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Why3
why3
Commits
a815ce53
Commit
a815ce53
authored
Oct 15, 2012
by
Asma Tafat-Bouzid
Browse files
rajout lemma appen_nil_l et cons_append
parent
43d448bf
Changes
9
Expand all
Show whitespace changes
Inline
Side-by-side
examples/hoare_logic/blocking_semantics5.mlw
View file @
a815ce53
...
...
@@ -408,6 +408,12 @@ theory FreshVariables
use import SemOp
use import list.Append
lemma Cons_append: forall a: 'a, l1 l2: list 'a.
Cons a (l1 ++ l2) = (Cons a l1) ++ l2
lemma Append_nil_l:
forall l: list 'a. Nil ++ l = l
(** substitution of a reference [x] by a logic variable [v]
warning: proper behavior only guaranted if [v] is fresh *)
...
...
@@ -605,7 +611,7 @@ fresh_in_fmla (fresh_from f) f
alors
sigma, pi |= f
*)
axiom abstract_effects_
gener
alize :
axiom abstract_effects_
speci
alize :
forall sigma:env, pi:stack, s:stmt, f:fmla.
eval_fmla sigma pi (abstract_effects s f) ->
eval_fmla sigma pi f
...
...
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_change_free_1.v
View file @
a815ce53
...
...
@@ -282,15 +282,22 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_nil_l
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
(
Nil
:
(
list
a
))
l
)
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
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
)))
|
((
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
*
)
...
...
@@ -305,12 +312,12 @@ Fixpoint msubst(f:fmla) (x:mident) (v:ident) {struct f}: fmla :=
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
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
*
)
...
...
@@ -370,14 +377,9 @@ Theorem eval_change_free : forall (f:fmla),
|
(
Fforall
i
d
f1
)
=>
True
end
.
destruct
f
;
auto
.
simpl
;
intros
H
sigma
pi
id
v
(
h1
&
h2
&
h3
).
simpl
;
intros
H
sigma
pi
id
v
(
h1
&
h2
&
h3
)
h
.
rewrite
eval_term_change_free
;
auto
.
assert
(
eval_fmla
sigma
(
infix_plpl
(
Nil
:
(
list
(
ident
*
value
)))
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
)))
f
=
eval_fmla
sigma
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
))
f
);
auto
.
rewrite
<-
H0
;
clear
H0
.
pattern
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
));
rewrite
<-
Append_nil_l
.
rewrite
eval_swap
;
auto
.
apply
H
;
auto
.
Qed
.
...
...
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_change_free_2.v
View file @
a815ce53
...
...
@@ -282,15 +282,22 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_nil_l
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
(
Nil
:
(
list
a
))
l
)
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
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
)))
|
((
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
*
)
...
...
@@ -305,12 +312,12 @@ Fixpoint msubst(f:fmla) (x:mident) (v:ident) {struct f}: fmla :=
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
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
*
)
...
...
@@ -369,12 +376,7 @@ Theorem eval_change_free : forall (f:fmla),
destruct
f
;
auto
.
simpl
;
intros
H
sigma
pi
id
v
(
h1
&
h2
&
h3
).
rewrite
eval_term_change_free
;
auto
.
assert
(
h
:
eval_fmla
sigma
(
infix_plpl
(
Nil
:
(
list
(
ident
*
value
)))
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
)))
f
=
eval_fmla
sigma
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
))
f
);
auto
.
rewrite
<-
h
.
pattern
(
Cons
(
i
,
eval_term
sigma
pi
t
)
(
Cons
(
id
,
v
)
pi
));
rewrite
<-
Append_nil_l
.
rewrite
eval_swap
;
auto
.
apply
H
;
auto
.
Qed
.
...
...
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_change_free_3.v
View file @
a815ce53
...
...
@@ -282,15 +282,22 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_nil_l
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
(
Nil
:
(
list
a
))
l
)
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
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
)))
|
((
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
*
)
...
...
@@ -305,12 +312,12 @@ Fixpoint msubst(f:fmla) (x:mident) (v:ident) {struct f}: fmla :=
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
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
*
)
...
...
@@ -393,12 +400,7 @@ simpl; apply H; auto.
(
*
TYbool
*
)
intro
b
.
assert
(
h
:
eval_fmla
sigma
(
Cons
(
i
,
Vbool
b
)
(
Cons
(
id
,
v
)
pi
))
f
);
auto
.
assert
(
eval_fmla
sigma
(
infix_plpl
(
Nil
:
(
list
(
ident
*
value
)))
(
Cons
(
i
,
Vbool
b
)
(
Cons
(
id
,
v
)
pi
)))
f
=
eval_fmla
sigma
(
Cons
(
i
,
Vbool
b
)
(
Cons
(
id
,
v
)
pi
))
f
);
auto
.
rewrite
<-
H0
.
pattern
(
Cons
(
i
,
Vbool
b
)
(
Cons
(
id
,
v
)
pi
));
rewrite
<-
Append_nil_l
.
rewrite
eval_swap
;
auto
.
simpl
;
apply
H
;
auto
.
Qed
.
...
...
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_change_free_4.v
View file @
a815ce53
...
...
@@ -282,15 +282,22 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_nil_l
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
(
Nil
:
(
list
a
))
l
)
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
r
:
mident
)
(
v
:
ident
),
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
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
)))
|
((
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
*
)
...
...
@@ -305,12 +312,12 @@ Fixpoint msubst(f:fmla) (x:mident) (v:ident) {struct f}: fmla :=
end
.
(
*
Why3
assumption
*
)
Fixpoint
fresh_in_term
(
x
:
ident
)
(
t
:
term
)
{
struct
t
}:
Prop
:=
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
*
)
...
...
@@ -393,9 +400,7 @@ apply H1.
(
*
Vbool
*
)
intro
b
.
rewrite
<-
(
H
_
_
id
v
);
auto
.
replace
(
Cons
(
id
,
v
)
(
Cons
(
i
,
Vbool
b
)
pi
))
with
(
infix_plpl
(
Nil
:
(
list
(
ident
*
value
)))
(
Cons
(
id
,
v
)
(
Cons
(
i
,
Vbool
b
)
pi
)))
by
auto
.
pattern
(
Cons
(
id
,
v
)
(
Cons
(
i
,
Vbool
b
)
pi
));
rewrite
<-
Append_nil_l
.
rewrite
eval_swap
;
auto
.
apply
H1
.
Qed
.
...
...
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_1.v
View file @
a815ce53
...
...
@@ -282,6 +282,13 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_l_nil1
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
l
(
Nil
:
(
list
a
)))
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
mident
)
(
v
:
ident
),
...
...
@@ -367,16 +374,13 @@ Theorem eval_swap : forall (f:fmla),
end
.
destruct
f
;
auto
.
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
))
by
(
simpl
;
auto
).
destruct
d
.
(
*
Void
*
)
ae
.
(
*
Int
*
)
intro
s
;
rewrite
h
;
ae
.
intro
;
rewrite
Cons_append
;
ae
.
(
*
Bool
*
)
intro
s
;
rewrite
h
;
ae
.
intro
;
rewrite
Cons_append
;
ae
.
Qed
.
examples/hoare_logic/blocking_semantics5/blocking_semantics5_FreshVariables_eval_swap_2.v
View file @
a815ce53
...
...
@@ -282,6 +282,13 @@ 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
))).
Axiom
Cons_append
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
a1
:
a
)
(
l1
:
(
list
a
))
(
l2
:
(
list
a
)),
((
Cons
a1
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a1
l1
)
l2
)).
Axiom
Append_l_nil1
:
forall
{
a
:
Type
}
{
a_WT
:
WhyType
a
}
,
forall
(
l
:
(
list
a
)),
((
infix_plpl
l
(
Nil
:
(
list
a
)))
=
l
).
Parameter
msubst_term
:
term
->
mident
->
ident
->
term
.
Axiom
msubst_term_def
:
forall
(
t
:
term
)
(
x
:
mident
)
(
v
:
ident
),
...
...
@@ -367,16 +374,13 @@ Theorem eval_swap : forall (f:fmla),
end
.
destruct
f
;
auto
.
intros
.
assert
(
h
:
forall
(
l1
l2
:
list
(
ident
*
value
))
(
a
:
(
ident
*
value
)),
(
Cons
a
(
infix_plpl
l1
l2
))
=
(
infix_plpl
(
Cons
a
l1
)
l2
))
by
(
simpl
;
auto
).
destruct
d
;
simpl
.
(
*
Void
*
)
ae
.
(
*
Int
*
)
intro
s
;
rewrite
h
;
ae
.
intro
;
rewrite
Cons_append
;
ae
.
(
*
Bool
*
)
intro
s
;
rewrite
h
;
ae
.
intro
;
rewrite
Cons_append
;
ae
.
Qed
.
examples/hoare_logic/blocking_semantics5/blocking_semantics5_WP_distrib_conj_1.v
View file @
a815ce53
...
...
@@ -539,7 +539,7 @@ Theorem distrib_conj : forall (s:stmt),
|
(
Swhile
t
f
s1
)
=>
True
end
.
destruct
s
;
auto
.
unfold
valid_fmla
;
simpl
;
ae
.
simpl
;
ae
.
(
*
intros
sigma
pi
p
q
.
intros
.
...
...
examples/hoare_logic/blocking_semantics5/why3session.xml
View file @
a815ce53
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