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Joshua Yanovski
iriscoq
Commits
54bd5cd8
Commit
54bd5cd8
authored
Sep 01, 2016
by
Joseph Tassarotti
Browse files
Double negation elimination for pure props holds under nnvs modality.
parent
f7afee85
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algebra/double_negation.v
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algebra/double_negation.v
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54bd5cd8
...
...
@@ 259,7 +259,6 @@ Proof.
by
apply
nnvs_k_trans
.
Qed
.
(
*
Now
that
we
have
shown
nnvs
has
all
of
the
desired
properties
of
rvs
,
we
go
further
and
show
it
is
in
fact
equivalent
to
rvs
!
The
direction
from
rvs
to
nnvs
is
similar
to
the
proof
of
...
...
@@ 301,16 +300,38 @@ Proof.
Qed
.
End
classical
.
(
*
We
might
wonder
whether
we
can
prove
an
adequacy
lemma
for
nnvs
.
We
could
combine
the
adequacy
lemma
for
rvs
with
the
previous
result
to
get
adquacy
for
nnvs
,
but
this
would
rely
on
the
classical
axiom
we
needed
to
prove
the
equivalence
!
Can
we
establish
adequacy
without
axioms
?
Unfortunately
not
,
because
adequacy
for
nnvs
would
imply
double
negation
elimination
,
which
is
classical
:
*
)
Lemma
nnvs_dne
φ
:
True
⊢
(
=
n
=>
(
■
(
¬¬
φ
→
φ
))
:
uPred
M
)
%
I
.
Proof
.
rewrite
/
uPred_nnvs
.
apply
forall_intro
=>
n
.
apply
wand_intro_l
.
rewrite
?
right_id
.
assert
(
∀
φ
,
¬¬¬¬φ
→
¬¬φ
)
by
naive_solver
.
assert
(
Hdne
:
¬¬
(
¬¬φ
→
φ
))
by
naive_solver
.
split
.
unseal
.
intros
n
'
??
Hvs
.
case
(
decide
(
n
'
<
n
)).

intros
.
move
:
laterN_small
.
unseal
.
naive_solver
.

intros
.
assert
(
n
≤
n
'
).
omega
.
exfalso
.
specialize
(
Hvs
n
'
∅
).
eapply
Hdne
.
intros
Hfal
.
eapply
laterN_big
;
eauto
.
unseal
.
rewrite
right_id
in
Hvs
*
;
naive_solver
.
Qed
.
(
*
Questions
:
1
)
Can
one
prove
an
adequacy
theorem
for
the
=
n
=>
modality
without
axioms
?
2
)
If
not
,
can
we
prove
a
weakened
form
of
adequacy
,
such
as
:
1
)
Can
we
prove
a
weakened
form
of
adequacy
for
nnvs
directly
,
such
as
:
Lemma
adequacy
'
φ
n
:
(
True
⊢
Nat
.
iter
n
(
λ
P
,
=
n
=>
▷
P
)
(
■
φ
))
→
¬¬
φ
.
One
idea
may
be
to
prove
a
limited
adequacy
theorem
for
each
nnvs_k
and
use
the
limiting
argument
we
did
for
transitivity
.
3
)
Do
the
basic
properties
of
the
=
r
=>
modality
(
rvs_intro
,
rvs_mono
,
rvs_trans
,
rvs_frame_r
,
2
)
Do
the
basic
properties
of
the
=
r
=>
modality
(
rvs_intro
,
rvs_mono
,
rvs_trans
,
rvs_frame_r
,
rvs_ownM_updateP
,
and
adequacy
)
uniquely
characterize
=
r
=>?
*
)
...
...
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