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stdpp
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b778f3eb
Commit
b778f3eb
authored
Jun 26, 2017
by
Robbert Krebbers
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Merge branch 'dfrumin/coq-stdpp-map_properties'
parents
9b01e3d8
cd0b7f48
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theories/fin_maps.v
theories/fin_maps.v
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theories/fin_maps.v
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b778f3eb
...
...
@@ -843,14 +843,6 @@ Proof.
rewrite
elem_of_map_to_list
in
Hj
;
simplify_option_eq
.
Qed
.
(** Properties of the zip_with function *)
Lemma
map_lookup_zip_with
{
A
B
C
}
(
f
:
A
→
B
→
C
)
m1
m2
i
:
map_zip_with
f
m1
m2
!!
i
=
x
←
m1
!!
i
;
y
←
m2
!!
i
;
Some
(
f
x
y
).
Proof
.
unfold
map_zip_with
.
rewrite
lookup_merge
by
done
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
).
Qed
.
(** ** Properties of conversion from collections *)
Section
map_of_to_collection
.
Context
{
A
:
Type
}
`
{
FinCollection
B
C
}.
...
...
@@ -1136,6 +1128,79 @@ Lemma insert_merge_r m1 m2 i x z :
Proof
.
by
intros
;
apply
partial_alter_merge_r
.
Qed
.
End
more_merge
.
(** Properties of the zip_with function *)
Lemma
map_lookup_zip_with
{
A
B
C
}
(
f
:
A
→
B
→
C
)
m1
m2
i
:
map_zip_with
f
m1
m2
!!
i
=
x
←
m1
!!
i
;
y
←
m2
!!
i
;
Some
(
f
x
y
).
Proof
.
unfold
map_zip_with
.
rewrite
lookup_merge
by
done
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
).
Qed
.
Lemma
map_zip_with_empty
{
A
B
C
}
(
f
:
A
→
B
→
C
)
:
map_zip_with
f
∅
∅
=
∅
.
Proof
.
unfold
map_zip_with
.
by
rewrite
merge_empty
by
done
.
Qed
.
Lemma
map_insert_zip_with
{
A
B
C
}
(
f
:
A
→
B
→
C
)
m1
m2
i
x
y
z
:
f
y
z
=
x
→
<[
i
:
=
x
]>(
map_zip_with
f
m1
m2
)
=
map_zip_with
f
(<[
i
:
=
y
]>
m1
)
(<[
i
:
=
z
]>
m2
).
Proof
.
intros
Hf
.
unfold
map_zip_with
.
erewrite
insert_merge
;
[
auto
|
by
compute
|
by
rewrite
Hf
].
Qed
.
Lemma
map_zip_with_fmap
{
A'
A
B'
B
C
}
(
f
:
A
→
B
→
C
)
(
g1
:
A'
→
A
)
(
g2
:
B'
→
B
)
m1
m2
:
map_zip_with
f
(
g1
<$>
m1
)
(
g2
<$>
m2
)
=
map_zip_with
(
λ
x
y
,
f
(
g1
x
)
(
g2
y
))
m1
m2
.
Proof
.
apply
map_eq
;
intro
i
.
rewrite
?map_lookup_zip_with
.
rewrite
?lookup_fmap
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
).
Qed
.
Lemma
map_zip_with_fmap_1
{
A'
A
B
C
}
(
f
:
A
→
B
→
C
)
(
g
:
A'
→
A
)
m1
m2
:
map_zip_with
f
(
g
<$>
m1
)
m2
=
map_zip_with
(
λ
x
y
,
f
(
g
x
)
y
)
m1
m2
.
Proof
.
rewrite
<-
(
map_fmap_id
m2
)
at
1
.
by
rewrite
map_zip_with_fmap
;
simpl
.
Qed
.
Lemma
map_zip_with_fmap_2
{
A
B'
B
C
}
(
f
:
A
→
B
→
C
)
(
g
:
B'
→
B
)
m1
m2
:
map_zip_with
f
m1
(
g
<$>
m2
)
=
map_zip_with
(
λ
x
y
,
f
x
(
g
y
))
m1
m2
.
Proof
.
rewrite
<-
(
map_fmap_id
m1
)
at
1
.
by
rewrite
map_zip_with_fmap
;
simpl
.
Qed
.
Lemma
map_fmap_zip_with
{
A
B
C
D
}
(
f
:
A
→
B
→
C
)
(
g
:
C
→
D
)
m1
m2
:
g
<$>
map_zip_with
f
m1
m2
=
map_zip_with
(
λ
x
y
,
g
(
f
x
y
))
m1
m2
.
Proof
.
apply
map_eq
;
intro
i
.
rewrite
lookup_fmap
.
rewrite
?map_lookup_zip_with
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
).
Qed
.
Lemma
map_zip_with_map_zip
{
A
B
C
}
(
f
:
A
→
B
→
C
)
m1
m2
:
map_zip_with
f
m1
m2
=
curry
f
<$>
map_zip
m1
m2
.
Proof
.
apply
map_eq
;
intro
i
.
rewrite
lookup_fmap
;
rewrite
?map_lookup_zip_with
;
rewrite
?lookup_fmap
.
by
destruct
(
m1
!!
i
),
(
m2
!!
i
).
Qed
.
Lemma
map_fmap_zip
{
A'
A
B'
B
}
(
g1
:
A'
→
A
)
(
g2
:
B'
→
B
)
m1
m2
:
map_zip
(
fmap
g1
m1
)
(
fmap
g2
m2
)
=
prod_map
g1
g2
<$>
map_zip
m1
m2
.
Proof
.
rewrite
map_zip_with_fmap
.
rewrite
map_zip_with_map_zip
.
generalize
(
map_zip
m1
m2
)
;
intro
m
.
apply
map_eq
;
intro
i
.
by
rewrite
?lookup_fmap
;
destruct
(
m
!!
i
)
as
[[
x1
x2
]|].
Qed
.
(** ** Properties on the [map_relation] relation *)
Section
Forall2
.
Context
{
A
B
}
(
R
:
A
→
B
→
Prop
)
(
P
:
A
→
Prop
)
(
Q
:
B
→
Prop
).
...
...
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