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Iris
Fairis
Commits
f1ea0187
Commit
f1ea0187
authored
Jan 17, 2016
by
Robbert Krebbers
Browse files
Weakest precondition.
parent
d442538d
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f1ea0187
Require
Export
iris
.
pviewshifts
.
Require
Import
iris
.
wsat
.
Local
Hint
Extern
10
(
_
≤
_
)
=>
omega
.
Local
Hint
Extern
100
(
@
eq
coPset
_
_
)
=>
eassumption

solve_elem_of
.
Local
Hint
Extern
100
(
_
∉
_
)
=>
solve_elem_of
.
Local
Hint
Extern
10
(
✓
{
_
}
_
)
=>
repeat
match
goal
with
H
:
wsat
_
_
_
_

_
=>
apply
wsat_valid
in
H
end
;
solve_validN
.
Record
wp_go
{
Σ
}
(
E
:
coPset
)
(
Q
Qfork
:
iexpr
Σ
→
nat
→
res
'
Σ
→
Prop
)
(
k
:
nat
)
(
rf
:
res
'
Σ
)
(
e1
:
iexpr
Σ
)
(
σ
1
:
istate
Σ
)
:=
{
wf_safe
:
∃
e2
σ
2
ef
,
prim_step
e1
σ
1
e2
σ
2
ef
;
wp_step
e2
σ
2
ef
:
prim_step
e1
σ
1
e2
σ
2
ef
→
∃
r2
r2
'
,
wsat
k
E
σ
2
(
r2
⋅
r2
'
⋅
rf
)
∧
Q
e2
k
r2
∧
∀
e
'
,
ef
=
Some
e
'
→
Qfork
e
'
k
r2
'
}
.
CoInductive
wp_pre
{
Σ
}
(
E
:
coPset
)
(
Q
:
ival
Σ
→
iProp
Σ
)
:
iexpr
Σ
→
nat
→
res
'
Σ
→
Prop
:=

wp_pre_0
e
r
:
wp_pre
E
Q
e
0
r

wp_pre_value
n
r
v
:
Q
v
n
r
→
wp_pre
E
Q
(
of_val
v
)
n
r

wp_pre_step
n
r1
e1
:
to_val
e1
=
None
→
(
∀
rf
k
Ef
σ
1
,
1
<
k
<
n
→
E
∩
Ef
=
∅
→
wsat
(
S
k
)
(
E
∪
Ef
)
σ
1
(
r1
⋅
rf
)
→
wp_go
(
E
∪
Ef
)
(
wp_pre
E
Q
)
(
wp_pre
coPset_all
(
λ
_
,
True
%
I
))
k
rf
e1
σ
1
)
→
wp_pre
E
Q
e1
n
r1
.
Program
Definition
wp
{
Σ
}
(
E
:
coPset
)
(
e
:
iexpr
Σ
)
(
Q
:
ival
Σ
→
iProp
Σ
)
:
iProp
Σ
:=
{
uPred_holds
:=
wp_pre
E
Q
e
}
.
Next
Obligation
.
intros
Σ
E
e
Q
r1
r2
n
Hwp
Hr
.
destruct
Hwp
as
[


n
r1
e2
?
Hgo
];
constructor
;
rewrite
?
Hr
;
auto
.
intros
rf
k
Ef
σ
1
?
;
rewrite

(
dist_le
_
_
_
_
Hr
);
naive_solver
.
Qed
.
Next
Obligation
.
constructor
.
Qed
.
Next
Obligation
.
intros
Σ
E
e
Q
r1
r2
n1
;
revert
Q
E
e
r1
r2
.
induction
n1
as
[
n1
IH
]
using
lt_wf_ind
;
intros
Q
E
e
r1
r1
'
n2
.
destruct
1
as
[


n1
r1
e1
?
Hgo
].
*
rewrite
Nat
.
le_0_r
;
intros
?
>
?
;
constructor
.
*
constructor
;
eauto
using
uPred_weaken
.
*
intros
[
rf
'
Hr
]
??
;
constructor
;
[
done

intros
rf
k
Ef
σ
1
???
].
destruct
(
Hgo
(
rf
'
⋅
rf
)
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
rewrite
?
(
associative
_
)
?
Hr
;
auto
;
constructor
;
[
done

].
intros
e2
σ
2
ef
?
;
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&?&?
);
auto
.
exists
r2
,
(
r2
'
⋅
rf
'
);
split_ands
;
eauto
10
using
(
IH
k
),
@
ra_included_l
.
by
rewrite
!
(
associative
_
)
(
associative
_
r2
).
Qed
.
Instance:
Params
(
@
wp
)
3.
Section
wp
.
Context
{
Σ
:
iParam
}
.
Implicit
Types
P
:
iProp
Σ
.
Implicit
Types
Q
:
ival
Σ
→
iProp
Σ
.
Implicit
Types
v
:
ival
Σ
.
Implicit
Types
e
:
iexpr
Σ
.
Lemma
wp_weaken
E1
E2
e
Q1
Q2
r
n
n
'
:
E1
⊆
E2
→
(
∀
v
r
n
'
,
n
'
≤
n
→
✓
{
n
'
}
r
→
Q1
v
n
'
r
→
Q2
v
n
'
r
)
→
n
'
≤
n
→
✓
{
n
'
}
r
→
wp
E1
e
Q1
n
'
r
→
wp
E2
e
Q2
n
'
r
.
Proof
.
intros
HE
HQ
;
revert
e
r
;
induction
n
'
as
[
n
'
IH
]
using
lt_wf_ind
;
intros
e
r
.
destruct
3
as
[


n
'
r
e1
?
Hgo
];
constructor
;
eauto
.
intros
rf
k
Ef
σ
1
???
.
assert
(
E2
∪
Ef
=
E1
∪
(
E2
∖
E1
∪
Ef
))
as
HE
'
.
{
by
rewrite
(
associative_L
_
)

union_difference_L
.
}
destruct
(
Hgo
rf
k
((
E2
∖
E1
)
∪
Ef
)
σ
1
)
as
[
Hsafe
Hstep
];
rewrite
?
HE
'
;
auto
.
split
;
[
done

intros
e2
σ
2
ef
?
].
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&?&?
);
auto
.
exists
r2
,
r2
'
;
split_ands
;
[
rewrite
HE
'

eapply
IH

];
eauto
.
Qed
.
Global
Instance
wp_ne
E
e
n
:
Proper
(
pointwise_relation
_
(
dist
n
)
==>
dist
n
)
(
wp
E
e
).
Proof
.
by
intros
Q
Q
'
HQ
;
split
;
apply
wp_weaken
with
n
;
try
apply
HQ
.
Qed
.
Global
Instance
wp_proper
E
e
:
Proper
(
pointwise_relation
_
(
≡
)
==>
(
≡
))
(
wp
E
e
).
Proof
.
by
intros
Q
Q
'
?
;
apply
equiv_dist
=>
n
;
apply
wp_ne
=>
v
;
apply
equiv_dist
.
Qed
.
Lemma
wp_value
E
Q
v
:
Q
v
⊑
wp
E
(
of_val
v
)
Q
.
Proof
.
by
constructor
.
Qed
.
Lemma
wp_mono
E
e
Q1
Q2
:
(
∀
v
,
Q1
v
⊑
Q2
v
)
→
wp
E
e
Q1
⊑
wp
E
e
Q2
.
Proof
.
by
intros
HQ
r
n
?
;
apply
wp_weaken
with
n
;
intros
;
try
apply
HQ
.
Qed
.
Lemma
wp_pvs
E
e
Q
:
pvs
E
E
(
wp
E
e
Q
)
⊑
wp
E
e
(
λ
v
,
pvs
E
E
(
Q
v
)).
Proof
.
intros
r
[

n
]
?
;
[
done

];
intros
Hvs
.
destruct
(
to_val
e
)
as
[
v

]
eqn
:
He
;
[
apply
of_to_val
in
He
;
subst

].
{
constructor
;
eapply
pvs_mono
,
Hvs
;
auto
;
clear
.
intros
r
n
?
;
inversion
1
as
[

???
He
];
simplify_equality
;
auto
.
by
rewrite
?
to_of_val
in
He
.
}
constructor
;
[
done

intros
rf
k
Ef
σ
1
???
].
destruct
(
Hvs
rf
(
S
k
)
Ef
σ
1
)
as
(
r
'
&
Hwp
&?
);
auto
.
inversion
Hwp
as
[

????
Hgo
];
subst
;
[
by
rewrite
to_of_val
in
He

].
destruct
(
Hgo
rf
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
auto
.
split
;
[
done

intros
e2
σ
2
ef
?
].
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&
Hwp
'
&?
);
auto
.
exists
r2
,
r2
'
;
split_ands
;
auto
.
eapply
wp_mono
,
Hwp
'
;
auto
using
pvs_intro
.
Qed
.
Lemma
wp_atomic
E1
E2
e
Q
:
E2
⊆
E1
→
atomic
e
→
pvs
E1
E2
(
wp
E2
e
(
λ
v
,
pvs
E2
E1
(
Q
v
)))
⊑
wp
E1
e
Q
.
Proof
.
intros
?
He
r
n
?
Hvs
;
constructor
;
eauto
using
atomic_not_value
.
intros
rf
k
Ef
σ
1
???
.
destruct
(
Hvs
rf
(
S
k
)
Ef
σ
1
)
as
(
r
'
&
Hwp
&?
);
auto
.
inversion
Hwp
as
[

????
Hgo
];
subst
;
[
by
destruct
(
atomic_of_val
v
)

].
destruct
(
Hgo
rf
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
clear
Hgo
;
auto
.
split
;
[
done

intros
e2
σ
2
ef
?
].
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&
Hwp
'
&?
);
clear
Hsafe
Hstep
;
auto
.
destruct
Hwp
'
as
[

k
r2
v
Hvs
'

k
r2
e2
Hgo
];
[
lia


destruct
(
atomic_step
e
σ
1
e2
σ
2
ef
);
naive_solver
].
destruct
(
Hvs
'
(
r2
'
⋅
rf
)
k
Ef
σ
2
)
as
(
r3
&
[]);
rewrite
?
(
associative
_
);
auto
.
by
exists
r3
,
r2
'
;
split_ands
;
[
rewrite

(
associative
_
)

constructor

].
Qed
.
Lemma
wp_mask_weaken
E1
E2
e
Q
:
E1
⊆
E2
→
wp
E1
e
Q
⊑
wp
E2
e
Q
.
Proof
.
by
intros
HE
r
n
?
;
apply
wp_weaken
with
n
.
Qed
.
Lemma
wp_frame_r
E
e
Q
R
:
(
wp
E
e
Q
★
R
)
⊑
wp
E
e
(
λ
v
,
Q
v
★
R
).
Proof
.
intros
r
'
n
Hvalid
(
r
&
rR
&
Hr
&
Hwp
&?
);
revert
Hvalid
.
rewrite
Hr
;
clear
Hr
;
revert
e
r
Hwp
.
induction
n
as
[
n
IH
]
using
lt_wf_ind
;
intros
e
r1
.
destruct
1
as
[


n
r
e
?
Hgo
];
constructor
;
[
exists
r
,
rR
;
eauto

auto

].
intros
rf
k
Ef
σ
1
???
;
destruct
(
Hgo
(
rR
⋅
rf
)
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
auto
.
{
by
rewrite
(
associative
_
).
}
split
;
[
done

intros
e2
σ
2
ef
?
].
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&?&?
);
auto
.
exists
(
r2
⋅
rR
),
r2
'
;
split_ands
;
auto
.
*
by
rewrite

(
associative
_
r2
)
(
commutative
_
rR
)
!
(
associative
_
)

(
associative
_
_
rR
).
*
apply
IH
;
eauto
using
uPred_weaken
.
Qed
.
Lemma
wp_frame_later_r
E
e
Q
R
:
to_val
e
=
None
→
(
wp
E
e
Q
★
▷
R
)
⊑
wp
E
e
(
λ
v
,
Q
v
★
R
).
Proof
.
intros
He
r
'
n
Hvalid
(
r
&
rR
&
Hr
&
Hwp
&?
);
revert
Hvalid
;
rewrite
Hr
;
clear
Hr
.
destruct
Hwp
as
[


[

n
]
r
e
?
Hgo
];
[
done

by
rewrite
to_of_val
in
He

done

].
constructor
;
[
done

intros
rf
k
Ef
σ
1
???
].
destruct
(
Hgo
(
rR
⋅
rf
)
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
rewrite
?
(
associative
_
);
auto
.
split
;
[
done

intros
e2
σ
2
ef
?
].
destruct
(
Hstep
e2
σ
2
ef
)
as
(
r2
&
r2
'
&?&?&?
);
auto
.
exists
(
r2
⋅
rR
),
r2
'
;
split_ands
;
auto
.
*
by
rewrite

(
associative
_
r2
)
(
commutative
_
rR
)
!
(
associative
_
)

(
associative
_
_
rR
).
*
apply
wp_frame_r
;
[
auto

exists
r2
,
rR
;
split_ands
;
auto
].
eapply
uPred_weaken
with
rR
n
;
eauto
.
Qed
.
Lemma
wp_bind
`
(
HK
:
is_ctx
K
)
E
e
Q
:
wp
E
e
(
λ
v
,
wp
E
(
K
(
of_val
v
))
Q
)
⊑
wp
E
(
K
e
)
Q
.
Proof
.
intros
r
n
;
revert
e
r
;
induction
n
as
[
n
IH
]
using
lt_wf_ind
;
intros
e
r
?
.
destruct
1
as
[


n
r
e
?
Hgo
];
[


constructor
];
auto
using
is_ctx_value
.
intros
rf
k
Ef
σ
1
???
;
destruct
(
Hgo
rf
k
Ef
σ
1
)
as
[
Hsafe
Hstep
];
auto
.
split
.
{
destruct
Hsafe
as
(
e2
&
σ
2
&
ef
&?
).
by
exists
(
K
e2
),
σ
2
,
ef
;
apply
is_ctx_step_preserved
.
}
intros
e2
σ
2
ef
?
.
destruct
(
is_ctx_step
_
HK
e
σ
1
e2
σ
2
ef
)
as
(
e2
'
&>&?
);
auto
.
destruct
(
Hstep
e2
'
σ
2
ef
)
as
(
r2
&
r2
'
&?&?&?
);
auto
.
exists
r2
,
r2
'
;
split_ands
;
try
eapply
IH
;
eauto
.
Qed
.
(
*
Derived
rules
*
)
Import
uPred
.
Global
Instance
wp_mono
'
E
e
:
Proper
(
pointwise_relation
_
(
⊑
)
==>
(
⊑
))
(
wp
E
e
).
Proof
.
by
intros
Q
Q
'
?
;
apply
wp_mono
.
Qed
.
Lemma
wp_frame_l
E
e
Q
R
:
(
R
★
wp
E
e
Q
)
⊑
wp
E
e
(
λ
v
,
R
★
Q
v
).
Proof
.
setoid_rewrite
(
commutative
_
R
);
apply
wp_frame_r
.
Qed
.
Lemma
wp_frame_later_l
E
e
Q
R
:
to_val
e
=
None
→
(
▷
R
★
wp
E
e
Q
)
⊑
wp
E
e
(
λ
v
,
R
★
Q
v
).
Proof
.
rewrite
(
commutative
_
(
▷
R
)
%
I
);
setoid_rewrite
(
commutative
_
R
).
apply
wp_frame_later_r
.
Qed
.
Lemma
wp_always_l
E
e
Q
R
:
(
□
R
∧
wp
E
e
Q
)
⊑
wp
E
e
(
λ
v
,
□
R
∧
Q
v
).
Proof
.
by
setoid_rewrite
always_and_sep_l
;
rewrite
wp_frame_l
.
Qed
.
Lemma
wp_always_r
E
e
Q
R
:
(
wp
E
e
Q
∧
□
R
)
⊑
wp
E
e
(
λ
v
,
Q
v
∧
□
R
).
Proof
.
by
setoid_rewrite
always_and_sep_r
;
rewrite
wp_frame_r
.
Qed
.
Lemma
wp_impl_l
E
e
Q1
Q2
:
((
□
∀
v
,
Q1
v
→
Q2
v
)
∧
wp
E
e
Q1
)
⊑
wp
E
e
Q2
.
Proof
.
rewrite
wp_always_l
;
apply
wp_mono
=>
v
.
by
rewrite
always_elim
(
forall_elim
_
v
)
impl_elim_l
.
Qed
.
Lemma
wp_impl_r
E
e
Q1
Q2
:
(
wp
E
e
Q1
∧
□
∀
v
,
Q1
v
→
Q2
v
)
⊑
wp
E
e
Q2
.
Proof
.
by
rewrite
(
commutative
_
)
wp_impl_l
.
Qed
.
End
wp
.
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