Skip to content
GitLab
Projects
Groups
Snippets
/
Help
Help
Support
Community forum
Keyboard shortcuts
?
Submit feedback
Contribute to GitLab
Sign in / Register
Toggle navigation
Menu
Open sidebar
Simon Spies
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
Changes
1
Hide whitespace changes
Inline
Sidebyside
iris/sts.v
View file @
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
.
Write
Preview
Supports
Markdown
0%
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Cancel
Please
register
or
sign in
to comment