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Why3
why3
Commits
c232ed5b
Commit
c232ed5b
authored
May 16, 2011
by
Jean-Christophe Filliâtre
Browse files
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Plain Diff
example decrease1: Coq proof and recursive version
parent
b7c7d6f9
Changes
3
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3 changed files
with
292 additions
and
11 deletions
+292
-11
examples/programs/decrease1.mlw
examples/programs/decrease1.mlw
+28
-11
examples/programs/decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v
...decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v
+100
-0
examples/programs/decrease1/why3session.xml
examples/programs/decrease1/why3session.xml
+164
-0
No files found.
examples/programs/decrease1.mlw
View file @
c232ed5b
...
...
@@ -18,21 +18,38 @@ module Decrease1
forall a: array int. decrease1 a ->
forall i j: int. 0 <= i <= j < length a -> a[j] >= a[i] + i - j
exception Found
int
exception Found
let search (a: array int) =
{ decrease1 a }
let i = ref 0 in
while !i < length a do
invariant { 0 <= i and
forall j: int. 0 <= j < i -> j < length a -> a[j] <> 0 }
variant { length a - i }
if get a !i = 0 then raise (Found !i);
if get a !i > 0 then i := !i + get a !i else i := !i + 1
done
{ forall j: int. 0 <= j < length a -> a[j] <> 0 }
| Found -> { 0 <= result < length a and a[result] = 0 and
forall j: int. 0 <= j < result -> a[j] <> 0 }
try
while !i < length a do
invariant { 0 <= i and
forall j: int. 0 <= j < i -> j < length a -> a[j] <> 0 }
variant { length a - i }
if get a !i = 0 then raise Found;
if get a !i > 0 then i := !i + get a !i else i := !i + 1
done;
-1
with Found ->
!i
end
{ (result = -1 and forall j: int. 0 <= j < length a -> a[j] <> 0)
or (0 <= result < length a and a[result] = 0 and
forall j: int. 0 <= j < result -> a[j] <> 0) }
let rec search_rec (a: array int) (i : int) =
{ decrease1 a and 0 <= i }
if i < length a then
if get a i = 0 then i
else if get a i > 0 then search_rec a (i + get a i)
else search_rec a (i + 1)
else
-1
{ (result = -1 and forall j: int. i <= j < length a -> a[j] <> 0)
or (i <= result < length a and a[result] = 0 and
forall j: int. i <= j < result -> a[j] <> 0) }
end
...
...
examples/programs/decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v
0 → 100644
View file @
c232ed5b
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Definition
unit
:=
unit
.
Parameter
ignore
:
forall
(
a
:
Type
),
a
->
unit
.
Implicit
Arguments
ignore
.
Parameter
label_
:
Type
.
Parameter
at1
:
forall
(
a
:
Type
),
a
->
label_
->
a
.
Implicit
Arguments
at1
.
Parameter
old
:
forall
(
a
:
Type
),
a
->
a
.
Implicit
Arguments
old
.
Definition
ref
(
a
:
Type
)
:=
a
.
Parameter
map
:
forall
(
a
:
Type
)
(
b
:
Type
),
Type
.
Parameter
get
:
forall
(
a
:
Type
)
(
b
:
Type
),
(
map
a
b
)
->
a
->
b
.
Implicit
Arguments
get
.
Parameter
set
:
forall
(
a
:
Type
)
(
b
:
Type
),
(
map
a
b
)
->
a
->
b
->
(
map
a
b
).
Implicit
Arguments
set
.
Axiom
Select_eq
:
forall
(
a
:
Type
)
(
b
:
Type
),
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
)
(
b
:
Type
),
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
create_const
:
forall
(
b
:
Type
)
(
a
:
Type
),
b
->
(
map
a
b
).
Set
Contextual
Implicit
.
Implicit
Arguments
create_const
.
Unset
Contextual
Implicit
.
Axiom
Const
:
forall
(
b
:
Type
)
(
a
:
Type
),
forall
(
b1
:
b
)
(
a1
:
a
),
((
get
(
create_const
(
b1
)
:
(
map
a
b
))
a1
)
=
b1
).
Inductive
array
(
a
:
Type
)
:=
|
mk_array
:
Z
->
(
map
Z
a
)
->
array
a
.
Implicit
Arguments
mk_array
.
Definition
elts
(
a
:
Type
)(
u
:
(
array
a
))
:
(
map
Z
a
)
:=
match
u
with
|
mk_array
_
elts1
=>
elts1
end
.
Implicit
Arguments
elts
.
Definition
length
(
a
:
Type
)(
u
:
(
array
a
))
:
Z
:=
match
u
with
|
mk_array
length1
_
=>
length1
end
.
Implicit
Arguments
length
.
Definition
mixfix_lbrb
(
a
:
Type
)(
a1
:
(
array
a
))
(
i
:
Z
)
:
a
:=
(
get
(
elts
a1
)
i
).
Implicit
Arguments
mixfix_lbrb
.
Definition
decrease1
(
a
:
(
array
Z
))
:
Prop
:=
forall
(
i
:
Z
),
((
0
%
Z
<=
i
)
%
Z
/
\
(
i
<
((
length
a
)
-
1
%
Z
)
%
Z
)
%
Z
)
->
(((
mixfix_lbrb
a
i
)
-
1
%
Z
)
%
Z
<=
(
mixfix_lbrb
a
(
i
+
1
%
Z
)
%
Z
))
%
Z
.
Theorem
decrease1_induction
:
forall
(
a
:
(
array
Z
)),
(
decrease1
a
)
->
forall
(
i
:
Z
)
(
j
:
Z
),
(((
0
%
Z
<=
i
)
%
Z
/
\
(
i
<=
j
)
%
Z
)
/
\
(
j
<
(
length
a
))
%
Z
)
->
((((
mixfix_lbrb
a
i
)
+
i
)
%
Z
-
j
)
%
Z
<=
(
mixfix_lbrb
a
j
))
%
Z
.
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
unfold
decrease1
.
intros
a
Ha
i
j
Hij
.
generalize
Hij
;
pattern
j
.
apply
(
Zlt_lower_bound_ind
_
i
).
2
:
omega
.
intuition
.
assert
(
x
=
i
\
/
i
<
x
)
%
Z
by
omega
.
destruct
H4
.
subst
x
.
ring_simplify
.
omega
.
apply
Zle_trans
with
(
mixfix_lbrb
a
(
x
-
1
)
-
1
)
%
Z
.
assert
(
i
<=
x
-
1
<
x
)
%
Z
by
omega
.
assert
(
0
<=
i
<=
x
-
1
/
\
x
-
1
<
length
a
)
%
Z
by
omega
.
generalize
(
H
(
x
-
1
)
%
Z
H8
H9
);
clear
H
;
intuition
.
apply
Zle_trans
with
(
mixfix_lbrb
a
(
x
-
1
+
1
))
%
Z
.
apply
(
Ha
(
x
-
1
)
%
Z
);
omega
.
ring_simplify
(
x
-
1
+
1
)
%
Z
.
omega
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/decrease1/why3session.xml
0 → 100644
View file @
c232ed5b
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session SYSTEM "why3session.dtd">
<why3session
name=
"examples/programs/decrease1/why3session.xml"
>
<file
name=
"../decrease1.mlw"
verified=
"true"
expanded=
"true"
>
<theory
name=
"Decrease1"
verified=
"true"
expanded=
"true"
>
<goal
name=
"decrease1_induction"
sum=
"8319c3299f6c5b391aec5943ca151429"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"coq"
timelimit=
"10"
edited=
"examples/programs/decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v"
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.75"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"unknown"
time=
"0.43"
/>
</proof>
</goal>
<goal
name=
"WP_search"
expl=
"correctness of search"
sum=
"5f9e29fcb501d5550c29bfbd1c8fd584"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"timeout"
time=
"10.12"
/>
</proof>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_search.1"
expl=
"loop invariant init"
sum=
"5d4fd28ee3c69be535f8ee535e6cb802"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.2"
expl=
"precondition"
sum=
"d315642af07745fcac77676c671ffee3"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.3"
expl=
"normal postcondition"
sum=
"fa8f0f929008e376cd5e388239a0320a"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.05"
/>
</proof>
</goal>
<goal
name=
"WP_search.4"
expl=
"precondition"
sum=
"026ddab308389db6e2db01c19192eb40"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.5"
expl=
"precondition"
sum=
"3f6bcaf82657cc2a967502da8329005b"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.6"
expl=
"loop invariant preservation"
sum=
"5bce229f4e8d1fc1fc6e04c42e6e408b"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"unknown"
time=
"3.17"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"10.19"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.10"
/>
</proof>
</goal>
<goal
name=
"WP_search.7"
expl=
"loop variant decreases"
sum=
"996216bff5e7480e0a1fe0e3968b8ffb"
proved=
"true"
expanded=
"true"
>
<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=
"0.02"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.04"
/>
</proof>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_search.7.1"
expl=
"correctness of search"
sum=
"d97f71408572fc6c730314dd5bebfe22"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.7.2"
expl=
"correctness of search"
sum=
"1f8eb20d029c95d115aa48530508032c"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
<proof
prover=
"yices"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
</goal>
</transf>
</goal>
<goal
name=
"WP_search.8"
expl=
"loop invariant preservation"
sum=
"fc6718c43629f048e462fa6063b11410"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.9"
expl=
"loop variant decreases"
sum=
"14c9dd5d9133e195e8e688d19a5d7e19"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search.10"
expl=
"normal postcondition"
sum=
"ff98d91de211d86ceddf8f37215c787c"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
</goal>
</transf>
</goal>
<goal
name=
"WP_search_rec"
expl=
"correctness of search_rec"
sum=
"359800e7e063b55fb2e4665ffe6c7a62"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"cvc3"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"unknown"
time=
"0.21"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"timeout"
time=
"10.22"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"timeout"
time=
"10.50"
/>
</proof>
<transf
name=
"split_goal"
proved=
"true"
expanded=
"true"
>
<goal
name=
"WP_search_rec.1"
expl=
"precondition"
sum=
"b862eaa14642db1bca4cfd29e71dc85a"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.2"
expl=
"normal postcondition"
sum=
"028a83456b4c241f325751fe75ff7cfe"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.3"
expl=
"precondition"
sum=
"58493f837aaa39a3890ba4c802e46109"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.4"
expl=
"precondition"
sum=
"99a18ac863afa1e6b4acfb99b4d0dae6"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.5"
expl=
"precondition"
sum=
"44a5c5d945aad7cfe2bcc7d4adf728f2"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.6"
expl=
"normal postcondition"
sum=
"699e81654d2a303c491cfab413eb3501"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"10.08"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"10"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.24"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.7"
expl=
"precondition"
sum=
"9453d56ae3a433905973837d2fab5f11"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.03"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.8"
expl=
"normal postcondition"
sum=
"179d0569c8256c2c7e5e5730ab534329"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.05"
/>
</proof>
</goal>
<goal
name=
"WP_search_rec.9"
expl=
"normal postcondition"
sum=
"a5e3a9553c1dbb830d1064e70335f3a9"
proved=
"true"
expanded=
"true"
>
<proof
prover=
"alt-ergo"
timelimit=
"10"
edited=
""
obsolete=
"true"
>
<result
status=
"valid"
time=
"0.02"
/>
</proof>
</goal>
</transf>
</goal>
</theory>
</file>
</why3session>
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