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4beb6889
Commit
4beb6889
authored
Jan 25, 2019
by
Ralf Jung
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theories/logatom/flat_combiner/atomic_sync.v
theories/logatom/flat_combiner/atomic_sync.v
+89
0
theories/logatom/flat_combiner/flat.v
theories/logatom/flat_combiner/flat.v
+301
0
theories/logatom/flat_combiner/misc.v
theories/logatom/flat_combiner/misc.v
+87
0
theories/logatom/flat_combiner/peritem.v
theories/logatom/flat_combiner/peritem.v
+88
0
theories/logatom/flat_combiner/simple_sync.v
theories/logatom/flat_combiner/simple_sync.v
+39
0
theories/logatom/flat_combiner/sync.v
theories/logatom/flat_combiner/sync.v
+28
0
theories/logatom/treiber.v
theories/logatom/treiber.v
+173
0
No files found.
theories/logatom/flat_combiner/atomic_sync.v
0 → 100644
View file @
4beb6889
From
iris
.
program_logic
Require
Export
weakestpre
hoare
atomic
.
From
iris
.
heap_lang
Require
Export
lang
proofmode
notation
.
From
iris
.
heap_lang
.
lib
Require
Import
spin_lock
.
From
iris
.
algebra
Require
Import
agree
frac
.
From
iris_atomic
Require
Import
sync
misc
.
(** The simple syncer spec in [sync.v] implies a logically atomic spec. *)
Definition
syncR
:
=
prodR
fracR
(
agreeR
valC
).
(* track the local knowledge of ghost state *)
Class
syncG
Σ
:
=
sync_tokG
:
>
inG
Σ
syncR
.
Definition
sync
Σ
:
gFunctors
:
=
#[
GFunctor
(
constRF
syncR
)].
Instance
subG_sync
Σ
{
Σ
}
:
subG
sync
Σ
Σ
→
syncG
Σ
.
Proof
.
by
intros
?%
subG_inG
.
Qed
.
Section
atomic_sync
.
Context
`
{
EqDecision
A
,
!
heapG
Σ
,
!
lockG
Σ
}.
Canonical
AC
:
=
leibnizC
A
.
Context
`
{!
inG
Σ
(
prodR
fracR
(
agreeR
AC
))}.
(* TODO: Rename and make opaque; the fact that this is a half should not be visible
to the user. *)
Definition
gHalf
(
γ
:
gname
)
g
:
iProp
Σ
:
=
own
γ
((
1
/
2
)%
Qp
,
to_agree
g
).
Definition
atomic_seq_spec
(
ϕ
:
A
→
iProp
Σ
)
α
β
(
f
x
:
val
)
:
=
(
∀
g
,
{{
ϕ
g
∗
α
g
}}
f
x
{{
v
,
∃
g'
,
ϕ
g'
∗
β
g
g'
v
}})%
I
.
(* TODO: Provide better masks. ∅ as inner mask is pretty weak. *)
Definition
atomic_synced
(
ϕ
:
A
→
iProp
Σ
)
γ
(
f
f'
:
val
)
:
=
(
□
∀
α
β
(
x
:
val
),
atomic_seq_spec
ϕ
α
β
f
x
→
<<<
∀
g
,
gHalf
γ
g
∗
□
α
g
>>>
f'
x
@
⊤
<<<
∃
v
,
∃
g'
,
gHalf
γ
g'
∗
β
g
g'
v
,
RET
v
>>>)%
I
.
(* TODO: Use our usual style with a generic postcondition. *)
(* TODO: We could get rid of the x, and hardcode a unit. That would
be no loss in expressiveness, but probably more annoying to apply.
How much more annoying? And how much to we gain in terms of things
becomign easier to prove? *)
(* This is really the core of the spec: It says that executing `s` on `f`
turns the sequential spec with f, α, β into the concurrent triple with f', α, β. *)
Definition
is_atomic_syncer
(
ϕ
:
A
→
iProp
Σ
)
γ
(
s
:
val
)
:
=
(
□
∀
(
f
:
val
),
WP
s
f
{{
f'
,
atomic_synced
ϕ
γ
f
f'
}})%
I
.
Definition
atomic_syncer_spec
(
mk_syncer
:
val
)
:
=
∀
(
g0
:
A
)
(
ϕ
:
A
→
iProp
Σ
),
{{{
ϕ
g0
}}}
mk_syncer
#()
{{{
γ
s
,
RET
s
;
gHalf
γ
g0
∗
is_atomic_syncer
ϕ
γ
s
}}}.
Lemma
atomic_spec
(
mk_syncer
:
val
)
:
mk_syncer_spec
mk_syncer
→
atomic_syncer_spec
mk_syncer
.
Proof
.
iIntros
(
Hsync
g0
ϕ
ret
)
"Hϕ Hret"
.
iMod
(
own_alloc
(((
1
/
2
)%
Qp
,
to_agree
g0
)
⋅
((
1
/
2
)%
Qp
,
to_agree
g0
)))
as
(
γ
)
"[Hg1 Hg2]"
.
{
by
rewrite
pair_op
agree_idemp
.
}
iApply
(
Hsync
(
∃
g
:
A
,
ϕ
g
∗
gHalf
γ
g
)%
I
with
"[Hg1 Hϕ]"
)=>//.
{
iExists
g0
.
by
iFrame
.
}
iNext
.
iIntros
(
s
)
"#Hsyncer"
.
iApply
"Hret"
.
iSplitL
"Hg2"
;
first
done
.
iIntros
"!#"
.
iIntros
(
f
).
iApply
wp_wand_r
.
iSplitR
;
first
by
iApply
"Hsyncer"
.
iIntros
(
f'
)
"#Hsynced {Hsyncer}"
.
iAlways
.
iIntros
(
α
β
x
)
"#Hseq"
.
change
(
ofe_car
AC
)
with
A
.
iApply
wp_atomic_intro
.
iIntros
(
Φ
'
)
"?"
.
(* TODO: Why can't I iApply "Hsynced"? *)
iSpecialize
(
"Hsynced"
$!
_
Φ
'
x
).
iApply
wp_wand_r
.
iSplitL
.

iApply
(
"Hsynced"
with
"[]"
)=>//
;
last
iAccu
.
iAlways
.
iIntros
"[HR HP]"
.
iDestruct
"HR"
as
(
g
)
"[Hϕ Hg1]"
.
(* we should view shift at this point *)
iApply
fupd_wp
.
iMod
"HP"
as
(?)
"[[Hg2 #Hα] [Hvs1 _]]"
.
iDestruct
(
m_frag_agree
with
"Hg1 Hg2"
)
as
%
Heq
.
subst
.
iMod
(
"Hvs1"
with
"[Hg2]"
)
as
"HP"
;
first
by
iFrame
.
iModIntro
.
iApply
wp_fupd
.
iApply
wp_wand_r
.
iSplitL
"Hϕ"
.
{
iApply
"Hseq"
.
iFrame
.
done
.
}
iIntros
(
v
)
"H"
.
iDestruct
"H"
as
(
g'
)
"[Hϕ' Hβ]"
.
iMod
(
"HP"
)
as
(
g''
)
"[[Hg'' _] [_ Hvs2]]"
.
iSpecialize
(
"Hvs2"
$!
v
).
iDestruct
(
m_frag_agree'
with
"Hg'' Hg1"
)
as
"[Hg %]"
.
subst
.
rewrite
Qp_div_2
.
iAssert
(==>
own
γ
(((
1
/
2
)%
Qp
,
to_agree
g'
)
⋅
((
1
/
2
)%
Qp
,
to_agree
g'
)))%
I
with
"[Hg]"
as
">[Hg1 Hg2]"
.
{
iApply
own_update
;
last
by
iAssumption
.
apply
cmra_update_exclusive
.
by
rewrite
pair_op
agree_idemp
.
}
iMod
(
"Hvs2"
with
"[Hg1 Hβ]"
).
{
iExists
g'
.
iFrame
.
}
iModIntro
.
iSplitL
"Hg2 Hϕ'"
;
last
done
.
iExists
g'
.
by
iFrame
.

iIntros
(?)
"?"
.
done
.
Qed
.
End
atomic_sync
.
theories/logatom/flat_combiner/flat.v
0 → 100644
View file @
4beb6889
This diff is collapsed.
Click to expand it.
theories/logatom/flat_combiner/misc.v
0 → 100644
View file @
4beb6889
(* Miscellaneous lemmas *)
From
iris
.
program_logic
Require
Export
weakestpre
.
From
iris
.
heap_lang
Require
Export
lang
proofmode
notation
.
From
iris
.
algebra
Require
Import
auth
frac
gmap
agree
.
From
iris
.
bi
Require
Import
fractional
.
From
iris
.
base_logic
Require
Import
auth
.
Import
uPred
.
Section
lemmas
.
Lemma
pair_l_frac_op'
(
p
q
:
Qp
)
(
g
g'
:
val
)
:
((
p
+
q
)%
Qp
,
to_agree
g'
)
~~>
(((
p
,
to_agree
g'
)
⋅
(
q
,
to_agree
g'
))).
Proof
.
by
rewrite
pair_op
agree_idemp
frac_op'
.
Qed
.
Lemma
pair_l_frac_op_1'
(
g
g'
:
val
)
:
(
1
%
Qp
,
to_agree
g'
)
~~>
(((
1
/
2
)%
Qp
,
to_agree
g'
)
⋅
((
1
/
2
)%
Qp
,
to_agree
g'
)).
Proof
.
by
rewrite
pair_op
agree_idemp
frac_op'
Qp_div_2
.
Qed
.
End
lemmas
.
Section
excl
.
Context
`
{!
inG
Σ
(
exclR
unitC
)}.
Lemma
excl_falso
γ
Q'
:
own
γ
(
Excl
())
∗
own
γ
(
Excl
())
⊢
Q'
.
Proof
.
iIntros
"[Ho1 Ho2]"
.
iCombine
"Ho1"
"Ho2"
as
"Ho"
.
iExFalso
.
by
iDestruct
(
own_valid
with
"Ho"
)
as
"%"
.
Qed
.
End
excl
.
Section
heap_extra
.
Context
`
{
heapG
Σ
}.
Lemma
bogus_heap
p
(
q1
q2
:
Qp
)
a
b
:
~((
q1
+
q2
)%
Qp
≤
1
%
Qp
)%
Qc
→
p
↦
{
q1
}
a
∗
p
↦
{
q2
}
b
⊢
False
.
Proof
.
iIntros
(?)
"Hp"
.
iDestruct
"Hp"
as
"[Hl Hr]"
.
iDestruct
(@
mapsto_valid_2
with
"Hl Hr"
)
as
%
H'
.
done
.
Qed
.
End
heap_extra
.
Section
big_op_later
.
Context
{
M
:
ucmraT
}.
Context
`
{
Countable
K
}
{
A
:
Type
}.
Implicit
Types
m
:
gmap
K
A
.
Implicit
Types
Φ
Ψ
:
K
→
A
→
uPred
M
.
Lemma
big_sepM_delete_later
Φ
m
i
x
:
m
!!
i
=
Some
x
→
▷
([
∗
map
]
k
↦
y
∈
m
,
Φ
k
y
)
⊣
⊢
▷
Φ
i
x
∗
▷
[
∗
map
]
k
↦
y
∈
delete
i
m
,
Φ
k
y
.
Proof
.
intros
?.
rewrite
big_sepM_delete
=>//.
apply
later_sep
.
Qed
.
Lemma
big_sepM_insert_later
Φ
m
i
x
:
m
!!
i
=
None
→
▷
([
∗
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
Φ
k
y
)
⊣
⊢
▷
Φ
i
x
∗
▷
[
∗
map
]
k
↦
y
∈
m
,
Φ
k
y
.
Proof
.
intros
?.
rewrite
big_sepM_insert
=>//.
apply
later_sep
.
Qed
.
End
big_op_later
.
Section
pair
.
Context
{
A
:
ofeT
}
`
{
EqDecision
A
,
!
OfeDiscrete
A
,
!
LeibnizEquiv
A
,
!
inG
Σ
(
prodR
fracR
(
agreeR
A
))}.
Lemma
m_frag_agree
γ
m
(
q1
q2
:
Qp
)
(
a1
a2
:
A
)
:
own
γ
m
(
q1
,
to_agree
a1
)

∗
own
γ
m
(
q2
,
to_agree
a2
)

∗
⌜
a1
=
a2
⌝
.
Proof
.
iIntros
"Ho Ho'"
.
destruct
(
decide
(
a1
=
a2
))
as
[>
Hneq
]=>//.
iCombine
"Ho"
"Ho'"
as
"Ho"
.
iDestruct
(
own_valid
with
"Ho"
)
as
%
Hvalid
.
exfalso
.
destruct
Hvalid
as
[
_
Hvalid
].
simpl
in
Hvalid
.
apply
agree_op_inv
in
Hvalid
.
apply
(
inj
to_agree
)
in
Hvalid
.
apply
Hneq
.
by
fold_leibniz
.
Qed
.
Lemma
m_frag_agree'
γ
m
(
q1
q2
:
Qp
)
(
a1
a2
:
A
)
:
own
γ
m
(
q1
,
to_agree
a1
)

∗
own
γ
m
(
q2
,
to_agree
a2
)

∗
own
γ
m
((
q1
+
q2
)%
Qp
,
to_agree
a1
)
∗
⌜
a1
=
a2
⌝
.
Proof
.
iIntros
"Ho Ho'"
.
iDestruct
(
m_frag_agree
with
"Ho Ho'"
)
as
%
Heq
.
subst
.
iCombine
"Ho"
"Ho'"
as
"Ho"
.
by
iFrame
.
Qed
.
End
pair
.
theories/logatom/flat_combiner/peritem.v
0 → 100644
View file @
4beb6889
From
iris
.
program_logic
Require
Export
weakestpre
.
From
iris
.
heap_lang
Require
Export
lang
.
From
iris
.
heap_lang
Require
Import
proofmode
notation
.
From
iris
.
proofmode
Require
Import
tactics
.
From
iris
.
algebra
Require
Import
frac
auth
gmap
csum
.
From
iris_atomic
Require
Export
treiber
misc
.
From
iris
.
base_logic
.
lib
Require
Import
invariants
.
Section
defs
.
Context
`
{
heapG
Σ
}
(
N
:
namespace
).
Context
(
R
:
val
→
iProp
Σ
)
`
{
∀
x
,
TimelessP
(
R
x
)}.
Fixpoint
is_list_R
(
hd
:
loc
)
(
xs
:
list
val
)
:
iProp
Σ
:
=
match
xs
with

[]
=>
(
∃
q
,
hd
↦
{
q
}
NONEV
)%
I

x
::
xs
=>
(
∃
(
hd'
:
loc
)
q
,
hd
↦
{
q
}
SOMEV
(
x
,
#
hd'
)
∗
inv
N
(
R
x
)
∗
is_list_R
hd'
xs
)%
I
end
.
Definition
is_bag_R
xs
s
:
=
(
∃
hd
:
loc
,
s
↦
#
hd
∗
is_list_R
hd
xs
)%
I
.
Lemma
dup_is_list_R
:
∀
xs
hd
,
is_list_R
hd
xs
⊢
==>
is_list_R
hd
xs
∗
is_list_R
hd
xs
.
Proof
.
induction
xs
as
[
y
xs'
IHxs'
].

iIntros
(
hd
)
"Hs"
.
simpl
.
iDestruct
"Hs"
as
(
q
)
"[Hhd Hhd']"
.
iSplitL
"Hhd"
;
eauto
.

iIntros
(
hd
)
"Hs"
.
simpl
.
iDestruct
"Hs"
as
(
hd'
q
)
"([Hhd Hhd'] & #Hinv & Hs')"
.
iDestruct
(
IHxs'
with
"[Hs']"
)
as
">[Hs1 Hs2]"
;
first
by
iFrame
.
iModIntro
.
iSplitL
"Hhd Hs1"
;
iExists
hd'
,
(
q
/
2
)%
Qp
;
by
iFrame
.
Qed
.
Definition
f_spec
(
R
:
iProp
Σ
)
(
f
:
val
)
(
Rf
:
iProp
Σ
)
x
:
=
{{{
inv
N
R
∗
Rf
}}}
f
x
{{{
RET
#()
;
Rf
}}}.
End
defs
.
Section
proofs
.
Context
`
{
heapG
Σ
}
(
N
:
namespace
).
Context
(
R
:
val
→
iProp
Σ
).
Definition
bag_inv
s
:
iProp
Σ
:
=
inv
N
(
∃
xs
,
is_bag_R
N
R
xs
s
)%
I
.
Lemma
new_bag_spec
:
{{{
True
}}}
new_stack
#()
{{{
s
,
RET
#
s
;
bag_inv
s
}}}.
Proof
.
iIntros
(
Φ
)
"_ HΦ"
.
iApply
wp_fupd
.
wp_lam
.
wp_bind
(
ref
NONE
)%
E
.
wp_alloc
l
as
"Hl"
.
wp_alloc
s
as
"Hs"
.
iAssert
((
∃
xs
,
is_bag_R
N
R
xs
s
))%
I
with
"[HΦ]"
as
"Hxs"
.
{
iFrame
.
iExists
[],
l
.
iFrame
.
simpl
.
eauto
.
}
iMod
(
inv_alloc
N
_
(
∃
xs
:
list
val
,
is_bag_R
N
R
xs
s
)%
I
with
"[HΦ]"
)
as
"#?"
;
first
eauto
.
iApply
"HΦ"
.
iFrame
"#"
.
done
.
Qed
.
Lemma
push_spec
(
s
:
loc
)
(
x
:
val
)
:
{{{
R
x
∗
bag_inv
s
}}}
push
#
s
x
{{{
RET
#()
;
inv
N
(
R
x
)
}}}.
Proof
.
iIntros
(
Φ
)
"(HRx & #?) HΦ"
.
iL
ö
b
as
"IH"
.
wp_rec
.
wp_let
.
wp_bind
(!
_
)%
E
.
iInv
N
as
"H1"
"Hclose"
.
iDestruct
"H1"
as
(
xs
hd
)
"[>Hs H1]"
.
wp_load
.
iMod
(
"Hclose"
with
"[Hs H1]"
).
{
iNext
.
iFrame
.
iExists
xs
,
hd
.
iFrame
.
}
iModIntro
.
wp_let
.
wp_alloc
l
as
"Hl"
.
wp_let
.
wp_bind
(
CAS
_
_
_
)%
E
.
iInv
N
as
"H1"
"Hclose"
.
iDestruct
"H1"
as
(
xs'
hd'
)
"[>Hs H1]"
.
destruct
(
decide
(
hd
=
hd'
))
as
[>
Hneq
].

wp_cas_suc
.
iMod
(
inv_alloc
N
_
(
R
x
)
with
"[HRx]"
)
as
"#HRx"
;
first
eauto
.
iMod
(
"Hclose"
with
"[Hs Hl H1]"
).
{
iNext
.
iFrame
.
iExists
(
x
::
xs'
),
l
.
iFrame
.
simpl
.
iExists
hd'
,
1
%
Qp
.
iFrame
.
by
iFrame
"#"
.
}
iModIntro
.
wp_if
.
by
iApply
"HΦ"
.

wp_cas_fail
.
iMod
(
"Hclose"
with
"[Hs H1]"
).
{
iNext
.
iFrame
.
iExists
(
xs'
),
hd'
.
iFrame
.
}
iModIntro
.
wp_if
.
iApply
(
"IH"
with
"HRx"
).
by
iNext
.
Qed
.
End
proofs
.
theories/logatom/flat_combiner/simple_sync.v
0 → 100644
View file @
4beb6889
(* Coarsegrained syncer *)
From
iris
.
program_logic
Require
Export
weakestpre
hoare
.
From
iris
.
heap_lang
Require
Export
lang
proofmode
notation
.
From
iris
.
heap_lang
.
lib
Require
Import
spin_lock
.
From
iris
.
algebra
Require
Import
frac
.
From
iris_atomic
Require
Import
sync
.
Import
uPred
.
Definition
mk_sync
:
val
:
=
λ
:
<>,
let
:
"l"
:
=
newlock
#()
in
λ
:
"f"
"x"
,
acquire
"l"
;;
let
:
"ret"
:
=
"f"
"x"
in
release
"l"
;;
"ret"
.
Section
syncer
.
Context
`
{!
heapG
Σ
,
!
lockG
Σ
}
(
N
:
namespace
).
Lemma
mk_sync_spec
:
mk_syncer_spec
mk_sync
.
Proof
.
iIntros
(
R
Φ
)
"HR HΦ"
.
wp_lam
.
wp_bind
(
newlock
_
).
iApply
(
newlock_spec
N
R
with
"[HR]"
)
;
first
done
.
iNext
.
iIntros
(
lk
γ
)
"#Hl"
.
wp_pures
.
iApply
"HΦ"
.
iIntros
"!#"
.
iIntros
(
f
).
wp_pures
.
iAlways
.
iIntros
(
P
Q
x
)
"#Hf !# HP"
.
wp_pures
.
wp_bind
(
acquire
_
).
iApply
(
acquire_spec
with
"Hl"
).
iNext
.
iIntros
"[Hlocked R]"
.
wp_seq
.
wp_bind
(
f
_
).
iSpecialize
(
"Hf"
with
"[R HP]"
)
;
first
by
iFrame
.
iApply
wp_wand_r
.
iSplitL
"Hf"
;
first
done
.
iIntros
(
v'
)
"[HR HQv]"
.
wp_let
.
wp_bind
(
release
_
).
iApply
(
release_spec
with
"[$Hl $HR $Hlocked]"
).
iNext
.
iIntros
"_"
.
by
wp_seq
.
Qed
.
End
syncer
.
theories/logatom/flat_combiner/sync.v
0 → 100644
View file @
4beb6889
(* Syncer spec *)
From
iris
.
program_logic
Require
Export
weakestpre
hoare
.
From
iris
.
heap_lang
Require
Export
lang
.
From
iris
.
heap_lang
Require
Import
proofmode
notation
.
Section
sync
.
Context
`
{!
heapG
Σ
}
(
N
:
namespace
).
(* TODO: We could get rid of the x, and hardcode a unit. That would
be no loss in expressiveness, but probably more annoying to apply.
How much more annoying? And how much to we gain in terms of things
becoming easier to prove? *)
Definition
synced
R
(
f
f'
:
val
)
:
=
(
□
∀
P
Q
(
x
:
val
),
{{
R
∗
P
}}
f
x
{{
v
,
R
∗
Q
v
}}
→
{{
P
}}
f'
x
{{
Q
}})%
I
.
(* Notice that `s f` is *unconditionally safe* to execute, and only
when executing the returned f' we have to provide a spec for f.
This is crucial. *)
(* TODO: Use our usual style with a generic postcondition. *)
Definition
is_syncer
(
R
:
iProp
Σ
)
(
s
:
val
)
:
=
(
□
∀
(
f
:
val
),
WP
s
f
{{
f'
,
synced
R
f
f'
}})%
I
.
Definition
mk_syncer_spec
(
mk_syncer
:
val
)
:
=
∀
(
R
:
iProp
Σ
),
{{{
R
}}}
mk_syncer
#()
{{{
s
,
RET
s
;
is_syncer
R
s
}}}.
End
sync
.
theories/logatom/treiber.v
0 → 100644
View file @
4beb6889
From
stdpp
Require
Import
namespaces
.
From
iris
.
program_logic
Require
Export
weakestpre
atomic
.
From
iris
.
heap_lang
Require
Export
lang
.
From
iris
.
heap_lang
Require
Import
proofmode
notation
.
From
iris
.
algebra
Require
Import
frac
auth
gmap
csum
.
Definition
new_stack
:
val
:
=
λ
:
<>,
ref
(
ref
NONE
).
Definition
push
:
val
:
=
rec
:
"push"
"s"
"x"
:
=
let
:
"hd"
:
=
!
"s"
in
let
:
"s'"
:
=
ref
SOME
(
"x"
,
"hd"
)
in
if
:
CAS
"s"
"hd"
"s'"
then
#()
else
"push"
"s"
"x"
.
Definition
pop
:
val
:
=
rec
:
"pop"
"s"
:
=
let
:
"hd"
:
=
!
"s"
in
match
:
!
"hd"
with
SOME
"cell"
=>
if
:
CAS
"s"
"hd"
(
Snd
"cell"
)
then
SOME
(
Fst
"cell"
)
else
"pop"
"s"

NONE
=>
NONE
end
.
Definition
iter
:
val
:
=
rec
:
"iter"
"hd"
"f"
:
=
match
:
!
"hd"
with
NONE
=>
#()

SOME
"cell"
=>
"f"
(
Fst
"cell"
)
;;
"iter"
(
Snd
"cell"
)
"f"
end
.
Section
proof
.
Context
`
{!
heapG
Σ
}
(
N
:
namespace
).
Fixpoint
is_list
(
hd
:
loc
)
(
xs
:
list
val
)
:
iProp
Σ
:
=
match
xs
with

[]
=>
(
∃
q
,
hd
↦
{
q
}
NONEV
)%
I

x
::
xs
=>
(
∃
(
hd'
:
loc
)
q
,
hd
↦
{
q
}
SOMEV
(
x
,
#
hd'
)
∗
is_list
hd'
xs
)%
I
end
.
Lemma
dup_is_list
:
∀
xs
hd
,
is_list
hd
xs
⊢
is_list
hd
xs
∗
is_list
hd
xs
.
Proof
.
induction
xs
as
[
y
xs'
IHxs'
].

iIntros
(
hd
)
"Hs"
.
simpl
.
iDestruct
"Hs"
as
(
q
)
"[Hhd Hhd']"
.
iSplitL
"Hhd"
;
eauto
.

iIntros
(
hd
)
"Hs"
.
simpl
.
iDestruct
"Hs"
as
(
hd'
q
)
"([Hhd Hhd'] & Hs')"
.
iDestruct
(
IHxs'
with
"[Hs']"
)
as
"[Hs1 Hs2]"
;
first
by
iFrame
.
iSplitL
"Hhd Hs1"
;
iExists
hd'
,
(
q
/
2
)%
Qp
;
by
iFrame
.
Qed
.
Lemma
uniq_is_list
:
∀
xs
ys
hd
,
is_list
hd
xs
∗
is_list
hd
ys
⊢
⌜
xs
=
ys
⌝
.
Proof
.
induction
xs
as
[
x
xs'
IHxs'
].

induction
ys
as
[
y
ys'
IHys'
].
+
auto
.
+
iIntros
(
hd
)
"(Hxs & Hys)"
.
simpl
.
iDestruct
"Hys"
as
(
hd'
?)
"(Hhd & Hys')"
.
iExFalso
.
iDestruct
"Hxs"
as
(?)
"Hhd'"
.
(* FIXME: If I dont use the @ here and below through this file, it loops. *)
by
iDestruct
(@
mapsto_agree
with
"[$Hhd] [$Hhd']"
)
as
%?.

induction
ys
as
[
y
ys'
IHys'
].
+
iIntros
(
hd
)
"(Hxs & Hys)"
.
simpl
.
iExFalso
.
iDestruct
"Hxs"
as
(?
?)
"(Hhd & _)"
.
iDestruct
"Hys"
as
(?)
"Hhd'"
.
by
iDestruct
(@
mapsto_agree
with
"[$Hhd] [$Hhd']"
)
as
%?.
+
iIntros
(
hd
)
"(Hxs & Hys)"
.
simpl
.
iDestruct
"Hxs"
as
(?
?)
"(Hhd & Hxs')"
.
iDestruct
"Hys"
as
(?
?)
"(Hhd' & Hys')"
.
iDestruct
(@
mapsto_agree
with
"[$Hhd] [$Hhd']"
)
as
%[=
Heq
].
subst
.
iDestruct
(
IHxs'
with
"[Hxs' Hys']"
)
as
"%"
;
first
by
iFrame
.
by
subst
.
Qed
.
Definition
is_stack
(
s
:
loc
)
xs
:
iProp
Σ
:
=
(
∃
hd
:
loc
,
s
↦
#
hd
∗
is_list
hd
xs
)%
I
.
Global
Instance
is_list_timeless
xs
hd
:
Timeless
(
is_list
hd
xs
).
Proof
.
generalize
hd
.
induction
xs
;
apply
_
.
Qed
.
Global
Instance
is_stack_timeless
xs
hd
:
Timeless
(
is_stack
hd
xs
).
Proof
.
generalize
hd
.
induction
xs
;
apply
_
.
Qed
.
Lemma
new_stack_spec
:
∀
(
Φ
:
val
→
iProp
Σ
),
(
∀
s
,
is_stack
s
[]

∗
Φ
#
s
)
⊢
WP
new_stack
#()
{{
Φ
}}.
Proof
.
iIntros
(
Φ
)
"HΦ"
.
wp_lam
.
wp_bind
(
ref
NONE
)%
E
.
wp_alloc
l
as
"Hl"
.
wp_alloc
l'
as
"Hl'"
.
iApply
"HΦ"
.
rewrite
/
is_stack
.
iExists
l
.
iFrame
.
by
iExists
1
%
Qp
.
Qed
.
Definition
push_triple
(
s
:
loc
)
(
x
:
val
)
:
=
<<<
∀
(
xs
:
list
val
)
(
hd
:
loc
),
s
↦
#
hd
∗
is_list
hd
xs
>>>
push
#
s
x
@
⊤
<<<
∃
hd'
:
loc
,
s
↦
#
hd'
∗
hd'
↦
SOMEV
(
x
,
#
hd
)
∗
is_list
hd
xs
,
RET
#()
>>>.
Lemma
push_atomic_spec
(
s
:
loc
)
(
x
:
val
)
:
push_triple
s
x
.
Proof
.
rewrite
/
push_triple
.
iApply
wp_atomic_intro
.
iIntros
(
Φ
)
"HP"
.
iL
ö
b
as
"IH"
.
wp_rec
.
wp_let
.
wp_bind
(!
_
)%
E
.
iMod
"HP"
as
(
xs
hd
)
"[[Hs Hhd] [Hvs' _]]"
.
wp_load
.
iMod
(
"Hvs'"
with
"[Hs Hhd]"
)
as
"HP"
;
first
by
iFrame
.
iModIntro
.
wp_let
.
wp_alloc
l
as
"Hl"
.
wp_let
.
wp_bind
(
CAS
_
_
_
)%
E
.
iMod
"HP"
as
(
xs'
hd'
)
"[[Hs Hhd'] Hvs']"
.
destruct
(
decide
(
hd
=
hd'
))
as
[>
Hneq
].
*
wp_cas_suc
.
iDestruct
"Hvs'"
as
"[_ Hvs']"
.
iMod
(
"Hvs'"
with
"[]"
)
as
"HQ"
.
{
by
iFrame
.
}
iModIntro
.
wp_if
.
eauto
.
*
wp_cas_fail
.
iDestruct
"Hvs'"
as
"[Hvs' _]"
.
iMod
(
"Hvs'"
with
"[]"
)
as
"HP"
;
first
by
iFrame
.
iModIntro
.
wp_if
.
by
iApply
"IH"
.
Qed
.
Definition
pop_triple
(
s
:
loc
)
:
=
<<<
∀
(
xs
:
list
val
)
(
hd
:
loc
),
s
↦
#
hd
∗
is_list
hd
xs
>>>
pop
#
s
@
⊤
<<<
match
xs
with
[]
=>
s
↦
#
hd
∗
is_list
hd
[]

x
::
xs'
=>
∃
q
(
hd'
:
loc
),
hd
↦
{
q
}
SOMEV
(
x
,
#
hd'
)
∗
s
↦
#
hd'
∗
is_list
hd'
xs'
end
,
RET
match
xs
with
[]
=>
NONEV

x
::
_
=>
SOMEV
x
end
>>>.
Lemma
pop_atomic_spec
(
s
:
loc
)
:
pop_triple
s
.
Proof
.
rewrite
/
pop_triple
.
iApply
wp_atomic_intro
.
iIntros
(
Φ
)
"HP"
.
iL
ö
b
as
"IH"
.
wp_rec
.
wp_bind
(!
_
)%
E
.
iMod
"HP"
as
(
xs
hd
)
"[[Hs Hhd] Hvs']"
.
destruct
xs
as
[
y'
xs'
].

simpl
.
wp_load
.
iDestruct
"Hvs'"
as
"[_ Hvs']"
.
iDestruct
"Hhd"
as
(
q
)
"[Hhd Hhd']"
.
iMod
(
"Hvs'"
with
"[Hhd]"
)
as
"HQ"
.
{
iFrame
.
eauto
.
}
iModIntro
.
wp_let
.
wp_load
.
wp_pures
.
eauto
.

simpl
.
iDestruct
"Hhd"
as
(
hd'
q
)
"([[Hhd1 Hhd2] Hhd'] & Hxs')"
.
iDestruct
(
dup_is_list
with
"[Hxs']"
)
as
"[Hxs1 Hxs2]"
;
first
by
iFrame
.
wp_load
.
iDestruct
"Hvs'"
as
"[Hvs' _]"
.
iMod
(
"Hvs'"
with
"[Hhd1 Hhd2 Hxs1]"
)
as
"HP"
.
{
iFrame
.
iExists
hd'
,
(
q
/
2
)%
Qp
.
by
iFrame
.
}
iModIntro
.
wp_let
.
wp_load
.
wp_match
.
wp_proj
.
wp_bind
(
CAS
_
_
_
).
iMod
"HP"
as
(
xs
hd''
)
"[[Hs Hhd''] Hvs']"
.
destruct
(
decide
(
hd
=
hd''
))
as
[>
Hneq
].
+
wp_cas_suc
.
iDestruct
"Hvs'"
as
"[_ Hvs']"
.
destruct
xs
as
[
x'
xs''
].
{
simpl
.
iDestruct
"Hhd''"
as
(?)
"H"
.
iExFalso
.
by
iDestruct
(@
mapsto_agree
with
"[$Hhd1] [$H]"
)
as
%?.
}
simpl
.
iDestruct
"Hhd''"
as
(
hd'''
?)
"(Hhd'' & Hxs'')"
.
iDestruct
(@
mapsto_agree
with
"[$Hhd1] [$Hhd'']"
)
as
%[=].
subst
.
iMod
(
"Hvs'"
with
"[]"
)
as
"HQ"
.
{
iExists
(
q
/
2
/
2
)%
Qp
,
_
.
iDestruct
(
uniq_is_list
with
"[Hxs1 Hxs'']"
)
as
"%"
;
first
by
iFrame
.
subst
.
repeat
(
iSplitR
"Hxs1 Hs"
;
first
done
).
iFrame
.
}
iModIntro
.
wp_if
.
wp_pures
.
eauto
.
+
wp_cas_fail
.
iDestruct
"Hvs'"
as
"[Hvs' _]"
.
iMod
(
"Hvs'"
with
"[]"
)
as
"HP"
;
first
by
iFrame
.
iModIntro
.
wp_if
.
by
iApply
"IH"
.
Qed
.
End
proof
.
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