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764f0e4b
Commit
764f0e4b
authored
Nov 18, 2015
by
Robbert Krebbers
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Big ops on RAs.
parent
dd061dcf
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iris/ra.v
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iris/ra.v
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764f0e4b
Require
Export
prelude
.
collections
prelude
.
relations
.
Require
Export
prelude
.
collections
prelude
.
relations
prelude
.
fin_maps
.
Class
Valid
(
A
:
Type
)
:
=
valid
:
A
→
Prop
.
Instance
:
Params
(@
valid
)
2
.
...
...
@@ 11,6 +11,14 @@ 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
.
...
...
@@ 58,6 +66,7 @@ Instance: Params (@ra_update) 3.
Section
ra
.
Context
`
{
RA
A
}.
Implicit
Types
x
y
z
:
A
.
Implicit
Types
xs
ys
zs
:
list
A
.
Global
Instance
ra_valid_proper'
:
Proper
((
≡
)
==>
iff
)
valid
.
Proof
.
by
intros
???
;
split
;
apply
ra_valid_proper
.
Qed
.
...
...
@@ 120,4 +129,43 @@ Lemma ra_unit_empty x : unit ∅ ≡ ∅.
Proof
.
by
rewrite
<(
ra_unit_l
∅
)
at
2
;
rewrite
(
right_id
_
_
).
Qed
.
Lemma
ra_empty_least
x
:
∅
≼
x
.
Proof
.
by
rewrite
ra_included_spec
;
exists
x
;
rewrite
(
left_id
_
_
).
Qed
.
(** * Big ops *)
Global
Instance
big_op_permutation
:
Proper
((
≡
ₚ
)
==>
(
≡
))
big_op
.
Proof
.
induction
1
as
[
x
xs1
xs2
?
IH

x
y
xs

xs1
xs2
xs3
]
;
simpl
;
auto
.
*
by
rewrite
IH
.
*
by
rewrite
!(
associative
_
),
(
commutative
_
x
).
*
by
transitivity
(
big_op
xs2
).
Qed
.
Global
Instance
big_op_proper
:
Proper
((
≡
)
==>
(
≡
))
big_op
.
Proof
.
by
induction
1
;
simpl
;
repeat
apply
(
_
:
Proper
(
_
==>
_
==>
_
)
op
).
Qed
.
Lemma
big_op_app
xs
ys
:
big_op
(
xs
++
ys
)
≡
big_op
xs
⋅
big_op
ys
.
Proof
.
induction
xs
as
[
x
xs
IH
]
;
simpl
;
[
by
rewrite
?(
left_id
_
_
)].
by
rewrite
IH
,
(
associative
_
).
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
.
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
→
_
).
Proof
.
intros
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
)
as
(
y
&?&
Hxy
)
;
auto
using
lookup_insert
.
by
rewrite
Hxy
,
<
big_opM_insert
,
insert_delete
by
auto
using
lookup_delete
.
Qed
.
End
ra
.
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