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54bd5cd8
Commit
54bd5cd8
authored
Sep 01, 2016
by
Joseph Tassarotti
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Double negation elimination for pure props holds under nnvs modality.
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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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