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Jeehoon Kang
iriscoq
Commits
e760dfb5
Commit
e760dfb5
authored
Nov 19, 2015
by
Robbert Krebbers
Browse files
More general RA bigop for finite maps.
parent
1d628d4b
Changes
1
Hide whitespace changes
Inline
Sidebyside
iris/ra.v
View file @
e760dfb5
...
...
@@ 11,14 +11,6 @@ Instance: Params (@op) 2.
Infix
"⋅"
:=
op
(
at
level
50
,
left
associativity
)
:
C_scope
.
Notation
"(⋅)"
:=
op
(
only
parsing
)
:
C_scope
.
Fixpoint
big_op
`
{
Op
A
,
Empty
A
}
(
xs
:
list
A
)
:
A
:=
match
xs
with
[]
=>
∅

x
::
xs
=>
x
⋅
big_op
xs
end
.
Arguments
big_op
_
_
_
!
_
/
.
Instance:
Params
(
@
big_op
)
3.
Definition
big_opM
`
{
FinMapToList
K
A
M
,
Op
A
,
Empty
A
}
(
m
:
M
)
:
A
:=
big_op
(
snd
<
$
>
map_to_list
m
).
Instance:
Params
(
@
big_opM
)
4.
Class
Included
(
A
:
Type
)
:=
included
:
relation
A
.
Instance:
Params
(
@
included
)
2.
Infix
"≼"
:=
included
(
at
level
70
)
:
C_scope
.
...
...
@@ 52,6 +44,16 @@ Class RAEmpty A `{Equiv A, Valid A, Op A, Empty A} : Prop := {
ra_empty_l
:>
LeftId
(
≡
)
∅
(
⋅
)
}
.
(
**
Big
ops
*
)
Fixpoint
big_op
`
{
Op
A
,
Empty
A
}
(
xs
:
list
A
)
:
A
:=
match
xs
with
[]
=>
∅

x
::
xs
=>
x
⋅
big_op
xs
end
.
Arguments
big_op
_
_
_
!
_
/
.
Instance:
Params
(
@
big_op
)
3.
Definition
big_opM
`
{
FinMapToList
K
A
M
,
Op
B
,
Empty
B
}
(
f
:
K
→
A
→
list
B
)
(
m
:
M
)
:
B
:=
big_op
(
map_to_list
m
≫
=
curry
f
).
Instance:
Params
(
@
big_opM
)
4.
(
**
Updates
*
)
Definition
ra_update_set
`
{
Op
A
,
Valid
A
}
(
x
:
A
)
(
P
:
A
→
Prop
)
:=
∀
z
,
valid
(
x
⋅
z
)
→
∃
y
,
P
y
∧
valid
(
y
⋅
z
).
...
...
@@ 147,21 +149,26 @@ Proof.
Qed
.
Context
`
{
FinMap
K
M
}
.
Lemma
big_opM_empty
:
big_opM
(
∅
:
M
A
)
≡
∅
.
Proof
.
unfold
big_opM
.
by
rewrite
map_to_list_empty
.
Qed
.
Lemma
big_opM_insert
(
m
:
M
A
)
i
x
:
m
!!
i
=
None
→
big_opM
(
<
[
i
:=
x
]
>
m
)
≡
x
⋅
big_opM
m
.
Proof
.
intros
?
;
unfold
big_opM
.
by
rewrite
map_to_list_insert
by
done
.
Qed
.
Lemma
big_opM_singleton
i
x
:
big_opM
(
{
[
i
,
x
]
}
:
M
A
)
≡
x
.
Context
`
{
Equiv
B
}
`
{!
Equivalence
((
≡
)
:
relation
B
)
}
(
f
:
K
→
B
→
list
A
).
Lemma
big_opM_empty
:
big_opM
f
(
∅
:
M
B
)
≡
∅
.
Proof
.
by
unfold
big_opM
;
rewrite
map_to_list_empty
.
Qed
.
Lemma
big_opM_insert
(
m
:
M
B
)
i
(
y
:
B
)
:
m
!!
i
=
None
→
big_opM
f
(
<
[
i
:=
y
]
>
m
)
≡
big_op
(
f
i
y
)
⋅
big_opM
f
m
.
Proof
.
intros
?
;
unfold
big_opM
.
by
rewrite
map_to_list_insert
,
bind_cons
,
big_op_app
by
done
.
Qed
.
Lemma
big_opM_singleton
i
(
y
:
B
)
:
big_opM
f
(
{
[
i
,
y
]
}
:
M
B
)
≡
big_op
(
f
i
y
).
Proof
.
unfold
singleton
,
map_singleton
.
rewrite
big_opM_insert
by
auto
using
lookup_empty
;
simpl
.
by
rewrite
big_opM_empty
,
(
right_id
_
_
).
Qed
.
Global
Instance
big_opM_proper
:
Proper
((
≡
)
==>
(
≡
))
(
big_opM
:
M
A
→
_
).
Global
Instance
big_opM_proper
:
(
∀
i
,
Proper
((
≡
)
==>
(
≡
))
(
f
i
))
→
Proper
((
≡
)
==>
(
≡
))
(
big_opM
f
:
M
B
→
A
).
Proof
.
intros
m1
;
induction
m1
as
[

i
x
m1
?
IH
]
using
map_ind
.
{
by
intros
m2
;
rewrite
(
symmetry_iff
(
≡
)),
map_equiv_empty
;
intros
>
.
}
intros
Hf
m1
;
induction
m1
as
[

i
x
m1
?
IH
]
using
map_ind
.
{
by
intros
m2
;
rewrite
(
symmetry_iff
(
≡
)
∅
),
map_equiv_empty
;
intros
>
.
}
intros
m2
Hm2
;
rewrite
big_opM_insert
by
done
.
rewrite
(
IH
(
delete
i
m2
))
by
(
by
rewrite
<
Hm2
,
delete_insert
).
destruct
(
map_equiv_lookup
(
<
[
i
:=
x
]
>
m1
)
m2
i
x
)
...
...
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