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David Swasey
coq-stdpp
Commits
ae7fba25
Commit
ae7fba25
authored
Oct 06, 2017
by
Hai Dang
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add more properties of intersection_with for fin_maps
parent
c60c5192
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theories/fin_maps.v
theories/fin_maps.v
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theories/fin_maps.v
View file @
ae7fba25
...
...
@@ -1124,6 +1124,16 @@ Proof.
intros
??.
apply
map_eq
.
intros
.
by
rewrite
!(
lookup_merge
f
),
lookup_empty
,
(
right_id_L
None
f
).
Qed
.
Global
Instance
:
LeftAbsorb
(=)
None
f
→
LeftAbsorb
(=)
(
∅
:
M
A
)
(
merge
f
).
Proof
.
intros
??.
apply
map_eq
.
intros
.
by
rewrite
!(
lookup_merge
f
),
lookup_empty
,
(
left_absorb_L
None
f
).
Qed
.
Global
Instance
:
RightAbsorb
(=)
None
f
→
RightAbsorb
(=)
(
∅
:
M
A
)
(
merge
f
).
Proof
.
intros
??.
apply
map_eq
.
intros
.
by
rewrite
!(
lookup_merge
f
),
lookup_empty
,
(
right_absorb_L
None
f
).
Qed
.
Lemma
merge_comm
m1
m2
:
(
∀
i
,
f
(
m1
!!
i
)
(
m2
!!
i
)
=
f
(
m2
!!
i
)
(
m1
!!
i
))
→
merge
f
m1
m2
=
merge
f
m2
m1
.
...
...
@@ -1725,18 +1735,72 @@ Lemma map_disjoint_of_list_zip_r_2 {A} (m : M A) is xs :
Proof
.
intro
.
by
rewrite
map_disjoint_of_list_zip_r
.
Qed
.
(** ** Properties of the [intersection_with] operation *)
Lemma
lookup_intersection_with
{
A
}
(
f
:
A
→
A
→
option
A
)
(
m1
m2
:
M
A
)
i
:
Section
intersection_with
.
Context
{
A
}
(
f
:
A
→
A
→
option
A
).
Implicit
Type
(
m
:
M
A
).
Global
Instance
:
LeftAbsorb
(@
eq
(
M
A
))
∅
(
intersection_with
f
).
Proof
.
unfold
intersection_with
,
map_intersection_with
.
apply
_
.
Qed
.
Global
Instance
:
RightAbsorb
(@
eq
(
M
A
))
∅
(
intersection_with
f
).
Proof
.
unfold
intersection_with
,
map_intersection_with
.
apply
_
.
Qed
.
Lemma
lookup_intersection_with
m1
m2
i
:
intersection_with
f
m1
m2
!!
i
=
intersection_with
f
(
m1
!!
i
)
(
m2
!!
i
).
Proof
.
by
rewrite
<-(
lookup_merge
_
).
Qed
.
Lemma
lookup_intersection_with_Some
{
A
}
(
f
:
A
→
A
→
option
A
)
(
m1
m2
:
M
A
)
i
z
:
Lemma
lookup_intersection_with_Some
m1
m2
i
z
:
intersection_with
f
m1
m2
!!
i
=
Some
z
↔
(
∃
x
y
,
m1
!!
i
=
Some
x
∧
m2
!!
i
=
Some
y
∧
f
x
y
=
Some
z
).
Proof
.
rewrite
lookup_intersection_with
.
destruct
(
m1
!!
i
),
(
m2
!!
i
)
;
compute
;
naive_solver
.
Qed
.
Lemma
intersection_with_comm
m1
m2
:
(
∀
i
x
y
,
m1
!!
i
=
Some
x
→
m2
!!
i
=
Some
y
→
f
x
y
=
f
y
x
)
→
intersection_with
f
m1
m2
=
intersection_with
f
m2
m1
.
Proof
.
intros
.
apply
(
merge_comm
_
).
intros
i
.
destruct
(
m1
!!
i
)
eqn
:
?,
(
m2
!!
i
)
eqn
:
?
;
simpl
;
eauto
.
Qed
.
Global
Instance
:
Comm
(=)
f
→
Comm
(@
eq
(
M
A
))
(
intersection_with
f
).
Proof
.
intros
???.
apply
intersection_with_comm
.
eauto
.
Qed
.
Lemma
intersection_with_idemp
m
:
(
∀
i
x
,
m
!!
i
=
Some
x
→
f
x
x
=
Some
x
)
→
intersection_with
f
m
m
=
m
.
Proof
.
intros
.
apply
(
merge_idemp
_
).
intros
i
.
destruct
(
m
!!
i
)
eqn
:
?
;
simpl
;
eauto
.
Qed
.
Lemma
alter_intersection_with
(
g
:
A
→
A
)
m1
m2
i
:
(
∀
x
y
,
m1
!!
i
=
Some
x
→
m2
!!
i
=
Some
y
→
g
<$>
f
x
y
=
f
(
g
x
)
(
g
y
))
→
alter
g
i
(
intersection_with
f
m1
m2
)
=
intersection_with
f
(
alter
g
i
m1
)
(
alter
g
i
m2
).
Proof
.
intros
.
apply
(
partial_alter_merge
_
).
destruct
(
m1
!!
i
)
eqn
:
?,
(
m2
!!
i
)
eqn
:
?
;
simpl
;
eauto
.
Qed
.
Lemma
delete_intersection_with
m1
m2
i
:
delete
i
(
intersection_with
f
m1
m2
)
=
intersection_with
f
(
delete
i
m1
)
(
delete
i
m2
).
Proof
.
by
apply
(
partial_alter_merge
_
).
Qed
.
Lemma
foldr_delete_intersection_with
(
m1
m2
:
M
A
)
is
:
foldr
delete
(
intersection_with
f
m1
m2
)
is
=
intersection_with
f
(
foldr
delete
m1
is
)
(
foldr
delete
m2
is
).
Proof
.
induction
is
;
simpl
.
done
.
by
rewrite
IHis
,
delete_intersection_with
.
Qed
.
Lemma
insert_intersection_with
m1
m2
i
x
y
z
:
f
x
y
=
Some
z
→
<[
i
:
=
z
]>(
intersection_with
f
m1
m2
)
=
intersection_with
f
(<[
i
:
=
x
]>
m1
)
(<[
i
:
=
y
]>
m2
).
Proof
.
by
intros
;
apply
(
partial_alter_merge
_
).
Qed
.
End
intersection_with
.
(** ** Properties of the [intersection] operation *)
Global
Instance
:
LeftAbsorb
(@
eq
(
M
A
))
∅
(
∩
)
:
=
_
.
Global
Instance
:
RightAbsorb
(@
eq
(
M
A
))
∅
(
∩
)
:
=
_
.
Global
Instance
:
Assoc
(@
eq
(
M
A
))
(
∩
).
Proof
.
intros
A
m1
m2
m3
.
unfold
intersection
,
map_intersection
,
intersection_with
,
map_intersection_with
.
apply
(
merge_assoc
_
).
intros
i
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
),
(
m3
!!
i
).
Qed
.
Global
Instance
:
IdemP
(@
eq
(
M
A
))
(
∩
).
Proof
.
intros
A
?.
by
apply
intersection_with_idemp
.
Qed
.
Lemma
lookup_intersection_Some
{
A
}
(
m1
m2
:
M
A
)
i
x
:
(
m1
∩
m2
)
!!
i
=
Some
x
↔
m1
!!
i
=
Some
x
∧
is_Some
(
m2
!!
i
).
Proof
.
...
...
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