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ddaf548e
Commit
ddaf548e
authored
May 28, 2014
by
Filip Sieczkowski
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Notations, erasure and viewshift definitions.
parent
368a2f5c
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iris.v
iris.v
+142
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masks.v
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iris.v
View file @
ddaf548e
...
...
@@ 34,7 +34,7 @@ Module Iris (RP RL : PCM_T) (C : CORE_LANG RP).
Local
Obligation
Tactic
:
=
intros
;
resp_set

eauto
with
typeclass_instances
.
Program
Definition
always
:
Props

n
>
Props
:
=
Program
Definition
box
:
Props

n
>
Props
:
=
n
[(
fun
p
=>
m
[(
fun
w
=>
mkUPred
(
fun
n
r
=>
p
w
n
(
pcm_unit
_
))
_
)])].
Next
Obligation
.
intros
n
m
r
s
HLe
_
Hp
;
rewrite
HLe
;
assumption
.
...
...
@@ 132,4 +132,145 @@ Module Iris (RP RL : PCM_T) (C : CORE_LANG RP).
Definition
ownL
(
r
:
RL
.
res
)
:
Props
:
=
own
(
pcm_unit
_
,
r
).
Notation
"□ p"
:
=
(
box
p
)
(
at
level
30
,
right
associativity
)
:
iris_scope
.
Notation
"⊤"
:
=
(
top
:
Props
)
:
iris_scope
.
Notation
"⊥"
:
=
(
bot
:
Props
)
:
iris_scope
.
Notation
"p ∧ q"
:
=
(
and
p
q
:
Props
)
(
at
level
40
,
left
associativity
)
:
iris_scope
.
Notation
"p ∨ q"
:
=
(
or
p
q
:
Props
)
(
at
level
50
,
left
associativity
)
:
iris_scope
.
Notation
"p * q"
:
=
(
sc
p
q
:
Props
)
(
at
level
40
,
left
associativity
)
:
iris_scope
.
Notation
"p → q"
:
=
(
BI
.
impl
p
q
:
Props
)
(
at
level
55
,
right
associativity
)
:
iris_scope
.
Notation
"p '*' q"
:
=
(
si
p
q
:
Props
)
(
at
level
55
,
right
associativity
)
:
iris_scope
.
Notation
"∀ x , p"
:
=
(
all
n
[(
fun
x
=>
p
)]
:
Props
)
(
at
level
60
,
x
ident
,
no
associativity
)
:
iris_scope
.
Notation
"∃ x , p"
:
=
(
all
n
[(
fun
x
=>
p
)]
:
Props
)
(
at
level
60
,
x
ident
,
no
associativity
)
:
iris_scope
.
Notation
"∀ x : T , p"
:
=
(
all
n
[(
fun
x
:
T
=>
p
)]
:
Props
)
(
at
level
60
,
x
ident
,
no
associativity
)
:
iris_scope
.
Notation
"∃ x : T , p"
:
=
(
all
n
[(
fun
x
:
T
=>
p
)]
:
Props
)
(
at
level
60
,
x
ident
,
no
associativity
)
:
iris_scope
.
Section
Erasure
.
Global
Instance
preo_unit
:
preoType
()
:
=
disc_preo
().
Local
Open
Scope
bi_scope
.
Local
Open
Scope
pcm_scope
.
(* XXX: logical state omitted, since it looks weird. Also, later over the whole deal. *)
Program
Definition
erasure
(
σ
:
state
)
(
m
:
mask
)
(
r
s
:
R
.
res
)
(
w
:
Wld
)
:
UPred
()
:
=
▹
(
mkUPred
(
fun
n
_
=>
erase_state
(
option_map
fst
(
Some
r
·
Some
s
))
σ
/\
forall
i
π
,
m
i
>
w
i
==
Some
π
>
(
ı
π
)
w
n
s
)
_
).
Next
Obligation
.
intros
n1
n2
_
_
HLe
_
[
HES
HRS
]
;
split
;
[
assumption

clear
HES
;
intros
].
rewrite
HLe
;
eauto
.
Qed
.
Global
Instance
erasure_equiv
σ
m
r
s
:
Proper
(
equiv
==>
equiv
)
(
erasure
σ
m
r
s
).
Proof
.
intros
w1
w2
EQw
[
n
]
[]
;
[
reflexivity
].
split
;
intros
[
HES
HRS
]
;
(
split
;
[
tauto

clear
HES
;
intros
?
?
HM
HLu
]).

rewrite
<
EQw
;
eapply
HRS
;
[
eassumption
].
change
(
w1
i
==
Some
π
)
;
rewrite
EQw
;
assumption
.

rewrite
EQw
;
eapply
HRS
;
[
eassumption
].
change
(
w2
i
==
Some
π
)
;
rewrite
<
EQw
;
assumption
.
Qed
.
Global
Instance
erasure_dist
n
σ
m
r
s
:
Proper
(
dist
n
==>
dist
n
)
(
erasure
σ
m
r
s
).
Proof
.
intros
w1
w2
EQw
[
n'
]
[]
HLt
;
[
reflexivity
]
;
destruct
n
as
[
n
]
;
[
now
inversion
HLt
].
split
;
intros
[
HES
HRS
]
;
(
split
;
[
tauto

clear
HES
;
intros
?
?
HM
HLu
]).

assert
(
EQ
π
:
=
EQw
i
)
;
specialize
(
HRS
i
)
;
rewrite
HLu
in
EQ
π
;
clear
HLu
.
destruct
(
w1
i
)
as
[
π
'
]
;
[
contradiction
]
;
do
3
red
in
EQ
π
.
apply
ı
in
EQ
π
;
apply
EQ
π
;
[
now
auto
with
arith
].
apply
(
met_morph_nonexp
_
_
(
ı
π
'
))
in
EQw
;
apply
EQw
;
[
now
auto
with
arith
].
apply
HRS
;
[
assumption

reflexivity
].

assert
(
EQ
π
:
=
EQw
i
)
;
specialize
(
HRS
i
)
;
rewrite
HLu
in
EQ
π
;
clear
HLu
.
destruct
(
w2
i
)
as
[
π
'
]
;
[
contradiction
]
;
do
3
red
in
EQ
π
.
apply
ı
in
EQ
π
;
apply
EQ
π
;
[
now
auto
with
arith
].
apply
(
met_morph_nonexp
_
_
(
ı
π
'
))
in
EQw
;
apply
EQw
;
[
now
auto
with
arith
].
apply
HRS
;
[
assumption

reflexivity
].
Qed
.
End
Erasure
.
Notation
" p @ k "
:
=
((
p
:
UPred
())
k
tt
)
(
at
level
60
,
no
associativity
).
Section
ViewShifts
.
Local
Open
Scope
mask_scope
.
Local
Open
Scope
pcm_scope
.
Local
Obligation
Tactic
:
=
intros
.
Program
Definition
preVS
(
m1
m2
:
mask
)
(
p
:
Props
)
(
w
:
Wld
)
:
UPred
R
.
res
:
=
mkUPred
(
fun
n
r
=>
forall
w1
s
rf
rc
mf
σ
k
(
HSub
:
w
⊑
w1
)
(
HLe
:
k
<=
n
)
(
HGt
:
k
>
0
)
(
HR
:
Some
rc
=
Some
r
·
Some
rf
)
(
HE
:
erasure
σ
(
m1
∪
mf
)
rc
s
w1
@
k
)
(
HD
:
mf
#
m1
∪
m2
),
exists
w2
rc'
r'
s'
,
w1
⊑
w2
/\
p
w2
k
r'
/\
Some
rc'
=
Some
r'
·
Some
rf
/\
erasure
σ
(
m2
∪
mf
)
rc'
s'
w2
@
k
)
_
.
Next
Obligation
.
intros
n1
n2
r1
r2
HLe
HSub
HP
;
intros
.
destruct
HSub
as
[
[
rd
]
HSub
]
;
[
erewrite
pcm_op_zero
in
HSub
by
eauto
with
typeclass_instances
;
discriminate
].
rewrite
(
comm
(
Commutative
:
=
pcm_op_comm
_
))
in
HSub
;
rewrite
<
HSub
in
HR
.
rewrite
<
(
assoc
(
Associative
:
=
pcm_op_assoc
_
))
in
HR
.
destruct
(
Some
rd
·
Some
rf
)
as
[
rf'
]
eqn
:
HR'
;
[
erewrite
(
comm
(
Commutative
:
=
pcm_op_comm
_
)),
pcm_op_zero
in
HR
by
apply
_;
discriminate
].
edestruct
(
HP
w1
s
rf'
rc
mf
σ
k
)
as
[
w2
[
rc'
[
r1'
[
s'
HH
]
]
]
]
;
try
eassumption
;
[
etransitivity
;
eassumption
]
;
clear

HR'
HH
.
destruct
HH
as
[
HW
[
HP
[
HR
HE
]
]
]
;
rewrite
<
HR'
in
HR
.
rewrite
(
assoc
(
Associative
:
=
pcm_op_assoc
_
))
in
HR
.
destruct
(
Some
r1'
·
Some
rd
)
as
[
r2'
]
eqn
:
HR''
;
[
erewrite
pcm_op_zero
in
HR
by
apply
_;
discriminate
].
exists
w2
rc'
r2'
s'
;
intuition
;
[].
eapply
uni_pred
,
HP
;
[
reflexivity
].
exists
(
Some
rd
)
;
rewrite
(
comm
(
Commutative
:
=
pcm_op_comm
_
))
;
assumption
.
Qed
.
Program
Definition
pvs
(
m1
m2
:
mask
)
:
Props

n
>
Props
:
=
n
[(
fun
p
=>
m
[(
preVS
m1
m2
p
)])].
Next
Obligation
.
intros
w1
w2
EQw
n
r
;
split
;
intros
HP
w2'
;
intros
.

eapply
HP
;
try
eassumption
;
[].
rewrite
EQw
;
assumption
.

eapply
HP
;
try
eassumption
;
[].
rewrite
<
EQw
;
assumption
.
Qed
.
Next
Obligation
.
intros
w1
w2
EQw
n'
r
HLt
;
destruct
n
as
[
n
]
;
[
now
inversion
HLt
]
;
split
;
intros
HP
w2'
;
intros
.

symmetry
in
EQw
;
assert
(
HDE
:
=
extend_dist
_
_
_
_
EQw
HSub
).
assert
(
HSE
:
=
extend_sub
_
_
_
_
EQw
HSub
)
;
specialize
(
HP
(
extend
w2'
w1
)).
edestruct
HP
as
[
w1''
[
rc'
[
r'
[
s'
[
HW
HH
]
]
]
]
]
;
try
eassumption
;
clear
HP
;
[

].
+
eapply
erasure_dist
,
HE
;
[
symmetry
;
eassumption

now
eauto
with
arith
].
+
symmetry
in
HDE
;
assert
(
HDE'
:
=
extend_dist
_
_
_
_
HDE
HW
).
assert
(
HSE'
:
=
extend_sub
_
_
_
_
HDE
HW
)
;
destruct
HH
as
[
HP
[
HR'
HE'
]
]
;
exists
(
extend
w1''
w2'
)
rc'
r'
s'
;
repeat
split
;
[
assumption


assumption
].
*
eapply
(
met_morph_nonexp
_
_
p
),
HP
;
[
symmetry
;
eassumption

now
eauto
with
arith
].
*
eapply
erasure_dist
,
HE'
;
[
symmetry
;
eassumption

now
eauto
with
arith
].

assert
(
HDE
:
=
extend_dist
_
_
_
_
EQw
HSub
)
;
assert
(
HSE
:
=
extend_sub
_
_
_
_
EQw
HSub
)
;
specialize
(
HP
(
extend
w2'
w2
)).
edestruct
HP
as
[
w1''
[
rc'
[
r'
[
s'
[
HW
HH
]
]
]
]
]
;
try
eassumption
;
clear
HP
;
[

].
+
eapply
erasure_dist
,
HE
;
[
symmetry
;
eassumption

now
eauto
with
arith
].
+
symmetry
in
HDE
;
assert
(
HDE'
:
=
extend_dist
_
_
_
_
HDE
HW
).
assert
(
HSE'
:
=
extend_sub
_
_
_
_
HDE
HW
)
;
destruct
HH
as
[
HP
[
HR'
HE'
]
]
;
exists
(
extend
w1''
w2'
)
rc'
r'
s'
;
repeat
split
;
[
assumption


assumption
].
*
eapply
(
met_morph_nonexp
_
_
p
),
HP
;
[
symmetry
;
eassumption

now
eauto
with
arith
].
*
eapply
erasure_dist
,
HE'
;
[
symmetry
;
eassumption

now
eauto
with
arith
].
Qed
.
Next
Obligation
.
intros
w1
w2
EQw
n
r
HP
w2'
;
intros
;
eapply
HP
;
try
eassumption
;
[].
etransitivity
;
eassumption
.
Qed
.
Next
Obligation
.
intros
p1
p2
EQp
w
n
r
;
split
;
intros
HP
w1
;
intros
.

setoid_rewrite
<
EQp
;
eapply
HP
;
eassumption
.

setoid_rewrite
EQp
;
eapply
HP
;
eassumption
.
Qed
.
Next
Obligation
.
intros
p1
p2
EQp
w
n'
r
HLt
;
split
;
intros
HP
w1
;
intros
.

edestruct
HP
as
[
w2
[
rc'
[
r'
[
s'
[
HW
[
HP'
[
HR'
HE'
]
]
]
]
]
]
]
;
try
eassumption
;
[].
clear
HP
;
repeat
eexists
;
try
eassumption
;
[].
apply
EQp
;
[
now
eauto
with
arith

assumption
].

edestruct
HP
as
[
w2
[
rc'
[
r'
[
s'
[
HW
[
HP'
[
HR'
HE'
]
]
]
]
]
]
]
;
try
eassumption
;
[].
clear
HP
;
repeat
eexists
;
try
eassumption
;
[].
apply
EQp
;
[
now
eauto
with
arith

assumption
].
Qed
.
Definition
vs
(
m1
m2
:
mask
)
(
p
q
:
Props
)
:
Props
:
=
□
(
p
→
pvs
m1
m2
q
).
End
ViewShifts
.
End
Iris
.
masks.v
View file @
ddaf548e
...
...
@@ 27,6 +27,7 @@ Notation "m1 ⊆ m2" := (mle m1 m2) (at level 70) : mask_scope.
Notation
"m1 ∩ m2"
:
=
(
fun
i
=>
(
m1
:
mask
)
i
/\
(
m2
:
mask
)
i
)
(
at
level
40
)
:
mask_scope
.
Notation
"m1 \ m2"
:
=
(
fun
i
=>
(
m1
:
mask
)
i
/\
~
(
m2
:
mask
)
i
)
(
at
level
30
)
:
mask_scope
.
Notation
"m1 ∪ m2"
:
=
(
fun
i
=>
(
m1
:
mask
)
i
\/
(
m2
:
mask
)
i
)
(
at
level
50
)
:
mask_scope
.
Notation
"m1 # m2"
:
=
(
mask_disj
m1
m2
)
(
at
level
70
)
:
mask_scope
.
Open
Scope
mask_scope
.
...
...
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