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PierreMarie Pédrot
Iris
Commits
d4c6321c
Commit
d4c6321c
authored
Dec 04, 2017
by
JacquesHenri Jourdan
Browse files
Remove plainly_exist_1 from the BI axioms.
parent
d831f1e2
Changes
4
Hide whitespace changes
Inline
Sidebyside
theories/base_logic/upred.v
View file @
d4c6321c
...
...
@@ 474,8 +474,6 @@ Proof.
unseal
;
split
=>
n
x
??
//.

(* (∀ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∀ a, Ψ a) *)
by
unseal
.

(* bi_plainly (∃ a, Ψ a) ⊢ ∃ a, bi_plainly (Ψ a) *)
by
unseal
.

(* bi_plainly ((P → Q) ∧ (Q → P)) ⊢ P ≡ Q *)
unseal
;
split
=>
n
x
?
/=
HPQ
;
split
=>
n'
x'
?
HP
;
split
;
eapply
HPQ
;
eauto
using
@
ucmra_unit_least
.
...
...
@@ 610,6 +608,11 @@ Proof.
Qed
.
Global
Instance
bupd_proper
:
Proper
((
≡
)
==>
(
≡
))
(@
uPred_bupd
M
)
:
=
ne_proper
_
.
(** PlainlyExist1BI *)
Lemma
uPred_plainly_exist_1
:
PlainlyExist1BI
(
uPredI
M
).
Proof
.
unfold
PlainlyExist1BI
.
by
unseal
.
Qed
.
(** Limits *)
Lemma
entails_lim
(
cP
cQ
:
chain
(
uPredC
M
))
:
(
∀
n
,
cP
n
⊢
cQ
n
)
→
compl
cP
⊢
compl
cQ
.
...
...
theories/bi/derived.v
View file @
d4c6321c
...
...
@@ 1183,15 +1183,18 @@ Proof.
apply
(
anti_symm
_
)
;
auto
using
plainly_forall_2
.
apply
forall_intro
=>
x
.
by
rewrite
(
forall_elim
x
).
Qed
.
Lemma
plainly_exist
{
A
}
(
Ψ
:
A
→
PROP
)
:
Lemma
plainly_exist_2
{
A
}
(
Ψ
:
A
→
PROP
)
:
(
∃
a
,
bi_plainly
(
Ψ
a
))
⊢
bi_plainly
(
∃
a
,
Ψ
a
).
Proof
.
apply
exist_elim
=>
x
.
by
rewrite
(
exist_intro
x
).
Qed
.
Lemma
plainly_exist
`
{
PlainlyExist1BI
PROP
}
{
A
}
(
Ψ
:
A
→
PROP
)
:
bi_plainly
(
∃
a
,
Ψ
a
)
⊣
⊢
∃
a
,
bi_plainly
(
Ψ
a
).
Proof
.
apply
(
anti_symm
_
)
;
auto
using
plainly_exist_1
.
apply
exist_elim
=>
x
.
by
rewrite
(
exist_intro
x
).
Qed
.
Proof
.
apply
(
anti_symm
_
)
;
auto
using
plainly_exist_1
,
plainly_exist_2
.
Qed
.
Lemma
plainly_and
P
Q
:
bi_plainly
(
P
∧
Q
)
⊣
⊢
bi_plainly
P
∧
bi_plainly
Q
.
Proof
.
rewrite
!
and_alt
plainly_forall
.
by
apply
forall_proper
=>
[].
Qed
.
Lemma
plainly_or
P
Q
:
bi_plainly
(
P
∨
Q
)
⊣
⊢
bi_plainly
P
∨
bi_plainly
Q
.
Lemma
plainly_or_2
P
Q
:
bi_plainly
P
∨
bi_plainly
Q
⊢
bi_plainly
(
P
∨
Q
).
Proof
.
rewrite
!
or_alt

plainly_exist_2
.
by
apply
exist_mono
=>
[].
Qed
.
Lemma
plainly_or
`
{
PlainlyExist1BI
PROP
}
P
Q
:
bi_plainly
(
P
∨
Q
)
⊣
⊢
bi_plainly
P
∨
bi_plainly
Q
.
Proof
.
rewrite
!
or_alt
plainly_exist
.
by
apply
exist_proper
=>
[].
Qed
.
Lemma
plainly_impl
P
Q
:
bi_plainly
(
P
→
Q
)
⊢
bi_plainly
P
→
bi_plainly
Q
.
Proof
.
...
...
@@ 1362,9 +1365,14 @@ Proof.
Qed
.
Lemma
affinely_plainly_and
P
Q
:
■
(
P
∧
Q
)
⊣
⊢
■
P
∧
■
Q
.
Proof
.
by
rewrite
plainly_and
affinely_and
.
Qed
.
Lemma
affinely_plainly_or
P
Q
:
■
(
P
∨
Q
)
⊣
⊢
■
P
∨
■
Q
.
Lemma
affinely_plainly_or_2
P
Q
:
■
P
∨
■
Q
⊢
■
(
P
∨
Q
).
Proof
.
by
rewrite

plainly_or_2
affinely_or
.
Qed
.
Lemma
affinely_plainly_or
`
{
PlainlyExist1BI
PROP
}
P
Q
:
■
(
P
∨
Q
)
⊣
⊢
■
P
∨
■
Q
.
Proof
.
by
rewrite
plainly_or
affinely_or
.
Qed
.
Lemma
affinely_plainly_exist
{
A
}
(
Φ
:
A
→
PROP
)
:
■
(
∃
x
,
Φ
x
)
⊣
⊢
∃
x
,
■
Φ
x
.
Lemma
affinely_plainly_exist_2
{
A
}
(
Φ
:
A
→
PROP
)
:
(
∃
x
,
■
Φ
x
)
⊢
■
(
∃
x
,
Φ
x
).
Proof
.
by
rewrite

plainly_exist_2
affinely_exist
.
Qed
.
Lemma
affinely_plainly_exist
`
{
PlainlyExist1BI
PROP
}
{
A
}
(
Φ
:
A
→
PROP
)
:
■
(
∃
x
,
Φ
x
)
⊣
⊢
∃
x
,
■
Φ
x
.
Proof
.
by
rewrite
plainly_exist
affinely_exist
.
Qed
.
Lemma
affinely_plainly_sep_2
P
Q
:
■
P
∗
■
Q
⊢
■
(
P
∗
Q
).
Proof
.
by
rewrite
affinely_sep_2
plainly_sep_2
.
Qed
.
...
...
@@ 1523,9 +1531,17 @@ Lemma plainly_if_pure p φ : bi_plainly_if p ⌜φ⌝ ⊣⊢ ⌜φ⌝.
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_pure
.
Qed
.
Lemma
plainly_if_and
p
P
Q
:
bi_plainly_if
p
(
P
∧
Q
)
⊣
⊢
bi_plainly_if
p
P
∧
bi_plainly_if
p
Q
.
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_and
.
Qed
.
Lemma
plainly_if_or
p
P
Q
:
bi_plainly_if
p
(
P
∨
Q
)
⊣
⊢
bi_plainly_if
p
P
∨
bi_plainly_if
p
Q
.
Lemma
plainly_if_or_2
p
P
Q
:
bi_plainly_if
p
P
∨
bi_plainly_if
p
Q
⊢
bi_plainly_if
p
(
P
∨
Q
).
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_or_2
.
Qed
.
Lemma
plainly_if_or
`
{
PlainlyExist1BI
PROP
}
p
P
Q
:
bi_plainly_if
p
(
P
∨
Q
)
⊣
⊢
bi_plainly_if
p
P
∨
bi_plainly_if
p
Q
.
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_or
.
Qed
.
Lemma
plainly_if_exist
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
(
bi_plainly_if
p
(
∃
a
,
Ψ
a
))
⊣
⊢
∃
a
,
bi_plainly_if
p
(
Ψ
a
).
Lemma
plainly_if_exist_2
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
(
∃
a
,
bi_plainly_if
p
(
Ψ
a
))
⊢
bi_plainly_if
p
(
∃
a
,
Ψ
a
).
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_exist_2
.
Qed
.
Lemma
plainly_if_exist
`
{
PlainlyExist1BI
PROP
}
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
bi_plainly_if
p
(
∃
a
,
Ψ
a
)
⊣
⊢
∃
a
,
bi_plainly_if
p
(
Ψ
a
).
Proof
.
destruct
p
;
simpl
;
auto
using
plainly_exist
.
Qed
.
Lemma
plainly_if_sep
`
{
PositiveBI
PROP
}
p
P
Q
:
bi_plainly_if
p
(
P
∗
Q
)
⊣
⊢
bi_plainly_if
p
P
∗
bi_plainly_if
p
Q
.
...
...
@@ 1550,9 +1566,15 @@ Lemma affinely_plainly_if_emp p : ■?p emp ⊣⊢ emp.
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_emp
.
Qed
.
Lemma
affinely_plainly_if_and
p
P
Q
:
■
?p
(
P
∧
Q
)
⊣
⊢
■
?p
P
∧
■
?p
Q
.
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_and
.
Qed
.
Lemma
affinely_plainly_if_or
p
P
Q
:
■
?p
(
P
∨
Q
)
⊣
⊢
■
?p
P
∨
■
?p
Q
.
Lemma
affinely_plainly_if_or_2
p
P
Q
:
■
?p
P
∨
■
?p
Q
⊢
■
?p
(
P
∨
Q
).
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_or_2
.
Qed
.
Lemma
affinely_plainly_if_or
`
{
PlainlyExist1BI
PROP
}
p
P
Q
:
■
?p
(
P
∨
Q
)
⊣
⊢
■
?p
P
∨
■
?p
Q
.
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_or
.
Qed
.
Lemma
affinely_plainly_if_exist
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
Lemma
affinely_plainly_if_exist_2
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
(
∃
a
,
■
?p
Ψ
a
)
⊢
■
?p
∃
a
,
Ψ
a
.
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_exist_2
.
Qed
.
Lemma
affinely_plainly_if_exist
`
{
PlainlyExist1BI
PROP
}
{
A
}
p
(
Ψ
:
A
→
PROP
)
:
(
■
?p
∃
a
,
Ψ
a
)
⊣
⊢
∃
a
,
■
?p
Ψ
a
.
Proof
.
destruct
p
;
simpl
;
auto
using
affinely_plainly_exist
.
Qed
.
Lemma
affinely_plainly_if_sep_2
p
P
Q
:
■
?p
P
∗
■
?p
Q
⊢
■
?p
(
P
∗
Q
).
...
...
@@ 1791,7 +1813,7 @@ Proof. apply plainly_emp_intro. Qed.
Global
Instance
and_plain
P
Q
:
Plain
P
→
Plain
Q
→
Plain
(
P
∧
Q
).
Proof
.
intros
.
by
rewrite
/
Plain
plainly_and
!
plain
.
Qed
.
Global
Instance
or_plain
P
Q
:
Plain
P
→
Plain
Q
→
Plain
(
P
∨
Q
).
Proof
.
intros
.
by
rewrite
/
Plain
plainly_or
!
plain
.
Qed
.
Proof
.
intros
.
by
rewrite
/
Plain

plainly_or
_2
!
plain
.
Qed
.
Global
Instance
forall_plain
{
A
}
(
Ψ
:
A
→
PROP
)
:
(
∀
x
,
Plain
(
Ψ
x
))
→
Plain
(
∀
x
,
Ψ
x
).
Proof
.
...
...
@@ 1800,7 +1822,7 @@ Qed.
Global
Instance
exist_plain
{
A
}
(
Ψ
:
A
→
PROP
)
:
(
∀
x
,
Plain
(
Ψ
x
))
→
Plain
(
∃
x
,
Ψ
x
).
Proof
.
intros
.
rewrite
/
Plain
plainly_exist
.
apply
exist_mono
=>
x
.
by
rewrite

plain
.
intros
.
rewrite
/
Plain

plainly_exist
_2
.
apply
exist_mono
=>
x
.
by
rewrite

plain
.
Qed
.
Global
Instance
internal_eq_plain
{
A
:
ofeT
}
(
a
b
:
A
)
:
...
...
@@ 1852,7 +1874,7 @@ Proof.
split
;
[
split
]
;
try
apply
_
.
apply
plainly_and
.
apply
plainly_pure
.
Qed
.
Global
Instance
bi_plainly_or_homomorphism
:
Global
Instance
bi_plainly_or_homomorphism
`
{
PlainlyExist1BI
PROP
}
:
MonoidHomomorphism
bi_or
bi_or
(
≡
)
(@
bi_plainly
PROP
).
Proof
.
split
;
[
split
]
;
try
apply
_
.
apply
plainly_or
.
apply
plainly_pure
.
...
...
@@ 2161,7 +2183,10 @@ Proof.
Qed
.
Lemma
except_0_later
P
:
◇
▷
P
⊢
▷
P
.
Proof
.
by
rewrite
/
bi_except_0

later_or
False_or
.
Qed
.
Lemma
except_0_plainly
P
:
◇
bi_plainly
P
⊣
⊢
bi_plainly
(
◇
P
).
Lemma
except_0_plainly_1
P
:
◇
bi_plainly
P
⊢
bi_plainly
(
◇
P
).
Proof
.
by
rewrite
/
bi_except_0

plainly_or_2

later_plainly
plainly_pure
.
Qed
.
Lemma
except_0_plainly
`
{
PlainlyExist1BI
PROP
}
P
:
◇
bi_plainly
P
⊣
⊢
bi_plainly
(
◇
P
).
Proof
.
by
rewrite
/
bi_except_0
plainly_or

later_plainly
plainly_pure
.
Qed
.
Lemma
except_0_persistently
P
:
◇
bi_persistently
P
⊣
⊢
bi_persistently
(
◇
P
).
Proof
.
...
...
@@ 2169,11 +2194,12 @@ Proof.
Qed
.
Lemma
except_0_affinely_2
P
:
bi_affinely
(
◇
P
)
⊢
◇
bi_affinely
P
.
Proof
.
rewrite
/
bi_affinely
except_0_and
.
auto
using
except_0_intro
.
Qed
.
Lemma
except_0_affinely_plainly_2
P
:
■
◇
P
⊢
◇
■
P
.
Lemma
except_0_affinely_plainly_2
`
{
PlainlyExist1BI
PROP
}
P
:
■
◇
P
⊢
◇
■
P
.
Proof
.
by
rewrite

except_0_plainly
except_0_affinely_2
.
Qed
.
Lemma
except_0_affinely_persistently_2
P
:
□
◇
P
⊢
◇
□
P
.
Proof
.
by
rewrite

except_0_persistently
except_0_affinely_2
.
Qed
.
Lemma
except_0_affinely_plainly_if_2
p
P
:
■
?p
◇
P
⊢
◇
■
?p
P
.
Lemma
except_0_affinely_plainly_if_2
`
{
PlainlyExist1BI
PROP
}
p
P
:
■
?p
◇
P
⊢
◇
■
?p
P
.
Proof
.
destruct
p
;
simpl
;
auto
using
except_0_affinely_plainly_2
.
Qed
.
Lemma
except_0_affinely_persistently_if_2
p
P
:
□
?p
◇
P
⊢
◇
□
?p
P
.
Proof
.
destruct
p
;
simpl
;
auto
using
except_0_affinely_persistently_2
.
Qed
.
...
...
@@ 2250,7 +2276,8 @@ Proof.

rewrite
/
bi_except_0
;
auto
.

apply
exist_elim
=>
x
.
rewrite
(
exist_intro
x
)
;
auto
.
Qed
.
Global
Instance
plainly_timeless
P
:
Timeless
P
→
Timeless
(
bi_plainly
P
).
Global
Instance
plainly_timeless
P
`
{
PlainlyExist1BI
PROP
}
:
Timeless
P
→
Timeless
(
bi_plainly
P
).
Proof
.
intros
.
rewrite
/
Timeless
/
bi_except_0
later_plainly_1
.
by
rewrite
(
timeless
P
)
/
bi_except_0
plainly_or
{
1
}
plainly_elim
.
...
...
theories/bi/interface.v
View file @
d4c6321c
...
...
@@ 107,8 +107,6 @@ Section bi_mixin.
bi_mixin_plainly_forall_2
{
A
}
(
Ψ
:
A
→
PROP
)
:
(
∀
a
,
bi_plainly
(
Ψ
a
))
⊢
bi_plainly
(
∀
a
,
Ψ
a
)
;
bi_mixin_plainly_exist_1
{
A
}
(
Ψ
:
A
→
PROP
)
:
bi_plainly
(
∃
a
,
Ψ
a
)
⊢
∃
a
,
bi_plainly
(
Ψ
a
)
;
bi_mixin_prop_ext
P
Q
:
bi_plainly
((
P
→
Q
)
∧
(
Q
→
P
))
⊢
bi_internal_eq
(
OfeT
PROP
prop_ofe_mixin
)
P
Q
;
...
...
@@ 331,6 +329,10 @@ Coercion sbi_valid {PROP : sbi} : PROP → Prop := bi_valid.
Arguments
bi_valid
{
_
}
_
%
I
:
simpl
never
.
Typeclasses
Opaque
bi_valid
.
Class
PlainlyExist1BI
(
PROP
:
bi
)
:
=
plainly_exist_1
A
(
Ψ
:
A
→
PROP
)
:
bi_plainly
(
∃
a
,
Ψ
a
)
⊢
∃
a
,
bi_plainly
(
Ψ
a
).
Arguments
plainly_exist_1
{
_
_
_
}
_
.
Module
bi
.
Section
bi_laws
.
Context
{
PROP
:
bi
}.
...
...
@@ 449,9 +451,6 @@ Proof. eapply bi_mixin_plainly_idemp_2, bi_bi_mixin. Qed.
Lemma
plainly_forall_2
{
A
}
(
Ψ
:
A
→
PROP
)
:
(
∀
a
,
bi_plainly
(
Ψ
a
))
⊢
bi_plainly
(
∀
a
,
Ψ
a
).
Proof
.
eapply
bi_mixin_plainly_forall_2
,
bi_bi_mixin
.
Qed
.
Lemma
plainly_exist_1
{
A
}
(
Ψ
:
A
→
PROP
)
:
bi_plainly
(
∃
a
,
Ψ
a
)
⊢
∃
a
,
bi_plainly
(
Ψ
a
).
Proof
.
eapply
bi_mixin_plainly_exist_1
,
bi_bi_mixin
.
Qed
.
Lemma
prop_ext
P
Q
:
bi_plainly
((
P
→
Q
)
∧
(
Q
→
P
))
⊢
P
≡
Q
.
Proof
.
eapply
(
bi_mixin_prop_ext
_
bi_entails
),
bi_bi_mixin
.
Qed
.
Lemma
persistently_impl_plainly
P
Q
:
...
...
theories/proofmode/class_instances.v
View file @
d4c6321c
...
...
@@ 495,7 +495,7 @@ Global Instance from_or_absorbingly P Q1 Q2 :
Proof
.
rewrite
/
FromOr
=>
<.
by
rewrite
absorbingly_or
.
Qed
.
Global
Instance
from_or_plainly
P
Q1
Q2
:
FromOr
P
Q1
Q2
→
FromOr
(
bi_plainly
P
)
(
bi_plainly
Q1
)
(
bi_plainly
Q2
).
Proof
.
rewrite
/
FromOr
=>
<.
by
rewrite
plainly_or
.
Qed
.
Proof
.
rewrite
/
FromOr
=>
<.
by
rewrite

plainly_or
_2
.
Qed
.
Global
Instance
from_or_persistently
P
Q1
Q2
:
FromOr
P
Q1
Q2
→
FromOr
(
bi_persistently
P
)
(
bi_persistently
Q1
)
(
bi_persistently
Q2
).
...
...
@@ 512,7 +512,7 @@ Proof. rewrite /IntoOr=>>. by rewrite affinely_or. Qed.
Global
Instance
into_or_absorbingly
P
Q1
Q2
:
IntoOr
P
Q1
Q2
→
IntoOr
(
bi_absorbingly
P
)
(
bi_absorbingly
Q1
)
(
bi_absorbingly
Q2
).
Proof
.
rewrite
/
IntoOr
=>>.
by
rewrite
absorbingly_or
.
Qed
.
Global
Instance
into_or_plainly
P
Q1
Q2
:
Global
Instance
into_or_plainly
`
{
PlainlyExist1BI
PROP
}
P
Q1
Q2
:
IntoOr
P
Q1
Q2
→
IntoOr
(
bi_plainly
P
)
(
bi_plainly
Q1
)
(
bi_plainly
Q2
).
Proof
.
rewrite
/
IntoOr
=>>.
by
rewrite
plainly_or
.
Qed
.
Global
Instance
into_or_persistently
P
Q1
Q2
:
...
...
@@ 534,7 +534,7 @@ Global Instance from_exist_absorbingly {A} P (Φ : A → PROP) :
Proof
.
rewrite
/
FromExist
=>
<.
by
rewrite
absorbingly_exist
.
Qed
.
Global
Instance
from_exist_plainly
{
A
}
P
(
Φ
:
A
→
PROP
)
:
FromExist
P
Φ
→
FromExist
(
bi_plainly
P
)
(
λ
a
,
bi_plainly
(
Φ
a
))%
I
.
Proof
.
rewrite
/
FromExist
=>
<.
by
rewrite
plainly_exist
.
Qed
.
Proof
.
rewrite
/
FromExist
=>
<.
by
rewrite

plainly_exist
_2
.
Qed
.
Global
Instance
from_exist_persistently
{
A
}
P
(
Φ
:
A
→
PROP
)
:
FromExist
P
Φ
→
FromExist
(
bi_persistently
P
)
(
λ
a
,
bi_persistently
(
Φ
a
))%
I
.
Proof
.
rewrite
/
FromExist
=>
<.
by
rewrite
persistently_exist
.
Qed
.
...
...
@@ 564,7 +564,7 @@ Qed.
Global
Instance
into_exist_absorbingly
{
A
}
P
(
Φ
:
A
→
PROP
)
:
IntoExist
P
Φ
→
IntoExist
(
bi_absorbingly
P
)
(
λ
a
,
bi_absorbingly
(
Φ
a
))%
I
.
Proof
.
rewrite
/
IntoExist
=>
HP
.
by
rewrite
HP
absorbingly_exist
.
Qed
.
Global
Instance
into_exist_plainly
{
A
}
P
(
Φ
:
A
→
PROP
)
:
Global
Instance
into_exist_plainly
`
{
PlainlyExist1BI
PROP
}
{
A
}
P
(
Φ
:
A
→
PROP
)
:
IntoExist
P
Φ
→
IntoExist
(
bi_plainly
P
)
(
λ
a
,
bi_plainly
(
Φ
a
))%
I
.
Proof
.
rewrite
/
IntoExist
=>
HP
.
by
rewrite
HP
plainly_exist
.
Qed
.
Global
Instance
into_exist_persistently
{
A
}
P
(
Φ
:
A
→
PROP
)
:
...
...
@@ 1051,7 +1051,7 @@ Proof. rewrite /IntoExcept0=> >. by rewrite except_0_affinely_2. Qed.
Global
Instance
into_except_0_absorbingly
P
Q
:
IntoExcept0
P
Q
→
IntoExcept0
(
bi_absorbingly
P
)
(
bi_absorbingly
Q
).
Proof
.
rewrite
/
IntoExcept0
=>
>.
by
rewrite
except_0_absorbingly
.
Qed
.
Global
Instance
into_except_0_plainly
P
Q
:
Global
Instance
into_except_0_plainly
`
{
PlainlyExist1BI
PROP
}
P
Q
:
IntoExcept0
P
Q
→
IntoExcept0
(
bi_plainly
P
)
(
bi_plainly
Q
).
Proof
.
rewrite
/
IntoExcept0
=>
>.
by
rewrite
except_0_plainly
.
Qed
.
Global
Instance
into_except_0_persistently
P
Q
:
...
...
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