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Simon Spies
Iris
Commits
75aed833
Commit
75aed833
authored
Feb 20, 2016
by
Ralf Jung
Browse files
strengthen STS to be able to take any number of steps at once
parent
0c221250
Changes
3
Hide whitespace changes
Inline
Side-by-side
algebra/sts.v
View file @
75aed833
...
...
@@ -29,6 +29,7 @@ Inductive step : relation (state sts * tokens sts) :=
(* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *)
prim_step
s1
s2
→
tok
s1
∩
T1
≡
∅
→
tok
s2
∩
T2
≡
∅
→
tok
s1
∪
T1
≡
tok
s2
∪
T2
→
step
(
s1
,
T1
)
(
s2
,
T2
).
Definition
steps
:
=
rtc
step
.
Inductive
frame_step
(
T
:
tokens
sts
)
(
s1
s2
:
state
sts
)
:
Prop
:
=
|
Frame_step
T1
T2
:
T1
∩
(
tok
s1
∪
T
)
≡
∅
→
step
(
s1
,
T1
)
(
s2
,
T2
)
→
frame_step
T
s1
s2
.
...
...
@@ -105,6 +106,16 @@ Proof.
-
eapply
Hstep
with
s1
,
Frame_step
with
T1
T2
;
auto
with
sts
.
-
set_solver
-
Hstep
Hs1
Hs2
.
Qed
.
Lemma
steps_closed
s1
s2
T1
T2
S
Tf
:
steps
(
s1
,
T1
)
(
s2
,
T2
)
→
closed
S
Tf
→
s1
∈
S
→
T1
∩
Tf
≡
∅
→
tok
s1
∩
T1
≡
∅
→
s2
∈
S
∧
T2
∩
Tf
≡
∅
∧
tok
s2
∩
T2
≡
∅
.
Proof
.
remember
(
s1
,
T1
)
as
sT1
.
remember
(
s2
,
T2
)
as
sT2
.
intros
Hsteps
.
revert
s1
T1
HeqsT1
s2
T2
HeqsT2
.
induction
Hsteps
as
[?|?
[
s'
T'
]
?
Hstep
Hsteps
IH
]
;
intros
;
subst
.
-
case
:
HeqsT2
=>?
?.
subst
.
done
.
-
eapply
step_closed
in
Hstep
;
[|
done
..].
destruct_conjs
.
eauto
.
Qed
.
(** ** Properties of the closure operators *)
Lemma
elem_of_up
s
T
:
s
∈
up
s
T
.
...
...
@@ -326,11 +337,22 @@ Lemma sts_op_auth_frag s S T :
Proof
.
intros
;
split
;
[
split
|
constructor
;
set_solver
]
;
simpl
.
-
intros
(?&?&?)
;
by
apply
closed_disjoint'
with
S
.
-
intros
;
split_and
?.
set_solver
+.
done
.
constructor
;
set_solver
.
-
intros
;
split_and
?.
+
set_solver
+.
+
done
.
+
constructor
;
set_solver
.
Qed
.
Lemma
sts_op_auth_frag_up
s
T
:
tok
s
∩
T
≡
∅
→
sts_auth
s
∅
⋅
sts_frag_up
s
T
≡
sts_auth
s
T
.
Proof
.
intros
;
apply
sts_op_auth_frag
;
auto
using
elem_of_up
,
closed_up
.
Qed
.
sts_auth
s
∅
⋅
sts_frag_up
s
T
≡
sts_auth
s
T
.
Proof
.
intros
;
split
;
[
split
|
constructor
;
set_solver
]
;
simpl
.
-
intros
(?&?&?).
apply
closed_disjoint'
with
(
up
s
T
)
;
first
done
.
apply
elem_of_up
.
-
intros
;
split_and
?.
+
set_solver
+.
+
by
apply
closed_up
.
+
constructor
;
last
set_solver
.
apply
elem_of_up
.
Qed
.
Lemma
sts_op_frag
S1
S2
T1
T2
:
T1
∩
T2
⊆
∅
→
sts
.
closed
S1
T1
→
sts
.
closed
S2
T2
→
...
...
@@ -344,10 +366,10 @@ Qed.
(** Frame preserving updates *)
Lemma
sts_update_auth
s1
s2
T1
T2
:
step
(
s1
,
T1
)
(
s2
,
T2
)
→
sts_auth
s1
T1
~~>
sts_auth
s2
T2
.
step
s
(
s1
,
T1
)
(
s2
,
T2
)
→
sts_auth
s1
T1
~~>
sts_auth
s2
T2
.
Proof
.
intros
?
;
apply
validity_update
;
inversion
3
as
[|?
S
?
Tf
|]
;
subst
.
destruct
(
step_closed
s1
s2
T1
T2
S
Tf
)
as
(?&?&?)
;
auto
.
destruct
(
step
s
_closed
s1
s2
T1
T2
S
Tf
)
as
(?&?&?)
;
auto
;
[]
.
repeat
(
done
||
constructor
).
Qed
.
...
...
algebra/upred.v
View file @
75aed833
...
...
@@ -983,9 +983,9 @@ Lemma always_entails_r P Q `{!AlwaysStable Q} : (P ⊑ Q) → P ⊑ (P ★ Q).
Proof
.
by
rewrite
-(
always_always
Q
)
;
apply
always_entails_r'
.
Qed
.
(* Derived lemmas that need a combination of the above *)
Lemma
l
ö
b_strong_sep
P
Q
:
(
▷
(
P
-
★
Q
)
★
P
)
⊑
Q
→
P
⊑
Q
.
Lemma
l
ö
b_strong_sep
P
Q
:
(
P
★
▷
(
P
-
★
Q
))
⊑
Q
→
P
⊑
Q
.
Proof
.
move
/
wand_intro_
r
=>
Hl
ö
b
.
rewrite
-[
P
](
left_id
True
(
∧
))%
I
.
move
/
wand_intro_
l
=>
Hl
ö
b
.
rewrite
-[
P
](
left_id
True
(
∧
))%
I
.
apply
impl_elim_l'
.
apply
:
always_entails
.
apply
l
ö
b_strong
.
rewrite
left_id
-
always_wand_impl
-
always_later
Hl
ö
b
.
done
.
Qed
.
...
...
program_logic/sts.v
View file @
75aed833
...
...
@@ -103,7 +103,7 @@ Section sts.
Qed
.
Lemma
sts_closing
E
γ
s
T
s'
T'
:
sts
.
step
(
s
,
T
)
(
s'
,
T'
)
→
sts
.
step
s
(
s
,
T
)
(
s'
,
T'
)
→
(
▷
φ
s'
★
own
γ
(
sts_auth
s
T
))
⊑
(|={
E
}=>
▷
sts_inv
γ
φ
★
sts_own
γ
s'
T'
).
Proof
.
intros
Hstep
.
rewrite
/
sts_inv
/
sts_own
-(
exist_intro
s'
).
...
...
@@ -112,7 +112,7 @@ Section sts.
rewrite
own_valid_l
discrete_validI
.
apply
const_elim_sep_l
=>
Hval
.
transitivity
(|={
E
}=>
own
γ
(
sts_auth
s'
T'
))%
I
.
{
by
apply
own_update
,
sts_update_auth
.
}
by
rewrite
-
own_op
sts_op_auth_frag_up
;
last
by
inversion_clear
Hstep
.
by
rewrite
-
own_op
sts_op_auth_frag_up
.
Qed
.
Context
{
V
}
(
fsa
:
FSA
Λ
(
globalF
Σ
)
V
)
`
{!
FrameShiftAssertion
fsaV
fsa
}.
...
...
@@ -123,7 +123,7 @@ Section sts.
P
⊑
(
sts_ownS
γ
S
T
★
∀
s
,
■
(
s
∈
S
)
★
▷
φ
s
-
★
fsa
(
E
∖
nclose
N
)
(
λ
x
,
∃
s'
T'
,
■
sts
.
step
(
s
,
T
)
(
s'
,
T'
)
★
▷
φ
s'
★
■
sts
.
step
s
(
s
,
T
)
(
s'
,
T'
)
★
▷
φ
s'
★
(
sts_own
γ
s'
T'
-
★
Ψ
x
)))
→
P
⊑
fsa
E
Ψ
.
Proof
.
...
...
@@ -152,7 +152,7 @@ Section sts.
P
⊑
(
sts_own
γ
s0
T
★
∀
s
,
■
(
s
∈
sts
.
up
s0
T
)
★
▷
φ
s
-
★
fsa
(
E
∖
nclose
N
)
(
λ
x
,
∃
s'
T'
,
■
(
sts
.
step
(
s
,
T
)
(
s'
,
T'
))
★
▷
φ
s'
★
■
(
sts
.
step
s
(
s
,
T
)
(
s'
,
T'
))
★
▷
φ
s'
★
(
sts_own
γ
s'
T'
-
★
Ψ
x
)))
→
P
⊑
fsa
E
Ψ
.
Proof
.
apply
sts_fsaS
.
Qed
.
...
...
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