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David Swasey
coq-stdpp
Commits
c15a1ba4
Commit
c15a1ba4
authored
Nov 21, 2016
by
Robbert Krebbers
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More properties and set_solver for filter.
parent
bee1e422
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theories/fin_collections.v
theories/fin_collections.v
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theories/fin_collections.v
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c15a1ba4
...
...
@@ -14,6 +14,7 @@ Definition collection_fold `{Elements A C} {B}
Instance
collection_filter
`
{
Elements
A
C
,
Empty
C
,
Singleton
A
C
,
Union
C
}
:
Filter
A
C
:
=
λ
P
_
X
,
of_list
(
filter
P
(
elements
X
)).
Typeclasses
Opaque
collection_filter
.
Section
fin_collection
.
Context
`
{
FinCollection
A
C
}.
...
...
@@ -211,12 +212,41 @@ Lemma minimal_exists_L R `{!LeibnizEquiv C, !Transitive R,
Proof
.
unfold_leibniz
.
apply
(
minimal_exists
R
).
Qed
.
(** * Filter *)
Lemma
elem_of_filter
(
P
:
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}
X
x
:
x
∈
filter
P
X
↔
P
x
∧
x
∈
X
.
Proof
.
unfold
filter
,
collection_filter
.
by
rewrite
elem_of_of_list
,
elem_of_list_filter
,
elem_of_elements
.
Qed
.
Section
filter
.
Context
(
P
:
A
→
Prop
)
`
{!
∀
x
,
Decision
(
P
x
)}.
Lemma
elem_of_filter
X
x
:
x
∈
filter
P
X
↔
P
x
∧
x
∈
X
.
Proof
.
unfold
filter
,
collection_filter
.
by
rewrite
elem_of_of_list
,
elem_of_list_filter
,
elem_of_elements
.
Qed
.
Global
Instance
set_unfold_filter
X
Q
:
SetUnfold
(
x
∈
X
)
Q
→
SetUnfold
(
x
∈
filter
P
X
)
(
P
x
∧
Q
).
Proof
.
intros
??
;
constructor
.
by
rewrite
elem_of_filter
,
(
set_unfold
(
x
∈
X
)
Q
).
Qed
.
Lemma
filter_empty
:
filter
P
(
∅
:
C
)
≡
∅
.
Proof
.
set_solver
.
Qed
.
Lemma
filter_union
X
Y
:
filter
P
(
X
∪
Y
)
≡
filter
P
X
∪
filter
P
Y
.
Proof
.
set_solver
.
Qed
.
Lemma
filter_singleton
x
:
P
x
→
filter
P
({[
x
]}
:
C
)
≡
{[
x
]}.
Proof
.
set_solver
.
Qed
.
Lemma
filter_singleton_not
x
:
¬
P
x
→
filter
P
({[
x
]}
:
C
)
≡
∅
.
Proof
.
set_solver
.
Qed
.
Section
leibniz_equiv
.
Context
`
{!
LeibnizEquiv
C
}.
Lemma
filter_empty_L
:
filter
P
(
∅
:
C
)
=
∅
.
Proof
.
set_solver
.
Qed
.
Lemma
filter_union_L
X
Y
:
filter
P
(
X
∪
Y
)
=
filter
P
X
∪
filter
P
Y
.
Proof
.
set_solver
.
Qed
.
Lemma
filter_singleton_L
x
:
P
x
→
filter
P
({[
x
]}
:
C
)
=
{[
x
]}.
Proof
.
set_solver
.
Qed
.
Lemma
filter_singleton_not_L
x
:
¬
P
x
→
filter
P
({[
x
]}
:
C
)
=
∅
.
Proof
.
set_solver
.
Qed
.
End
leibniz_equiv
.
End
filter
.
(** * Decision procedures *)
Global
Instance
set_Forall_dec
`
(
P
:
A
→
Prop
)
...
...
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