->
(
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
]
[
a_WT
]].
(
*
Why3
assumption
*
)
Definition
elts
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
v
:
(
array
a
))
:
(
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
get1
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
i
:
Z
)
:
a
:=
(
get
(
elts
a1
)
i
).
(
*
Why3
assumption
*
)
Definition
set1
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
i
:
Z
)
(
v
:
a
)
:
(
array
a
)
:=
(
mk_array
(
length
a1
)
(
set
(
elts
a1
)
i
v
)).
(
*
Why3
assumption
*
)
Definition
make
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
n
:
Z
)
(
v
:
a
)
:
(
array
a
)
:=
(
mk_array
n
(
const
v
:
(
map
Z
a
))).
(
*
Why3
assumption
*
)
Definition
map_eq_sub
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
forall
(
i
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
((
get
a1
i
)
=
(
get
a2
i
)).
(
*
Why3
assumption
*
)
Definition
exchange
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
))
(
i
:
Z
)
(
j
:
Z
)
:
Prop
:=
((
get
a1
i
)
=
(
get
a2
j
))
/
\
(((
get
a2
i
)
=
(
get
a1
j
))
/
\
forall
(
k
:
Z
),
((
~
(
k
=
i
))
/
\
~
(
k
=
j
))
->
((
get
a1
k
)
=
(
get
a2
k
))).
Axiom
exchange_set
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
map
Z
a
)),
forall
(
i
:
Z
)
(
j
:
Z
),
(
exchange
a1
(
set
(
set
a1
i
(
get
a1
j
))
j
(
get
a1
i
))
i
j
).
(
*
Why3
assumption
*
)
Inductive
permut_sub
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
:
(
map
Z
a
)
->
(
map
Z
a
)
->
Z
->
Z
->
Prop
:=
|
permut_refl
:
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
map_eq_sub
a1
a2
l
u
)
->
(
permut_sub
a1
a2
l
u
)
|
permut_sym
:
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut_sub
a1
a2
l
u
)
->
(
permut_sub
a2
a1
l
u
)
|
permut_trans
:
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
))
(
a3
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut_sub
a1
a2
l
u
)
->
((
permut_sub
a2
a3
l
u
)
->
(
permut_sub
a1
a3
l
u
))
|
permut_exchange
:
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
)
(
i
:
Z
)
(
j
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
(((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
->
((
exchange
a1
a2
i
j
)
->
(
permut_sub
a1
a2
l
u
))).
Axiom
permut_weakening
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l1
:
Z
)
(
r1
:
Z
)
(
l2
:
Z
)
(
r2
:
Z
),
(((
l1
<=
l2
)
%
Z
/
\
(
l2
<=
r2
)
%
Z
)
/
\
(
r2
<=
r1
)
%
Z
)
->
((
permut_sub
a1
a2
l2
r2
)
->
(
permut_sub
a1
a2
l1
r1
)).
Axiom
permut_eq
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut_sub
a1
a2
l
u
)
->
forall
(
i
:
Z
),
((
i
<
l
)
%
Z
\
/
(
u
<=
i
)
%
Z
)
->
((
get
a2
i
)
=
(
get
a1
i
)).
Axiom
permut_exists
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
map
Z
a
))
(
a2
:
(
map
Z
a
)),
forall
(
l
:
Z
)
(
u
:
Z
),
(
permut_sub
a1
a2
l
u
)
->
forall
(
i
:
Z
),
((
l
<=
i
)
%
Z
/
\
(
i
<
u
)
%
Z
)
->
exists
j
:
Z
,
((
l
<=
j
)
%
Z
/
\
(
j
<
u
)
%
Z
)
/
\
((
get
a2
i
)
=
(
get
a1
j
)).
(
*
Why3
assumption
*
)
Definition
exchange1
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
i
:
Z
)
(
j
:
Z
)
:
Prop
:=
(
exchange
(
elts
a1
)
(
elts
a2
)
i
j
).
(
*
Why3
assumption
*
)
Definition
permut_sub1
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
(
permut_sub
(
elts
a1
)
(
elts
a2
)
l
u
).
(
*
Why3
assumption
*
)
Definition
permut
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
:
Prop
:=
((
length
a1
)
=
(
length
a2
))
/
\
(
permut_sub
(
elts
a1
)
(
elts
a2
)
0
%
Z
(
length
a1
)).
Axiom
exchange_permut
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
i
:
Z
)
(
j
:
Z
),
(
exchange1
a1
a2
i
j
)
->
(((
length
a1
)
=
(
length
a2
))
->
(((
0
%
Z
<=
i
)
%
Z
/
\
(
i
<
(
length
a1
))
%
Z
)
->
(((
0
%
Z
<=
j
)
%
Z
/
\
(
j
<
(
length
a1
))
%
Z
)
->
(
permut
a1
a2
)))).
Axiom
permut_sym1
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
array
a
))
(
a2
:
(
array
a
)),
(
permut
a1
a2
)
->
(
permut
a2
a1
).
Axiom
permut_trans1
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
a3
:
(
array
a
)),
(
permut
a1
a2
)
->
((
permut
a2
a3
)
->
(
permut
a1
a3
)).
(
*
Why3
assumption
*
)
Definition
array_eq_sub
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
(
map_eq_sub
(
elts
a1
)
(
elts
a2
)
l
u
).
(
*
Why3
assumption
*
)
Definition
array_eq
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
:
Prop
:=
((
length
a1
)
=
(
length
a2
))
/
\
(
array_eq_sub
a1
a2
0
%
Z
(
length
a1
)).
Axiom
array_eq_sub_permut
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
array
a
))
(
a2
:
(
array
a
))
(
l
:
Z
)
(
u
:
Z
),
(
array_eq_sub
a1
a2
l
u
)
->
(
permut_sub1
a1
a2
l
u
).
Axiom
array_eq_permut
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
(
array
a
))
(
a2
:
(
array
a
)),
(
array_eq
a1
a2
)
->
(
permut
a1
a2
).
Axiom
elt
:
Type
.
Parameter
elt_WhyType
:
WhyType
elt
.
Existing
Instance
elt_WhyType
.
Parameter
le
:
elt
->
elt
->
Prop
.
(
*
Why3
assumption
*
)
Definition
sorted_sub
(
a
:
(
map
Z
elt
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
forall
(
i1
:
Z
)
(
i2
:
Z
),
(((
l
<=
i1
)
%
Z
/
\
(
i1
<=
i2
)
%
Z
)
/
\
(
i2
<
u
)
%
Z
)
->
(
le
(
get
a
i1
)
(
get
a
i2
)).
Axiom
le_refl
:
forall
(
x
:
elt
),
(
le
x
x
).
Axiom
le_asym
:
forall
(
x
:
elt
)
(
y
:
elt
),
(
~
(
le
x
y
))
->
(
le
y
x
).
Axiom
le_trans
:
forall
(
x
:
elt
)
(
y
:
elt
)
(
z
:
elt
),
((
le
x
y
)
/
\
(
le
y
z
))
->
(
le
x
z
).
(
*
Why3
assumption
*
)
Definition
sorted_sub1
(
a
:
(
array
elt
))
(
l
:
Z
)
(
u
:
Z
)
:
Prop
:=
(
sorted_sub
(
elts
a
)
l
u
).
(
*
Why3
assumption
*
)
Definition
sorted
(
a
:
(
array
elt
))
:
Prop
:=
(
sorted_sub
(
elts
a
)
0
%
Z
(
length
a
)).
Require
Import
Why3
.
Ltac
ae
:=
why3
"alt-ergo"
timelimit
3.
(
*
Why3
goal
*
)
Theorem
WP_parameter_insertion_sort
:
forall
(
a
:
Z
),
forall
(
a1
:
(
map
Z
elt
)),
let
a2
:=
(
mk_array
a
a1
)
in
((
1
%
Z
<=
(
a
-
1
%
Z
)
%
Z
)
%
Z
->
forall
(
a3
:
(
map
Z
elt
)),
forall
(
i
:
Z
),
((
1
%
Z
<=
i
)
%
Z
/
\
(
i
<=
(
a
-
1
%
Z
)
%
Z
)
%
Z
)
->
(((
sorted_sub
a3
0
%
Z
i
)
/
\
(
permut
a2
(
mk_array
a
a3
)))
->
(((
0
%
Z
<=
i
)
%
Z
/
\
(
i
<
a
)
%
Z
)
->
let
v
:=
(
get
a3
i
)
in
forall
(
j
:
Z
)
(
a4
:
(
map
Z
elt
)),
(((((
0
%
Z
<=
j
)
%
Z
/
\
(
j
<=
i
)
%
Z
)
/
\
(
permut
a2
(
mk_array
a
(
set
a4
j
v
))))
/
\
forall
(
k1
:
Z
)
(
k2
:
Z
),
(((
0
%
Z
<=
k1
)
%
Z
/
\
(
k1
<=
k2
)
%
Z
)
/
\
(
k2
<=
i
)
%
Z
)
->
((
~
(
k1
=
j
))
->
((
~
(
k2
=
j
))
->
(
le
(
get
a4
k1
)
(
get
a4
k2
)))))
/
\
forall
(
k
:
Z
),
(((
j
+
1
%
Z
)
%
Z
<=
k
)
%
Z
/
\
(
k
<=
i
)
%
Z
)
->
(
le
v
(
get
a4
k
)))
->
((
0
%
Z
<
j
)
%
Z
->
(((
0
%
Z
<=
(
j
-
1
%
Z
)
%
Z
)
%
Z
/
\
((
j
-
1
%
Z
)
%
Z
<
a
)
%
Z
)
->
((
~
(
le
(
get
a4
(
j
-
1
%
Z
)
%
Z
)
v
))
->
(((
0
%
Z
<=
(
j
-
1
%
Z
)
%
Z
)
%
Z
/
\
((
j
-
1
%
Z
)
%
Z
<
a
)
%
Z
)
->
(((
0
%
Z
<=
j
)
%
Z
/
\
(
j
<
a
)
%
Z
)
->
forall
(
a5
:
(
map
Z
elt
)),
(
a5
=
(
set
a4
j
(
get
a4
(
j
-
1
%
Z
)
%
Z
)))
->
((
exchange
(
set
a4
j
v
)
(
set
a5
(
j
-
1
%
Z
)
%
Z
v
)
(
j
-
1
%
Z
)
%
Z
j
)
->
forall
(
j1
:
Z
),
(
j1
=
(
j
-
1
%
Z
)
%
Z
)
->
(
permut
a2
(
mk_array
a
(
set
a5
j1
v
)))))))))))).
intros
a
a1
a2
h1
a3
i
(
h2
,
h3
)
(
h4
,
h5
)
(
h6
,
h7
)
v
j
a4
((((
h8
,
h9
),
h10
),
h11
),
h12
)
h13
(
h14
,
h15
)
h16
(
h17
,
h18
)
(
h19
,
h20
)
a5
h21
h22
j1
h23
.
unfold
permut
in
*
.
simpl
;
split
;
auto
.
simpl
in
h10
.
destruct
h10
as
(
h10a
&
h10b
).
apply
permut_trans
with
(
1
:=
h10b
).
ae
.
Qed
.
examples/programs/insertion_sort/why3session.xml
View file @
c6ede020
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examples/programs/insertion_sort_dumb.mlw
View file @
c6ede020
...
...
@@ -42,4 +42,56 @@ module InsertionSortDumb
done
end
\ No newline at end of file
end
module InsertionSortDumbGen
use import int.Int
use import ref.Ref
use import ref.Refint
use import array.Array
use import array.ArrayPermut
clone import map.MapSorted as M
axiom le_refl: forall x:elt. le x x
axiom le_asym: forall x y:elt. not (le x y) -> le y x
axiom le_trans: forall x y z:elt. le x y /\ le y z -> le x z
predicate sorted_sub (a : array elt) (l u : int) =
M.sorted_sub a.elts l u
predicate sorted (a : array elt) =
M.sorted_sub a.elts 0 a.length
let sort (a:array elt)
ensures { sorted a }
ensures { permut (old a) a }
=
'Init:
for i = 0 to a.length - 1 do
invariant { permut (at a 'Init) a }
invariant { sorted_sub a 0 i }
let j = ref i in
while !j > 0 && not (le a[!j-1] a[!j]) do
invariant { 0 <= !j <= i }
invariant { permut (at a 'Init) a }
invariant { sorted_sub a 0 !j }
invariant { sorted_sub a !j (i+1) }
invariant { forall k1 k2:int. 0 <= k1 < !j /\ !j+1 <= k2 <= i ->
le a[k1] a[k2] }
'L:
let b = !j - 1 in
let t = a[!j] in
a[!j] <- a[b];
a[b] <- t;
assert { exchange (at a 'L) a (!j-1) !j };
decr j
done
done
end
examples/programs/insertion_sort_dumb/why3session.xml
View file @
c6ede020
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