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Why3
why3
Commits
e9c6e1ae
Commit
e9c6e1ae
authored
Oct 12, 2012
by
MARCHE Claude
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preuves avec time limit de 30 secondes
parent
adeceeea
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3 changed files
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251 additions
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273 deletions
+251
-273
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_1.v
...mantics5/blocking_semantics5_FreshVariables_eval_swap_1.v
+39
-50
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_2.v
...mantics5/blocking_semantics5_FreshVariables_eval_swap_2.v
+37
-48
examples/hoare_logic/blocking_semantics5/why3session.xml
examples/hoare_logic/blocking_semantics5/why3session.xml
+175
-175
No files found.
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_1.v
View file @
e9c6e1ae
...
...
@@ -202,14 +202,14 @@ Inductive one_step : (map mident value) -> (list (ident* value)%type) -> stmt
value
)
%
type
))
(
f
:
fmla
),
(
eval_fmla
sigma
pi
f
)
->
(
one_step
sigma
pi
(
Sassert
f
)
sigma
pi
Sskip
)
|
one_step_while_true
:
forall
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
eval_fmla
sigma
pi
inv
)
->
(
((
eval_term
sigma
pi
cond
)
=
(
Vbool
true
))
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
(
Sseq
body
(
Swhile
cond
inv
body
)))
)
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
(
eval_fmla
sigma
pi
inv
)
/
\
((
eval_term
sigma
pi
cond
)
=
(
Vbool
true
))
)
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
(
Sseq
body
(
Swhile
cond
inv
body
)))
|
one_step_while_false
:
forall
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
eval_fmla
sigma
pi
inv
)
->
(
((
eval_term
sigma
pi
cond
)
=
(
Vbool
false
))
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
Sskip
)
)
.
(
(
eval_fmla
sigma
pi
inv
)
/
\
((
eval_term
sigma
pi
cond
)
=
(
Vbool
false
))
)
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
Sskip
).
(
*
Why3
assumption
*
)
Inductive
many_steps
:
(
map
mident
value
)
->
(
list
(
ident
*
value
)
%
type
)
...
...
@@ -230,7 +230,7 @@ Axiom steps_non_neg : forall (sigma1:(map mident value)) (sigma2:(map mident
(
0
%
Z
<=
n
)
%
Z
.
(
*
Why3
assumption
*
)
Definition
reducible
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
Definition
reduc
t
ible
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
s
:
stmt
)
:
Prop
:=
exists
sigma
'
:
(
map
mident
value
),
exists
pi
'
:
(
list
(
ident
*
value
)
%
type
),
exists
s
'
:
stmt
,
(
one_step
sigma
pi
s
sigma
'
pi
'
s
'
).
...
...
@@ -282,13 +282,35 @@ Axiom mem_decomp : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (l:(list
a
)),
(
mem
x
l
)
->
exists
l1
:
(
list
a
),
exists
l2
:
(
list
a
),
(
l
=
(
infix_plpl
l1
(
Cons
x
l2
))).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
mident
)
(
v
:
ident
),
match
t
with
|
((
Tvalue
_
)
|
(
Tvar
_
))
=>
((
msubst_term
t
x
v
)
=
t
)
|
(
Tderef
y
)
=>
((
x
=
y
)
->
((
msubst_term
t
x
v
)
=
(
Tvar
v
)))
/
\
((
~
(
x
=
y
))
->
((
msubst_term
t
x
v
)
=
t
))
|
(
Tbin
t1
op
t2
)
=>
((
msubst_term
t
x
v
)
=
(
Tbin
(
msubst_term
t1
x
v
)
op
(
msubst_term
t2
x
v
)))
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
Fixpoint
msubst
(
f
:
fmla
)
(
x
:
mident
)
(
v
:
ident
)
{
struct
f
}:
fmla
:=
match
f
with
|
(
Fterm
e
)
=>
(
Fterm
(
msubst_term
e
x
v
))
|
(
Fand
f1
f2
)
=>
(
Fand
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Fnot
f1
)
=>
(
Fnot
(
msubst
f1
x
v
))
|
(
Fimplies
f1
f2
)
=>
(
Fimplies
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Flet
y
t
f1
)
=>
(
Flet
y
(
msubst_term
t
x
v
)
(
msubst
f1
x
v
))
|
(
Fforall
y
ty
f1
)
=>
(
Fforall
y
ty
(
msubst
f1
x
v
))
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
id
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
match
t
with
|
(
Tvalue
_
)
=>
True
|
(
Tvar
i
)
=>
~
(
x
=
i
)
|
(
Tvar
i
)
=>
~
(
id
=
i
)
|
(
Tderef
_
)
=>
True
|
(
Tbin
t1
_
t2
)
=>
(
fresh_in_term
x
t1
)
/
\
(
fresh_in_term
x
t2
)
|
(
Tbin
t1
_
t2
)
=>
(
fresh_in_term
id
t1
)
/
\
(
fresh_in_term
id
t2
)
end
.
(
*
Why3
assumption
*
)
...
...
@@ -303,28 +325,6 @@ Fixpoint fresh_in_fmla(id:ident) (f:fmla) {struct f}: Prop :=
|
(
Fforall
y
ty
f1
)
=>
(
~
(
id
=
y
))
/
\
(
fresh_in_fmla
id
f1
)
end
.
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
match
t
with
|
((
Tvalue
_
)
|
(
Tvar
_
))
=>
((
msubst_term
t
r
v
)
=
t
)
|
(
Tderef
x
)
=>
((
r
=
x
)
->
((
msubst_term
t
r
v
)
=
(
Tvar
v
)))
/
\
((
~
(
r
=
x
))
->
((
msubst_term
t
r
v
)
=
t
))
|
(
Tbin
t1
op
t2
)
=>
((
msubst_term
t
r
v
)
=
(
Tbin
(
msubst_term
t1
r
v
)
op
(
msubst_term
t2
r
v
)))
end
.
(
*
Why3
assumption
*
)
Fixpoint
msubst
(
f
:
fmla
)
(
x
:
mident
)
(
v
:
ident
)
{
struct
f
}:
fmla
:=
match
f
with
|
(
Fterm
e
)
=>
(
Fterm
(
msubst_term
e
x
v
))
|
(
Fand
f1
f2
)
=>
(
Fand
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Fnot
f1
)
=>
(
Fnot
(
msubst
f1
x
v
))
|
(
Fimplies
f1
f2
)
=>
(
Fimplies
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Flet
y
t
f1
)
=>
(
Flet
y
(
msubst_term
t
x
v
)
(
msubst
f1
x
v
))
|
(
Fforall
y
ty
f1
)
=>
(
Fforall
y
ty
(
msubst
f1
x
v
))
end
.
Axiom
eval_msubst_term
:
forall
(
e
:
term
)
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
x
:
mident
)
(
v
:
ident
),
(
fresh_in_term
v
e
)
->
((
eval_term
sigma
pi
(
msubst_term
e
x
v
))
=
(
eval_term
(
set
sigma
x
...
...
@@ -343,7 +343,7 @@ Axiom eval_swap_term : forall (t:term) (sigma:(map mident value)) (pi:(list
Require
Import
Why3
.
Ltac
ae
:=
why3
"alt-ergo"
timelimit
3
.
Ltac
ae
:=
why3
"alt-ergo"
timelimit
5
.
(
*
Why3
goal
*
)
Theorem
eval_swap
:
forall
(
f
:
fmla
),
...
...
@@ -366,28 +366,17 @@ Theorem eval_swap : forall (f:fmla),
f
))
end
.
destruct
f
;
auto
.
intros
.
simpl
in
*
.
simpl
;
intros
.
assert
(
h
:
forall
(
l1
l2
:
list
(
ident
*
value
))
(
a
:
(
ident
*
value
)),
(
Cons
a
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a
l1
)
l2
)).
intros
.
induction
l1
.
simpl
;
auto
.
simpl
;
auto
.
destruct
d
;
auto
.
(
Cons
a
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a
l1
)
l2
))
by
(
simpl
;
auto
).
destruct
d
.
(
*
Void
*
)
ae
.
(
*
rewrite
h
.
rewrite
h
in
H1
.
apply
H
;
auto
.
*
)
(
*
Int
*
)
intros
.
rewrite
h
.
ae
.
intros
;
rewrite
h
;
ae
.
(
*
Bool
*
)
intros
.
rewrite
h
.
ae
.
intros
;
rewrite
h
;
ae
.
Qed
.
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_2.v
View file @
e9c6e1ae
...
...
@@ -202,14 +202,14 @@ Inductive one_step : (map mident value) -> (list (ident* value)%type) -> stmt
value
)
%
type
))
(
f
:
fmla
),
(
eval_fmla
sigma
pi
f
)
->
(
one_step
sigma
pi
(
Sassert
f
)
sigma
pi
Sskip
)
|
one_step_while_true
:
forall
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
eval_fmla
sigma
pi
inv
)
->
(
((
eval_term
sigma
pi
cond
)
=
(
Vbool
true
))
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
(
Sseq
body
(
Swhile
cond
inv
body
)))
)
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
(
eval_fmla
sigma
pi
inv
)
/
\
((
eval_term
sigma
pi
cond
)
=
(
Vbool
true
))
)
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
(
Sseq
body
(
Swhile
cond
inv
body
)))
|
one_step_while_false
:
forall
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
cond
:
term
)
(
inv
:
fmla
)
(
body
:
stmt
),
(
eval_fmla
sigma
pi
inv
)
->
(
((
eval_term
sigma
pi
cond
)
=
(
Vbool
false
))
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
Sskip
)
)
.
(
(
eval_fmla
sigma
pi
inv
)
/
\
((
eval_term
sigma
pi
cond
)
=
(
Vbool
false
))
)
->
(
one_step
sigma
pi
(
Swhile
cond
inv
body
)
sigma
pi
Sskip
).
(
*
Why3
assumption
*
)
Inductive
many_steps
:
(
map
mident
value
)
->
(
list
(
ident
*
value
)
%
type
)
...
...
@@ -230,7 +230,7 @@ Axiom steps_non_neg : forall (sigma1:(map mident value)) (sigma2:(map mident
(
0
%
Z
<=
n
)
%
Z
.
(
*
Why3
assumption
*
)
Definition
reducible
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
Definition
reduc
t
ible
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
s
:
stmt
)
:
Prop
:=
exists
sigma
'
:
(
map
mident
value
),
exists
pi
'
:
(
list
(
ident
*
value
)
%
type
),
exists
s
'
:
stmt
,
(
one_step
sigma
pi
s
sigma
'
pi
'
s
'
).
...
...
@@ -282,13 +282,35 @@ Axiom mem_decomp : forall {a:Type} {a_WT:WhyType a}, forall (x:a) (l:(list
a
)),
(
mem
x
l
)
->
exists
l1
:
(
list
a
),
exists
l2
:
(
list
a
),
(
l
=
(
infix_plpl
l1
(
Cons
x
l2
))).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
mident
)
(
v
:
ident
),
match
t
with
|
((
Tvalue
_
)
|
(
Tvar
_
))
=>
((
msubst_term
t
x
v
)
=
t
)
|
(
Tderef
y
)
=>
((
x
=
y
)
->
((
msubst_term
t
x
v
)
=
(
Tvar
v
)))
/
\
((
~
(
x
=
y
))
->
((
msubst_term
t
x
v
)
=
t
))
|
(
Tbin
t1
op
t2
)
=>
((
msubst_term
t
x
v
)
=
(
Tbin
(
msubst_term
t1
x
v
)
op
(
msubst_term
t2
x
v
)))
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
Fixpoint
msubst
(
f
:
fmla
)
(
x
:
mident
)
(
v
:
ident
)
{
struct
f
}:
fmla
:=
match
f
with
|
(
Fterm
e
)
=>
(
Fterm
(
msubst_term
e
x
v
))
|
(
Fand
f1
f2
)
=>
(
Fand
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Fnot
f1
)
=>
(
Fnot
(
msubst
f1
x
v
))
|
(
Fimplies
f1
f2
)
=>
(
Fimplies
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Flet
y
t
f1
)
=>
(
Flet
y
(
msubst_term
t
x
v
)
(
msubst
f1
x
v
))
|
(
Fforall
y
ty
f1
)
=>
(
Fforall
y
ty
(
msubst
f1
x
v
))
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
id
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
match
t
with
|
(
Tvalue
_
)
=>
True
|
(
Tvar
i
)
=>
~
(
x
=
i
)
|
(
Tvar
i
)
=>
~
(
id
=
i
)
|
(
Tderef
_
)
=>
True
|
(
Tbin
t1
_
t2
)
=>
(
fresh_in_term
x
t1
)
/
\
(
fresh_in_term
x
t2
)
|
(
Tbin
t1
_
t2
)
=>
(
fresh_in_term
id
t1
)
/
\
(
fresh_in_term
id
t2
)
end
.
(
*
Why3
assumption
*
)
...
...
@@ -303,28 +325,6 @@ Fixpoint fresh_in_fmla(id:ident) (f:fmla) {struct f}: Prop :=
|
(
Fforall
y
ty
f1
)
=>
(
~
(
id
=
y
))
/
\
(
fresh_in_fmla
id
f1
)
end
.
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
match
t
with
|
((
Tvalue
_
)
|
(
Tvar
_
))
=>
((
msubst_term
t
r
v
)
=
t
)
|
(
Tderef
x
)
=>
((
r
=
x
)
->
((
msubst_term
t
r
v
)
=
(
Tvar
v
)))
/
\
((
~
(
r
=
x
))
->
((
msubst_term
t
r
v
)
=
t
))
|
(
Tbin
t1
op
t2
)
=>
((
msubst_term
t
r
v
)
=
(
Tbin
(
msubst_term
t1
r
v
)
op
(
msubst_term
t2
r
v
)))
end
.
(
*
Why3
assumption
*
)
Fixpoint
msubst
(
f
:
fmla
)
(
x
:
mident
)
(
v
:
ident
)
{
struct
f
}:
fmla
:=
match
f
with
|
(
Fterm
e
)
=>
(
Fterm
(
msubst_term
e
x
v
))
|
(
Fand
f1
f2
)
=>
(
Fand
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Fnot
f1
)
=>
(
Fnot
(
msubst
f1
x
v
))
|
(
Fimplies
f1
f2
)
=>
(
Fimplies
(
msubst
f1
x
v
)
(
msubst
f2
x
v
))
|
(
Flet
y
t
f1
)
=>
(
Flet
y
(
msubst_term
t
x
v
)
(
msubst
f1
x
v
))
|
(
Fforall
y
ty
f1
)
=>
(
Fforall
y
ty
(
msubst
f1
x
v
))
end
.
Axiom
eval_msubst_term
:
forall
(
e
:
term
)
(
sigma
:
(
map
mident
value
))
(
pi
:
(
list
(
ident
*
value
)
%
type
))
(
x
:
mident
)
(
v
:
ident
),
(
fresh_in_term
v
e
)
->
((
eval_term
sigma
pi
(
msubst_term
e
x
v
))
=
(
eval_term
(
set
sigma
x
...
...
@@ -367,27 +367,16 @@ Theorem eval_swap : forall (f:fmla),
end
.
destruct
f
;
auto
.
intros
.
simpl
in
*
.
assert
(
h
:
forall
(
l1
l2
:
list
(
ident
*
value
))
(
a
:
(
ident
*
value
)),
(
Cons
a
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a
l1
)
l2
)).
intros
.
induction
l1
.
simpl
;
auto
.
simpl
;
auto
.
destruct
d
;
auto
.
(
Cons
a
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a
l1
)
l2
))
by
(
simpl
;
auto
).
destruct
d
;
simpl
.
(
*
Void
*
)
ae
.
(
*
rewrite
h
.
rewrite
h
in
H1
.
apply
H
;
auto
.
*
)
(
*
Int
*
)
intros
.
rewrite
h
.
ae
.
intros
;
rewrite
h
;
ae
.
(
*
Bool
*
)
intros
.
rewrite
h
.
ae
.
intros
;
rewrite
h
;
ae
.
Qed
.
examples/hoare_logic/blocking_semantics5/why3session.xml
View file @
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