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Simon Spies
stdpp
Commits
6aac4455
Commit
6aac4455
authored
Aug 27, 2013
by
Robbert Krebbers
Browse files
Prove decidability of quantification over finite types.
parent
be8dfddb
Changes
1
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Inline
Side-by-side
theories/finite.v
View file @
6aac4455
...
...
@@ -129,6 +129,27 @@ Lemma bijective_card `{Finite A} `{Finite B} (f : A → B)
`
{!
Injective
(=)
(=)
f
}
`
{!
Surjective
(=)
f
}
:
card
A
=
card
B
.
Proof
.
apply
finite_bijective
.
eauto
.
Qed
.
(** Decidability of quantification over finite types *)
Section
forall_exists
.
Context
`
{
Finite
A
}
(
P
:
A
→
Prop
)
`
{
∀
x
,
Decision
(
P
x
)}.
Lemma
Forall_finite
:
Forall
P
(
enum
A
)
↔
(
∀
x
,
P
x
).
Proof
.
rewrite
Forall_forall
.
intuition
auto
using
elem_of_enum
.
Qed
.
Lemma
Exists_finite
:
Exists
P
(
enum
A
)
↔
(
∃
x
,
P
x
).
Proof
.
rewrite
Exists_exists
.
naive_solver
eauto
using
elem_of_enum
.
Qed
.
Global
Instance
forall_dec
:
Decision
(
∀
x
,
P
x
).
Proof
.
refine
(
cast_if
(
decide
(
Forall
P
(
enum
A
))))
;
abstract
by
rewrite
<-
Forall_finite
.
Defined
.
Global
Instance
exists_dec
:
Decision
(
∃
x
,
P
x
).
Proof
.
refine
(
cast_if
(
decide
(
Exists
P
(
enum
A
))))
;
abstract
by
rewrite
<-
Exists_finite
.
Defined
.
End
forall_exists
.
(** Instances *)
Section
enc_finite
.
Context
`
{
∀
x
y
:
A
,
Decision
(
x
=
y
)}.
...
...
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