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Simon Spies
Iris
Commits
3c70bf99
Commit
3c70bf99
authored
Jan 04, 2017
by
Ralf Jung
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Merge branch 'master' of
https://gitlab.mpi-sws.org/FP/iris-coq
parents
fa25e2cc
991952fe
Changes
3
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3 changed files
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58 additions
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7 deletions
+58
-7
theories/base_logic/lib/invariants.v
theories/base_logic/lib/invariants.v
+27
-0
theories/base_logic/lib/wsat.v
theories/base_logic/lib/wsat.v
+24
-0
theories/prelude/sorting.v
theories/prelude/sorting.v
+7
-7
No files found.
theories/base_logic/lib/invariants.v
View file @
3c70bf99
...
...
@@ -41,6 +41,33 @@ Proof.
-
rewrite
/
uPred_except_0
;
eauto
.
Qed
.
Lemma
inv_alloc_open
N
E
P
:
↑
N
⊆
E
→
True
={
E
,
E
∖↑
N
}=
∗
inv
N
P
∗
(
▷
P
={
E
∖↑
N
,
E
}=
∗
True
).
Proof
.
rewrite
inv_eq
/
inv_def
fupd_eq
/
fupd_def
.
iIntros
(
Sub
)
"[Hw HE]"
.
iMod
(
ownI_alloc_open
(
∈
↑
N
)
P
with
"Hw"
)
as
(
i
)
"(% & Hw & #Hi & HD)"
.
-
intros
Ef
.
exists
(
coPpick
(
↑
N
∖
coPset
.
of_gset
Ef
)).
rewrite
-
coPset
.
elem_of_of_gset
comm
-
elem_of_difference
.
apply
coPpick_elem_of
=>
Hfin
.
eapply
nclose_infinite
,
(
difference_finite_inv
_
_
),
Hfin
.
apply
of_gset_finite
.
-
iAssert
(
ownE
{[
i
]}
∗
ownE
(
↑
N
∖
{[
i
]})
∗
ownE
(
E
∖
↑
N
))%
I
with
"[HE]"
as
"(HEi & HEN\i & HE\N)"
.
{
rewrite
-
?ownE_op
;
[|
set_solver
|
set_solver
].
rewrite
assoc_L
.
rewrite
<-!
union_difference_L
;
try
done
;
set_solver
.
}
iModIntro
.
rewrite
/
uPred_except_0
.
iRight
.
iFrame
.
iSplitL
"Hw HEi"
.
+
by
iApply
"Hw"
.
+
iSplitL
"Hi"
;
[
eauto
|].
iIntros
"HP [Hw HE\N]"
.
iDestruct
(
ownI_close
with
"[$Hw $Hi $HP $HD]"
)
as
"[? HEi]"
.
iModIntro
.
iRight
.
iFrame
.
iSplitL
;
[|
done
].
iCombine
"HEi"
"HEN\i"
as
"HEN"
.
iCombine
"HEN"
"HE\N"
as
"HE"
.
rewrite
-
?ownE_op
;
[|
set_solver
|
set_solver
].
rewrite
<-!
union_difference_L
;
try
done
;
set_solver
.
Qed
.
Lemma
inv_open
E
N
P
:
↑
N
⊆
E
→
inv
N
P
={
E
,
E
∖↑
N
}=
∗
▷
P
∗
(
▷
P
={
E
∖↑
N
,
E
}=
∗
True
).
Proof
.
...
...
theories/base_logic/lib/wsat.v
View file @
3c70bf99
...
...
@@ -142,4 +142,28 @@ Proof.
iApply
(
big_sepM_insert
_
I
)
;
first
done
.
iFrame
"HI"
.
iLeft
.
by
rewrite
/
ownD
;
iFrame
.
Qed
.
Lemma
ownI_alloc_open
φ
P
:
(
∀
E
:
gset
positive
,
∃
i
,
i
∉
E
∧
φ
i
)
→
wsat
==
∗
∃
i
,
⌜φ
i
⌝
∗
(
ownE
{[
i
]}
-
∗
wsat
)
∗
ownI
i
P
∗
ownD
{[
i
]}.
Proof
.
iIntros
(
Hfresh
)
"Hw"
.
iDestruct
"Hw"
as
(
I
)
"[? HI]"
.
iMod
(
own_empty
(
gset_disjUR
positive
)
disabled_name
)
as
"HD"
.
iMod
(
own_updateP
with
"HD"
)
as
"HD"
.
{
apply
(
gset_disj_alloc_empty_updateP_strong'
(
λ
i
,
I
!!
i
=
None
∧
φ
i
)).
intros
E
.
destruct
(
Hfresh
(
E
∪
dom
_
I
))
as
(
i
&
[?
HIi
%
not_elem_of_dom
]%
not_elem_of_union
&
?)
;
eauto
.
}
iDestruct
"HD"
as
(
X
)
"[Hi HD]"
;
iDestruct
"Hi"
as
%(
i
&
->
&
HIi
&
?).
iMod
(
own_update
with
"Hw"
)
as
"[Hw HiP]"
.
{
eapply
auth_update_alloc
,
(
alloc_singleton_local_update
_
i
(
invariant_unfold
P
))
;
last
done
.
by
rewrite
/=
lookup_fmap
HIi
.
}
iModIntro
;
iExists
i
;
iSplit
;
[
done
|].
rewrite
/
ownI
;
iFrame
"HiP"
.
rewrite
-/(
ownD
_
).
iFrame
"HD"
.
iIntros
"HE"
.
iExists
(<[
i
:
=
P
]>
I
)
;
iSplitL
"Hw"
.
{
by
rewrite
fmap_insert
insert_singleton_op
?lookup_fmap
?HIi
.
}
iApply
(
big_sepM_insert
_
I
)
;
first
done
.
iFrame
"HI"
.
by
iRight
.
Qed
.
End
wsat
.
theories/prelude/sorting.v
View file @
3c70bf99
...
...
@@ -114,7 +114,7 @@ End sorted.
(** ** Correctness of merge sort *)
Section
merge_sort_correct
.
Context
{
A
}
(
R
:
relation
A
)
`
{
∀
x
y
,
Decision
(
R
x
y
)}
`
{!
Total
R
}
.
Context
{
A
}
(
R
:
relation
A
)
`
{
∀
x
y
,
Decision
(
R
x
y
)}.
Lemma
list_merge_cons
x1
x2
l1
l2
:
list_merge
R
(
x1
::
l1
)
(
x2
::
l2
)
=
...
...
@@ -127,7 +127,7 @@ Section merge_sort_correct.
destruct
1
as
[|
x1
l1
IH1
],
1
as
[|
x2
l2
IH2
]
;
rewrite
?list_merge_cons
;
simpl
;
repeat
case_decide
;
auto
.
Qed
.
Lemma
Sorted_list_merge
l1
l2
:
Lemma
Sorted_list_merge
`
{!
Total
R
}
l1
l2
:
Sorted
R
l1
→
Sorted
R
l2
→
Sorted
R
(
list_merge
R
l1
l2
).
Proof
.
intros
Hl1
.
revert
l2
.
induction
Hl1
as
[|
x1
l1
IH1
]
;
...
...
@@ -158,7 +158,7 @@ Section merge_sort_correct.
|
Some
l
::
st
=>
l
++
merge_stack_flatten
st
end
.
Lemma
Sorted_merge_list_to_stack
st
l
:
Lemma
Sorted_merge_list_to_stack
`
{!
Total
R
}
st
l
:
merge_stack_Sorted
st
→
Sorted
R
l
→
merge_stack_Sorted
(
merge_list_to_stack
R
st
l
).
Proof
.
...
...
@@ -172,7 +172,7 @@ Section merge_sort_correct.
revert
l
.
induction
st
as
[|[
l'
|]
st
IH
]
;
intros
l
;
simpl
;
auto
.
by
rewrite
IH
,
merge_Permutation
,
(
assoc_L
_
),
(
comm
(++)
l
).
Qed
.
Lemma
Sorted_merge_stack
st
:
Lemma
Sorted_merge_stack
`
{!
Total
R
}
st
:
merge_stack_Sorted
st
→
Sorted
R
(
merge_stack
R
st
).
Proof
.
induction
1
;
simpl
;
auto
using
Sorted_list_merge
.
Qed
.
Lemma
merge_stack_Permutation
st
:
merge_stack
R
st
≡
ₚ
merge_stack_flatten
st
.
...
...
@@ -180,7 +180,7 @@ Section merge_sort_correct.
induction
st
as
[|[]
?
IH
]
;
intros
;
simpl
;
auto
.
by
rewrite
merge_Permutation
,
IH
.
Qed
.
Lemma
Sorted_merge_sort_aux
st
l
:
Lemma
Sorted_merge_sort_aux
`
{!
Total
R
}
st
l
:
merge_stack_Sorted
st
→
Sorted
R
(
merge_sort_aux
R
st
l
).
Proof
.
revert
st
.
induction
l
;
simpl
;
...
...
@@ -194,11 +194,11 @@ Section merge_sort_correct.
-
rewrite
IH
,
merge_list_to_stack_Permutation
;
simpl
.
by
rewrite
Permutation_middle
.
Qed
.
Lemma
Sorted_merge_sort
l
:
Sorted
R
(
merge_sort
R
l
).
Lemma
Sorted_merge_sort
`
{!
Total
R
}
l
:
Sorted
R
(
merge_sort
R
l
).
Proof
.
apply
Sorted_merge_sort_aux
.
by
constructor
.
Qed
.
Lemma
merge_sort_Permutation
l
:
merge_sort
R
l
≡
ₚ
l
.
Proof
.
unfold
merge_sort
.
by
rewrite
merge_sort_aux_Permutation
.
Qed
.
Lemma
StronglySorted_merge_sort
`
{!
Transitive
R
}
l
:
Lemma
StronglySorted_merge_sort
`
{!
Transitive
R
,
!
Total
R
}
l
:
StronglySorted
R
(
merge_sort
R
l
).
Proof
.
auto
using
Sorted_StronglySorted
,
Sorted_merge_sort
.
Qed
.
End
merge_sort_correct
.
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