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PierreMarie Pédrot
Iris
Commits
562b2c2b
Commit
562b2c2b
authored
Sep 22, 2016
by
Robbert Krebbers
Browse files
Reorganize big ops for CMRA based on those for uPred.
parent
d1ef32dd
Changes
2
Hide whitespace changes
Inline
Sidebyside
algebra/cmra.v
View file @
562b2c2b
...
...
@@ 244,7 +244,7 @@ Global Instance cmra_pcore_proper' : Proper ((≡) ==> (≡)) (@pcore A _).
Proof
.
apply
(
ne_proper
_
).
Qed
.
Global
Instance
cmra_op_ne'
n
:
Proper
(
dist
n
==>
dist
n
==>
dist
n
)
(@
op
A
_
).
Proof
.
intros
x1
x2
Hx
y1
y2
Hy
.
by
rewrite
Hy
(
comm
_
x1
)
Hx
(
comm
_
y2
).
Qed
.
Global
Instance
ra_op_proper'
:
Proper
((
≡
)
==>
(
≡
)
==>
(
≡
))
(@
op
A
_
).
Global
Instance
cm
ra_op_proper'
:
Proper
((
≡
)
==>
(
≡
)
==>
(
≡
))
(@
op
A
_
).
Proof
.
apply
(
ne_proper_2
_
).
Qed
.
Global
Instance
cmra_validN_ne'
:
Proper
(
dist
n
==>
iff
)
(@
validN
A
_
n
)

1
.
Proof
.
by
split
;
apply
cmra_validN_ne
.
Qed
.
...
...
algebra/cmra_big_op.v
View file @
562b2c2b
From
iris
.
algebra
Require
Export
cmra
list
.
From
iris
.
prelude
Require
Import
gmap
.
From
iris
.
prelude
Require
Import
functions
gmap
.
Fixpoint
big_op
{
A
:
ucmraT
}
(
xs
:
list
A
)
:
A
:
=
(** The operator [ [⋅] Ps ] folds [⋅] over the list [Ps]. This operator is not a
quantifier, so it binds strongly.
Apart from that, we define the following big operators with binders build in:
 The operator [ [⋅ list] k ↦ x ∈ l, P ] folds over a list [l]. The binder [x]
refers to each element at index [k].
 The operator [ [⋅ map] k ↦ x ∈ m, P ] folds over a map [m]. The binder [x]
refers to each element at index [k].
 The operator [ [⋅ set] x ∈ X, P ] folds over a set [m]. The binder [x] refers
to each element.
Since these big operators are like quantifiers, they have the same precedence as
[∀] and [∃]. *)
(** * Big ops over lists *)
(* This is the basic building block for other big ops *)
Fixpoint
big_op
{
M
:
ucmraT
}
(
xs
:
list
M
)
:
M
:
=
match
xs
with
[]
=>
∅

x
::
xs
=>
x
⋅
big_op
xs
end
.
Arguments
big_op
_
!
_
/.
Instance
:
Params
(@
big_op
)
1
.
Definition
big_opM
`
{
Countable
K
}
{
A
:
ucmraT
}
(
m
:
gmap
K
A
)
:
A
:
=
big_op
(
snd
<$>
map_to_list
m
).
Instance
:
Params
(@
big_opM
)
4
.
Notation
"'[⋅]' xs"
:
=
(
big_op
xs
)
(
at
level
20
)
:
C_scope
.
(** * Other big ops *)
Definition
big_opL
{
M
:
ucmraT
}
{
A
}
(
l
:
list
A
)
(
f
:
nat
→
A
→
M
)
:
M
:
=
[
⋅
]
(
imap
f
l
).
Instance
:
Params
(@
big_opL
)
2
.
Typeclasses
Opaque
big_opL
.
Notation
"'[⋅' 'list' ] k ↦ x ∈ l , P"
:
=
(
big_opL
l
(
λ
k
x
,
P
))
(
at
level
200
,
l
at
level
10
,
k
,
x
at
level
1
,
right
associativity
,
format
"[⋅ list ] k ↦ x ∈ l , P"
)
:
C_scope
.
Notation
"'[⋅' 'list' ] x ∈ l , P"
:
=
(
big_opL
l
(
λ
_
x
,
P
))
(
at
level
200
,
l
at
level
10
,
x
at
level
1
,
right
associativity
,
format
"[⋅ list ] x ∈ l , P"
)
:
C_scope
.
Definition
big_opM
{
M
:
ucmraT
}
`
{
Countable
K
}
{
A
}
(
m
:
gmap
K
A
)
(
f
:
K
→
A
→
M
)
:
M
:
=
[
⋅
]
(
curry
f
<$>
map_to_list
m
).
Instance
:
Params
(@
big_opM
)
6
.
Typeclasses
Opaque
big_opM
.
Notation
"'[⋅' 'map' ] k ↦ x ∈ m , P"
:
=
(
big_opM
m
(
λ
k
x
,
P
))
(
at
level
200
,
m
at
level
10
,
k
,
x
at
level
1
,
right
associativity
,
format
"[⋅ map ] k ↦ x ∈ m , P"
)
:
C_scope
.
Definition
big_opS
{
M
:
ucmraT
}
`
{
Countable
A
}
(
X
:
gset
A
)
(
f
:
A
→
M
)
:
M
:
=
[
⋅
]
(
f
<$>
elements
X
).
Instance
:
Params
(@
big_opS
)
5
.
Typeclasses
Opaque
big_opS
.
Notation
"'[⋅' 'set' ] x ∈ X , P"
:
=
(
big_opS
X
(
λ
x
,
P
))
(
at
level
200
,
X
at
level
10
,
x
at
level
1
,
right
associativity
,
format
"[⋅ set ] x ∈ X , P"
)
:
C_scope
.
(** * Properties about big ops *)
Section
big_op
.
Context
{
A
:
ucmraT
}.
Implicit
Types
xs
:
list
A
.
Context
{
M
:
ucmraT
}.
Implicit
Types
xs
:
list
M
.
(** * Big ops *)
Lemma
big_op_nil
:
big_op
(@
nil
A
)
=
∅
.
Global
Instance
big_op_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
big_op
M
).
Proof
.
by
induction
1
;
simpl
;
repeat
apply
(
_
:
Proper
(
_
==>
_
==>
_
)
op
).
Qed
.
Global
Instance
big_op_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
big_op
M
)
:
=
ne_proper
_
.
Lemma
big_op_nil
:
[
⋅
]
(@
nil
M
)
=
∅
.
Proof
.
done
.
Qed
.
Lemma
big_op_cons
x
xs
:
big_op
(
x
::
xs
)
=
x
⋅
big_op
xs
.
Lemma
big_op_cons
x
xs
:
[
⋅
]
(
x
::
xs
)
=
x
⋅
[
⋅
]
xs
.
Proof
.
done
.
Qed
.
Global
Instance
big_op_permutation
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
big_op
A
).
Lemma
big_op_app
xs
ys
:
[
⋅
]
(
xs
++
ys
)
≡
[
⋅
]
xs
⋅
[
⋅
]
ys
.
Proof
.
induction
xs
as
[
x
xs
IH
]
;
simpl
;
first
by
rewrite
?left_id
.
by
rewrite
IH
assoc
.
Qed
.
Lemma
big_op_mono
xs
ys
:
Forall2
(
≼
)
xs
ys
→
[
⋅
]
xs
≼
[
⋅
]
ys
.
Proof
.
induction
1
as
[
x
y
xs
ys
Hxy
?
IH
]
;
simpl
;
eauto
using
cmra_mono
.
Qed
.
Global
Instance
big_op_permutation
:
Proper
((
≡
ₚ
)
==>
(
≡
))
(@
big_op
M
).
Proof
.
induction
1
as
[
x
xs1
xs2
?
IH

x
y
xs

xs1
xs2
xs3
]
;
simpl
;
auto
.

by
rewrite
IH
.

by
rewrite
!
assoc
(
comm
_
x
).

by
trans
(
big_op
xs2
).
Qed
.
Global
Instance
big_op_ne
n
:
Proper
(
dist
n
==>
dist
n
)
(@
big_op
A
).
Proof
.
by
induction
1
;
simpl
;
repeat
apply
(
_
:
Proper
(
_
==>
_
==>
_
)
op
).
Qed
.
Global
Instance
big_op_proper
:
Proper
((
≡
)
==>
(
≡
))
big_op
:
=
ne_proper
_
.
Lemma
big_op_app
xs
ys
:
big_op
(
xs
++
ys
)
≡
big_op
xs
⋅
big_op
ys
.
Proof
.
induction
xs
as
[
x
xs
IH
]
;
simpl
;
first
by
rewrite
?left_id
.
by
rewrite
IH
assoc
.
Qed
.
Lemma
big_op_contains
xs
ys
:
xs
`
contains
`
ys
→
big_op
xs
≼
big_op
ys
.
Lemma
big_op_contains
xs
ys
:
xs
`
contains
`
ys
→
[
⋅
]
xs
≼
[
⋅
]
ys
.
Proof
.
intros
[
xs'
>]%
contains_Permutation
.
rewrite
big_op_app
;
apply
cmra_included_l
.
Qed
.
Lemma
big_op_delete
xs
i
x
:
xs
!!
i
=
Some
x
→
x
⋅
big_op
(
delete
i
xs
)
≡
big_op
xs
.
Lemma
big_op_delete
xs
i
x
:
xs
!!
i
=
Some
x
→
x
⋅
[
⋅
]
delete
i
xs
≡
[
⋅
]
xs
.
Proof
.
by
intros
;
rewrite
{
2
}(
delete_Permutation
xs
i
x
).
Qed
.
Context
`
{
Countable
K
}.
Implicit
Types
m
:
gmap
K
A
.
Lemma
big_opM_empty
:
big_opM
(
∅
:
gmap
K
A
)
≡
∅
.
Proof
.
unfold
big_opM
.
by
rewrite
map_to_list_empty
.
Qed
.
Lemma
big_opM_insert
m
i
x
:
m
!!
i
=
None
→
big_opM
(<[
i
:
=
x
]>
m
)
≡
x
⋅
big_opM
m
.
Proof
.
intros
?
;
by
rewrite
/
big_opM
map_to_list_insert
.
Qed
.
Lemma
big_opM_delete
m
i
x
:
m
!!
i
=
Some
x
→
x
⋅
big_opM
(
delete
i
m
)
≡
big_opM
m
.
Proof
.
intros
.
rewrite
{
2
}(
insert_id
m
i
x
)
//

insert_delete
.
by
rewrite
big_opM_insert
?lookup_delete
.
Qed
.
Lemma
big_opM_singleton
i
x
:
big_opM
({[
i
:
=
x
]}
:
gmap
K
A
)
≡
x
.
Lemma
big_sep_elem_of
xs
x
:
x
∈
xs
→
x
≼
[
⋅
]
xs
.
Proof
.
rewrite

insert_empty
big_opM_insert
/=
;
last
auto
using
lookup_empty
.
by
rewrite
big_opM_empty
right_id
.
Qed
.
Global
Instance
big_opM_proper
:
Proper
((
≡
)
==>
(
≡
))
(
big_opM
:
gmap
K
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
//.
rewrite
(
IH
(
delete
i
m2
))
;
last
by
rewrite

Hm2
delete_insert
.
destruct
(
map_equiv_lookup_l
(<[
i
:
=
x
]>
m1
)
m2
i
x
)
as
(
y
&?&
Hxy
)
;
auto
using
lookup_insert
.
rewrite
Hxy

big_opM_insert
;
last
auto
using
lookup_delete
.
by
rewrite
insert_delete
insert_id
.
Qed
.
Lemma
big_opM_lookup_valid
n
m
i
x
:
✓
{
n
}
big_opM
m
→
m
!!
i
=
Some
x
→
✓
{
n
}
x
.
Proof
.
intros
Hm
?
;
revert
Hm
;
rewrite
(
big_opM_delete
_
i
x
)
//.
apply
cmra_validN_op_l
.
intros
[
i
?]%
elem_of_list_lookup
.
rewrite

big_op_delete
//.
apply
cmra_included_l
.
Qed
.
(** ** Big ops over lists *)
Section
list
.
Context
{
A
:
Type
}.
Implicit
Types
l
:
list
A
.
Implicit
Types
f
g
:
nat
→
A
→
M
.
Lemma
big_opL_mono
f
g
l
:
(
∀
k
y
,
l
!!
k
=
Some
y
→
f
k
y
≼
g
k
y
)
→
([
⋅
list
]
k
↦
y
∈
l
,
f
k
y
)
≼
[
⋅
list
]
k
↦
y
∈
l
,
g
k
y
.
Proof
.
intros
Hf
.
apply
big_op_mono
.
revert
f
g
Hf
.
induction
l
as
[
x
l
IH
]=>
f
g
Hf
;
first
constructor
.
rewrite
!
imap_cons
;
constructor
;
eauto
.
Qed
.
Lemma
big_opL_proper
f
g
l
:
(
∀
k
y
,
l
!!
k
=
Some
y
→
f
k
y
≡
g
k
y
)
→
([
⋅
list
]
k
↦
y
∈
l
,
f
k
y
)
≡
([
⋅
list
]
k
↦
y
∈
l
,
g
k
y
).
Proof
.
intros
Hf
;
apply
big_op_proper
.
revert
f
g
Hf
.
induction
l
as
[
x
l
IH
]=>
f
g
Hf
;
first
constructor
.
rewrite
!
imap_cons
;
constructor
;
eauto
.
Qed
.
Global
Instance
big_opL_ne
l
n
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
dist
n
))
==>
(
dist
n
))
(
big_opL
(
M
:
=
M
)
l
).
Proof
.
intros
f
g
Hf
.
apply
big_op_ne
.
revert
f
g
Hf
.
induction
l
as
[
x
l
IH
]=>
f
g
Hf
;
first
constructor
.
rewrite
!
imap_cons
;
constructor
.
by
apply
Hf
.
apply
IH
=>
n'
;
apply
Hf
.
Qed
.
Global
Instance
big_opL_proper'
l
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
≡
))
==>
(
≡
))
(
big_opL
(
M
:
=
M
)
l
).
Proof
.
intros
f1
f2
Hf
.
by
apply
big_opL_proper
;
intros
;
last
apply
Hf
.
Qed
.
Global
Instance
big_opL_mono'
l
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
≼
))
==>
(
≼
))
(
big_opL
(
M
:
=
M
)
l
).
Proof
.
intros
f1
f2
Hf
.
by
apply
big_opL_mono
;
intros
;
last
apply
Hf
.
Qed
.
Lemma
big_opL_nil
f
:
([
⋅
list
]
k
↦
y
∈
nil
,
f
k
y
)
=
∅
.
Proof
.
done
.
Qed
.
Lemma
big_opL_cons
f
x
l
:
([
⋅
list
]
k
↦
y
∈
x
::
l
,
f
k
y
)
=
f
0
x
⋅
[
⋅
list
]
k
↦
y
∈
l
,
f
(
S
k
)
y
.
Proof
.
by
rewrite
/
big_opL
imap_cons
.
Qed
.
Lemma
big_opL_singleton
f
x
:
([
⋅
list
]
k
↦
y
∈
[
x
],
f
k
y
)
≡
f
0
x
.
Proof
.
by
rewrite
big_opL_cons
big_opL_nil
right_id
.
Qed
.
Lemma
big_opL_app
f
l1
l2
:
([
⋅
list
]
k
↦
y
∈
l1
++
l2
,
f
k
y
)
≡
([
⋅
list
]
k
↦
y
∈
l1
,
f
k
y
)
⋅
([
⋅
list
]
k
↦
y
∈
l2
,
f
(
length
l1
+
k
)
y
).
Proof
.
by
rewrite
/
big_opL
imap_app
big_op_app
.
Qed
.
Lemma
big_opL_lookup
f
l
i
x
:
l
!!
i
=
Some
x
→
f
i
x
≼
[
⋅
list
]
k
↦
y
∈
l
,
f
k
y
.
Proof
.
intros
.
rewrite
(
take_drop_middle
l
i
x
)
//
big_opL_app
big_opL_cons
.
rewrite
Nat
.
add_0_r
take_length_le
;
eauto
using
lookup_lt_Some
,
Nat
.
lt_le_incl
.
eapply
transitivity
,
cmra_included_r
;
eauto
using
cmra_included_l
.
Qed
.
Lemma
big_opL_elem_of
(
f
:
A
→
M
)
l
x
:
x
∈
l
→
f
x
≼
[
⋅
list
]
y
∈
l
,
f
y
.
Proof
.
intros
[
i
?]%
elem_of_list_lookup
;
eauto
using
(
big_opL_lookup
(
λ
_
,
f
)).
Qed
.
Lemma
big_opL_fmap
{
B
}
(
h
:
A
→
B
)
(
f
:
nat
→
B
→
M
)
l
:
([
⋅
list
]
k
↦
y
∈
h
<$>
l
,
f
k
y
)
≡
([
⋅
list
]
k
↦
y
∈
l
,
f
k
(
h
y
)).
Proof
.
by
rewrite
/
big_opL
imap_fmap
.
Qed
.
Lemma
big_opL_opL
f
g
l
:
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
⋅
g
k
x
)
≡
([
⋅
list
]
k
↦
x
∈
l
,
f
k
x
)
⋅
([
⋅
list
]
k
↦
x
∈
l
,
g
k
x
).
Proof
.
revert
f
g
;
induction
l
as
[
x
l
IH
]=>
f
g
.
{
by
rewrite
!
big_opL_nil
left_id
.
}
rewrite
!
big_opL_cons
IH
.
by
rewrite
!
assoc
(
assoc
_
(
g
_
_
))
[(
g
_
_
⋅
_
)]
comm
!
assoc
.
Qed
.
End
list
.
(** ** Big ops over finite maps *)
Section
gmap
.
Context
`
{
Countable
K
}
{
A
:
Type
}.
Implicit
Types
m
:
gmap
K
A
.
Implicit
Types
f
g
:
K
→
A
→
M
.
Lemma
big_opM_mono
f
g
m1
m2
:
m1
⊆
m2
→
(
∀
k
x
,
m2
!!
k
=
Some
x
→
f
k
x
≼
g
k
x
)
→
([
⋅
map
]
k
↦
x
∈
m1
,
f
k
x
)
≼
[
⋅
map
]
k
↦
x
∈
m2
,
g
k
x
.
Proof
.
intros
HX
Hf
.
trans
([
⋅
map
]
k
↦
x
∈
m2
,
f
k
x
).

by
apply
big_op_contains
,
fmap_contains
,
map_to_list_contains
.

apply
big_op_mono
,
Forall2_fmap
,
Forall_Forall2
.
apply
Forall_forall
=>
[
i
x
]
?
/=.
by
apply
Hf
,
elem_of_map_to_list
.
Qed
.
Lemma
big_opM_proper
f
g
m
:
(
∀
k
x
,
m
!!
k
=
Some
x
→
f
k
x
≡
g
k
x
)
→
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
≡
([
⋅
map
]
k
↦
x
∈
m
,
g
k
x
).
Proof
.
intros
Hf
.
apply
big_op_proper
,
equiv_Forall2
,
Forall2_fmap
,
Forall_Forall2
.
apply
Forall_forall
=>
[
i
x
]
?
/=.
by
apply
Hf
,
elem_of_map_to_list
.
Qed
.
Global
Instance
big_opM_ne
m
n
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
dist
n
))
==>
(
dist
n
))
(
big_opM
(
M
:
=
M
)
m
).
Proof
.
intros
f1
f2
Hf
.
apply
big_op_ne
,
Forall2_fmap
.
apply
Forall_Forall2
,
Forall_true
=>
[
i
x
]
;
apply
Hf
.
Qed
.
Global
Instance
big_opM_proper'
m
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
≡
))
==>
(
≡
))
(
big_opM
(
M
:
=
M
)
m
).
Proof
.
intros
f1
f2
Hf
.
by
apply
big_opM_proper
;
intros
;
last
apply
Hf
.
Qed
.
Global
Instance
big_opM_mono'
m
:
Proper
(
pointwise_relation
_
(
pointwise_relation
_
(
≼
))
==>
(
≼
))
(
big_opM
(
M
:
=
M
)
m
).
Proof
.
intros
f1
f2
Hf
.
by
apply
big_opM_mono
;
intros
;
last
apply
Hf
.
Qed
.
Lemma
big_opM_empty
f
:
([
⋅
map
]
k
↦
x
∈
∅
,
f
k
x
)
=
∅
.
Proof
.
by
rewrite
/
big_opM
map_to_list_empty
.
Qed
.
Lemma
big_opM_insert
f
m
i
x
:
m
!!
i
=
None
→
([
⋅
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
f
k
y
)
≡
f
i
x
⋅
[
⋅
map
]
k
↦
y
∈
m
,
f
k
y
.
Proof
.
intros
?.
by
rewrite
/
big_opM
map_to_list_insert
.
Qed
.
Lemma
big_opM_delete
f
m
i
x
:
m
!!
i
=
Some
x
→
([
⋅
map
]
k
↦
y
∈
m
,
f
k
y
)
≡
f
i
x
⋅
[
⋅
map
]
k
↦
y
∈
delete
i
m
,
f
k
y
.
Proof
.
intros
.
rewrite

big_opM_insert
?lookup_delete
//.
by
rewrite
insert_delete
insert_id
.
Qed
.
Lemma
big_opM_lookup
f
m
i
x
:
m
!!
i
=
Some
x
→
f
i
x
≼
[
⋅
map
]
k
↦
y
∈
m
,
f
k
y
.
Proof
.
intros
.
rewrite
big_opM_delete
//.
apply
cmra_included_l
.
Qed
.
Lemma
big_opM_singleton
f
i
x
:
([
⋅
map
]
k
↦
y
∈
{[
i
:
=
x
]},
f
k
y
)
≡
f
i
x
.
Proof
.
rewrite

insert_empty
big_opM_insert
/=
;
last
auto
using
lookup_empty
.
by
rewrite
big_opM_empty
right_id
.
Qed
.
Lemma
big_opM_fmap
{
B
}
(
h
:
A
→
B
)
(
f
:
K
→
B
→
M
)
m
:
([
⋅
map
]
k
↦
y
∈
h
<$>
m
,
f
k
y
)
≡
([
⋅
map
]
k
↦
y
∈
m
,
f
k
(
h
y
)).
Proof
.
rewrite
/
big_opM
map_to_list_fmap

list_fmap_compose
.
f_equiv
;
apply
reflexive_eq
,
list_fmap_ext
.
by
intros
[].
done
.
Qed
.
Lemma
big_opM_insert_override
(
f
:
K
→
M
)
m
i
x
y
:
m
!!
i
=
Some
x
→
([
⋅
map
]
k
↦
_
∈
<[
i
:
=
y
]>
m
,
f
k
)
≡
([
⋅
map
]
k
↦
_
∈
m
,
f
k
).
Proof
.
intros
.
rewrite

insert_delete
big_opM_insert
?lookup_delete
//.
by
rewrite

big_opM_delete
.
Qed
.
Lemma
big_opM_fn_insert
{
B
}
(
g
:
K
→
A
→
B
→
M
)
(
f
:
K
→
B
)
m
i
(
x
:
A
)
b
:
m
!!
i
=
None
→
([
⋅
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
g
k
y
(<[
i
:
=
b
]>
f
k
))
≡
(
g
i
x
b
⋅
[
⋅
map
]
k
↦
y
∈
m
,
g
k
y
(
f
k
)).
Proof
.
intros
.
rewrite
big_opM_insert
//
fn_lookup_insert
.
apply
cmra_op_proper'
,
big_opM_proper
;
auto
=>
k
y
?.
by
rewrite
fn_lookup_insert_ne
;
last
set_solver
.
Qed
.
Lemma
big_opM_fn_insert'
(
f
:
K
→
M
)
m
i
x
P
:
m
!!
i
=
None
→
([
⋅
map
]
k
↦
y
∈
<[
i
:
=
x
]>
m
,
<[
i
:
=
P
]>
f
k
)
≡
(
P
⋅
[
⋅
map
]
k
↦
y
∈
m
,
f
k
).
Proof
.
apply
(
big_opM_fn_insert
(
λ
_
_
,
id
)).
Qed
.
Lemma
big_opM_opM
f
g
m
:
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
⋅
g
k
x
)
≡
([
⋅
map
]
k
↦
x
∈
m
,
f
k
x
)
⋅
([
⋅
map
]
k
↦
x
∈
m
,
g
k
x
).
Proof
.
rewrite
/
big_opM
.
induction
(
map_to_list
m
)
as
[[
i
x
]
l
IH
]
;
csimpl
;
rewrite
?right_id
//.
by
rewrite
IH
!
assoc
(
assoc
_
(
g
_
_
))
[(
g
_
_
⋅
_
)]
comm
!
assoc
.
Qed
.
End
gmap
.
(** ** Big ops over finite sets *)
Section
gset
.
Context
`
{
Countable
A
}.
Implicit
Types
X
:
gset
A
.
Implicit
Types
f
:
A
→
M
.
Lemma
big_opS_mono
f
g
X
Y
:
X
⊆
Y
→
(
∀
x
,
x
∈
Y
→
f
x
≼
g
x
)
→
([
⋅
set
]
x
∈
X
,
f
x
)
≼
[
⋅
set
]
x
∈
Y
,
g
x
.
Proof
.
intros
HX
Hf
.
trans
([
⋅
set
]
x
∈
Y
,
f
x
).

by
apply
big_op_contains
,
fmap_contains
,
elements_contains
.

apply
big_op_mono
,
Forall2_fmap
,
Forall_Forall2
.
apply
Forall_forall
=>
x
?
/=.
by
apply
Hf
,
elem_of_elements
.
Qed
.
Lemma
big_opS_proper
f
g
X
Y
:
X
≡
Y
→
(
∀
x
,
x
∈
X
→
x
∈
Y
→
f
x
≡
g
x
)
→
([
⋅
set
]
x
∈
X
,
f
x
)
≡
([
⋅
set
]
x
∈
Y
,
g
x
).
Proof
.
intros
HX
Hf
.
trans
([
⋅
set
]
x
∈
Y
,
f
x
).

apply
big_op_permutation
.
by
rewrite
HX
.

apply
big_op_proper
,
equiv_Forall2
,
Forall2_fmap
,
Forall_Forall2
.
apply
Forall_forall
=>
x
?
/=.
apply
Hf
;
first
rewrite
HX
;
by
apply
elem_of_elements
.
Qed
.
Lemma
big_opS_ne
X
n
:
Proper
(
pointwise_relation
_
(
dist
n
)
==>
dist
n
)
(
big_opS
(
M
:
=
M
)
X
).
Proof
.
intros
f1
f2
Hf
.
apply
big_op_ne
,
Forall2_fmap
.
apply
Forall_Forall2
,
Forall_true
=>
x
;
apply
Hf
.
Qed
.
Lemma
big_opS_proper'
X
:
Proper
(
pointwise_relation
_
(
≡
)
==>
(
≡
))
(
big_opS
(
M
:
=
M
)
X
).
Proof
.
intros
f1
f2
Hf
.
apply
big_opS_proper
;
naive_solver
.
Qed
.
Lemma
big_opS_mono'
X
:
Proper
(
pointwise_relation
_
(
≼
)
==>
(
≼
))
(
big_opS
(
M
:
=
M
)
X
).
Proof
.
intros
f1
f2
Hf
.
apply
big_opS_mono
;
naive_solver
.
Qed
.
Lemma
big_opS_empty
f
:
([
⋅
set
]
x
∈
∅
,
f
x
)
=
∅
.
Proof
.
by
rewrite
/
big_opS
elements_empty
.
Qed
.
Lemma
big_opS_insert
f
X
x
:
x
∉
X
→
([
⋅
set
]
y
∈
{[
x
]}
∪
X
,
f
y
)
≡
(
f
x
⋅
[
⋅
set
]
y
∈
X
,
f
y
).
Proof
.
intros
.
by
rewrite
/
big_opS
elements_union_singleton
.
Qed
.
Lemma
big_opS_fn_insert
{
B
}
(
f
:
A
→
B
→
M
)
h
X
x
b
:
x
∉
X
→
([
⋅
set
]
y
∈
{[
x
]}
∪
X
,
f
y
(<[
x
:
=
b
]>
h
y
))
≡
(
f
x
b
⋅
[
⋅
set
]
y
∈
X
,
f
y
(
h
y
)).
Proof
.
intros
.
rewrite
big_opS_insert
//
fn_lookup_insert
.
apply
cmra_op_proper'
,
big_opS_proper
;
auto
=>
y
??.
by
rewrite
fn_lookup_insert_ne
;
last
set_solver
.
Qed
.
Lemma
big_opS_fn_insert'
f
X
x
P
:
x
∉
X
→
([
⋅
set
]
y
∈
{[
x
]}
∪
X
,
<[
x
:
=
P
]>
f
y
)
≡
(
P
⋅
[
⋅
set
]
y
∈
X
,
f
y
).
Proof
.
apply
(
big_opS_fn_insert
(
λ
y
,
id
)).
Qed
.
Lemma
big_opS_delete
f
X
x
:
x
∈
X
→
([
⋅
set
]
y
∈
X
,
f
y
)
≡
f
x
⋅
[
⋅
set
]
y
∈
X
∖
{[
x
]},
f
y
.
Proof
.
intros
.
rewrite

big_opS_insert
;
last
set_solver
.
by
rewrite

union_difference_L
;
last
set_solver
.
Qed
.
Lemma
big_opS_elem_of
f
X
x
:
x
∈
X
→
f
x
≼
[
⋅
set
]
y
∈
X
,
f
y
.
Proof
.
intros
.
rewrite
big_opS_delete
//.
apply
cmra_included_l
.
Qed
.
Lemma
big_opS_singleton
f
x
:
([
⋅
set
]
y
∈
{[
x
]},
f
y
)
≡
f
x
.
Proof
.
intros
.
by
rewrite
/
big_opS
elements_singleton
/=
right_id
.
Qed
.
Lemma
big_opS_opS
f
g
X
:
([
⋅
set
]
y
∈
X
,
f
y
⋅
g
y
)
≡
([
⋅
set
]
y
∈
X
,
f
y
)
⋅
([
⋅
set
]
y
∈
X
,
g
y
).
Proof
.
rewrite
/
big_opS
.
induction
(
elements
X
)
as
[
x
l
IH
]
;
csimpl
;
first
by
rewrite
?right_id
.
by
rewrite
IH
!
assoc
(
assoc
_
(
g
_
))
[(
g
_
⋅
_
)]
comm
!
assoc
.
Qed
.
End
gset
.
End
big_op
.
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