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Iris
Iris
Commits
06417e80
Commit
06417e80
authored
Nov 16, 2015
by
Robbert Krebbers
Browse files
STS can now have tokens of any type with decidable equality.
parent
b1d4cb1d
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iris/sts.v
iris/sts.v
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iris/sts.v
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06417e80
Require
Export
iris
.
ra
.
Require
Import
prelude
.
sets
prelude
.
stringmap
iris
.
dra
.
Require
Import
prelude
.
sets
prelude
.
listset
iris
.
dra
.
Local
Arguments
valid
_
_
!
_
/.
Local
Arguments
op
_
_
!
_
!
_
/.
Local
Arguments
unit
_
_
!
_
/.
Module
sts
.
Inductive
t
{
A
}
(
R
:
relation
A
)
(
tok
:
A
→
st
ring
set
)
:
=

auth
:
A
→
st
ring
set
→
t
R
tok

frag
:
set
A
→
st
ring
set
→
t
R
tok
.
Arguments
auth
{
_
_
_
}
_
_
.
Arguments
frag
{
_
_
_
}
_
_
.
Inductive
t
{
A
B
}
(
R
:
relation
A
)
(
tok
:
A
→
li
stset
B
)
:
=

auth
:
A
→
li
stset
B
→
t
R
tok

frag
:
set
A
→
li
stset
B
→
t
R
tok
.
Arguments
auth
{
_
_
_
_
}
_
_
.
Arguments
frag
{
_
_
_
_
}
_
_
.
Section
sts_core
.
Context
{
A
}
(
R
:
relation
A
)
(
tok
:
A
→
stringset
).
Context
{
A
B
:
Type
}
`
{
∀
x
y
:
B
,
Decision
(
x
=
y
)}.
Context
(
R
:
relation
A
)
(
tok
:
A
→
listset
B
).
Inductive
sts_equiv
:
Equiv
(
t
R
tok
)
:
=

auth_equiv
s
T1
T2
:
T1
=
T2
→
auth
s
T1
≡
auth
s
T2

frag_equiv
S1
S2
T1
T2
:
T1
=
T2
→
S1
≡
S2
→
frag
S1
T1
≡
frag
S2
T2
.

auth_equiv
s
T1
T2
:
T1
≡
T2
→
auth
s
T1
≡
auth
s
T2

frag_equiv
S1
S2
T1
T2
:
T1
≡
T2
→
S1
≡
S2
→
frag
S1
T1
≡
frag
S2
T2
.
Global
Existing
Instance
sts_equiv
.
Inductive
step
:
relation
(
A
*
st
ring
set
)
:
=
Inductive
step
:
relation
(
A
*
li
stset
B
)
:
=

Step
s1
s2
T1
T2
:
R
s1
s2
→
tok
s1
∩
T1
=
∅
→
tok
s2
∩
T2
=
∅
→
tok
s1
∪
T1
=
tok
s2
∪
T2
→
R
s1
s2
→
tok
s1
∩
T1
≡
∅
→
tok
s2
∩
T2
≡
∅
→
tok
s1
∪
T1
≡
tok
s2
∪
T2
→
step
(
s1
,
T1
)
(
s2
,
T2
).
Hint
Resolve
Step
.
Inductive
frame_step
(
T
:
st
ring
set
)
(
s1
s2
:
A
)
:
Prop
:
=
Inductive
frame_step
(
T
:
li
stset
B
)
(
s1
s2
:
A
)
:
Prop
:
=

Frame_step
T1
T2
:
T1
∩
(
tok
s1
∪
T
)
=
∅
→
step
(
s1
,
T1
)
(
s2
,
T2
)
→
frame_step
T
s1
s2
.
T1
∩
(
tok
s1
∪
T
)
≡
∅
→
step
(
s1
,
T1
)
(
s2
,
T2
)
→
frame_step
T
s1
s2
.
Hint
Resolve
Frame_step
.
Record
closed
(
T
:
st
ring
set
)
(
S
:
set
A
)
:
Prop
:
=
Closed
{
closed_disjoint
s
:
s
∈
S
→
tok
s
∩
T
=
∅
;
Record
closed
(
T
:
li
stset
B
)
(
S
:
set
A
)
:
Prop
:
=
Closed
{
closed_disjoint
s
:
s
∈
S
→
tok
s
∩
T
≡
∅
;
closed_step
s1
s2
:
s1
∈
S
→
frame_step
T
s1
s2
→
s2
∈
S
}.
Lemma
closed_steps
S
T
s1
s2
:
closed
T
S
→
s1
∈
S
→
rtc
(
frame_step
T
)
s1
s2
→
s2
∈
S
.
Proof
.
induction
3
;
eauto
using
closed_step
.
Qed
.
Global
Instance
sts_valid
:
Valid
(
t
R
tok
)
:
=
λ
x
,
match
x
with
auth
s
T
=>
tok
s
∩
T
=
∅

frag
S'
T
=>
closed
T
S'
end
.
Definition
up
(
T
:
st
ring
set
)
(
s
:
A
)
:
set
A
:
=
mkSet
(
rtc
(
frame_step
T
)
s
).
Definition
up_set
(
T
:
st
ring
set
)
(
S
:
set
A
)
:
set
A
:
=
S
≫
=
up
T
.
match
x
with
auth
s
T
=>
tok
s
∩
T
≡
∅

frag
S'
T
=>
closed
T
S'
end
.
Definition
up
(
T
:
li
stset
B
)
(
s
:
A
)
:
set
A
:
=
mkSet
(
rtc
(
frame_step
T
)
s
).
Definition
up_set
(
T
:
li
stset
B
)
(
S
:
set
A
)
:
set
A
:
=
S
≫
=
up
T
.
Global
Instance
sts_unit
:
Unit
(
t
R
tok
)
:
=
λ
x
,
match
x
with

frag
S'
_
=>
frag
(
up_set
∅
S'
)
∅

auth
s
_
=>
frag
(
up
∅
s
)
∅
end
.
Inductive
sts_disjoint
:
Disjoint
(
t
R
tok
)
:
=

frag_frag_disjoint
S1
S2
T1
T2
:
T1
∩
T2
=
∅
→
frag
S1
T1
⊥
frag
S2
T2

auth_frag_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
=
∅
→
auth
s
T1
⊥
frag
S
T2

frag_auth_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
=
∅
→
frag
S
T1
⊥
auth
s
T2
.

frag_frag_disjoint
S1
S2
T1
T2
:
T1
∩
T2
≡
∅
→
frag
S1
T1
⊥
frag
S2
T2

auth_frag_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
≡
∅
→
auth
s
T1
⊥
frag
S
T2

frag_auth_disjoint
s
S
T1
T2
:
s
∈
S
→
T1
∩
T2
≡
∅
→
frag
S
T1
⊥
auth
s
T2
.
Global
Existing
Instance
sts_disjoint
.
Global
Instance
sts_op
:
Op
(
t
R
tok
)
:
=
λ
x1
x2
,
match
x1
,
x2
with
...
...
@@ 68,8 +69,8 @@ Global Instance sts_minus : Minus (t R tok) := λ x1 x2,

auth
s
T1
,
auth
_
T2
=>
frag
(
up
(
T1
∖
T2
)
s
)
(
T1
∖
T2
)
end
.
Hint
Extern
5
(
_
≡
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
5
(
@
eq
stringset
_
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
5
(
equiv
(
A
:
=
set
_
)
_
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
5
(
eq
uiv
(
A
:
=
listset
_
)
_
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
5
(
_
∈
_
)
=>
esolve_elem_of
:
sts
.
Hint
Extern
5
(
_
⊆
_
)
=>
esolve_elem_of
:
sts
.
Instance
:
Equivalence
((
≡
)
:
relation
(
t
R
tok
)).
...
...
@@ 79,14 +80,17 @@ Proof.
*
by
destruct
1
;
constructor
.
*
destruct
1
;
inversion_clear
1
;
constructor
;
etransitivity
;
eauto
.
Qed
.
Instance
closed_proper'
T
:
Proper
((
≡
)
==>
impl
)
(
closed
T
).
Instance
framestep_proper
:
Proper
((
≡
)
==>
(=)
==>
(=)
==>
impl
)
frame_step
.
Proof
.
intros
??
HT
??
<
??
<
;
destruct
1
;
econstructor
;
eauto
with
sts
.
Qed
.
Instance
closed_proper'
:
Proper
((
≡
)
==>
(
≡
)
==>
impl
)
closed
.
Proof
.
intros
??
HS
;
destruct
1
;
constructor
;
intros
until
0
;
rewrite
<
?HS
;
eauto
.
intros
??
HT
??
HS
;
destruct
1
;
constructor
;
intros
until
0
;
rewrite
<
?HS
,
<
?HT
;
eauto
.
Qed
.
Instance
closed_proper
T
:
Proper
((
≡
)
==>
iff
)
(
closed
T
)
.
Proof
.
by
intros
???
;
split
;
apply
closed_proper'
.
Qed
.
Instance
closed_proper
:
Proper
((
≡
)
==>
(
≡
)
==>
iff
)
closed
.
Proof
.
by
split
;
apply
closed_proper'
.
Qed
.
Lemma
closed_op
T1
T2
S1
S2
:
closed
T1
S1
→
closed
T2
S2
→
T1
∩
T2
=
∅
→
closed
(
T1
∪
T2
)
(
S1
∩
S2
).
closed
T1
S1
→
closed
T2
S2
→
T1
∩
T2
≡
∅
→
closed
(
T1
∪
T2
)
(
S1
∩
S2
).
Proof
.
intros
[?
Hstep1
]
[?
Hstep2
]
?
;
split
;
[
esolve_elem_of
].
intros
s3
s4
;
rewrite
!
elem_of_intersection
;
intros
[??]
[
T
??]
;
split
.
...
...
@@ 96,19 +100,21 @@ Qed.
Lemma
closed_all
:
closed
∅
set_all
.
Proof
.
split
;
auto
with
sts
.
Qed
.
Hint
Resolve
closed_all
:
sts
.
Instance
up_preserving
:
Proper
(
flip
(
⊆
)
==>
(=)
==>
(
⊆
))
up
.
Instance
up_preserving
:
Proper
(
flip
(
⊆
)
==>
(=)
==>
(
⊆
))
up
.
Proof
.
intros
T
T'
HT
s
?
<
;
apply
elem_of_subseteq
.
induction
1
as
[
s1
s2
s3
[
T1
T2
]]
;
[
constructor
].
eapply
rtc_l
;
[
eapply
Frame_step
with
T1
T2
]
;
eauto
with
sts
.
Qed
.
Instance
up_set_proper
T
:
Proper
((
≡
)
==>
(
≡
))
(
up_set
T
).
Proof
.
intros
S1
S2
HS
;
unfold
up_set
;
auto
with
sts
.
Qed
.
Instance
up_proper
:
Proper
((
≡
)
==>
(=)
==>
(
≡
))
up
.
Proof
.
by
intros
??
[??]
???
;
split
;
apply
up_preserving
.
Qed
.
Instance
up_set_proper
:
Proper
((
≡
)
==>
(
≡
)
==>
(
≡
))
up_set
.
Proof
.
by
intros
T1
T2
HT
S1
S2
HS
;
unfold
up_set
;
rewrite
HS
,
HT
.
Qed
.
Lemma
elem_of_up
s
T
:
s
∈
up
T
s
.
Proof
.
constructor
.
Qed
.
Lemma
subseteq_up_set
S
T
:
S
⊆
up_set
T
S
.
Proof
.
intros
s
?
;
apply
elem_of_bind
;
eauto
using
elem_of_up
.
Qed
.
Lemma
closed_up_set
S
T
:
(
∀
s
,
s
∈
S
→
tok
s
∩
T
=
∅
)
→
closed
T
(
up_set
T
S
).
Lemma
closed_up_set
S
T
:
(
∀
s
,
s
∈
S
→
tok
s
∩
T
≡
∅
)
→
closed
T
(
up_set
T
S
).
Proof
.
intros
HS
;
unfold
up_set
;
split
.
*
intros
s
;
rewrite
!
elem_of_bind
;
intros
(
s'
&
Hstep
&
Hs'
).
...
...
@@ 120,9 +126,9 @@ Proof.
Qed
.
Lemma
closed_up_set_empty
S
:
closed
∅
(
up_set
∅
S
).
Proof
.
eauto
using
closed_up_set
with
sts
.
Qed
.
Lemma
closed_up
s
T
:
tok
s
∩
T
=
∅
→
closed
T
(
up
T
s
).
Lemma
closed_up
s
T
:
tok
s
∩
T
≡
∅
→
closed
T
(
up
T
s
).
Proof
.
intros
.
rewrite
<(
collection_bind_singleton
_
s
).
intros
;
rewrite
<(
collection_bind_singleton
(
up
T
)
s
).
apply
closed_up_set
;
auto
with
sts
.
Qed
.
Lemma
closed_up_empty
s
:
closed
∅
(
up
∅
s
).
...
...
@@ 149,13 +155,13 @@ Proof.
*
by
do
2
destruct
1
;
constructor
;
setoid_subst
.
*
by
do
2
destruct
1
;
inversion_clear
1
;
econstructor
;
setoid_subst
.
*
assert
(
∀
T
T'
S
s
,
closed
T
S
→
s
∈
S
→
tok
s
∩
T'
=
∅
→
tok
s
∩
(
T
∪
T'
)
=
∅
).
closed
T
S
→
s
∈
S
→
tok
s
∩
T'
≡
∅
→
tok
s
∩
(
T
∪
T'
)
≡
∅
).
{
intros
S
T
T'
s
[??]
;
esolve_elem_of
.
}
destruct
3
;
simpl
in
*
;
auto
using
closed_op
with
sts
.
*
intros
[]
;
simpl
;
eauto
using
closed_up
,
closed_up_set
with
sts
.
*
destruct
3
;
simpl
in
*
;
setoid_subst
;
eauto
using
closed_up
with
sts
.
eapply
closed_up_set
;
eauto
2
using
closed_disjoint
with
sts
.
*
intros
[]
[]
[]
;
constructor
;
rewrite
?(
associative
_L
_
)
;
auto
with
sts
.
*
intros
[]
[]
[]
;
constructor
;
rewrite
?(
associative
_
)
;
auto
with
sts
.
*
destruct
4
;
inversion_clear
1
;
constructor
;
auto
with
sts
.
*
destruct
4
;
inversion_clear
1
;
constructor
;
auto
with
sts
.
*
destruct
1
;
constructor
;
auto
with
sts
.
...
...
@@ 168,21 +174,20 @@ Proof.
+
by
rewrite
(
up_closed
(
up_set
_
_
))
by
auto
using
closed_up_set
with
sts
.
*
destruct
3
as
[
S1
S2

]
;
simpl
;
match
goal
with

_
≼
frag
?S
_
=>
apply
frag_frag_included
with
S
end
;
rewrite
?difference_diag_L
;
eauto
using
closed_up_empty
,
closed_up_set_empty
;
unfold
up_set
;
esolve_elem_of
.
*
destruct
3
as
[
S1
S2
T1
T2

]
;
econstructor
;
eauto
with
sts
.
by
replace
((
T1
∪
T2
)
∖
T1
)
with
T2
by
esolve_elem_of
.
by
setoid_
replace
((
T1
∪
T2
)
∖
T1
)
with
T2
by
esolve_elem_of
.
*
destruct
3
;
constructor
;
eauto
using
elem_of_up
with
sts
.
*
destruct
3
as
[
S1
S2
T1
T2
S'

]
;
constructor
;
rewrite
?(
commutative
_L
_
(
_
∖
_
)),
<
?union_difference
_L
;
auto
with
sts
.
rewrite
?(
commutative
_
(
_
∖
_
)),
<
?union_difference
;
auto
with
sts
.
assert
(
S2
⊆
up_set
(
T2
∖
T1
)
S2
)
by
eauto
using
subseteq_up_set
.
assert
(
up_set
(
T2
∖
T1
)
(
S1
∩
S'
)
⊆
S'
)
by
eauto
using
up_set_subseteq
.
esolve_elem_of
.
Qed
.
Lemma
step_closed
s1
s2
T1
T2
S
Tf
:
step
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
Tf
S
→
s1
∈
S
→
T1
∩
Tf
=
∅
→
s2
∈
S
∧
T2
∩
Tf
=
∅
∧
tok
s2
∩
T2
=
∅
.
step
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
Tf
S
→
s1
∈
S
→
T1
∩
Tf
≡
∅
→
s2
∈
S
∧
T2
∩
Tf
≡
∅
∧
tok
s2
∩
T2
≡
∅
.
Proof
.
inversion_clear
1
as
[????
HR
Hs1
Hs2
]
;
intros
[?
Hstep
]
??
;
split_ands
;
auto
.
*
eapply
Hstep
with
s1
,
Frame_step
with
T1
T2
;
auto
with
sts
.
...
...
@@ 192,7 +197,8 @@ End sts_core.
End
sts
.
Section
sts_ra
.
Context
{
A
}
(
R
:
relation
A
)
(
tok
:
A
→
stringset
).
Context
{
A
B
:
Type
}
`
{
∀
x
y
:
B
,
Decision
(
x
=
y
)}.
Context
(
R
:
relation
A
)
(
tok
:
A
→
listset
B
).
Definition
sts
:
=
validity
(
valid
:
sts
.
t
R
tok
→
Prop
).
Global
Instance
sts_unit
:
Unit
sts
:
=
validity_unit
_
.
...
...
@@ 200,14 +206,14 @@ Global Instance sts_op : Op sts := validity_op _.
Global
Instance
sts_included
:
Included
sts
:
=
validity_included
_
.
Global
Instance
sts_minus
:
Minus
sts
:
=
validity_minus
_
.
Global
Instance
sts_ra
:
RA
sts
:
=
validity_ra
_
.
Definition
sts_auth
(
s
:
A
)
(
T
:
st
ring
set
)
:
sts
:
=
to_validity
(
sts
.
auth
s
T
).
Definition
sts_frag
(
S
:
set
A
)
(
T
:
st
ring
set
)
:
sts
:
=
Definition
sts_auth
(
s
:
A
)
(
T
:
li
stset
B
)
:
sts
:
=
to_validity
(
sts
.
auth
s
T
).
Definition
sts_frag
(
S
:
set
A
)
(
T
:
li
stset
B
)
:
sts
:
=
to_validity
(
sts
.
frag
S
T
).
Lemma
sts_update
s1
s2
T1
T2
:
sts
.
step
R
tok
(
s1
,
T1
)
(
s2
,
T2
)
→
sts_auth
s1
T1
⇝
sts_auth
s2
T2
.
Proof
.
intros
?
;
apply
dra_update
;
inversion
3
as
[?
S
?
Tf
]
;
subst
.
destruct
(
sts
.
step_closed
R
tok
s1
s2
T1
T2
S
Tf
)
as
(?&?&?)
;
auto
.
by
repeat
constructor
.
repeat
(
done

constructor
)
.
Qed
.
End
sts_ra
.
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