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ac1a1276
Commit
ac1a1276
authored
Dec 03, 2017
by
Robbert Krebbers
Browse files
Merge branch 'master' into gen_proofmode
parents
c6b84830
e88994af
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#5709
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theories/bi/derived.v
View file @
ac1a1276
...
...
@@ 1072,7 +1072,7 @@ Proof.
by
rewrite
(
comm
bi_and
)
persistently_and_emp_elim
wand_elim_l
.
Qed
.
Lemma
wand_impl
_persistently_2
P
Q
:
(
bi_persistently
P

∗
Q
)
⊢
(
bi_persistently
P
→
Q
).
Lemma
impl_wand
_persistently_2
P
Q
:
(
bi_persistently
P

∗
Q
)
⊢
(
bi_persistently
P
→
Q
).
Proof
.
apply
impl_intro_l
.
by
rewrite
persistently_and_sep_l_1
wand_elim_r
.
Qed
.
Section
persistently_affinely_bi
.
...
...
@@ 1098,9 +1098,9 @@ Section persistently_affinely_bi.
by
rewrite

persistently_and_sep_r
persistently_elim
impl_elim_r
.
Qed
.
Lemma
wand_impl
_persistently
P
Q
:
(
bi_persistently
P

∗
Q
)
⊣
⊢
(
bi_persistently
P
→
Q
).
Lemma
impl_wand
_persistently
P
Q
:
(
bi_persistently
P
→
Q
)
⊣
⊢
(
bi_persistently
P

∗
Q
).
Proof
.
apply
(
anti_symm
(
⊢
)).
apply
wand_impl_persistently_2
.
by
rewrite

impl_wand_1
.
apply
(
anti_symm
(
⊢
)).
by
rewrite

impl_wand_1
.
apply
impl_wand_persistently_2
.
Qed
.
Lemma
wand_alt
P
Q
:
(
P

∗
Q
)
⊣
⊢
∃
R
,
R
∗
bi_persistently
(
P
∗
R
→
Q
).
...
...
@@ 1267,7 +1267,7 @@ Proof.
by
rewrite
(
comm
bi_and
)
plainly_and_emp_elim
wand_elim_l
.
Qed
.
Lemma
wand_impl
_plainly_2
P
Q
:
(
bi_plainly
P

∗
Q
)
⊢
(
bi_plainly
P
→
Q
).
Lemma
impl_wand
_plainly_2
P
Q
:
(
bi_plainly
P

∗
Q
)
⊢
(
bi_plainly
P
→
Q
).
Proof
.
apply
impl_intro_l
.
by
rewrite
plainly_and_sep_l_1
wand_elim_r
.
Qed
.
Section
plainly_affinely_bi
.
...
...
@@ 1291,9 +1291,9 @@ Section plainly_affinely_bi.
by
rewrite

plainly_and_sep_r
plainly_elim
impl_elim_r
.
Qed
.
Lemma
wand_impl
_plainly
P
Q
:
(
bi_plainly
P

∗
Q
)
⊣
⊢
(
bi_plainly
P
→
Q
).
Lemma
impl_wand
_plainly
P
Q
:
(
bi_plainly
P
→
Q
)
⊣
⊢
(
bi_plainly
P

∗
Q
).
Proof
.
apply
(
anti_symm
(
⊢
)).
by
rewrite
wand_impl_plainly_2
.
by
rewrite

impl_wand_
1
.
apply
(
anti_symm
(
⊢
)).
by
rewrite

impl_wand_1
.
by
rewrite
impl_wand_
plainly_2
.
Qed
.
End
plainly_affinely_bi
.
...
...
@@ 1343,11 +1343,11 @@ Proof.
by
rewrite

persistently_and_affinely_sep_l
affinely_and_r
affinely_and
idemp
.
Qed
.
Lemma
wand_impl
_affinely_persistently
P
Q
:
(
□
P

∗
Q
)
⊣
⊢
(
bi_persistently
P
→
Q
).
Lemma
impl_wand
_affinely_persistently
P
Q
:
(
bi_persistently
P
→
Q
)
⊣
⊢
(
□
P

∗
Q
).
Proof
.
apply
(
anti_symm
(
⊢
)).

apply
impl_intro_l
.
by
rewrite
persistently_and_affinely_sep_l
wand_elim_r
.

apply
wand_intro_l
.
by
rewrite

persistently_and_affinely_sep_l
impl_elim_r
.

apply
impl_intro_l
.
by
rewrite
persistently_and_affinely_sep_l
wand_elim_r
.
Qed
.
(* The combined affinely plainly modality *)
...
...
@@ 1388,8 +1388,8 @@ Proof.
by
rewrite

plainly_and_affinely_sep_l
affinely_and_r
affinely_and
idemp
.
Qed
.
Lemma
wand_impl
_affinely_plainly
P
Q
:
(
■
P

∗
Q
)
⊣
⊢
(
bi_plainly
P
→
Q
).
Proof
.
by
rewrite
(
persistently_plainly
P
)
wand_impl
_affinely_persistently
.
Qed
.
Lemma
impl_wand
_affinely_plainly
P
Q
:
(
bi_plainly
P
→
Q
)
⊣
⊢
(
■
P

∗
Q
).
Proof
.
by
rewrite
(
persistently_plainly
P
)
impl_wand
_affinely_persistently
.
Qed
.
(* Conditional affinely modality *)
Global
Instance
affinely_if_ne
p
:
NonExpansive
(@
bi_affinely_if
PROP
p
).
...
...
@@ 1778,7 +1778,7 @@ Global Instance wand_persistent P Q :
Plain
P
→
Persistent
Q
→
Absorbing
Q
→
Persistent
(
P

∗
Q
).
Proof
.
intros
.
rewrite
/
Persistent
{
2
}(
plain
P
).
trans
(
bi_persistently
(
bi_plainly
P
→
Q
)).

rewrite

persistently_impl_plainly

wand_impl
_affinely_plainly
(
persistent
Q
).

rewrite

persistently_impl_plainly
impl_wand
_affinely_plainly
(
persistent
Q
).
by
rewrite
affinely_plainly_elim
.

apply
persistently_mono
,
wand_intro_l
.
by
rewrite
sep_and
impl_elim_r
.
Qed
.
...
...
@@ 1817,7 +1817,7 @@ Global Instance wand_plain P Q :
Plain
P
→
Plain
Q
→
Absorbing
Q
→
Plain
(
P

∗
Q
).
Proof
.
intros
.
rewrite
/
Plain
{
2
}(
plain
P
).
trans
(
bi_plainly
(
bi_plainly
P
→
Q
)).

rewrite

plainly_impl_plainly

wand_impl
_affinely_plainly
(
plain
Q
).

rewrite

plainly_impl_plainly
impl_wand
_affinely_plainly
(
plain
Q
).
by
rewrite
affinely_plainly_elim
.

apply
plainly_mono
,
wand_intro_l
.
by
rewrite
sep_and
impl_elim_r
.
Qed
.
...
...
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