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stdpp
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c134e48c
Commit
c134e48c
authored
Jan 25, 2017
by
Robbert Krebbers
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Fold operation on finite maps.
parent
a89c6d78
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2
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48 additions
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2 deletions
+48
-2
theories/fin_maps.v
theories/fin_maps.v
+46
-0
theories/list.v
theories/list.v
+2
-2
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theories/fin_maps.v
View file @
c134e48c
...
...
@@ -112,6 +112,11 @@ Definition map_imap `{∀ A, Insert K A (M A), ∀ A, Empty (M A),
∀
A
,
FinMapToList
K
A
(
M
A
)}
{
A
B
}
(
f
:
K
→
A
→
option
B
)
(
m
:
M
A
)
:
M
B
:
=
map_of_list
(
omap
(
λ
ix
,
(
fst
ix
,)
<$>
curry
f
ix
)
(
map_to_list
m
)).
(* Folds a function [f] over a map. The order in which the function is called
is unspecified. *)
Definition
map_fold
`
{
FinMapToList
K
A
M
}
{
B
}
(
f
:
K
→
A
→
B
→
B
)
(
b
:
B
)
:
M
→
B
:
=
foldr
(
curry
f
)
b
∘
map_to_list
.
(** * Theorems *)
Section
theorems
.
Context
`
{
FinMap
K
M
}.
...
...
@@ -814,6 +819,47 @@ Proof.
-
by
apply
lt_wf
.
Qed
.
(** ** The fold operation *)
Lemma
map_fold_empty
{
A
B
}
(
f
:
K
→
A
→
B
→
B
)
(
b
:
B
)
:
map_fold
f
b
∅
=
b
.
Proof
.
unfold
map_fold
;
simpl
.
by
rewrite
map_to_list_empty
.
Qed
.
Lemma
map_fold_insert
{
A
B
}
(
R
:
relation
B
)
`
{!
PreOrder
R
}
(
f
:
K
→
A
→
B
→
B
)
(
b
:
B
)
(
i
:
K
)
(
x
:
A
)
(
m
:
M
A
)
:
(
∀
j
z
,
Proper
(
R
==>
R
)
(
f
j
z
))
→
(
∀
j1
j2
z1
z2
y
,
R
(
f
j1
z1
(
f
j2
z2
y
))
(
f
j2
z2
(
f
j1
z1
y
)))
→
m
!!
i
=
None
→
R
(
map_fold
f
b
(<[
i
:
=
x
]>
m
))
(
f
i
x
(
map_fold
f
b
m
)).
Proof
.
intros
.
unfold
map_fold
;
simpl
.
assert
(
∀
kz
,
Proper
(
R
==>
R
)
(
curry
f
kz
))
by
(
intros
[]
;
apply
_
).
trans
(
foldr
(
curry
f
)
b
((
i
,
x
)
::
map_to_list
m
))
;
[|
done
].
eapply
(
foldr_permutation
R
(
curry
f
)
b
),
map_to_list_insert
;
auto
.
intros
[]
[]
;
simpl
;
eauto
.
Qed
.
Lemma
map_fold_ind
{
A
B
}
(
P
:
B
→
M
A
→
Prop
)
(
f
:
K
→
A
→
B
→
B
)
(
b
:
B
)
:
P
b
∅
→
(
∀
i
x
m
r
,
m
!!
i
=
None
→
P
r
m
→
P
(
f
i
x
r
)
(<[
i
:
=
x
]>
m
))
→
∀
m
,
P
(
map_fold
f
b
m
)
m
.
Proof
.
intros
Hemp
Hinsert
.
cut
(
∀
l
,
NoDup
l
→
∀
m
,
(
∀
i
x
,
m
!!
i
=
Some
x
↔
(
i
,
x
)
∈
l
)
→
P
(
foldr
(
curry
f
)
b
l
)
m
).
{
intros
help
?.
apply
help
;
[
apply
NoDup_map_to_list
|].
intros
i
x
.
by
rewrite
elem_of_map_to_list
.
}
induction
1
as
[|[
i
x
]
l
??
IH
]
;
simpl
.
{
intros
m
Hm
.
cut
(
m
=
∅
)
;
[
by
intros
->|].
apply
map_empty
;
intros
i
.
apply
eq_None_not_Some
;
intros
[
x
[]%
Hm
%
elem_of_nil
].
}
intros
m
Hm
.
assert
(
m
!!
i
=
Some
x
)
by
(
apply
Hm
;
by
left
).
rewrite
<-(
insert_id
m
i
x
),
<-
insert_delete
by
done
.
apply
Hinsert
;
auto
using
lookup_delete
.
apply
IH
.
intros
j
y
.
rewrite
lookup_delete_Some
,
Hm
.
split
.
-
by
intros
[?
[[=
??]|?]%
elem_of_cons
].
-
intros
?
;
split
;
[
intros
->|
by
right
].
assert
(
m
!!
j
=
Some
y
)
by
(
apply
Hm
;
by
right
).
naive_solver
.
Qed
.
(** ** Properties of the [map_Forall] predicate *)
Section
map_Forall
.
Context
{
A
}
(
P
:
K
→
A
→
Prop
).
...
...
theories/list.v
View file @
c134e48c
...
...
@@ -3210,8 +3210,8 @@ Definition foldr_app := @fold_right_app.
Lemma foldl_app {A B} (f : A → B → A) (l k : list B) (a : A) :
foldl f a (l ++ k) = foldl f (foldl f a l) k.
Proof. revert a. induction l; simpl; auto. Qed.
Lemma foldr_permutation {A B} (R : relation B) `{!
Equivalence
R}
(f : A → B → B) (b : B) `{!Proper (
(=) ==>
R ==> R)
f
}
Lemma foldr_permutation {A B} (R : relation B) `{!
PreOrder
R}
(f : A → B → B) (b : B) `{!
∀ x,
Proper (R ==> R)
(f x)
}
(Hf : ∀ a1 a2 b, R (f a1 (f a2 b)) (f a2 (f a1 b))) :
Proper ((≡ₚ) ==> R) (foldr f b).
Proof. induction 1; simpl; [done|by f_equiv|apply Hf|etrans; eauto]. Qed.
...
...
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