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Amin Timany
iris-coq
Commits
a314151d
Commit
a314151d
authored
8 years ago
by
Robbert Krebbers
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Simplify weakestpre_fix construction.
parent
1c48ea12
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program_logic/weakestpre_fix.v
+23
-41
23 additions, 41 deletions
program_logic/weakestpre_fix.v
with
23 additions
and
41 deletions
program_logic/weakestpre_fix.v
+
23
−
41
View file @
a314151d
...
@@ -13,11 +13,10 @@ wp on paper. We show that the two versions are equivalent. *)
...
@@ -13,11 +13,10 @@ wp on paper. We show that the two versions are equivalent. *)
Section
def
.
Section
def
.
Context
{
Λ
:
language
}
{
Σ
:
iFunctor
}
.
Context
{
Λ
:
language
}
{
Σ
:
iFunctor
}
.
Local
Notation
iProp
:=
(
iProp
Λ
Σ
)
.
Local
Notation
iProp
:=
(
iProp
Λ
Σ
)
.
Local
Notation
coPsetC
:=
(
leibnizC
(
coPset
))
.
Program
Definition
wp_pre
Program
Definition
wp_pre
(
wp
:
coPset
C
-
n
>
expr
C
Λ
-
n
>
(
val
C
Λ
-
n
>
iProp
)
-
n
>
iProp
)
(
wp
:
coPset
-
c
>
expr
Λ
-
c
>
(
val
Λ
-
c
>
iProp
)
-
c
>
iProp
)
:
(
E
:
coPset
)
(
e1
:
expr
Λ
)
(
Φ
:
val
C
Λ
-
n
>
iProp
)
:
iProp
:=
coPset
-
c
>
expr
Λ
-
c
>
(
val
Λ
-
c
>
iProp
)
-
c
>
iProp
:=
λ
E
e1
Φ
,
{|
uPred_holds
n
r1
:=
∀
k
Ef
σ1
rf
,
{|
uPred_holds
n
r1
:=
∀
k
Ef
σ1
rf
,
0
≤
k
<
n
→
E
⊥
Ef
→
0
≤
k
<
n
→
E
⊥
Ef
→
wsat
(
S
k
)
(
E
∪
Ef
)
σ1
(
r1
⋅
rf
)
→
wsat
(
S
k
)
(
E
∪
Ef
)
σ1
(
r1
⋅
rf
)
→
...
@@ -46,49 +45,32 @@ Next Obligation.
...
@@ -46,49 +45,32 @@ Next Obligation.
Qed
.
Qed
.
Next
Obligation
.
repeat
intro
;
eauto
.
Qed
.
Next
Obligation
.
repeat
intro
;
eauto
.
Qed
.
Lemma
wp_pre_contractive'
n
E
e
Φ1
Φ2
r
Local
Instance
pre_wp_contractive
:
Contractive
wp_pre
.
(
wp1
wp2
:
coPsetC
-
n
>
exprC
Λ
-
n
>
(
valC
Λ
-
n
>
iProp
)
-
n
>
iProp
)
:
(
∀
i
:
nat
,
i
<
n
→
wp1
≡
{
i
}
≡
wp2
)
→
Φ1
≡
{
n
}
≡
Φ2
→
wp_pre
wp1
E
e
Φ1
n
r
→
wp_pre
wp2
E
e
Φ2
n
r
.
Proof
.
Proof
.
intros
HI
HΦ
Hwp
k
Ef
σ1
rf
???
.
assert
(
∀
n
E
e
Φ
r
destruct
(
Hwp
k
Ef
σ1
rf
)
as
[
Hval
Hstep
];
auto
.
(
wp1
wp2
:
coPset
-
c
>
expr
Λ
-
c
>
(
val
Λ
-
c
>
iProp
)
-
c
>
iProp
),
split
.
(
∀
i
:
nat
,
i
<
n
→
wp1
≡
{
i
}
≡
wp2
)
→
{
intros
v
?
.
destruct
(
Hval
v
)
as
(
r2
&
?
&
?);
auto
.
wp_pre
wp1
E
e
Φ
n
r
→
wp_pre
wp2
E
e
Φ
n
r
)
as
help
.
exists
r2
.
split
;
[
apply
HΦ
|];
auto
.
}
{
intros
n
E
e
Φ
r
wp1
wp2
HI
Hwp
k
Ef
σ1
rf
???
.
intros
??
.
destruct
Hstep
as
[
Hred
Hpstep
];
auto
.
destruct
(
Hwp
k
Ef
σ1
rf
)
as
[
Hval
Hstep
];
auto
.
split
;
[
done
|]=>
e2
σ2
ef
?
.
split
;
first
done
.
destruct
(
Hpstep
e2
σ2
ef
)
as
(
r2
&
r2'
&
?
&
?
&
?);
[
done
..|]
.
intros
??
.
destruct
Hstep
as
[
Hred
Hpstep
];
auto
.
exists
r2
,
r2'
;
split_and
?;
auto
.
split
;
[
done
|]=>
e2
σ2
ef
?
.
-
apply
HI
with
k
;
auto
.
destruct
(
Hpstep
e2
σ2
ef
)
as
(
r2
&
r2'
&
?
&
?
&
?);
[
done
..|]
.
assert
(
wp1
E
e2
Φ2
≡
{
n
}
≡
wp1
E
e2
Φ1
)
as
HwpΦ
by
(
by
rewrite
HΦ
)
.
exists
r2
,
r2'
;
split_and
?;
auto
.
apply
HwpΦ
;
auto
.
-
apply
HI
with
k
;
auto
.
-
destruct
ef
as
[
ef
|];
simpl
in
*
;
last
done
.
-
destruct
ef
as
[
ef
|];
simpl
in
*
;
last
done
.
apply
HI
with
k
;
auto
.
apply
HI
with
k
;
auto
.
}
Qed
.
split
;
split
;
eapply
help
;
auto
using
(
symmetry
(
R
:=
dist
_))
.
Instance
wp_pre_ne
n
wp
E
e
:
Proper
(
dist
n
==>
dist
n
)
(
wp_pre
wp
E
e
)
.
Proof
.
split
;
split
;
eapply
wp_pre_contractive'
;
eauto
using
dist_le
,
(
symmetry
(
R
:=
dist
_))
.
Qed
.
Definition
wp_preC
(
wp
:
coPsetC
-
n
>
exprC
Λ
-
n
>
(
valC
Λ
-
n
>
iProp
)
-
n
>
iProp
)
:
coPsetC
-
n
>
exprC
Λ
-
n
>
(
valC
Λ
-
n
>
iProp
)
-
n
>
iProp
:=
CofeMor
(
λ
E
:
coPsetC
,
CofeMor
(
λ
e
:
exprC
Λ
,
CofeMor
(
wp_pre
wp
E
e
)))
.
Local
Instance
pre_wp_contractive
:
Contractive
wp_preC
.
Proof
.
split
;
split
;
eapply
wp_pre_contractive'
;
auto
using
(
symmetry
(
R
:=
dist
_))
.
Qed
.
Qed
.
Definition
wp_fix
:
coPset
C
-
n
>
expr
C
Λ
-
n
>
(
val
C
Λ
-
n
>
iProp
)
-
n
>
iProp
:=
Definition
wp_fix
:
coPset
→
expr
Λ
→
(
val
Λ
→
iProp
)
→
iProp
:=
fixpoint
wp_pre
C
.
fixpoint
wp_pre
.
Lemma
wp_fix_unfold
E
e
Φ
:
wp_fix
E
e
Φ
⊣⊢
wp_pre
C
wp_fix
E
e
Φ
.
Lemma
wp_fix_unfold
E
e
Φ
:
wp_fix
E
e
Φ
⊣⊢
wp_pre
wp_fix
E
e
Φ
.
Proof
.
by
rewrite
/
wp_fix
-
fixpoint_unfold
.
Qed
.
Proof
.
apply
(
fixpoint_unfold
wp_pre
)
.
Qed
.
Lemma
wp_fix_correct
E
e
(
Φ
:
val
C
Λ
-
n
>
iProp
)
:
wp_fix
E
e
Φ
⊣⊢
wp
E
e
Φ
.
Lemma
wp_fix_correct
E
e
(
Φ
:
val
Λ
→
iProp
)
:
wp_fix
E
e
Φ
⊣⊢
wp
E
e
Φ
.
Proof
.
Proof
.
split
=>
n
r
Hr
.
rewrite
wp_eq
/
wp_def
{
2
}
/
uPred_holds
.
split
=>
n
r
Hr
.
rewrite
wp_eq
/
wp_def
{
2
}
/
uPred_holds
.
split
;
revert
r
E
e
Φ
Hr
.
split
;
revert
r
E
e
Φ
Hr
.
...
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