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Dan Frumin
ReLoCv1
Commits
a8fa1982
Commit
a8fa1982
authored
May 26, 2016
by
Amin Timany
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Add the basic monoid need for the stack example
parent
77f2e6be
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F_mu_ref_par/examples/stack/rules.v
F_mu_ref_par/examples/stack/rules.v
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F_mu_ref_par/examples/stack/rules.v
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a8fa1982
From
iris
.
proofmode
Require
Import
invariants
ghost_ownership
tactics
.
From
F_mu_ref_par
Require
Import
lang
examples
.
lock
typing
logrel_binary
fundamental_binary
rules_binary
rules
.
From
iris
.
algebra
Require
Import
gmap
dec_agree
auth
.
From
iris
.
program_logic
Require
Import
ownership
auth
.
Import
uPred
.
Definition
stackR
:
cmraT
:=
gmapR
loc
(
dec_agreeR
val
).
Class
stackG
Σ
:=
StackG
{
stack_inG
:>
authG
lang
Σ
stackR
;
stack_name
:
gname
}
.
Section
Rules
.
Context
{
Σ
:
gFunctors
}
{
istk
:
stackG
Σ
}
.
Lemma
stackR_self_op
(
h
:
stackR
)
:
h
≡
h
⋅
h
.
Proof
.
intros
i
.
rewrite
lookup_op
.
match
goal
with

?
A
≡
?
B
⋅
?
B
=>
change
B
with
A
;
destruct
A
as
[
c

]
end
;
trivial
.
destruct
c
as
[
c

];
unfold
op
,
cmra_op
;
simpl
;
unfold
op
,
cmra_op
;
simpl
;
trivial
;
destruct
decide
;
trivial
;
tauto
.
Qed
.
Lemma
stackR_get_auth
(
h
:
stackR
)
:
(
●
h
)
~~>
(
●
h
⋅
◯
h
).
Proof
.
intros
n
z
H1
.
destruct
z
as
[[
ze


]
zo
];
unfold
validN
,
cmra_validN
in
*
;
simpl
in
*
;
trivial
.
destruct
H1
as
[
H1
H2
];
split
;
trivial
.
destruct
H1
as
[
u
H1
].
eexists
u
.
rewrite
H1
.
rewrite
?
cmra_unit_left_id
.
rewrite

assoc
.
by
rewrite

stackR_self_op
.
Qed
.
Lemma
dec_agree_valid_op_eq
(
x
y
:
dec_agreeR
val
)
:
✓
(
Some
x
⋅
Some
y
)
→
x
=
y
.
Proof
.
intros
H1
.
destruct
x
as
[
x

];
destruct
y
as
[
y

];
trivial
;
cbv

[
decide
]
in
H1
;
try
destruct
decide
;
subst
;
simpl
;
intuition
trivial
.
Qed
.
Lemma
stackR_auth_is_subheap
(
h
h
'
:
stackR
)
:
✓
(
●
h
⋅
◯
h
'
)
→
∀
i
x
,
h
'
!!
i
≡
Some
x
→
h
!!
i
≡
Some
x
.
Proof
.
intros
H1
i
x
H2
.
destruct
H1
as
[
H11
H12
];
simpl
in
H11
.
specialize
(
H11
1
).
destruct
H11
as
[
z
H11
].
revert
H11
;
rewrite
cmra_unit_left_id
=>
H11
.
eapply
cmra_extend
in
H11
;
[

by
apply
cmra_valid_validN
].
destruct
H11
as
[[
z1
z2
]
[
H31
[
H32
H33
]]];
simpl
in
*
.
specialize
(
H32
i
).
assert
(
H4
:
✓
(
z1
⋅
z2
))
by
(
by
rewrite

H31
).
rewrite
H31
.
rewrite
lookup_op
.
specialize
(
H4
i
).
rewrite
?
lookup_op
in
H4
.
revert
H32
;
rewrite
H2
=>
H32
.
match
type
of
H32
with
?
C
≡
{
_
}
≡
_
=>
match
goal
with

?
A
⋅
?
B
≡
_
=>
change
C
with
A
in
*
;
destruct
A
as
[
a

];
inversion
H32
;
subst
end
end
.
match
type
of
H32
with
?
C
≡
{
_
}
≡
_
=>
match
goal
with

?
A
⋅
?
B
≡
_
=>
destruct
B
end
end
.

set
(
H5
:=
dec_agree_valid_op_eq
_
_
H4
);
clearbody
H5
.
subst
.
inversion
H1
;
subst
.
destruct
x
as
[
x

];
cbv

[
decide
];
try
destruct
decide
;
constructor
;
intuition
trivial
.

inversion
H1
;
subst
.
constructor
;
trivial
.
Qed
.
Lemma
stackR_alloc
(
h
:
stackR
)
(
i
:
loc
)
(
v
:
val
)
:
h
!!
i
=
None
→
(
●
h
)
~~>
(
●
(
{
[
i
:=
DecAgree
v
]
}
⋅
h
)).
Proof
.
intros
H1
n
z
H2
.
destruct
z
as
[[
ze


]
zo
];
unfold
validN
,
cmra_validN
in
*
;
simpl
in
*
;
trivial
.
destruct
H2
as
[
H21
H22
];
split
.

destruct
H21
as
[
u
H21
].
eexists
(
{
[
i
:=
DecAgree
v
]
}
⋅
u
).
rewrite
H21
.
rewrite
?
cmra_unit_left_id
.
rewrite
?
assoc
.
by
rewrite
(
comm
_
zo
).

intros
j
.
rewrite
lookup_op
.
destruct
(
decide
(
i
=
j
))
as
[

Hneq
];
subst
.
+
rewrite
H1
.
rewrite
lookup_singleton
;
constructor
.
+
rewrite
lookup_singleton_ne
;
trivial
.
specialize
(
H22
j
).
revert
H22
.
match
goal
with

✓
{
_
}
?
B
→
✓
{
_
}
(
_
⋅
?
A
)
=>
change
B
with
A
;
destruct
A
;
by
try
constructor
end
.
Qed
.
End
Rules
.
\ No newline at end of file
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