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Why3
why3
Commits
a09408ee
Commit
a09408ee
authored
Sep 29, 2011
by
MARCHE Claude
Browse files
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Coq proofs for euler001
parent
c6032b96
Changes
8
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Showing
8 changed files
with
403 additions
and
150 deletions
+403
-150
examples/programs/euler001.mlw
examples/programs/euler001.mlw
+14
-10
examples/programs/euler001/euler001_DivModHints_mod_div_unique_1.v
...programs/euler001/euler001_DivModHints_mod_div_unique_1.v
+26
-0
examples/programs/euler001/euler001_DivModHints_mod_succ_1_1.v
...les/programs/euler001/euler001_DivModHints_mod_succ_1_1.v
+37
-0
examples/programs/euler001/euler001_DivModHints_mod_succ_2_1.v
...les/programs/euler001/euler001_DivModHints_mod_succ_2_1.v
+38
-0
examples/programs/euler001/euler001_SumMultiple_Closed_formula_n_3_1.v
...rams/euler001/euler001_SumMultiple_Closed_formula_n_3_1.v
+83
-0
examples/programs/euler001/euler001_SumMultiple_div_minus1_2_1.v
...s/programs/euler001/euler001_SumMultiple_div_minus1_2_1.v
+50
-0
examples/programs/euler001/euler001_SumMultiple_div_minus1_2_2.v
...s/programs/euler001/euler001_SumMultiple_div_minus1_2_2.v
+43
-0
examples/programs/euler001/why3session.xml
examples/programs/euler001/why3session.xml
+112
-140
No files found.
examples/programs/euler001.mlw
View file @
a09408ee
...
...
@@ -13,8 +13,8 @@ theory DivModHints
use import int.ComputerDivision
lemma mod_div_unique :
forall x y q r:int. y > 0 /\ x = q*y + r /\ 0 <= r < y ->
r = mod x y /\ q = div x y
forall x y q r:int.
x >= 0 /\
y > 0 /\ x = q*y + r /\ 0 <= r < y ->
q = div x y /\ r = mod x y
lemma mod_succ_1 :
forall x y:int. x >= 0 /\ y > 0 ->
...
...
@@ -72,21 +72,25 @@ theory SumMultiple
forall n:int.
mod n 15 = 0 <-> mod n 3 = 0 /\ mod n 5 = 0
lemma triangle_numbers:
forall n:int.
2 * div (n*(n+1)) 2 = n*(n+1)
lemma Closed_formula_n:
forall n:int. n >
0 -> p (n-1)
->
mod
n 3 <> 0 /\ mod n 5 <> 0 -> p n
forall n:int. n >
= 0 -> p n
->
mod
(n+1) 3 <> 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
lemma Closed_formula_n_3:
forall n:int. n >
0 -> p (n-1)
->
mod
n 3 = 0 /\ mod n 5 <> 0 -> p n
forall n:int. n >
= 0 -> p n
->
mod
(n+1) 3 = 0 /\ mod (n+1) 5 <> 0 -> p (n+1)
lemma Closed_formula_n_5:
forall n:int. n >
0 -> p (n-1)
->
mod
n 3 <> 0 /\ mod n 5 = 0 -> p n
forall n:int. n >
= 0 -> p n
->
mod
(n+1) 3 <> 0 /\ mod (n+1) 5 = 0 -> p (n+1)
lemma Closed_formula_n_15:
forall n:int. n >
0 -> p (n-1)
->
mod
n 3 = 0 /\ mod n 5 = 0 -> p n
forall n:int. n >
= 0 -> p n
->
mod
(n+1) 3 = 0 /\ mod (n+1) 5 = 0 -> p (n+1)
clone int.Induction as I with predicate p = p
...
...
examples/programs/euler001/euler001_DivModHints_mod_div_unique_1.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
ZOdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
mod_div_unique
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
q
:
Z
)
(
r
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
((
0
%
Z
<
y
)
%
Z
/
\
((
x
=
((
q
*
y
)
%
Z
+
r
)
%
Z
)
/
\
((
0
%
Z
<=
r
)
%
Z
/
\
(
r
<
y
)
%
Z
))))
->
((
q
=
(
ZOdiv
x
y
))
/
\
(
r
=
(
ZOmod
x
y
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
q
r
(
H1
,(
H2
,(
H3
,(
H4
,
H5
)))).
apply
ZOdiv_mod_unique_full
.
2
:
rewrite
H3
;
ring
.
red
.
left
.
rewrite
Zabs_eq
;
auto
with
zarith
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/euler001_DivModHints_mod_succ_1_1.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
ZOdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
Axiom
mod_div_unique
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
q
:
Z
)
(
r
:
Z
),
((
0
%
Z
<
y
)
%
Z
/
\
((
x
=
((
q
*
y
)
%
Z
+
r
)
%
Z
)
/
\
((
0
%
Z
<=
r
)
%
Z
/
\
(
r
<
y
)
%
Z
)))
->
((
r
=
(
ZOmod
x
y
))
/
\
(
q
=
(
ZOdiv
x
y
))).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
Open
Scope
Z_scope
.
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
mod_succ_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
((
ZOmod
x
y
)
+
1
%
Z
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
)
H
.
assert
(
h
:
y
>
0
)
by
omega
.
generalize
(
ZO_div_mod_eq
x
y
);
intro
h1
.
generalize
(
ZO_div_mod_eq
(
x
+
1
)
y
);
intro
h2
.
assert
(
h3
:
x
=
y
*
((
x
+
1
)
/
y
)
+
((
x
+
1
)
mod
y
-
1
))
by
omega
.
generalize
(
mod_div_unique
x
y
((
x
+
1
)
/
y
)
((
x
+
1
)
mod
y
-
1
)).
intuition
.
destruct
H1
;
auto
with
zarith
.
rewrite
h3
at
1.
ring
.
assert
(
0
<=
(
x
+
1
)
mod
y
<
y
).
apply
ZOmod_lt_pos_pos
;
omega
.
omega
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/euler001_DivModHints_mod_succ_2_1.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
ZOdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
Axiom
mod_div_unique
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
q
:
Z
)
(
r
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
((
0
%
Z
<
y
)
%
Z
/
\
((
x
=
((
q
*
y
)
%
Z
+
r
)
%
Z
)
/
\
((
0
%
Z
<=
r
)
%
Z
/
\
(
r
<
y
)
%
Z
))))
->
((
q
=
(
ZOdiv
x
y
))
/
\
(
r
=
(
ZOmod
x
y
))).
Axiom
mod_succ_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
((
ZOmod
x
y
)
+
1
%
Z
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
mod_succ_2
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
)
->
((
ZOmod
x
y
)
=
(
y
-
1
%
Z
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
)
H
.
generalize
(
ZO_div_mod_eq
x
y
);
intro
h1
.
generalize
(
ZO_div_mod_eq
(
x
+
1
)
y
);
intro
h2
.
assert
(
h3
:
x
=
y
*
((
x
+
1
)
/
y
-
1
)
+
((
x
+
1
)
mod
y
+
y
-
1
)).
ring_simplify
;
omega
.
rewrite
H
in
h3
.
generalize
(
mod_div_unique
x
y
((
x
+
1
)
/
y
-
1
)
(
0
+
y
-
1
)).
intuition
.
destruct
H1
;
auto
with
zarith
.
rewrite
h3
at
1.
ring
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/euler001_SumMultiple_Closed_formula_n_3_1.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
ZOdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
Parameter
sum_multiple_3_5_lt
:
Z
->
Z
.
Axiom
SumEmpty
:
((
sum_multiple_3_5_lt
0
%
Z
)
=
0
%
Z
).
Axiom
SumNo
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
(((
~
((
ZOmod
n
3
%
Z
)
=
0
%
Z
))
/
\
~
((
ZOmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
(
sum_multiple_3_5_lt
n
))).
Axiom
SumYes
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((((
ZOmod
n
3
%
Z
)
=
0
%
Z
)
\
/
((
ZOmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
((
sum_multiple_3_5_lt
n
)
+
n
)
%
Z
)).
Definition
closed_formula
(
n
:
Z
)
:
Z
:=
let
n3
:=
(
ZOdiv
n
3
%
Z
)
in
let
n5
:=
(
ZOdiv
n
5
%
Z
)
in
let
n15
:=
(
ZOdiv
n
15
%
Z
)
in
(
ZOdiv
((((
3
%
Z
*
n3
)
%
Z
*
(
n3
+
1
%
Z
)
%
Z
)
%
Z
+
((
5
%
Z
*
n5
)
%
Z
*
(
n5
+
1
%
Z
)
%
Z
)
%
Z
)
%
Z
-
((
15
%
Z
*
n15
)
%
Z
*
(
n15
+
1
%
Z
)
%
Z
)
%
Z
)
%
Z
2
%
Z
).
Definition
p
(
n
:
Z
)
:
Prop
:=
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
(
closed_formula
n
)).
Axiom
Closed_formula_0
:
(
p
0
%
Z
).
Axiom
mod_div_unique
:
forall
(
x
:
Z
)
(
y
:
Z
)
(
q
:
Z
)
(
r
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
((
0
%
Z
<
y
)
%
Z
/
\
((
x
=
((
q
*
y
)
%
Z
+
r
)
%
Z
)
/
\
((
0
%
Z
<=
r
)
%
Z
/
\
(
r
<
y
)
%
Z
))))
->
((
q
=
(
ZOdiv
x
y
))
/
\
(
r
=
(
ZOmod
x
y
))).
Axiom
mod_succ_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
((
ZOmod
x
y
)
+
1
%
Z
)
%
Z
)).
Axiom
mod_succ_2
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
)
->
((
ZOmod
x
y
)
=
(
y
-
1
%
Z
)
%
Z
)).
Axiom
div_succ_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
)
->
((
ZOdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
((
ZOdiv
x
y
)
+
1
%
Z
)
%
Z
)).
Axiom
div_succ_2
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
ZOmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
ZOdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
(
ZOdiv
x
y
))).
Axiom
mod_15
:
forall
(
n
:
Z
),
((
ZOmod
n
15
%
Z
)
=
0
%
Z
)
<->
(((
ZOmod
n
3
%
Z
)
=
0
%
Z
)
/
\
((
ZOmod
n
5
%
Z
)
=
0
%
Z
)).
Axiom
triangle_numbers
:
forall
(
n
:
Z
),
((
2
%
Z
*
(
ZOdiv
(
n
*
(
n
+
1
%
Z
)
%
Z
)
%
Z
2
%
Z
))
%
Z
=
(
n
*
(
n
+
1
%
Z
)
%
Z
)
%
Z
).
Axiom
Closed_formula_n
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((
p
n
)
->
(((
~
((
ZOmod
(
n
+
1
%
Z
)
%
Z
3
%
Z
)
=
0
%
Z
))
/
\
~
((
ZOmod
(
n
+
1
%
Z
)
%
Z
5
%
Z
)
=
0
%
Z
))
->
(
p
(
n
+
1
%
Z
)
%
Z
))).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
Closed_formula_n_3
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((
p
n
)
->
((((
ZOmod
(
n
+
1
%
Z
)
%
Z
3
%
Z
)
=
0
%
Z
)
/
\
~
((
ZOmod
(
n
+
1
%
Z
)
%
Z
5
%
Z
)
=
0
%
Z
))
->
(
p
(
n
+
1
%
Z
)
%
Z
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
n
Hn
Hind
(
H3
,
H5
).
unfold
p
in
*
.
rewrite
SumYes
;
auto
with
zarith
.
rewrite
Hind
;
clear
Hind
.
unfold
closed_formula
.
rewrite
(
div_succ_1
n
3
);
auto
with
zarith
.
rewrite
(
div_succ_2
n
5
);
auto
with
zarith
.
rewrite
(
div_succ_2
n
15
);
auto
with
zarith
.
2
:
generalize
(
mod_15
(
n
+
1
));
intuition
.
ring_simplify
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/euler001_SumMultiple_div_minus1_2_1.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
Zdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
Parameter
sum_multiple_3_5_lt
:
Z
->
Z
.
Axiom
SumEmpty
:
((
sum_multiple_3_5_lt
0
%
Z
)
=
0
%
Z
).
Axiom
SumNo
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
(((
~
((
Zmod
n
3
%
Z
)
=
0
%
Z
))
/
\
~
((
Zmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
(
sum_multiple_3_5_lt
n
))).
Axiom
SumYes
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((((
Zmod
n
3
%
Z
)
=
0
%
Z
)
\
/
((
Zmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
((
sum_multiple_3_5_lt
n
)
+
n
)
%
Z
)).
Axiom
div_minus1_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(((
Zmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
)
->
((
Zdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
((
Zdiv
x
y
)
+
1
%
Z
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
div_minus1_2
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
Zmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
Zdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
(
Zdiv
x
y
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
)
H
.
pose
(
q
:=
(
x
/
y
)
%
Z
).
pose
(
r
:=
(
x
mod
y
)
%
Z
).
assert
(
h2
:
(
x
=
y
*
q
+
r
)
%
Z
).
apply
(
Z_div_mod_eq
x
y
);
auto
with
zarith
.
assert
(
h3
:
(
0
<=
r
<
y
-
1
)
%
Z
).
admit
.
rewrite
h2
.
rewrite
Zmult_comm
.
rewrite
<-
Zplus_assoc
.
rewrite
Z_div_plus_full_l
;
auto
with
zarith
.
rewrite
Z_div_plus_full_l
;
auto
with
zarith
.
rewrite
Zdiv_small
;
auto
with
zarith
.
rewrite
Zdiv_small
;
auto
with
zarith
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/euler001_SumMultiple_div_minus1_2_2.v
0 → 100644
View file @
a09408ee
(
*
This
file
is
generated
by
Why3
'
s
Coq
driver
*
)
(
*
Beware
!
Only
edit
allowed
sections
below
*
)
Require
Import
ZArith
.
Require
Import
Rbase
.
Require
Import
Zdiv
.
Axiom
Abs_le
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
Zabs
x
)
<=
y
)
%
Z
<->
(((
-
y
)
%
Z
<=
x
)
%
Z
/
\
(
x
<=
y
)
%
Z
).
Parameter
sum_multiple_3_5_lt
:
Z
->
Z
.
Axiom
SumEmpty
:
((
sum_multiple_3_5_lt
0
%
Z
)
=
0
%
Z
).
Axiom
SumNo
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
(((
~
((
Zmod
n
3
%
Z
)
=
0
%
Z
))
/
\
~
((
Zmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
(
sum_multiple_3_5_lt
n
))).
Axiom
SumYes
:
forall
(
n
:
Z
),
(
0
%
Z
<=
n
)
%
Z
->
((((
Zmod
n
3
%
Z
)
=
0
%
Z
)
\
/
((
Zmod
n
5
%
Z
)
=
0
%
Z
))
->
((
sum_multiple_3_5_lt
(
n
+
1
%
Z
)
%
Z
)
=
((
sum_multiple_3_5_lt
n
)
+
n
)
%
Z
)).
Axiom
div_minus1_1
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
(((
Zmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
)
->
((
Zdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
((
Zdiv
x
y
)
+
1
%
Z
)
%
Z
)).
(
*
YOU
MAY
EDIT
THE
CONTEXT
BELOW
*
)
(
*
DO
NOT
EDIT
BELOW
*
)
Theorem
div_minus1_2
:
forall
(
x
:
Z
)
(
y
:
Z
),
((
0
%
Z
<=
x
)
%
Z
/
\
(
0
%
Z
<
y
)
%
Z
)
->
((
~
((
Zmod
(
x
+
1
%
Z
)
%
Z
y
)
=
0
%
Z
))
->
((
Zdiv
(
x
+
1
%
Z
)
%
Z
y
)
=
(
Zdiv
x
y
))).
(
*
YOU
MAY
EDIT
THE
PROOF
BELOW
*
)
intros
x
y
(
Hx
,
Hy
)
H
.
pose
(
q
:=
(
x
/
y
)
%
Z
).
pose
(
r
:=
(
x
mod
y
)
%
Z
).
assert
(
h2
:
(
x
=
y
*
q
+
r
)
%
Z
).
apply
(
Z_div_mod_eq
x
y
);
auto
with
zarith
.
assert
(
h3
:
(
0
<
r
<=
y
-
1
)
%
Z
).
admit
.
Qed
.
(
*
DO
NOT
EDIT
BELOW
*
)
examples/programs/euler001/why3session.xml
View file @
a09408ee
...
...
@@ -52,47 +52,26 @@
expanded=
"true"
>
<theory
name=
"DivModHints"
verified=
"
fals
e"
verified=
"
tru
e"
expanded=
"true"
>
<goal
name=
"mod_div_unique"
sum=
"
dad7f26312903f3c0ce7ea31f08a8a94
"
proved=
"
fals
e"
sum=
"
f9938f131fc09755c1d3c9cb8bea2a18
"
proved=
"
tru
e"
expanded=
"true"
shape=
"ainfix =V2adivV0V1Aainfix =V3amodV0V1Iainfix <V3V1Aainfix <=c0V3Aainfix =V0ainfix +ainfix *V2V1V3Aainfix >V1c0F"
>
<proof
prover=
"simplify"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"unknown"
time=
"0.05"
/>
</proof>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.13"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"unknown"
time=
"0.37"
/>
</proof>
shape=
"ainfix =V3amodV0V1Aainfix =V2adivV0V1Iainfix <V3V1Aainfix <=c0V3Aainfix =V0ainfix +ainfix *V2V1V3Aainfix >V1c0Aainfix >=V0c0F"
>
<proof
prover=
"
z3
"
prover=
"
coq
"
timelimit=
"5"
edited=
""
edited=
"
euler001_DivModHints_mod_div_unique_1.v
"
obsolete=
"false"
>
<result
status=
"
timeout"
time=
"5.08
"
/>
<result
status=
"
valid"
time=
"1.11
"
/>
</proof>
</goal>
<goal
name=
"mod_succ_1"
sum=
"
0a9a4fed2057820683962116afbcf691
"
proved=
"
fals
e"
sum=
"
6f6edb13a7196ddd82da5b51501ddbf0
"
proved=
"
tru
e"
expanded=
"true"
shape=
"ainfix =amodainfix +V0c1V1ainfix +amodV0V1c1Iainfix =amodainfix +V0c1V1c0NIainfix >V1c0Aainfix >=V0c0F"
>
<proof
...
...
@@ -100,103 +79,40 @@
timelimit=
"5"
edited=
"euler001_DivModHints_mod_succ_1_1.v"
obsolete=
"false"
>
<result
status=
"unknown"
time=
"1.54"
/>
</proof>
<proof
prover=
"simplify"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.07"
/>
</proof>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.08"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.08"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.09"
/>
</proof>
<proof
prover=
"gappa"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"unknown"
time=
"0.02"
/>
</proof>
<proof
prover=
"vampire"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.01"
/>
</proof>
<proof
prover=
"spass"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.06"
/>
<result
status=
"valid"
time=
"1.33"
/>
</proof>
</goal>
<goal
name=
"mod_succ_2"
sum=
"
d10d5aa2a6f8a1ece4e07b571cd6ab5a
"
proved=
"
fals
e"
sum=
"
7104913925174b07351cfd6bb290b525
"
proved=
"
tru
e"
expanded=
"true"
shape=
"ainfix =amodV0V1ainfix -V1c1Iainfix =amodainfix +V0c1V1c0Iainfix >V1c0Aainfix >=V0c0F"
>
<proof
prover=
"alt-ergo"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.08"
/>
</proof>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.09"
/>
</proof>
<proof
prover=
"z3"
prover=
"coq"
timelimit=
"5"
edited=
""
edited=
"
euler001_DivModHints_mod_succ_2_1.v
"
obsolete=
"false"
>
<result
status=
"
timeout"
time=
"5.10
"
/>
<result
status=
"
valid"
time=
"1.29
"
/>
</proof>
</goal>
<goal
name=
"div_succ_1"
sum=
"
773313eff1da137611f6b6782b46b107
"
sum=
"
2fdd276de2d8937784e2bd42924d0a83
"
proved=
"true"
expanded=
"
fals
e"
expanded=
"
tru
e"
shape=
"ainfix =adivainfix +V0c1V1ainfix +adivV0V1c1Iainfix =amodainfix +V0c1V1c0Iainfix >V1c0Aainfix >=V0c0F"
>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.
10
"
/>
<result
status=
"valid"
time=
"0.
08
"
/>
</proof>
</goal>
<goal
name=
"div_succ_2"
sum=
"
500683558e28d63d764f874cc487b206
"
sum=
"
af570b10585b322352a5f00bc9f9fd12
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =adivainfix +V0c1V1adivV0V1Iainfix =amodainfix +V0c1V1c0NIainfix >V1c0Aainfix >=V0c0F"
>
...
...
@@ -205,21 +121,21 @@
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"1.
28
"
/>
<result
status=
"valid"
time=
"1.
33
"
/>
</proof>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
9
"
/>
<result
status=
"valid"
time=
"0.0
8
"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.
75
"
/>
<result
status=
"valid"
time=
"0.
49
"
/>
</proof>
</goal>
</theory>
...
...
@@ -238,7 +154,7 @@
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.
07
"
/>
<result
status=
"valid"
time=
"0.
12
"
/>
</proof>
<proof
prover=
"alt-ergo"
...
...
@@ -252,12 +168,12 @@
timelimit=
"2"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.0
7
"
/>
<result
status=
"valid"
time=
"0.0
9
"
/>
</proof>
</goal>
<goal
name=
"mod_15"
sum=
"
071e467d1502abbcf3367fb8f2269e3c
"
sum=
"
7c10cad2c344640bfeff4b3a6a8fb07d
"
proved=
"true"
expanded=
"false"
shape=
"ainfix =amodV0c5c0Aainfix =amodV0c3c0qainfix =amodV0c15c0F"
>
...
...
@@ -266,160 +182,216 @@
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.
37
"
/>
<result
status=
"valid"
time=
"0.
53
"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"
3
.05"
/>
<result
status=
"valid"
time=
"
2
.05"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.10"
/>
<result
status=
"timeout"
time=
"5.08"
/>
</proof>
</goal>
<goal
name=
"triangle_numbers"
sum=
"eab47529f1d5363b4583acc38168c78f"
proved=
"true"
expanded=
"true"
shape=
"ainfix =ainfix *c2adivainfix *V0ainfix +V0c1c2ainfix *V0ainfix +V0c1F"
>
<proof
prover=
"cvc3"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"timeout"
time=
"5.09"
/>
</proof>
<proof
prover=
"alt-ergo"
timelimit=
"5"
edited=
""
obsolete=
"false"
>
<result
status=
"valid"
time=
"0.04"
/>
</proof>
<proof
prover=
"z3"
timelimit=
"5"
edited=
""