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Tej Chajed
iris
Commits
4b897d12
Commit
4b897d12
authored
Nov 01, 2018
by
Dan Frumin
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Add `big_opM_union` and `big_sepM_union`.
parent
8004a1a6
Changes
2
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18 additions
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+18
-0
theories/algebra/big_op.v
theories/algebra/big_op.v
+12
-0
theories/bi/big_op.v
theories/bi/big_op.v
+6
-0
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theories/algebra/big_op.v
View file @
4b897d12
...
...
@@ -241,6 +241,18 @@ Section gmap.
([^
o
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
<[
i
:
=
P
]>
f
k
)
≡
(
P
`
o
`
[^
o
map
]
k
↦
y
∈
m
,
f
k
).
Proof
.
apply
(
big_opM_fn_insert
(
λ
_
_
,
id
)).
Qed
.
Lemma
big_opM_union
f
m1
m2
:
m1
##
ₘ
m2
→
([^
o
map
]
k
↦
y
∈
m1
∪
m2
,
f
k
y
)
≡
([^
o
map
]
k
↦
y
∈
m1
,
f
k
y
)
`
o
`
([^
o
map
]
k
↦
y
∈
m2
,
f
k
y
).
Proof
.
intros
.
induction
m1
as
[|
i
x
m
?
IH
]
using
map_ind
.
{
by
rewrite
big_opM_empty
!
left_id
.
}
decompose_map_disjoint
.
rewrite
-
insert_union_l
!
big_opM_insert
//
;
last
by
apply
lookup_union_None
.
rewrite
-
assoc
IH
//.
Qed
.
Lemma
big_opM_opM
f
g
m
:
([^
o
map
]
k
↦
x
∈
m
,
f
k
x
`
o
`
g
k
x
)
≡
([^
o
map
]
k
↦
x
∈
m
,
f
k
x
)
`
o
`
([^
o
map
]
k
↦
x
∈
m
,
g
k
x
).
...
...
theories/bi/big_op.v
View file @
4b897d12
...
...
@@ -626,6 +626,12 @@ Section gmap.
([
∗
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
<[
i
:
=
P
]>
Φ
k
)
⊣
⊢
(
P
∗
[
∗
map
]
k
↦
y
∈
m
,
Φ
k
).
Proof
.
apply
big_opM_fn_insert'
.
Qed
.
Lemma
big_sepM_union
Φ
m1
m2
:
m1
##
ₘ
m2
→
([
∗
map
]
k
↦
y
∈
m1
∪
m2
,
Φ
k
y
)
⊣
⊢
([
∗
map
]
k
↦
y
∈
m1
,
Φ
k
y
)
∗
([
∗
map
]
k
↦
y
∈
m2
,
Φ
k
y
).
Proof
.
apply
big_opM_union
.
Qed
.
Lemma
big_sepM_sepM
Φ
Ψ
m
:
([
∗
map
]
k
↦
x
∈
m
,
Φ
k
x
∗
Ψ
k
x
)
⊣
⊢
([
∗
map
]
k
↦
x
∈
m
,
Φ
k
x
)
∗
([
∗
map
]
k
↦
x
∈
m
,
Ψ
k
x
).
...
...
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