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stdpp
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e18fe0f6
Commit
e18fe0f6
authored
Feb 20, 2019
by
Robbert Krebbers
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Alternative versions of finite/infinite predicates in terms of sets.
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theories/fin_sets.v
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e18fe0f6
...
...
@@ -325,4 +325,21 @@ Proof.
refine
(
cast_if
(
decide
(
Exists
P
(
elements
X
))))
;
by
rewrite
set_Exists_elements
.
Defined
.
(** Alternative versions of finite and infinite predicates *)
Lemma
pred_finite_set
(
P
:
A
→
Prop
)
:
pred_finite
P
↔
(
∃
X
:
C
,
∀
x
,
P
x
→
x
∈
X
).
Proof
.
split
.
-
intros
[
xs
Hfin
].
exists
(
list_to_set
xs
).
set_solver
.
-
intros
[
X
Hfin
].
exists
(
elements
X
).
set_solver
.
Qed
.
Lemma
pred_infinite_set
(
P
:
A
→
Prop
)
:
pred_infinite
P
↔
(
∀
X
:
C
,
∃
x
,
P
x
∧
x
∉
X
).
Proof
.
split
.
-
intros
Hinf
X
.
destruct
(
Hinf
(
elements
X
)).
set_solver
.
-
intros
Hinf
xs
.
destruct
(
Hinf
(
list_to_set
xs
)).
set_solver
.
Qed
.
End
fin_set
.
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