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David Swasey
coqstdpp
Commits
20f1b822
Commit
20f1b822
authored
Sep 29, 2017
by
Robbert Krebbers
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Make coding style more consistent.
parent
29f26e4e
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theories/fin_maps.v
theories/fin_maps.v
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theories/fin_maps.v
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20f1b822
...
...
@@ 1008,62 +1008,48 @@ Qed.
(** ** The filter operation *)
Section
map_Filter
.
Context
{
A
}
(
P
:
K
*
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}.
Implicit
Types
m
:
M
A
.
Lemma
map_filter_lookup_Some
:
∀
m
k
v
,
filter
P
m
!!
k
=
Some
v
↔
m
!!
k
=
Some
v
∧
P
(
k
,
v
).
Lemma
map_filter_lookup_Some
m
i
x
:
filter
P
m
!!
i
=
Some
x
↔
m
!!
i
=
Some
x
∧
P
(
i
,
x
).
Proof
.
apply
(
map_fold_ind
(
λ
m1
m2
,
∀
k
v
,
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
∧
P
_
)).

setoid_rewrite
lookup_empty
.
naive_solver
.

intros
k
v
m
m'
Hm
Eq
k'
v'
.
case_match
;
case
(
decide
(
k'
=
k
))
as
[>?].
revert
m
i
x
.
apply
(
map_fold_ind
(
λ
m1
m2
,
∀
i
x
,
m1
!!
i
=
Some
x
↔
m2
!!
i
=
Some
x
∧
P
_
))
;
intros
i
x
.

rewrite
lookup_empty
.
naive_solver
.

intros
m
m'
Hm
Eq
j
y
.
case_decide
;
case
(
decide
(
j
=
i
))
as
[>?].
+
rewrite
2
!
lookup_insert
.
naive_solver
.
+
do
2
(
rewrite
lookup_insert_ne
;
[
auto
]).
by
apply
Eq
.
+
rewrite
Eq
,
Hm
,
lookup_insert
.
split
;
[
naive_solver
].
destruct
1
as
[
Eq'
].
inversion
Eq'
.
by
subst
.
+
rewrite
!
lookup_insert_ne
by
done
.
by
apply
Eq
.
+
rewrite
Eq
,
Hm
,
lookup_insert
.
naive_solver
.
+
by
rewrite
lookup_insert_ne
.
Qed
.
Lemma
map_filter_lookup_None
:
∀
m
k
,
filter
P
m
!!
k
=
None
↔
m
!!
k
=
None
∨
∀
v
,
m
!!
k
=
Some
v
→
¬
P
(
k
,
v
).
Lemma
map_filter_lookup_None
m
i
:
filter
P
m
!!
i
=
None
↔
m
!!
i
=
None
∨
∀
x
,
m
!!
i
=
Some
x
→
¬
P
(
i
,
x
).
Proof
.
intros
m
k
.
rewrite
eq_None_not_Some
.
unfold
is_Some
.
rewrite
eq_None_not_Some
.
unfold
is_Some
.
setoid_rewrite
map_filter_lookup_Some
.
naive_solver
.
Qed
.
Lemma
map_filter_lookup_eq
uiv
m1
m2
:
(
∀
k
v
,
P
(
k
,
v
)
→
m1
!!
k
=
Some
v
↔
m2
!!
k
=
Some
v
)
→
filter
P
m1
=
filter
P
m2
.
Lemma
map_filter_lookup_eq
m1
m2
:
(
∀
k
x
,
P
(
k
,
x
)
→
m1
!!
k
=
Some
x
↔
m2
!!
k
=
Some
x
)
→
filter
P
m1
=
filter
P
m2
.
Proof
.
intros
HP
.
apply
map_eq
.
intros
k
.
destruct
(
filter
P
m2
!!
k
)
as
[
v2
]
eqn
:
Hv2
;
[
apply
map_filter_lookup_Some
in
Hv2
as
[
Hv2
HP2
]
;
specialize
(
HP
k
v2
HP2
)

apply
map_filter_lookup_None
;
right
;
intros
v
EqS
ISP
;
apply
map_filter_lookup_None
in
Hv2
as
[
Hv2

Hv2
]].

apply
map_filter_lookup_Some
.
by
rewrite
HP
.

specialize
(
HP
_
_
ISP
).
rewrite
HP
,
Hv2
in
EqS
.
naive_solver
.

apply
(
Hv2
v
)
;
[
by
apply
HP

done
].
intros
HP
.
apply
map_eq
.
intros
i
.
apply
option_eq
;
intros
x
.
rewrite
!
map_filter_lookup_Some
.
naive_solver
.
Qed
.
Lemma
map_filter_lookup_insert
m
k
v
:
P
(
k
,
v
)
→
<[
k
:
=
v
]>
(
filter
P
m
)
=
filter
P
(<[
k
:
=
v
]>
m
).
Lemma
map_filter_lookup_insert
m
i
x
:
P
(
i
,
x
)
→
<[
i
:
=
x
]>
(
filter
P
m
)
=
filter
P
(<[
i
:
=
x
]>
m
).
Proof
.
intros
HP
.
apply
map_eq
.
intros
k'
.
case
(
decide
(
k'
=
k
))
as
[>?]
;
[
rewrite
lookup_insert

rewrite
lookup_insert_ne
;
[
auto
]].

symmetry
.
apply
map_filter_lookup_Some
.
by
rewrite
lookup_insert
.

destruct
(
filter
P
(<[
k
:
=
v
]>
m
)
!!
k'
)
eqn
:
Hk
;
revert
Hk
;
[
rewrite
map_filter_lookup_Some
,
lookup_insert_ne
;
[
by
auto
]
;
by
rewrite
<
map_filter_lookup_Some

rewrite
map_filter_lookup_None
,
lookup_insert_ne
;
[
auto
]
;
by
rewrite
<
map_filter_lookup_None
].
intros
HP
.
apply
map_eq
.
intros
j
.
apply
option_eq
;
intros
y
.
destruct
(
decide
(
j
=
i
))
as
[>?].

rewrite
map_filter_lookup_Some
,
!
lookup_insert
.
naive_solver
.

rewrite
lookup_insert_ne
,
!
map_filter_lookup_Some
,
lookup_insert_ne
by
done
.
naive_solver
.
Qed
.
Lemma
map_filter_empty
:
filter
P
∅
=
∅
.
Lemma
map_filter_empty
:
filter
P
(
∅
:
M
A
)
=
∅
.
Proof
.
apply
map_fold_empty
.
Qed
.
End
map_Filter
.
(** ** Properties of the [map_Forall] predicate *)
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