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Rodolphe Lepigre
Iris
Commits
8930527a
Commit
8930527a
authored
Mar 07, 2016
by
Ralf Jung
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establish some properties of STSs without tokens
parent
3059c657
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algebra/sts.v
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8930527a
...
...
@@ -450,3 +450,78 @@ Proof.
Qed
.
End
stsRA
.
(** STSs without tokens: Some stuff is simpler *)
Module
sts_notok
.
Structure
stsT
:
=
STS
{
state
:
Type
;
prim_step
:
relation
state
;
}.
Arguments
STS
{
_
}
_
.
Arguments
prim_step
{
_
}
_
_
.
Notation
states
sts
:
=
(
set
(
state
sts
)).
Canonical
sts_notok
(
sts
:
stsT
)
:
sts
.
stsT
:
=
sts
.
STS
(
token
:
=
Empty_set
)
(@
prim_step
sts
)
(
λ
_
,
∅
).
Section
sts
.
Context
{
sts
:
stsT
}.
Implicit
Types
s
:
state
sts
.
Implicit
Types
S
:
states
sts
.
Notation
prim_steps
:
=
(
rtc
prim_step
).
Lemma
sts_step
s1
s2
:
prim_step
s1
s2
→
sts
.
step
(
s1
,
∅
)
(
s2
,
∅
).
Proof
.
intros
.
split
;
set_solver
.
Qed
.
Lemma
sts_steps
s1
s2
:
prim_steps
s1
s2
→
sts
.
steps
(
s1
,
∅
)
(
s2
,
∅
).
Proof
.
induction
1
;
eauto
using
sts_step
,
rtc_refl
,
rtc_l
.
Qed
.
Lemma
frame_prim_step
T
s1
s2
:
sts
.
frame_step
T
s1
s2
→
prim_step
s1
s2
.
Proof
.
inversion
1
as
[???
Hstep
].
inversion_clear
Hstep
.
done
.
Qed
.
Lemma
prim_frame_step
T
s1
s2
:
prim_step
s1
s2
→
sts
.
frame_step
T
s1
s2
.
Proof
.
intros
Hstep
.
apply
sts
.
Frame_step
with
∅
∅
;
first
set_solver
.
by
apply
sts_step
.
Qed
.
Lemma
mk_closed
S
:
(
∀
s1
s2
,
s1
∈
S
→
prim_step
s1
s2
→
s2
∈
S
)
→
sts
.
closed
S
∅
.
Proof
.
intros
?.
constructor
;
first
by
set_solver
.
intros
????.
eauto
using
frame_prim_step
.
Qed
.
End
sts
.
Notation
steps
:
=
(
rtc
prim_step
).
End
sts_notok
.
Coercion
sts_notok
.
sts_notok
:
sts_notok
.
stsT
>->
sts
.
stsT
.
Notation
sts_notokT
:
=
sts_notok
.
stsT
.
Notation
STS_NoTok
:
=
sts_notok
.
STS
.
Section
sts_notokRA
.
Import
sts_notok
.
Context
{
sts
:
sts_notokT
}.
Implicit
Types
s
:
state
sts
.
Implicit
Types
S
:
states
sts
.
Lemma
sts_notok_update_auth
s1
s2
:
rtc
prim_step
s1
s2
→
sts_auth
s1
∅
~~>
sts_auth
s2
∅
.
Proof
.
intros
.
by
apply
sts_update_auth
,
sts_steps
.
Qed
.
End
sts_notokRA
.
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