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89a00a27
Commit
89a00a27
authored
Nov 06, 2019
by
Robbert Krebbers
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Move `monPred_at` lemmas up, so we can use them for other lemmas.
parent
048c1078
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theories/bi/monpred.v
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theories/bi/monpred.v
View file @
89a00a27
...
...
@@ -370,6 +370,54 @@ Local Notation BiIndexBottom := (@BiIndexBottom I).
Implicit
Types
i
:
I
.
Implicit
Types
P
Q
:
monPred
.
(** monPred_at unfolding laws *)
Lemma
monPred_at_pure
i
(
φ
:
Prop
)
:
monPred_at
⌜φ⌝
i
⊣
⊢
⌜φ⌝
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_emp
i
:
monPred_at
emp
i
⊣
⊢
emp
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_and
i
P
Q
:
(
P
∧
Q
)
i
⊣
⊢
P
i
∧
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_or
i
P
Q
:
(
P
∨
Q
)
i
⊣
⊢
P
i
∨
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_impl
i
P
Q
:
(
P
→
Q
)
i
⊣
⊢
∀
j
,
⌜
i
⊑
j
⌝
→
P
j
→
Q
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_forall
{
A
}
i
(
Φ
:
A
→
monPred
)
:
(
∀
x
,
Φ
x
)
i
⊣
⊢
∀
x
,
Φ
x
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_exist
{
A
}
i
(
Φ
:
A
→
monPred
)
:
(
∃
x
,
Φ
x
)
i
⊣
⊢
∃
x
,
Φ
x
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_sep
i
P
Q
:
(
P
∗
Q
)
i
⊣
⊢
P
i
∗
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_wand
i
P
Q
:
(
P
-
∗
Q
)
i
⊣
⊢
∀
j
,
⌜
i
⊑
j
⌝
→
P
j
-
∗
Q
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_persistently
i
P
:
(<
pers
>
P
)
i
⊣
⊢
<
pers
>
(
P
i
).
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_in
i
j
:
monPred_at
(
monPred_in
j
)
i
⊣
⊢
⌜
j
⊑
i
⌝
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_objectively
i
P
:
(<
obj
>
P
)
i
⊣
⊢
∀
j
,
P
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_subjectively
i
P
:
(<
subj
>
P
)
i
⊣
⊢
∃
j
,
P
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_persistently_if
i
p
P
:
(<
pers
>
?p
P
)
i
⊣
⊢
<
pers
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_persistently
.
Qed
.
Lemma
monPred_at_affinely
i
P
:
(<
affine
>
P
)
i
⊣
⊢
<
affine
>
(
P
i
).
Proof
.
by
rewrite
/
bi_affinely
monPred_at_and
monPred_at_emp
.
Qed
.
Lemma
monPred_at_affinely_if
i
p
P
:
(<
affine
>
?p
P
)
i
⊣
⊢
<
affine
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_affinely
.
Qed
.
Lemma
monPred_at_intuitionistically
i
P
:
(
□
P
)
i
⊣
⊢
□
(
P
i
).
Proof
.
by
rewrite
/
bi_intuitionistically
monPred_at_affinely
monPred_at_persistently
.
Qed
.
Lemma
monPred_at_intuitionistically_if
i
p
P
:
(
□
?p
P
)
i
⊣
⊢
□
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_intuitionistically
.
Qed
.
Lemma
monPred_at_absorbingly
i
P
:
(<
absorb
>
P
)
i
⊣
⊢
<
absorb
>
(
P
i
).
Proof
.
by
rewrite
/
bi_absorbingly
monPred_at_sep
monPred_at_pure
.
Qed
.
Lemma
monPred_at_absorbingly_if
i
p
P
:
(<
absorb
>
?p
P
)
i
⊣
⊢
<
absorb
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_absorbingly
.
Qed
.
Lemma
monPred_wand_force
i
P
Q
:
(
P
-
∗
Q
)
i
-
∗
(
P
i
-
∗
Q
i
).
Proof
.
unseal
.
rewrite
bi
.
forall_elim
bi
.
pure_impl_forall
bi
.
forall_elim
//.
Qed
.
Lemma
monPred_impl_force
i
P
Q
:
(
P
→
Q
)
i
-
∗
(
P
i
→
Q
i
).
Proof
.
unseal
.
rewrite
bi
.
forall_elim
bi
.
pure_impl_forall
bi
.
forall_elim
//.
Qed
.
(** Instances *)
Global
Instance
monPred_at_mono
:
Proper
((
⊢
)
==>
(
⊑
)
==>
(
⊢
))
monPred_at
.
...
...
@@ -422,6 +470,9 @@ Global Instance monPred_bi_embed : BiEmbed PROP monPredI :=
Global
Instance
monPred_bi_embed_emp
:
BiEmbedEmp
PROP
monPredI
.
Proof
.
split
.
by
unseal
.
Qed
.
Lemma
monPred_at_embed
i
(
P
:
PROP
)
:
monPred_at
⎡
P
⎤
i
⊣
⊢
P
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_emp_unfold
:
emp
%
I
=
⎡
emp
:
PROP
⎤
%
I
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_pure_unfold
:
bi_pure
=
λ
φ
,
⎡
⌜
φ
⌝
:
PROP
⎤
%
I
.
...
...
@@ -469,56 +520,6 @@ Proof. rewrite monPred_subjectively_unfold. apply _. Qed.
Global
Instance
monPred_subjectively_affine
P
:
Affine
P
→
Affine
(<
subj
>
P
).
Proof
.
rewrite
monPred_subjectively_unfold
.
apply
_
.
Qed
.
(** monPred_at unfolding laws *)
Lemma
monPred_at_embed
i
(
P
:
PROP
)
:
monPred_at
⎡
P
⎤
i
⊣
⊢
P
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_pure
i
(
φ
:
Prop
)
:
monPred_at
⌜φ⌝
i
⊣
⊢
⌜φ⌝
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_emp
i
:
monPred_at
emp
i
⊣
⊢
emp
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_and
i
P
Q
:
(
P
∧
Q
)
i
⊣
⊢
P
i
∧
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_or
i
P
Q
:
(
P
∨
Q
)
i
⊣
⊢
P
i
∨
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_impl
i
P
Q
:
(
P
→
Q
)
i
⊣
⊢
∀
j
,
⌜
i
⊑
j
⌝
→
P
j
→
Q
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_forall
{
A
}
i
(
Φ
:
A
→
monPred
)
:
(
∀
x
,
Φ
x
)
i
⊣
⊢
∀
x
,
Φ
x
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_exist
{
A
}
i
(
Φ
:
A
→
monPred
)
:
(
∃
x
,
Φ
x
)
i
⊣
⊢
∃
x
,
Φ
x
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_sep
i
P
Q
:
(
P
∗
Q
)
i
⊣
⊢
P
i
∗
Q
i
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_wand
i
P
Q
:
(
P
-
∗
Q
)
i
⊣
⊢
∀
j
,
⌜
i
⊑
j
⌝
→
P
j
-
∗
Q
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_persistently
i
P
:
(<
pers
>
P
)
i
⊣
⊢
<
pers
>
(
P
i
).
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_in
i
j
:
monPred_at
(
monPred_in
j
)
i
⊣
⊢
⌜
j
⊑
i
⌝
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_objectively
i
P
:
(<
obj
>
P
)
i
⊣
⊢
∀
j
,
P
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_subjectively
i
P
:
(<
subj
>
P
)
i
⊣
⊢
∃
j
,
P
j
.
Proof
.
by
unseal
.
Qed
.
Lemma
monPred_at_persistently_if
i
p
P
:
(<
pers
>
?p
P
)
i
⊣
⊢
<
pers
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_persistently
.
Qed
.
Lemma
monPred_at_affinely
i
P
:
(<
affine
>
P
)
i
⊣
⊢
<
affine
>
(
P
i
).
Proof
.
by
rewrite
/
bi_affinely
monPred_at_and
monPred_at_emp
.
Qed
.
Lemma
monPred_at_affinely_if
i
p
P
:
(<
affine
>
?p
P
)
i
⊣
⊢
<
affine
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_affinely
.
Qed
.
Lemma
monPred_at_intuitionistically
i
P
:
(
□
P
)
i
⊣
⊢
□
(
P
i
).
Proof
.
by
rewrite
/
bi_intuitionistically
monPred_at_affinely
monPred_at_persistently
.
Qed
.
Lemma
monPred_at_intuitionistically_if
i
p
P
:
(
□
?p
P
)
i
⊣
⊢
□
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_intuitionistically
.
Qed
.
Lemma
monPred_at_absorbingly
i
P
:
(<
absorb
>
P
)
i
⊣
⊢
<
absorb
>
(
P
i
).
Proof
.
by
rewrite
/
bi_absorbingly
monPred_at_sep
monPred_at_pure
.
Qed
.
Lemma
monPred_at_absorbingly_if
i
p
P
:
(<
absorb
>
?p
P
)
i
⊣
⊢
<
absorb
>
?p
(
P
i
).
Proof
.
destruct
p
=>//=.
apply
monPred_at_absorbingly
.
Qed
.
Lemma
monPred_wand_force
i
P
Q
:
(
P
-
∗
Q
)
i
-
∗
(
P
i
-
∗
Q
i
).
Proof
.
unseal
.
rewrite
bi
.
forall_elim
bi
.
pure_impl_forall
bi
.
forall_elim
//.
Qed
.
Lemma
monPred_impl_force
i
P
Q
:
(
P
→
Q
)
i
-
∗
(
P
i
→
Q
i
).
Proof
.
unseal
.
rewrite
bi
.
forall_elim
bi
.
pure_impl_forall
bi
.
forall_elim
//.
Qed
.
(* Laws for monPred_objectively and of Objective. *)
Lemma
monPred_objectively_elim
P
:
<
obj
>
P
⊢
P
.
Proof
.
rewrite
monPred_objectively_unfold
.
unseal
.
split
=>?.
apply
bi
.
forall_elim
.
Qed
.
...
...
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