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c905411d
Commit
c905411d
authored
Feb 25, 2016
by
Robbert Krebbers
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Simplify barrier protocol proofs.
parent
2d9c5f33
Pipeline
#156
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barrier/protocol.v
barrier/protocol.v
+25
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barrier/protocol.v
View file @
c905411d
...
...
@@ 17,12 +17,9 @@ Inductive prim_step : relation state :=

ChangeI
p
I2
I1
:
prim_step
(
State
p
I1
)
(
State
p
I2
)

ChangePhase
I
:
prim_step
(
State
Low
I
)
(
State
High
I
).
Definition
change_tok
(
I
:
gset
gname
)
:
set
token
:
=
{[
t

match
t
with
Change
i
=>
i
∉
I

Send
=>
False
end
]}.
Definition
send_tok
(
p
:
phase
)
:
set
token
:
=
match
p
with
Low
=>
∅

High
=>
{[
Send
]}
end
.
Definition
tok
(
s
:
state
)
:
set
token
:
=
change_tok
(
state_I
s
)
∪
send_tok
(
state_phase
s
).
{[
t

∃
i
,
t
=
Change
i
∧
i
∉
state_I
s
]}
∪
(
if
state_phase
s
is
High
then
{[
Send
]}
else
∅
).
Global
Arguments
tok
!
_
/.
Canonical
Structure
sts
:
=
sts
.
STS
prim_step
tok
.
...
...
@@ 35,30 +32,22 @@ Definition low_states : set state := {[ s  state_phase s = Low ]}.
Lemma
i_states_closed
i
:
sts
.
closed
(
i_states
i
)
{[
Change
i
]}.
Proof
.
split
.

intros
[
p
I
].
rewrite
/=
/
i_states
/
change_tok
.
destruct
p
;
set_solver
.

(* 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". *)
intros
s1
s2
Hs1
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
;
simpl
in
*
;
last
done
.
move
:
Hs1
Hdisj
Htok
.
rewrite
elem_of_equiv_empty
elem_of_equiv
.
move
=>
?
/(
_
(
Change
i
))
Hdisj
/(
_
(
Change
i
))
;
move
:
Hdisj
.
rewrite
elem_of_intersection
elem_of_union
!
elem_of_mkSet
.
intros
;
apply
dec_stable
.
destruct
p
;
set_solver
.
split
;
first
(
intros
[[]
I
]
;
set_solver
).
(* 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". *)
intros
s1
s2
Hs1
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
as
[[]
??]
;
done

set_solver
.
Qed
.
Lemma
low_states_closed
:
sts
.
closed
low_states
{[
Send
]}.
Proof
.
split
.

intros
[
p
I
].
rewrite
/
low_states
.
set_solver
.

intros
s1
s2
Hs1
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
;
simpl
in
*
;
first
by
destruct
p
.
exfalso
;
apply
dec_stable
;
set_solver
.
split
;
first
(
intros
[??]
;
set_solver
).
intros
s1
s2
Hs1
[
T1
T2
Hdisj
Hstep'
].
inversion_clear
Hstep'
as
[?
?
?
?
Htrans
_
_
Htok
].
destruct
Htrans
as
[[]
??]
;
done

set_solver
.
Qed
.
(* Proof that we can take the steps we need. *)
...
...
@@ 70,12 +59,8 @@ Lemma wait_step i I :
sts
.
steps
(
State
High
I
,
{[
Change
i
]})
(
State
High
(
I
∖
{[
i
]}),
∅
).
Proof
.
intros
.
apply
rtc_once
.
constructor
;
first
constructor
;
rewrite
/=
/
change_tok
;
[
set_solver
by
eauto
..].
(* TODO this proof is rather annoying. *)
apply
elem_of_equiv
=>
t
.
rewrite
!
elem_of_union
.
rewrite
!
elem_of_mkSet
/
change_tok
/=.
destruct
t
as
[
j
]
;
last
set_solver
.
rewrite
elem_of_difference
elem_of_singleton
.
constructor
;
first
constructor
;
[
set_solver
..].
apply
elem_of_equiv
=>[
j
]
;
last
set_solver
.
destruct
(
decide
(
i
=
j
))
;
set_solver
.
Qed
.
...
...
@@ 85,17 +70,14 @@ Lemma split_step p i i1 i2 I :
(
State
p
I
,
{[
Change
i
]})
(
State
p
({[
i1
]}
∪
({[
i2
]}
∪
(
I
∖
{[
i
]}))),
{[
Change
i1
;
Change
i2
]}).
Proof
.
intros
.
apply
rtc_once
.
constructor
;
first
constructor
;
simpl
.
intros
.
apply
rtc_once
.
constructor
;
first
constructor
.

destruct
p
;
set_solver
.

destruct
p
;
set_solver
.
(* This gets annoying... and I think I can see a pattern with all these proofs. Automatable? *)

apply
elem_of_equiv
=>
t
.
destruct
t
;
last
set_solver
.
rewrite
!
elem_of_mkSet
!
not_elem_of_union
!
not_elem_of_singleton
not_elem_of_difference
elem_of_singleton
!(
inj_iff
Change
).
destruct
p
;
naive_solver
.

apply
elem_of_equiv
=>
t
.
destruct
t
as
[
j
]
;
last
set_solver
.
rewrite
!
elem_of_mkSet
!
not_elem_of_union
!
not_elem_of_singleton
not_elem_of_difference
elem_of_singleton
!(
inj_iff
Change
).
destruct
(
decide
(
i1
=
j
))
as
[>]
;
first
tauto
.
destruct
(
decide
(
i2
=
j
))
as
[>]
;
intuition
.

apply
elem_of_equiv
=>
/=
[
j
]
;
last
set_solver
.
set_unfold
;
rewrite
!(
inj_iff
Change
).
assert
(
Change
j
∈
match
p
with
Low
=>
∅

High
=>
{[
Send
]}
end
↔
False
)
as
>
by
(
destruct
p
;
set_solver
).
destruct
(
decide
(
i1
=
j
))
as
[>]
;
first
naive_solver
.
destruct
(
decide
(
i2
=
j
))
as
[>]
;
first
naive_solver
.
destruct
(
decide
(
i
=
j
))
as
[>]
;
naive_solver
.
Qed
.
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