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William Mansky
Iris
Commits
e06423f4
Commit
e06423f4
authored
8 years ago
by
Robbert Krebbers
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Prove more later properties in the logic.
parent
bd15a981
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algebra/upred.v
+32
-15
32 additions, 15 deletions
algebra/upred.v
with
32 additions
and
15 deletions
algebra/upred.v
+
32
−
15
View file @
e06423f4
...
...
@@ -731,12 +731,6 @@ Lemma pure_elim_r φ Q R : (φ → Q ⊢ R) → Q ∧ ■ φ ⊢ R.
Proof
.
intros
;
apply
pure_elim
with
φ
;
eauto
.
Qed
.
Lemma
pure_equiv
(
φ
:
Prop
)
:
φ
→
■
φ
⊣⊢
True
.
Proof
.
intros
;
apply
(
anti_symm
_);
auto
using
pure_intro
.
Qed
.
Lemma
pure_alt
φ
:
■
φ
⊣⊢
∃
_
:
φ
,
True
.
Proof
.
apply
(
anti_symm
_)
.
-
eapply
pure_elim
;
eauto
=>
H
.
rewrite
-
(
exist_intro
H
);
auto
.
-
by
apply
exist_elim
,
pure_intro
.
Qed
.
Lemma
eq_refl'
{
A
:
cofeT
}
(
a
:
A
)
P
:
P
⊢
a
≡
a
.
Proof
.
rewrite
(
True_intro
P
)
.
apply
eq_refl
.
Qed
.
...
...
@@ -750,6 +744,23 @@ Proof.
apply
(
eq_rewrite
P
Q
(
λ
Q
,
P
↔
Q
))
%
I
;
first
solve_proper
;
auto
using
iff_refl
.
Qed
.
Lemma
pure_alt
φ
:
■
φ
⊣⊢
∃
_
:
φ
,
True
.
Proof
.
apply
(
anti_symm
_)
.
-
eapply
pure_elim
;
eauto
=>
H
.
rewrite
-
(
exist_intro
H
);
auto
.
-
by
apply
exist_elim
,
pure_intro
.
Qed
.
Lemma
and_alt
P
Q
:
P
∧
Q
⊣⊢
∀
b
:
bool
,
if
b
then
P
else
Q
.
Proof
.
apply
(
anti_symm
_);
first
apply
forall_intro
=>
-
[];
auto
.
apply
and_intro
.
by
rewrite
(
forall_elim
true
)
.
by
rewrite
(
forall_elim
false
)
.
Qed
.
Lemma
or_alt
P
Q
:
P
∨
Q
⊣⊢
∃
b
:
bool
,
if
b
then
P
else
Q
.
Proof
.
apply
(
anti_symm
_);
last
apply
exist_elim
=>
-
[];
auto
.
apply
or_elim
.
by
rewrite
-
(
exist_intro
true
)
.
by
rewrite
-
(
exist_intro
false
)
.
Qed
.
(* BI connectives *)
Lemma
sep_mono
P
P'
Q
Q'
:
(
P
⊢
Q
)
→
(
P'
⊢
Q'
)
→
P
★
P'
⊢
Q
★
Q'
.
Proof
.
...
...
@@ -1020,11 +1031,7 @@ Proof.
unseal
;
split
=>
n
x
?
HP
;
induction
n
as
[|
n
IH
];
[
by
apply
HP
|]
.
apply
HP
,
IH
,
uPred_closed
with
(
S
n
);
eauto
using
cmra_validN_S
.
Qed
.
Lemma
later_and
P
Q
:
▷
(
P
∧
Q
)
⊣⊢
▷
P
∧
▷
Q
.
Proof
.
unseal
;
split
=>
-
[|
n
]
x
;
by
split
.
Qed
.
Lemma
later_or
P
Q
:
▷
(
P
∨
Q
)
⊣⊢
▷
P
∨
▷
Q
.
Proof
.
unseal
;
split
=>
-
[|
n
]
x
;
simpl
;
tauto
.
Qed
.
Lemma
later_forall
{
A
}
(
Φ
:
A
→
uPred
M
)
:
(
▷
∀
a
,
Φ
a
)
⊣⊢
(
∀
a
,
▷
Φ
a
)
.
Lemma
later_forall_2
{
A
}
(
Φ
:
A
→
uPred
M
)
:
(
∀
a
,
▷
Φ
a
)
⊢
▷
∀
a
,
Φ
a
.
Proof
.
unseal
;
by
split
=>
-
[|
n
]
x
.
Qed
.
Lemma
later_exist_false
{
A
}
(
Φ
:
A
→
uPred
M
)
:
(
▷
∃
a
,
Φ
a
)
⊢
▷
False
∨
(
∃
a
,
▷
Φ
a
)
.
...
...
@@ -1059,13 +1066,17 @@ Proof. intros P Q; apply later_mono. Qed.
Lemma
later_intro
P
:
P
⊢
▷
P
.
Proof
.
rewrite
-
(
and_elim_l
(
▷
P
)
P
)
-
(
löb
(
▷
P
∧
P
))
later_and
.
apply
impl_intro_l
;
auto
.
rewrite
-
(
and_elim_l
(
▷
P
)
P
)
-
(
löb
(
▷
P
∧
P
))
.
apply
impl_intro_l
.
by
rewrite
{
1
}(
and_elim_r
(
▷
P
))
.
Qed
.
Lemma
later_True
:
▷
True
⊣⊢
True
.
Proof
.
apply
(
anti_symm
(
⊢
));
auto
using
later_intro
.
Qed
.
Lemma
later_impl
P
Q
:
▷
(
P
→
Q
)
⊢
▷
P
→
▷
Q
.
Proof
.
apply
impl_intro_l
;
rewrite
-
later_and
;
eauto
using
impl_elim
.
Qed
.
Lemma
later_forall
{
A
}
(
Φ
:
A
→
uPred
M
)
:
(
▷
∀
a
,
Φ
a
)
⊣⊢
(
∀
a
,
▷
Φ
a
)
.
Proof
.
apply
(
anti_symm
_);
auto
using
later_forall_2
.
apply
forall_intro
=>
x
.
by
rewrite
(
forall_elim
x
)
.
Qed
.
Lemma
later_exist
`{
Inhabited
A
}
(
Φ
:
A
→
uPred
M
)
:
▷
(
∃
a
,
Φ
a
)
⊣⊢
(
∃
a
,
▷
Φ
a
)
.
Proof
.
...
...
@@ -1073,6 +1084,12 @@ Proof.
rewrite
later_exist_false
.
apply
or_elim
;
last
done
.
rewrite
-
(
exist_intro
inhabitant
);
auto
.
Qed
.
Lemma
later_and
P
Q
:
▷
(
P
∧
Q
)
⊣⊢
▷
P
∧
▷
Q
.
Proof
.
rewrite
!
and_alt
later_forall
.
by
apply
forall_proper
=>
-
[]
.
Qed
.
Lemma
later_or
P
Q
:
▷
(
P
∨
Q
)
⊣⊢
▷
P
∨
▷
Q
.
Proof
.
rewrite
!
or_alt
later_exist
.
by
apply
exist_proper
=>
-
[]
.
Qed
.
Lemma
later_impl
P
Q
:
▷
(
P
→
Q
)
⊢
▷
P
→
▷
Q
.
Proof
.
apply
impl_intro_l
;
rewrite
-
later_and
;
eauto
using
impl_elim
.
Qed
.
Lemma
later_wand
P
Q
:
▷
(
P
-★
Q
)
⊢
▷
P
-★
▷
Q
.
Proof
.
apply
wand_intro_r
;
rewrite
-
later_sep
;
eauto
using
wand_elim_l
.
Qed
.
Lemma
later_iff
P
Q
:
▷
(
P
↔
Q
)
⊢
▷
P
↔
▷
Q
.
...
...
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