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Jonas Kastberg
iris
Commits
86b8e9ed
Commit
86b8e9ed
authored
Feb 16, 2016
by
Ralf Jung
Browse files
define the set of low states and prove it closed
parent
1109ca07
Changes
1
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Inline
Side-by-side
barrier/barrier.v
View file @
86b8e9ed
...
...
@@ -3,9 +3,11 @@ From heap_lang Require Export derived heap wp_tactics notation.
Definition
newchan
:
=
(
λ
:
""
,
ref
'
0
)%
L
.
Definition
signal
:
=
(
λ
:
"x"
,
"x"
<-
'
1
)%
L
.
Definition
wait
:
=
(
rec
:
"wait"
"x"
:
=
if
:
!
"x"
=
'
1
then
'
()
else
"wait"
"x"
)%
L
.
Definition
wait
:
=
(
rec
:
"wait"
"x"
:
=
if
:
!
"x"
=
'
1
then
'
()
else
"wait"
"x"
)%
L
.
(** The STS describing the main barrier protocol. *)
(** The STS describing the main barrier protocol. Every state has an index-set
associated with it. These indices are actually [gname], because we use them
with saved propositions. *)
Module
barrier_proto
.
Inductive
phase
:
=
Low
|
High
.
Record
stateT
:
=
State
{
state_phase
:
phase
;
state_I
:
gset
gname
}.
...
...
@@ -27,6 +29,7 @@ Module barrier_proto.
Definition
sts
:
=
sts
.
STS
trans
tok
.
(* The set of states containing some particular i *)
Definition
i_states
(
i
:
gname
)
:
set
stateT
:
=
mkSet
(
λ
s
,
i
∈
state_I
s
).
...
...
@@ -34,17 +37,21 @@ Module barrier_proto.
sts
.
closed
sts
(
i_states
i
)
{[
Change
i
]}.
Proof
.
split
.
-
apply
(
non_empty_inhabited
(
State
Low
{[
i
]})).
rewrite
!
mkSet_elem_of
/=.
-
apply
(
non_empty_inhabited
(
State
Low
{[
i
]})).
rewrite
!
mkSet_elem_of
/=.
apply
lookup_singleton
.
-
move
=>[
p
I
].
rewrite
/=
/
tok
!
mkSet_elem_of
/=
=>
HI
.
move
=>
s'
/
elem_of_intersection
.
rewrite
!
mkSet_elem_of
/=.
move
=>[[
Htok
|
Htok
]
?
]
;
subst
s'
;
first
done
.
destruct
p
;
done
.
-
move
=>
s1
s2
.
rewrite
!
mkSet_elem_of
/==>
Hs1
Hstep
.
-
(* If we do the destruct of the states early, and then inversion
on the proof of a transition, it doesn't work - we do not obtain
the equalities we need. So we destruct the states late, because this
means we can use "destruct" instead of "inversion". *)
move
=>
s1
s2
.
rewrite
!
mkSet_elem_of
/==>
Hs1
Hstep
.
(* We probably want some helper lemmas for this... *)
inversion_clear
Hstep
as
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
Htok1
Htok2
Htok
].
destruct
Htrans
;
last
done
;
move
:
Hs1
Hdisj
Htok1
Htok2
Htok
.
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
;
last
done
;
move
:
Hs1
Hdisj
Htok
.
rewrite
/=
/
tok
/=.
intros
.
apply
dec_stable
.
assert
(
Change
i
∉
change_tokens
I1
)
as
HI1
...
...
@@ -55,6 +62,25 @@ Module barrier_proto.
-
solve_elem_of
+
Htok
Hdisj
HI1
/
discriminate
.
}
done
.
Qed
.
(* The set of low states *)
Definition
low_states
:
set
stateT
:
=
mkSet
(
λ
s
,
if
state_phase
s
is
Low
then
True
else
False
).
Lemma
low_states_closed
:
sts
.
closed
sts
low_states
{[
Send
]}.
Proof
.
split
.
-
apply
(
non_empty_inhabited
(
State
Low
∅
)).
by
rewrite
!
mkSet_elem_of
/=.
-
move
=>[
p
I
].
rewrite
/=
/
tok
!
mkSet_elem_of
/=
=>
HI
.
destruct
p
;
last
done
.
solve_elem_of
+
/
discriminate
.
-
move
=>
s1
s2
.
rewrite
!
mkSet_elem_of
/==>
Hs1
Hstep
.
inversion_clear
Hstep
as
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
;
move
:
Hs1
Hdisj
Htok
=>/=
;
first
by
destruct
p
.
rewrite
/=
/
tok
/=.
intros
.
solve_elem_of
+
Hdisj
Htok
.
Qed
.
End
barrier_proto
.
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