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Iris
c
Commits
4193161c
Commit
4193161c
authored
Oct 04, 2018
by
Léon Gondelman
Browse files
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Plain Diff
vcgen and dcexpr correctness fixed (vcg_solver remains)
parent
d3b68845
Changes
2
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2 changed files
with
128 additions
and
78 deletions
+128
78
theories/vcgen/dcexpr.v
theories/vcgen/dcexpr.v
+11
5
theories/vcgen/vcgen.v
theories/vcgen/vcgen.v
+117
73
No files found.
theories/vcgen/dcexpr.v
View file @
4193161c
...
@@ 383,11 +383,17 @@ Qed.
...
@@ 383,11 +383,17 @@ Qed.
Global
Instance
dcexpr_closed
X
E
de
:
Global
Instance
dcexpr_closed
X
E
de
:
dcexpr_wf
X
E
de
→
dcexpr_wf
X
E
de
→
Closed
X
(
cdexpr_interp
E
de
).
Closed
X
(
dcexpr_interp
E
de
).
Proof
.
Admitted
.
Proof
.
(* induction de; simpl; try solve_closed. rewrite /Closed /=.
revert
X
.
induction
de
;
intros
;
unfold
Closed
in
*
;
simplify_eq
/=
;
split_and. change (Closed [] a_ret). solve_closed. apply (dexpr_closed E d).
try
(
destruct_and
!
;
split_and
;
first
split_and
;
[
apply
is_closed_of_val

by
apply
IHde2

by
apply
IHde1
])
;
Qed. *)
try
(
destruct_and
!
;
split_and
;
first
split_and
;
[
apply
is_closed_of_val

by
apply
IHde1

by
apply
IHde2
]).

split_and
;
eauto
using
is_closed_of_val
.
by
apply
dexpr_closed
.

split_and
;
[
apply
is_closed_of_val

by
apply
IHde
].

split_and
;
[
apply
is_closed_of_val

by
apply
IHde
].

split_and
;
first
split_and
;
[
apply
is_closed_of_val

apply
is_closed_of_val

by
apply
IHde
].

by
apply
W
.
is_closed_correct
.
Qed
.
(** * Reification of C syntax *)
(** * Reification of C syntax *)
(** ** LocLookup *)
(** ** LocLookup *)
...
...
theories/vcgen/vcgen.v
View file @
4193161c
...
@@ 724,14 +724,12 @@ Section vcg_spec.
...
@@ 724,14 +724,12 @@ Section vcg_spec.
(
λ
v
,
⌜
v
=
dval_interp
E
dv
⌝
∧
denv_interp
E
mNew
).
(
λ
v
,
⌜
v
=
dval_interp
E
dv
⌝
∧
denv_interp
E
mNew
).
Proof
.
Proof
.
iIntros
(
Hsp
Hwf
)
"Hm"
.
iIntros
(
Hsp
Hwf
)
"Hm"
.
iPoseProof
vcg_sp_correct'
as
"Hawp"
;
first
eassumption
.
iPoseProof
vcg_sp_correct'
as
"Hawp"
;
first
eassumption
;
first
done
.
pose
(
vcg_sp_length
_
_
_
_
_
_
_
Hsp
)
as
Hlen
.
pose
(
vcg_sp_length
_
_
_
_
_
_
_
Hsp
)
as
Hlen
;
assert
(
∃
m'
,
ms
=
[
m'
])
as
[
m'
>]=>/=.
assert
(
∃
m'
,
ms
=
[
m'
])
as
[
m'
>]=>/=.
{
destruct
ms
as
[
m'
[
m''
ms''
]]
;
eauto
;
inversion
Hlen
.
}
{
destruct
ms
as
[
m'
[
m''
ms''
]]
;
eauto
;
inversion
Hlen
.
}
{
done
.
}
rewrite
denv_merge_nil_r
.
iDestruct
(
"Hawp"
with
"Hm"
)
as
"[$ $]"
.
Admitted
.
Qed
.
(* rewrite denv_merge_nil_r. iDestruct ("Hawp" with "Hm") as "[$ $]".
Qed.*)
Lemma
vcg_sp'_correct
E
de
m
m'
mNew
dv
R
:
Lemma
vcg_sp'_correct
E
de
m
m'
mNew
dv
R
:
vcg_sp'
E
m
de
=
Some
(
m'
,
mNew
,
dv
)
→
vcg_sp'
E
m
de
=
Some
(
m'
,
mNew
,
dv
)
→
...
@@ 748,24 +746,41 @@ Section vcg_spec.
...
@@ 748,24 +746,41 @@ Section vcg_spec.
pose
(
vcg_sp_length
_
_
_
_
_
_
_
Hsp
)
as
Hlen
.
pose
(
vcg_sp_length
_
_
_
_
_
_
_
Hsp
)
as
Hlen
.
assert
(
ms
=
[])
as
>
by
(
destruct
ms
;
eauto
;
inversion
Hlen
)
;
simplify_eq
.
assert
(
ms
=
[])
as
>
by
(
destruct
ms
;
eauto
;
inversion
Hlen
)
;
simplify_eq
.
rewrite
vcg_sp_correct
;
last
eassumption
;
simplify_eq
;
last
done
.
simpl
.
rewrite
vcg_sp_correct
;
last
eassumption
;
simplify_eq
;
last
done
.
simpl
.
rewrite
denv_merge_nil_r
.
iFrame
.
eauto
422
with
iFrame
.
simplify_option_eq
.
rewrite
denv_merge_nil_r
.
clear
Hsp
Hlen
.
simplify_eq
/=.
Admitted
.
by
iFrame
.
Qed
.
Lemma
vcg_sp_wf'
E
de
ms
ms'
mNew
dv
n
:
Lemma
vcg_sp_wf'
E
de
ms
ms'
mNew
dv
n
:
Forall
(
denv_wf
E
)
ms
→
Forall
(
denv_wf
E
)
ms
→
dcexpr_wf
E
de
→
dcexpr_wf
[]
E
de
→
vcg_sp
E
ms
de
n
=
Some
(
ms'
,
mNew
,
dv
)
→
vcg_sp
E
ms
de
n
=
Some
(
ms'
,
mNew
,
dv
)
→
Forall
(
denv_wf
E
)
ms'
∧
denv_wf
E
mNew
∧
dval_wf
E
dv
.
Forall
(
denv_wf
E
)
ms'
∧
denv_wf
E
mNew
∧
dval_wf
E
dv
.
Proof
.
Admitted
.
Proof
.
(* revert ms ms' mNew dv. induction de;
revert
de
ms
ms'
mNew
dv
.
induction
n
as
[
n
IH
]
;
intros ms ms' mNew dv Hwfms Hwfde Hsp; simplify_eq/=; eauto.
intros
de
ms
ms'
mNew
dv
Hwfms
Hwfde
Hsp
;
simplify_eq
/=
;
eauto
.
{
destruct
de
;
simplify_option_eq
.
split_and
?
;
auto
;
first
by
destruct
E
.
by
eapply
vcg_eval_dexpr_wf
.
}
destruct
de
;
simplify_eq
/=.

simplify_option_eq
.
split_and
?
;
auto
;
first
by
destruct
E
.

simplify_option_eq
.
split_and
?
;
auto
;
first
by
destruct
E
.
by
eapply
vcg_eval_dexpr_wf
.
by
eapply
vcg_eval_dexpr_wf
.

destruct
(
vcg_sp
E
ms
de1
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.
destruct
(
vcg_sp
E
(
mNew1
::
ms1
)
(
dce_subst
E
s
dv1
de2
))
as
[[[
ms2
mNew2
]
dv2
]]
eqn
:
Hsp2
;
simplify_eq
/=.
destruct
ms2
as
[
t
ms2
]
;
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
assert
(
Forall
(
denv_wf
E
)
(
mNew1
::
ms1
))
as
Hwfms2
.
{
apply
Forall_cons
.
split
;
eauto
.
}
assert
(
dcexpr_wf
[]
E
(
dce_subst
E
s
dv1
de2
))
as
Hwfsubst
.
by
apply
dce_subst_dcexpr_wf
.
destruct
(
IH
(
dce_subst
E
s
dv1
de2
)
_
_
_
_
Hwfms2
Hwfsubst
Hsp2
)
as
(
Hall
&?&?).
apply
Forall_cons
in
Hall
.
destruct
Hall
.
repeat
split
;
eauto
using
denv_wf_merge
.

destruct
(
vcg_sp
E
ms
de
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.

destruct
(
vcg_sp
E
ms
de
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.
destruct
(
is_dloc
_
_
)
as
[
i
]
eqn
:
Hdl
;
try
apply
is_dloc_Some
in
Hdl
as
>
;
simplify_eq
/=.
destruct
(
is_dloc
_
_
)
as
[
i
]
eqn
:
Hdl
;
try
apply
is_dloc_Some
in
Hdl
as
>
;
simplify_eq
/=.
destruct
(
denv_delete_frac_2
i
ms1
mNew1
)
destruct
(
denv_delete_frac_2
i
ms1
mNew1
)
as
[[[[
ms2
mNew2
]
q
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
as
[[[[
ms2
mNew2
]
q
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
destruct (IHde _ _ _ _ Hwfms Hwfde Hsp1) as (?&?&?).
destruct
(
IH
de
_
_
_
_
Hwfms
Hwfde
Hsp1
)
as
(?&?&?).
eapply
denv_wf_delete_frac_2
in
Hout1
;
eauto
.
eapply
denv_wf_delete_frac_2
in
Hout1
;
eauto
.
destruct
Hout1
as
(?&?&?).
destruct
Hout1
as
(?&?&?).
repeat
split
;
eauto
using
denv_wf_insert
.
repeat
split
;
eauto
using
denv_wf_insert
.
...
@@ 777,8 +792,8 @@ Section vcg_spec.
...
@@ 777,8 +792,8 @@ Section vcg_spec.
destruct
(
denv_delete_full_2
i
ms2
(
denv_merge
mNew1
mNew2
))
destruct
(
denv_delete_full_2
i
ms2
(
denv_merge
mNew1
mNew2
))
as
[[[
ms3
mNew3
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
as
[[[
ms3
mNew3
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct (IHde1 _ _ _ _ Hwfms Hwfde1 Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
destruct (IHde2 _ _ _ _ Hwfms1 Hwfde2 Hsp2) as (?&?&?).
destruct
(
IH
de2
_
_
_
_
Hwfms1
Hwfde2
Hsp2
)
as
(?&?&?).
eapply
denv_wf_delete_full_2
in
Hout1
;
try
eassumption
.
eapply
denv_wf_delete_full_2
in
Hout1
;
try
eassumption
.
destruct
Hout1
as
(?&?&?).
destruct
Hout1
as
(?&?&?).
repeat
split
;
eauto
using
denv_wf_insert
.
repeat
split
;
eauto
using
denv_wf_insert
.
...
@@ 789,8 +804,8 @@ Section vcg_spec.
...
@@ 789,8 +804,8 @@ Section vcg_spec.
simplify_eq
/=.
simplify_eq
/=.
destruct
(
dcbin_op_eval
E
c
dv1
dv2
)
eqn
:
Hboe
;
simplify_eq
/=.
destruct
(
dcbin_op_eval
E
c
dv1
dv2
)
eqn
:
Hboe
;
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct (IHde1 _ _ _ _ Hwfms Hwfde1 Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
destruct (IHde2 _ _ _ _ Hwfms1 Hwfde2 Hsp2) as (?&?&?).
destruct
(
IH
de2
_
_
_
_
Hwfms1
Hwfde2
Hsp2
)
as
(?&?&?).
repeat
split
;
eauto
.
apply
denv_wf_merge
;
eauto
.
repeat
split
;
eauto
.
apply
denv_wf_merge
;
eauto
.
eapply
(
dcbin_op_eval_Some_wf
_
dv1
dv2
)
;
eauto
.
eapply
(
dcbin_op_eval_Some_wf
_
dv1
dv2
)
;
eauto
.

destruct
(
vcg_sp
E
ms
de1
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;

destruct
(
vcg_sp
E
ms
de1
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
...
@@ 802,8 +817,8 @@ Section vcg_spec.
...
@@ 802,8 +817,8 @@ Section vcg_spec.
as
[[[
ms3
mNew3
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
as
[[[
ms3
mNew3
]
dv1
]]
eqn
:
Hout1
;
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct
(
dcbin_op_eval
E
c
dv1
dv2
)
eqn
:
Hboe
;
simplify_eq
/=.
destruct
(
dcbin_op_eval
E
c
dv1
dv2
)
eqn
:
Hboe
;
simplify_eq
/=.
destruct (IHde1 _ _ _ _ Hwfms Hwfde1 Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
destruct (IHde2 _ _ _ _ Hwfms1 Hwfde2 Hsp2) as (?&?&?).
destruct
(
IH
de2
_
_
_
_
Hwfms1
Hwfde2
Hsp2
)
as
(?&?&?).
eapply
denv_wf_delete_full_2
in
Hout1
;
try
eassumption
.
eapply
denv_wf_delete_full_2
in
Hout1
;
try
eassumption
.
destruct
Hout1
as
(?&?&?).
destruct
Hout1
as
(?&?&?).
repeat
split
;
eauto
using
denv_wf_insert
,
denv_wf_merge
.
repeat
split
;
eauto
using
denv_wf_insert
,
denv_wf_merge
.
...
@@ 811,7 +826,7 @@ Section vcg_spec.
...
@@ 811,7 +826,7 @@ Section vcg_spec.
eapply
(
dcbin_op_eval_Some_wf
_
dv
dv2
)
;
eauto
.
eapply
(
dcbin_op_eval_Some_wf
_
dv
dv2
)
;
eauto
.
by
eapply
denv_wf_merge
.
by
eapply
denv_wf_merge
.

destruct
(
vcg_sp
E
ms
de
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.

destruct
(
vcg_sp
E
ms
de
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.
destruct (IHde _ _ _ _ Hwfms Hwfde Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de
_
_
_
_
Hwfms
Hwfde
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
destruct
(
dun_op_eval
E
u
dv1
)
as
[?]
eqn
:
Hop
;
simplify_eq
/=.
destruct
(
dun_op_eval
E
u
dv1
)
as
[?]
eqn
:
Hop
;
simplify_eq
/=.
repeat
split
;
eauto
.
repeat
split
;
eauto
.
eapply
dun_op_eval_Some_wf
;
eauto
.
eapply
dun_op_eval_Some_wf
;
eauto
.
...
@@ 820,36 +835,37 @@ Section vcg_spec.
...
@@ 820,36 +835,37 @@ Section vcg_spec.
as
[[[
ms2
mNew2
]
dv2
]]
eqn
:
Hsp2
;
simplify_eq
/=.
as
[[[
ms2
mNew2
]
dv2
]]
eqn
:
Hsp2
;
simplify_eq
/=.
destruct
ms2
as
[
t
ms2
]
;
simplify_eq
/=.
destruct
ms2
as
[
t
ms2
]
;
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct (IHde1 _ _ _ _ Hwfms Hwfde1 Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
assert
(
Forall
(
denv_wf
E
)
(
denv_unlock
mNew1
::
ms1
))
as
Hwfms2
.
assert
(
Forall
(
denv_wf
E
)
(
denv_unlock
mNew1
::
ms1
))
as
Hwfms2
.
{
apply
Forall_cons
.
split
;
eauto
.
by
apply
denv_wf_unlock
.
}
{
apply
Forall_cons
.
split
;
eauto
.
by
apply
denv_wf_unlock
.
}
destruct (IHde2 _ _ _ _ Hwfms2 Hwfde2 Hsp2) as (Hall&?&?).
destruct
(
IH
de2
_
_
_
_
Hwfms2
Hwfde2
Hsp2
)
as
(
Hall
&?&?).
apply
Forall_cons
in
Hall
.
destruct
Hall
.
apply
Forall_cons
in
Hall
.
destruct
Hall
.
repeat
split
;
eauto
using
denv_wf_merge
.
repeat
split
;
eauto
using
denv_wf_merge
.

destruct
(
vcg_sp
E
ms
de1
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.

destruct
(
vcg_sp
E
ms
de1
)
as
[[[
ms1
mNew1
]
dv1
]]
eqn
:
Hsp1
;
simplify_eq
/=.
destruct
(
vcg_sp
E
ms1
de2
)
as
[[[
ms2
mNew2
]
dv2
]]
eqn
:
Hsp2
;
destruct
(
vcg_sp
E
ms1
de2
)
as
[[[
ms2
mNew2
]
dv2
]]
eqn
:
Hsp2
;
simplify_eq
/=.
simplify_eq
/=.
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
apply
andb_True
in
Hwfde
.
destruct
Hwfde
as
[
Hwfde1
Hwfde2
].
destruct (IHde1 _ _ _ _ Hwfms Hwfde1 Hsp1) as (Hwfms1&HwfNew1&Hwfdv1).
destruct
(
IH
de1
_
_
_
_
Hwfms
Hwfde1
Hsp1
)
as
(
Hwfms1
&
HwfNew1
&
Hwfdv1
).
destruct (IHde2 _ _ _ _ Hwfms1 Hwfde2 Hsp2) as (?&?&?).
destruct
(
IH
de2
_
_
_
_
Hwfms1
Hwfde2
Hsp2
)
as
(?&?&?).
repeat
split
;
eauto
.
apply
denv_wf_merge
;
eauto
.
repeat
split
;
eauto
.
apply
denv_wf_merge
;
eauto
.
Qed.
*)
Qed
.
Lemma
vcg_sp'_wf
E
de
m
m'
mNew
dv
:
Lemma
vcg_sp'_wf
E
de
m
m'
mNew
dv
:
denv_wf
E
m
→
denv_wf
E
m
→
dcexpr_wf
E
de
→
dcexpr_wf
[]
E
de
→
vcg_sp'
E
m
de
=
Some
(
m'
,
mNew
,
dv
)
→
vcg_sp'
E
m
de
=
Some
(
m'
,
mNew
,
dv
)
→
denv_wf
E
m'
∧
denv_wf
E
mNew
∧
dval_wf
E
dv
.
denv_wf
E
m'
∧
denv_wf
E
mNew
∧
dval_wf
E
dv
.
Proof
.
Admitted
.
Proof
.
(*
rewrite /vcg_sp'. intros Hm Hde Hsp'.
rewrite
/
vcg_sp'
.
intros
Hm
Hde
Hsp'
.
assert
(
Forall
(
denv_wf
E
)
[
m
])
as
Hms
by
(
econstructor
;
eauto
).
assert
(
Forall
(
denv_wf
E
)
[
m
])
as
Hms
by
(
econstructor
;
eauto
).
destruct (vcg_sp E [m] de) as [[[ms ?mNew] ?dv]] eqn:Hsp; simplify_eq/=.
destruct
(
vcg_sp
E
[
m
]
de
)
as
[[[
ms
?mNew
]
?dv
]]
eqn
:
Hsp
;
simplify_eq
.
destruct ms as [?m' ms]; simplify_eq/=.
destruct
ms
as
[
?m'
ms
]
;
simplify_eq
.
pose (vcg_sp_length _ _ _ _ _ _ Hsp) as Hlen.
pose
(
vcg_sp_length
_
_
_
_
_
_
_
Hsp
)
as
Hlen
.
assert (ms = []) as > by (destruct ms; eauto; inversion Hlen).
assert
(
ms
=
[])
as
>
by
(
destruct
ms
;
eauto
;
inversion
Hlen
).
simpl
in
Hsp'
.
destruct (vcg_sp_wf' E de [m] [m'] mNew _ Hms Hde Hsp) as (Hm'&?&?).
inversion
Hsp'
.
subst
.
destruct
(
vcg_sp_wf'
E
de
[
m
]
[
m'
]
mNew
_
_
Hms
Hde
Hsp
)
as
(
Hm'
&?&?).
repeat
split
;
eauto
.
by
inversion
Hm'
.
repeat
split
;
eauto
.
by
inversion
Hm'
.
Qed.
*)
Qed
.
Lemma
vcg_wp_awp_continuation_correct
R
E
m
de
Φ
:
Lemma
vcg_wp_awp_continuation_correct
R
E
m
de
Φ
:
denv_wf
E
m
→
denv_wf
E
m
→
...
@@ 1008,47 +1024,75 @@ Section vcg_spec.
...
@@ 1008,47 +1024,75 @@ Section vcg_spec.
iSplit
.
iPureIntro
.
by
trans
E'
.
eauto
with
iFrame
.
iSplit
.
iPureIntro
.
by
trans
E'
.
eauto
with
iFrame
.
Qed
.
Qed
.
Lemma
vcg_wp_correct
R
E
m
de
Φ
:
Lemma
vcg_wp_correct
R
E
m
de
Φ
n
:
dcexpr_wf
E
de
→
dcexpr_wf
[]
E
de
→
denv_wf
E
m
→
denv_wf
E
m
→
denv_interp
E
m

∗
denv_interp
E
m

∗
vcg_wp
E
m
de
R
Φ
1024
(*TODO: Fix this *)

∗
vcg_wp
E
m
de
R
Φ
n

∗
awp
(
dcexpr_interp
E
de
)
R
(
vcg_wp_continuation
E
Φ
).
awp
(
dcexpr_interp
E
de
)
R
(
vcg_wp_continuation
E
Φ
).
Proof
.
Proof
.
Admitted
.
(*
revert
de
Φ
E
m
.
induction
n
as
[
n
IH
]
;
intros
de
Φ
E
m
Hwf
;
iIntros
(
Hmwf
)
"Hm Hwp"
.
revert Φ E m. induction de; intros Φ E m Hwf; iIntros (Hmwf) "Hm Hwp".
{
simpl
;
case_match
;
eauto
with
iFrame
.
simpl
.
case_match
.
+
iApply
awp_ret
.
iApply
wp_wand
;
first
by
iApply
vcg_eval_dexpr_correct
.
iIntros
(?
>).
iExists
E
,
_
,
m
.
repeat
iSplit
;
eauto
with
iFrame
.
iPureIntro
.
by
eapply
vcg_eval_dexpr_wf
.
+
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
destruct
de
;
eauto
with
iFrame
.

simpl
.
case_match
.

simpl
.
case_match
.
+
iApply
awp_ret
.
iApply
wp_wand
;
first
by
iApply
vcg_eval_dexpr_correct
.
+
iApply
awp_ret
.
iApply
wp_wand
;
first
by
iApply
vcg_eval_dexpr_correct
.
iIntros
(?
>).
iExists
E
,
_
,
m
.
repeat
iSplit
;
eauto
with
iFrame
.
iIntros
(?
>).
iExists
E
,
_
,
m
.
repeat
iSplit
;
eauto
with
iFrame
.
iPureIntro
.
by
eapply
vcg_eval_dexpr_wf
.
iPureIntro
.
by
eapply
vcg_eval_dexpr_wf
.
+
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
+
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).

simpl
in
Hwf
;
apply
andb_prop_elim
in
Hwf
as
[
Hwf1
Hwf2
].
rewrite
(
IH
de1
)
//.
iSpecialize
(
"Hm"
with
"Hwp"
)
;
fold
vcg_wp
.
assert
(
Closed
[
s
]
(
dcexpr_interp
E
de2
)).
{
by
apply
dcexpr_closed
.
}
iApply
(
awp_bind
with
"[Hm]"
)
;
fold
dcexpr_interp
.
{
exists
(
λ
:
s
,
dcexpr_interp
E
de2
)%
V
.
by
unlock
.
}
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v
)
"Hex"
.
awp_let
.
iDestruct
"Hex"
as
(
Enew
dv
m'
Hpre
Hpre'
Hm'wf
)
"(% & Hm & Hwp)"
.
assert
(
dcexpr_wf
[]
Enew
(
dce_subst
Enew
s
dv
de2
))
as
Hwfs
.
{
apply
dce_subst_dcexpr_wf
;
first
done
.
by
eapply
dcexpr_wf_mono
.
}
rewrite
(
IH
(
dce_subst
Enew
s
dv
de2
))
//.
iSpecialize
(
"Hm"
with
"Hwp"
).
rewrite
(
dcexpr_interp_mono
_
E
Enew
_
Hwf2
Hpre'
).
rewrite
Hpre

dce_subst_subst_comm
.
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v1
)
"Hex"
.
iDestruct
"Hex"
as
(
Ef
dvf
mf
Hpref
Hpre3
Hyp4
Hyp5
)
"(Hm & Hwp)"
.
simpl
.
iExists
Ef
,
dvf
,
mf
;
iSplit
;
first
done
.
iSplit
.
iPureIntro
.
trans
Enew
;
done
.
iSplit
;
first
done
.
iFrame
.
iPureIntro
;
done
.

simpl
in
*.
apply
andb_prop_elim
in
Hwf
as
[
Hwf1
Hwf2
].

simpl
in
*.
apply
andb_prop_elim
in
Hwf
as
[
Hwf1
Hwf2
].
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?
&
?
&
?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?
&
?
&
?).
iDestruct (vcg_sp'_correct with "Hm") as "[Hm' Hde2]";
first
done.
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde2]"
;
try
done
.
clear
Heqsp2
Heqsp
.
clear
Heqsp2
Heqsp
.
iDestruct (IHde1 with "Hm' Hwp") as "Hde1"; try done.
iDestruct
(
IH
de1
with
"Hm' Hwp"
)
as
"Hde1"
;
try
done
.
iApply
(
a_alloc_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iApply
(
a_alloc_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iDestruct
1
as
(
E'
dw
m''
>
Hpre
?
?)
"(Hm' & H)"
.
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
dw
m''
>
Hpre
?
?)
"(Hm' & H)"
.
iIntros
"[> HmNew]"
.
rewrite
(
dval_interp_mono
E
E'
)
//.
rewrite
(
dval_interp_mono
E
E'
)
//.
rewrite
/
vcg_wp_alloc
mapsto_wand_list_spec
.
iCombine
"HmNew Hm'"
as
"Hm'"
.
rewrite
/
vcg_wp_alloc
mapsto_wand_list_spec
.
iCombine
"HmNew Hm'"
as
"Hm'"
.
rewrite
(
denv_interp_mono
E
E'
)
//
denv_merge_interp
.
rewrite
(
denv_interp_mono
E
E'
)
//
denv_merge_interp
.
iDestruct ("H" with "Hm'") as (n
>) "HΦ". iExists n
; iSplit; first done.
iDestruct
(
"H"
with
"Hm'"
)
as
(
n
'
>)
"HΦ"
.
iExists
n'
;
iSplit
;
first
done
.
iIntros
(
l
)
"Hl"
.
by
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"HΦ"
.
iIntros
(
l
)
"Hl"
.
by
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"HΦ"
.
+
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
+
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde1]"
;
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde1]"
;
first
done. clear Heqsp.
try
done
.
clear
Heqsp
.
iDestruct (IHde2 with "Hm' Hwp") as "Hde2"; try done.
iDestruct
(
IH
de2
with
"Hm' Hwp"
)
as
"Hde2"
;
try
done
.
iApply
(
a_alloc_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iApply
(
a_alloc_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
d_new
m1
>
???)
"[Hm' H]"
.
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
d_new
m1
>
???)
"[Hm' H]"
.
rewrite
/
vcg_wp_alloc
mapsto_wand_list_spec
.
iCombine
"HmNew Hm'"
as
"Hm'"
.
rewrite
/
vcg_wp_alloc
mapsto_wand_list_spec
.
iCombine
"HmNew Hm'"
as
"Hm'"
.
rewrite
(
denv_interp_mono
E
E'
)
//
denv_merge_interp
.
rewrite
(
denv_interp_mono
E
E'
)
//
denv_merge_interp
.
rewrite
!(
dval_interp_mono
E
E'
)
//
.
rewrite
!(
dval_interp_mono
E
E'
)
//
.
iDestruct ("H" with "Hm'") as (n
>) "HΦ". iExists n
; iSplit; first done.
iDestruct
(
"H"
with
"Hm'"
)
as
(
n
'
>)
"HΦ"
.
iExists
n'
;
iSplit
;
first
done
.
iIntros
(
l
)
"Hl"
.
by
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"HΦ"
.
iIntros
(
l
)
"Hl"
.
by
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"HΦ"
.
 rewrite
IHde
//. iRename "Hm" into "Hawp".

rewrite
(
IH
de
)
//.
iRename
"Hm"
into
"Hawp"
.
iSpecialize
(
"Hawp"
with
"Hwp"
)
;
simpl
.
iSpecialize
(
"Hawp"
with
"Hwp"
)
;
simpl
.
iApply
(
a_load_spec_exists_frac
with
"[Hawp]"
).
iApply
(
a_load_spec_exists_frac
with
"[Hawp]"
).
iApply
(
awp_wand
with
"Hawp"
).
iApply
(
awp_wand
with
"Hawp"
).
...
@@ 1068,9 +1112,9 @@ Section vcg_spec.
...
@@ 1068,9 +1112,9 @@ Section vcg_spec.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?
&
?
&
?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?
&
?
&
?).
iDestruct (vcg_sp'_correct with "Hm") as "[Hm' Hde2]";
first
done.
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde2]"
;
try
done
.
clear
Heqsp2
Heqsp
.
clear
Heqsp2
Heqsp
.
iDestruct (IHde1 with "Hm' Hwp") as "Hde1"; try done.
iDestruct
(
IH
de1
with
"Hm' Hwp"
)
as
"Hde1"
;
try
done
.
iApply
(
a_store_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iApply
(
a_store_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iDestruct
1
as
(
E'
dw
m''
?
Hpre
?
?)
"(Hm' & H)"
.
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
dw
m''
?
Hpre
?
?)
"(Hm' & H)"
.
iIntros
"[> HmNew]"
.
rewrite
(
dval_interp_mono
E
E'
)
//.
subst
v1
.
rewrite
(
dval_interp_mono
E
E'
)
//.
subst
v1
.
...
@@ 1082,8 +1126,8 @@ Section vcg_spec.
...
@@ 1082,8 +1126,8 @@ Section vcg_spec.
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"H"
.
done
.
iApply
vcg_wp_continuation_mono
;
last
by
iApply
"H"
.
done
.
+
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
+
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde1]"
;
iDestruct
(
vcg_sp'_correct
with
"Hm"
)
as
"[Hm' Hde1]"
;
first
done. clear Heqsp.
try
done
.
clear
Heqsp
.
iDestruct (IHde2 with "Hm' Hwp") as "Hde2"; try done.
iDestruct
(
IH
de2
with
"Hm' Hwp"
)
as
"Hde2"
;
try
done
.
iApply
(
a_store_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iApply
(
a_store_spec
with
"Hde1 Hde2"
).
iIntros
"!>"
(
v1
v2
).
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
d_new
m1
?
?
?)
"(% & Hm' & H)"
.
iIntros
"[> HmNew]"
.
iDestruct
1
as
(
E'
d_new
m1
?
?
?)
"(% & Hm' & H)"
.
rewrite
(
dval_interp_mono
E
E'
)
//.
subst
v2
.
rewrite
(
dval_interp_mono
E
E'
)
//.
subst
v2
.
...
@@ 1098,11 +1142,11 @@ Section vcg_spec.
...
@@ 1098,11 +1142,11 @@ Section vcg_spec.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
rewrite
IHde1
; [ done  done]. iSpecialize ("Hm'" with "Hwp").
rewrite
(
IH
de1
)
;
[
done

done
].
iSpecialize
(
"Hm'"
with
"Hwp"
).
iApply
(
a_bin_op_spec
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iApply
(
a_bin_op_spec
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew)"
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew)"
.
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
...
@@ 1112,10 +1156,10 @@ Section vcg_spec.
...
@@ 1112,10 +1156,10 @@ Section vcg_spec.
eauto
using
dval_wf_mono
,
denv_wf_merge
,
denv_wf_mono
.
eauto
using
dval_wf_mono
,
denv_wf_merge
,
denv_wf_mono
.
{
iApply
(
denv_interp_mono
with
"HmNew"
)
;
eauto
.
}
{
iApply
(
denv_interp_mono
with
"HmNew"
)
;
eauto
.
}
iExists
w
.
iSplit
;
first
done
.
by
iApply
vcg_wp_continuation_mono
.
iExists
w
.
iSplit
;
first
done
.
by
iApply
vcg_wp_continuation_mono
.
+ iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
+
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.
rewrite
IHde2
; [ done  done]. iSpecialize ("Hm'" with "Hwp").
rewrite
(
IH
de2
)
;
[
done

done
].
iSpecialize
(
"Hm'"
with
"Hwp"
).
iRename
"Hm'"
into
"Hde2"
.
iRename
"Hm'"
into
"Hde2"
.
iApply
(
a_bin_op_spec
with
"Hde1 Hde2"
)
;
fold
dcexpr_interp
.
iApply
(
a_bin_op_spec
with
"Hde1 Hde2"
)
;
fold
dcexpr_interp
.
iNext
.
iIntros
(
v1
v2
)
"(> & HmNew) Hex"
.
iNext
.
iIntros
(
v1
v2
)
"(> & HmNew) Hex"
.
...
@@ 1131,11 +1175,11 @@ Section vcg_spec.
...
@@ 1131,11 +1175,11 @@ Section vcg_spec.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
rewrite
IHde1
; [ done  done]. iSpecialize ("Hm'" with "Hwp").
rewrite
(
IH
de1
)
;
[
done

done
].
iSpecialize
(
"Hm'"
with
"Hwp"
).
iApply
(
a_pre_bin_op_spec
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iApply
(
a_pre_bin_op_spec
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew) $"
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew) $"
.
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
...
@@ 1149,10 +1193,10 @@ Section vcg_spec.
...
@@ 1149,10 +1193,10 @@ Section vcg_spec.
iExists
l
,
v
,
w
.
iFrame
.
repeat
iSplit
;
eauto
.
iExists
l
,
v
,
w
.
iFrame
.
repeat
iSplit
;
eauto
.
*
iPureIntro
.
rewrite
(
dval_interp_mono
E
E'
)
;
auto
.
*
iPureIntro
.
rewrite
(
dval_interp_mono
E
E'
)
;
auto
.
*
rewrite
(
vcg_wp_continuation_mono
E
E'
)
;
auto
.
*
rewrite
(
vcg_wp_continuation_mono
E
E'
)
;
auto
.
+ iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
+
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.
rewrite
IHde2
; [ done  done]. iSpecialize ("Hm'" with "Hwp").
rewrite
(
IH
de2
)
;
[
done

done
].
iSpecialize
(
"Hm'"
with
"Hwp"
).
iRename
"Hm'"
into
"Hde2"
.
iRename
"Hm'"
into
"Hde2"
.
iApply
(
a_pre_bin_op_spec
with
"Hde1 Hde2"
)
;
fold
dcexpr_interp
.
iApply
(
a_pre_bin_op_spec
with
"Hde1 Hde2"
)
;
fold
dcexpr_interp
.
iNext
.
iIntros
(
v1
v2
)
"(> & HmNew) Hex $"
.
iNext
.
iIntros
(
v1
v2
)
"(> & HmNew) Hex $"
.
...
@@ 1167,19 +1211,19 @@ Section vcg_spec.
...
@@ 1167,19 +1211,19 @@ Section vcg_spec.
iExists
l
,
v
,
w
.
iFrame
.
repeat
iSplit
;
eauto
.
iExists
l
,
v
,
w
.
iFrame
.
repeat
iSplit
;
eauto
.
*
iPureIntro
.
rewrite
(
dval_interp_mono
E
E'
)
;
auto
.
*
iPureIntro
.
rewrite
(
dval_interp_mono
E
E'
)
;
auto
.
*
rewrite
(
vcg_wp_continuation_mono
E
E'
)
;
auto
.
*
rewrite
(
vcg_wp_continuation_mono
E
E'
)
;
auto
.
 rewrite
IHde
//. iApply a_un_op_spec. iSpecialize ("Hm" with "Hwp").

rewrite
(
IH
de
)
//.
iApply
a_un_op_spec
.
iSpecialize
(
"Hm"
with
"Hwp"
).
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v
)
"Hex"
.
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v
)
"Hex"
.
iDestruct
"Hex"
as
(
E'
dv
m'
>
Hpre'
Hm'wf
)
"(% & Hm & Hwp)"
.
iDestruct
"Hex"
as
(
E'
dv
m'
>
Hpre'
Hm'wf
)
"(% & Hm & Hwp)"
.
iDestruct
(
vcg_wp_un_op_correct
with
"Hm Hwp"
)
as
(
w
?)
"Hcont"
;
auto
.
iDestruct
(
vcg_wp_un_op_correct
with
"Hm Hwp"
)
as
(
w
?)
"Hcont"
;
auto
.
iExists
w
.
iSplit
;
first
done
.
by
iApply
vcg_wp_continuation_mono
.
iExists
w
.
iSplit
;
first
done
.
by
iApply
vcg_wp_continuation_mono
.

simpl
in
Hwf
;
apply
andb_prop_elim
in
Hwf
as
[
Hwf1
Hwf2
].

simpl
in
Hwf
;
apply
andb_prop_elim
in
Hwf
as
[
Hwf1
Hwf2
].
rewrite
IHde1
//. iSpecialize ("Hm" with "Hwp"); fold vcg_wp.
rewrite
(
IH
de1
)
//.
iSpecialize
(
"Hm"
with
"Hwp"
)
;
fold
vcg_wp
.
iApply (a_sequence_spec' with "[Hm]"); fold dcexpr_interp.
iApply
(
a_sequence_spec'
with
"[Hm]"
)
;
fold
dcexpr_interp
.
by
apply
dcexpr_closed
.
iNext
.
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v
)
"Hex"
.
iNext
.
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v
)
"Hex"
.
iDestruct
"Hex"
as
(
Enew
dv
m'
Hpre
Hpre'
Hm'wf
)
"(% & Hm & Hwp)"
.
simpl
.
iDestruct
"Hex"
as
(
Enew
dv
m'
Hpre
Hpre'
Hm'wf
)
"(% & Hm & Hwp)"
.
simpl
.
rewrite
denv_unlock_interp
.
rewrite
denv_unlock_interp
.
iModIntro. rewrite
IHde2
//. iSpecialize ("Hm" with "Hwp").
iModIntro
.
rewrite
(
IH
de2
)
//.
iSpecialize
(
"Hm"
with
"Hwp"
).
specialize (dcexpr_interp_mono E Enew de2 Hwf2 Hpre').
specialize
(
dcexpr_interp_mono
[]
E
Enew
de2
Hwf2
Hpre'
).
intro
Heq
;
rewrite
Heq
.
iApply
(
awp_wand
with
"Hm"
).
intro
Heq
;
rewrite
Heq
.
iApply
(
awp_wand
with
"Hm"
).
iIntros
(
v1
)
"Hex"
.
iIntros
(
v1
)
"Hex"
.
iDestruct
"Hex"
as
(
Ef
dvf
mf
Hpref
Hpre3
Hyp4
Hyp5
)
"(Hm & Hwp)"
.
simpl
.
iDestruct
"Hex"
as
(
Ef
dvf
mf
Hpref
Hpre3
Hyp4
Hyp5
)
"(Hm & Hwp)"
.
simpl
.
...
@@ 1193,11 +1237,11 @@ Section vcg_spec.
...
@@ 1193,11 +1237,11 @@ Section vcg_spec.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
destruct
(
vcg_sp'
E
m
de1
)
as
[[[
m'
mNew
]
dv1
]]
eqn
:
Heqsp
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
+
destruct
(
vcg_sp'
E
m
de2
)
as
[[[
m'
mNew
]
dv2
]]
eqn
:
Heqsp2
;
last
first
.
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
{
by
iApply
(
vcg_wp_awp_continuation_correct
with
"Hm Hwp"
).
}
iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf2
Heqsp2
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde2]"
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
iClear
"Hsp"
;
clear
Heqsp2
Heqsp
.
rewrite
IHde1
; [ done  done]. iSpecialize ("Hm'" with "Hwp").
rewrite
(
IH
de1
)
;
[
done

done
].
iSpecialize
(
"Hm'"
with
"Hwp"
).
iApply
(
awp_par
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iApply
(
awp_par
with
"Hm' Hde2"
)
;
fold
dcexpr_interp
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew)"
.
iNext
.
iIntros
(
v1
v2
)
"Hex (> & HmNew)"
.
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
iDestruct
"Hex"
as
(
E'
dv1
m''
Heq
Hpre
Hm'wf
)
"(% & Hm' & Hwp)"
;
...
@@ 1209,10 +1253,10 @@ Section vcg_spec.
...
@@ 1209,10 +1253,10 @@ Section vcg_spec.
eauto
using
dval_wf_mono
,
denv_wf_mono
,
denv_wf_merge
).
eauto
using
dval_wf_mono
,
denv_wf_mono
,
denv_wf_merge
).
iFrame
.
rewrite
(
denv_interp_mono
E
)
//.
iFrame
.
rewrite
(
denv_interp_mono
E
)
//.
iApply
denv_merge_interp
.
iFrame
.
iApply
denv_merge_interp
.
iFrame
.
+ iPoseProof (vcg_sp'_correct) as "Hsp"; first eassumption.
+
iPoseProof
(
vcg_sp'_correct
)
as
"Hsp"
;
first
eassumption
.
done
.
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
specialize
(
vcg_sp'_wf
_
_
_
_
_
_
Hmwf
Hwf1
Heqsp
)
as
(?&?&?).
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.
iDestruct
(
"Hsp"
with
"Hm"
)
as
"[Hm' Hde1]"
;
iClear
"Hsp"
;
clear
Heqsp
.